The Pennsylvania State University the Graduate School College of Engineering MAGNETICALLY INDUCED ACTUATION and OPTIMIZATION OF

The Pennsylvania State University

The Graduate School

College of Engineering

MAGNETICALLY INDUCED ACTUATION AND OPTIMIZATION

OF THE MIURA-ORI STRUCTURE

A Thesis in

Mechanical Engineering

Brett M. Cowan

Submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Science

December 2015

The thesis of Brett M. Cowan was reviewed and approved* by the following:

Paris vonLockette Associate Professor of Mechanical Engineering Thesis Advisor

Zoubeida Ounaies Professor of Mechanical Engineering Dorothy Quiggle Career Development Professor

Karen Thole Department Head of Mechanical and Nuclear Engineering Professor of Mechanical Engineering

*Signatures are on file in the Graduate School

Abstract

Origami engineering is an emerging field that attempts to apply origami principles to engineering applications. One application is the folding/unfolding of origami structures by way of external stimuli, such as thermal fields, electrical fields, and/or magnetic fields, for active systems. This research aims to actuate the Miura-ori pattern from an initial flat state using neodymium magnets on an elastomer substrate within a magnetic field to assess performance characteristics versus magnet placement and orientation. Additionally, proof-of-concept devices using magneto-active elastomers (MAEs) patches will be studied. The MAE material consists of magnetic particles embedded and aligned within a silicon elastomer substrate then cured. In the presence of a magnetic field, both the neodymium magnets and MAE material align with the field, causing a magnetic moment and thus, magnetic work. In this work, the Miura-ori pattern was fabricated from a silicone elastomer substrate with prescribed, reduced-thickness creases and removed material at crease vertex points. Four magnetization orientation configurations of the Miura-ori pattern were generated and fabricated by attaching neodymium magnets to the Miura-ori substrates. The prototypes were tested within a magnetic field ranging from 0 – 240 mT and selected crease fold angles were measured at each field strength. Theoretical magnetic work for each configuration was calculated based on an origami folding model from the Miura-ori’s initial flat state to its completely folded state. These calculations were applied to a design space visualization program to determine the magnetization orientation for each configuration that resulted in the maximum possible theoretical work achieved. Each configuration was analyzed and compared in relation to its experimentally determined overall actuation, experimentally determined ability to follow the ideal folding behavior of the Miura-ori pattern, and the theoretical normalized work for fixed and varied magnetization orientations. The configuration with the highest overall rating of the aforementioned criteria was selected to be tested with the magnetization orientations that resulted in its maximum possible theoretical work. The configuration with the maximum theoretical normalized work was fabricated with attached neodymium magnets. A similar configuration with slightly different magnetization orientations resulting in an offset theoretical normalized work was also tested, and was fabricated using two methods: attached neodymium magnets and embedded MAE patches. The MAE patches were created using a 30% volume fraction of 325 mesh barium hexaferrite particles mixed with Dow Sylgard 184 silicone rubber compound at a 10:1 base to catalyst ratio and cured within a uniform (0.7 T) magnetic field in a prescribed alignment. Both sets of prototypes were tested using the same experimental setup as was used for the original four configurations and were compared using the same criterion. Configuration I*, which had magnetization orientations that maximized the theoretical normalized work, outperformed all other configurations in a

iii weighted sum model that additionally accounted for idealness and actuation. The method used to determine favorable magnetization orientations could potentially be applied to other origami structures investigated for magnetic actuation. In addition, the model used to calculate theoretical normalized work can be the basis of more comprehensive model that could include concepts such as crease and panel stiffness and magnetic saturation.

Table of Contents List of Tables ...... vi List of Figures ...... vii Nomenclature ...... ix Acknowledgements ...... x

Chapter 1. Introduction ...... 1 1.1 Problem statement ...... 1 1.2 Literature review ...... 1 1.2.1 Origami engineering ...... 1 1.2.2 Properties and research of the Miura-ori ...... 3 1.2.3 Magnetorheological/Magneto-active elastomers ...... 7 1.2.4 Neodymium magnets ...... 10 1.2.5 Actuation of origami structures ...... 11 1.3 Research objectives ...... 14

Chapter 2. Methodology ...... 15 2.1 Miura-ori substrate design ...... 15 2.2 Magnet orientation determination ...... 17 2.3 Substrate fabrication ...... 21 2.4 Experimental setup ...... 23 2.5 Magnetic work analysis ...... 26

Chapter 3. Results and discussion ...... 32 3.1 Experimental data analysis ...... 32 3.2 Design space exploration ...... 40 3.3 Configuration optimization ...... 46 3.4 Maximum work configuration fabrication and analysis ...... 49

Chapter 4. Conclusions ...... 58

References ...... 61 Appendix A: Prototype selection data for initial four configurations ...... 64 Appendix B: MATLAB code for experimental theoretical normalized work ...... 67 Appendix C: Experimental data of initial four configurations ...... 107 Appendix D: Fminsearch/ATSV MATLAB code ...... 121 Appendix E: Configuration I* and I** prototype selection and experimental data ...... 138

List of Tables

Table 2.1. Results of the thought-experiment. Region ii has all creases folding in the correct direction ... 19 Table 3.1 List of prototypes and their respective batch within each configuration ...... 33 Table 3.2 Table 3.1 Maximum average fold angles (240 mT external field strength) in degrees for ...... 35 Table 3.3 Average deviation values 퐷̅ for each configuration when comparing mountain vs. valley ...... 38 Table 3.4 MATLAB’s Fminsearch results for maximizing the theoretical normalized magnetic work ..... 41 Table 3.5 Preference Sampler results for the symmetry case for the maximum normalized work of ...... 44 Table 3.6 Percent difference comparison between the symmetry cases of Fminsearch and ATSV ...... 45 Table 3.7 Preference Sampler results for independent case for the maximum normalized work of ...... 46 Table 3.8 Actuation, Ideal behavior, and Theoretical Norm. Work from fixed magnetization ...... 47 Table 3.9 Actuation, Ideal behavior, and Theoretical Norm. Work from varying magnetization ...... 48 Table 3.10 Weighted sum model and the respective criterion of Actuation, Ideal behavior, and ...... 57 Table A.1. Excel Statistical Analysis output of panel thickness data and calculation of the upper and ..... 64 Table A.2. Half-thickness model: crease panel thickness data and the selection of suitable ...... 65 Table A.3. One-third-thickness model: crease panel thickness data and the selection of the suitable ...... 66 Table A.4 Experimental data of the configuration I prototypes ...... 109 Table A.5 Experimental data of the configuration II prototypes ...... 112 Table A.6 Experimental data of the configuration III prototypes ...... 115 Table A.7 Experimental data of the configuration IV prototypes ...... 118 Table A.8 Miura-ori substrate selection data for the configuration I* and configuration I** ...... 138 Table A.9 Experimental data of the configuration I* - Neodymium prototypes ...... 140 Table A.10 Experimental data of the configuration I** - Neodymium prototypes ...... 143 Table A.11 Experimental data of the configuration I** - MAE prototypes ...... 146

List of Figures

Figure 1.1 Two approaches for rigid-foldable origami with thickness from Tachi [9], (a) the axis ...... 3 Figure 1.2 The Bi-axial shortening of a plane into Miura’s developable double corrugation (DDC) ...... 4 Figure 1.3 Loci of two thumbs in unfolding a map from Miura [12] of (a) orthogonal folding ...... 5 Figure 1.4 Unit cell geometry of a folded Miura-ori sheet from Schenk [15]. The parallelogram ...... 6 Figure 1.5 Designations of (a) soft magnetic particles and (b) hard magnetic particles. (c) Four ...... 8 Figure 1.6 Magnetic work as a function of magnetic field strength for each class of sample from ...... 9 Figure 1.7 (a) Displacement change and (b) block force change of H – MREs with different particle ...... 10 Figure 1.8 (a) The mountain (solid lines) and valley (dotted lines) crease pattern for the Miura-ori ...... 12 Figure 1.9 Schematic of the optimal orientation of magnetic material for (a) the waterbomb base ...... 13 Figure 2.1 Miura-ori design with mountain folds being the dashed red lines and the valley folds ...... 15 Figure 2.2 Top (left) and Isometric (right) views of the first Miura-ori design modeled in SolidWorks ... 16 Figure 2.3 Top (left) and Isometric (right) views of the second Miura-ori design modeled in ...... 17 Figure 2.4 2D Miura-ori panel dimensions for both designs, with the creases centered on the orange ...... 17 Figure 2.5 An arbitrary Miura-ori panel divided into four regions with example torque vector and ...... 18 Figure 2.6 Magnet configuration I and the magnetic field direction, H ...... 20 Figure 2.7 Magnet configuration II and the magnetic field direction, H ...... 20 Figure 2.8 Magnet configuration III and the magnetic field direction, H ...... 21 Figure 2.9 Magnet configuration IV and the magnetic field direction, H ...... 21 Figure 2.10 Representative bottom (left) and top lattice (right) mold for Miura substrate casting ...... 22 Figure 2.11 End-product of Miura substrate fabrication with attached Delrin sheets cut to fit each ...... 23 Figure 2.12 Acrylic test stand suspending the Teflon sheet. The circular base of the stand allows it to .... 24 Figure 2.13 Experimental setup within the big magnet. The Gaussmeter probe extends underneath ...... 25 Figure 2.14 Designated crease numbering of the Miura-ori. The creases highlighted in green ...... 25 Figure 2.15 Image capture of the folding crease 1 of a configuration IV prototype. The black ...... 26 Figure 2.16 Representation of the crease vector 퐶푖. The points 푝푟 and 푝푞 are located at adjacent ...... 27 Figure 2.17 Panel movements during the folding process with the blue lines representing the ...... 28 Figure 2.18 Folded cellular metamaterial of individual Miura-ori sheets with alternating unit cell ...... 29 Figure 2.19 Directionality of the magnetic field H (bisector of crease 6’s fold angle) throughout ...... 30 Figure 3.1 configuration II prototype within the Walker Scientific 7H electromagnetic subjected to ...... 32 Figure 3.2 Average Fold Angle vs. Applied Field Strength results for all prototypes of ...... 34 Figure 3.3 Bar chart representation of the maximum average fold angles for each crease and ...... 35 Figure 3.4 Vertical Crease Averages and Horizontal Crease Averages compared to the ideal behavior ... 37 Figure 3.5 Magnetic work done per total magnetic energy potential as a function of Horizontal angle .... 40 Figure 3.6 Panel and magnetization orientation numbering for the Miura-ori. Simplification by ...... 41 Figure 3.7 Trade spaces for (a) configuration I, (b) configuration II, (c) configuration III, and (d) ...... 43 Figure 3.8 Preference Sampler trade space results for (a) configuration I, (b) configuration II, (c) ...... 44 Figure 3.9 Weighted sum performance values for the fixed magnetization orientations...... 48 Figure 3.10 Weighted sum performance values for each case of the varied magnetization orientations ... 49 Figure 3.11 Magnetization orientation directions and corresponding initial magnetic torque directions ... 50 Figure 3.12 3-layer MAE patch mold for Panel 4. The acrylic layers are connected by plastic screws ..... 51 Figure 3.13 MAE patch layer dimensions of (a) Panels 2 and 8, (b) Panel 4, and (c) Panel 6. All ...... 51 Figure 3.14 An embedded MAE prototype of configuration I** ...... 52 Figure 3.15 Neodymium configuration I* prototype within the Walker Scientific 7H ...... 53 Figure 3.16 Average Fold Angle vs. Applied Field Strength results for all prototypes of configuration ... 54 Figure 3.17 (a) Normalized actuation and (b) idealness factor as a function of field strength for all ...... 55 Figure 3.18 Weighted sum performance values for all configurations and theoretical normalized ...... 57 Figure 4.1 Miura-ori design with mountain folds being the dashed red lines and the valley folds being .. 59 Figure A.1 Configuration I prototype within the Walker Scientific 7H electromagnet subjected to ...... 107

vii

Figure A.2 Configuration III prototype within the Walker Scientific 7H electromagnet subjected to ..... 108 Figure A.3 Configuration IV prototype within the Walker Scientific 7H electromagnet subjected to ..... 108 Figure A.4 Neodymium configuration I** prototype within the Walker Scientific 7H electromagnet .... 139 Figure A.5 MAE configuration I** prototype within the Walker Scientific 7H electromagnet ...... 139

viii

Nomenclature

MRE ...... Magnetorheological Elastomer MAE ...... Magneto-Active Elastomer HTC ...... Half-thickness Crease OTTC ...... One-third Thickness Crease ROF ...... Rigid Origami Folder

Acknowledgements

I would like to thank my advisor Dr. Paris von Lockette for his academic and life guidance throughout my graduate studies. I would like to thank my thesis reader, Dr. Zoubeida Ounaies, and Dr. Mary Frecker for their valuable comments and criticisms during my weekly updates of my research progress. There are several other people who aided in my completion of my graduate studies. First, I would like to thank Zhonghua Xi at George Mason University for generating a 3 x 3 Miura-ori pattern within the Rigid Origami Folder model that I could base calculations on. I would like to thank Landen Bowen for helping me extract the necessary information from the Rigid Origami Folder model and teaching me how to use the ARL Trade Space Visualizer (ATSV). I would also like to thank my fellow MACS lab mates for their constant support and assistance. Lastly, thank you to my family and friends for believing in me and encouraging me throughout my time as a graduate student. We gratefully acknowledge the support of the National Science Foundation grant number 1240459 and the Air Force Office of Scientific Research. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Chapter 1. Introduction

1.1 Problem statement Origami, the art of paper folding, has seen increased use in a variety of fields. The ability of origami structures to actively fold and unfold under the influence of external stimuli is of great interest in the field of origami engineering. With the use of smart materials, responsive origami-inspired structures would reduce complexity, increase robustness, and potentially reduce cost as opposed to a mechanical system possessing many moving parts. This work aims to actuate a specific origami structure, the Miura- ori, from an initial flat state using an external magnetic field with small neodymium magnets surface mounted to a flexible elastomer substrate. To assess the performance of the Miura-ori structures, the amount of magnetic torque work applied to each crease of the Miura-ori and how it affects folding behavior will be analyzed theoretically and experimentally. Simulation results of the magnetic torque work will be compared to actuation performance of experimental prototypes that utilize neodymium magnets, the combined results of which will be utilized to adjust the orientation of each magnet to increase the magnetic torque work performed and to achieve a final deformed shape that best approximates the ideal Miura-ori, which is detailed in Section 1.2.2. In this manner, the design space of magnet placement and orientation, and thus the magnetic torque vector placement and orientation, may be explored systematically. After a magnetic placement and orientation that maximizes calculated performance metrics is found, a Miura-ori prototype will be created with attached neodymium magnets to test the performance of the new optimum configuration. In addition, a Miura-ori proof-of-concept prototype with patches of magneto-active elastomer (MAE) embedded in the elastomer substrate will be tested. Magneto-active elastomer material itself is a particulate-filled composite of barium hexaferrite powder and elastomer matrix.

1.2 Literature review This portion contains related works on origami engineering, the Miura-ori and its properties, magnetorheological/magneto-active elastomers, and actuation of origami structures.

1.2.1 Origami engineering Origami engineering, which applies concepts of origami to that of engineering applications, is a relatively new research topic that has seen widespread interest in various facets of society. Current structures that use origami principles include stents, safety features such as automobile airbags, and potentially solar arrays for satellites [1-3]. Traditional origami, which was established in Japan by the 7th century, involves a single unaltered piece of paper that can be folded into a variety of different shapes [4].

In origami engineering, however, media used for these designs come from a variety of materials and are alterable (i.e. notches, cuts, etc.) [1]. Furthermore, the desired origami patterns can be generated through deformation of the sheet resulting from external stimuli as opposed to traditional by-hand folding. Examples of such active origami structures include deformable wheel robots [5,6], soft robots capable of locomotion [7], and pneumatic actuators [8]. The nature of the folds in an origami structure is of increasing importance as media beyond paper are employed. In traditional paper origami, sharp creases are generated to distinguish where the paper will fold. The overall crease pattern on a sheet of paper determines the final folded structure. When paper is replaced by other materials, however, it may no longer be possible to create sharp creases. The efficacy of folding various materials results in the necessary distinction between bending vs. folding. Lauff et al. studied this distinction through an extensive literature search [1]. To aid in distinguishing between bending and folding, two general assumptions concerning folding in origami engineering were made: (1) folding occurs along a predetermined path and (2) causes plastic or reversible deformation at and near the creases. Additionally, bending was reported as being more reliant on the material properties, while folding relied on both material properties and the properties of the crease. As a result of the defined differences between bending and folding, Lauff et al. proposed new definitions for both terms: bending is a non-localized deformation that results in curvature while folding is localized deformation along a crease that results in curvature. When discussing the deformation of origami structures in this document, these definitions will be employed. The materials used in origami engineering may bend, fold, or have a combination of bending and folding along their prescribed creases. Localized deformation, i.e. folding, is the driving force behind origami and thus bending is a somewhat unwanted alternative. Bending could cause difficulties in origami structures achieving their final folded structures. One method of biasing the material in an origami structure toward folding is to make it rigid-foldable. Rigid-foldable origami, or rigid origami, is continuously transformable along its folds without deformation by bending or folding of any facet [9], which are the individual panels generated from the crease pattern. In other words, rigid origami will only fold across its prescribed creases and will not bend or fold a facet or panel which would alter its predetermined structure. This is accomplished by utilizing stiff panels and hinges, which are also incorporated in the design of kinetic structures. Tachi investigated the previous methods proposed to solve the issue of thickening panels of symmetric Miura-ori vertex and slidable hinges, but each could not be applied to non-symmetric or non-flat-foldable vertices and some origami patterns, respectively. Therefore, a method that applied to any origami structure and any vertex design with finite thickness panels was required.

One method introduced to align the rigid and ideal motions required the use of tapered panels. With this method, the origami structure can follow its kinetic motion of rigid origami but is unable to achieve its maximum folding angle due to the finite thickness of each panel. This is shown in Figure 1.1(b). The folding angle can be improved by further trimming the panels while still maintaining the desired thickness.

Figure 1.1 Two approaches for rigid-foldable origami with thickness from Tachi [9], (a) the axis of rotation is shifted to the surface of the origami structure. (b) Material is removed from the panels, specifically where the folds occur. In (b), the middle grey line is the ideal origami without thickness.

In this thesis, the design of the Miura-ori experimental prototypes will attempt to promote local deformation at a crease and will utilize tapered panels. Notches cast in the substrate along crease lines will be employed to localize deformation and thereby promote folding when actuated within a magnetic field. Additionally, the panels of the Miura-ori will be tapered, resulting in crease regions with removed material that are easier to fold with the tradeoff of prohibiting the Miura-ori from reaching its maximum fold angle and hence its completely folded state.

1.2.2 The Miura-ori pattern and its properties The Miura-ori pattern was chosen as the focus of this work due to its advantageous qualities. The Miura-ori has a simple, symmetric, and repeating crease pattern, allowing for a relatively easy fabrication on a large scale. The pattern can be modified to not only adjust its size, but the ease in which it can fold and unfold by altering the dimensions of the parallelograms that comprise the pattern. The Miura-ori has garnered interest in the engineering community for its possible applications in load absorption and as actuators in control systems, which will be discussed later in this section. It is believed that the Miura-ori pattern was first discovered by Koryo Miura in 1970 by observing wrinkles in old people’s brows and photographs taken by spacecraft of Earth’s surface [10]. However, it

3 was not until 1980 that Miura introduced it as a solution to deployable structures in space, most notably solar panels and solar sails. In this work, Miura discussed the disadvantages of orthogonal folding, which is the process of folding in one direction and then in a direction perpendicular to the first. This process of folding, when considering sheets of membrane with a finite thickness, introduces maximum tensile stresses on the outer surface and orthogonal to the fold. When the second fold is introduced, another tensile stress occurs perpendicular to the new fold, in addition to creating a stress concentration at the node, which is formed when two or more folds intersect one another. These tensile stresses and stress concentration will eventually result in the formation of cracks and failure of the sheet [11]. In order to reduce these stresses and simplify the folding process, Miura believed that the following conditions were necessary for a deployable space structure. 1. Isometric condition must be held unchanged throughout the process. 2. Fold line is to be a two-dimensional tessellation of a plane by repetition of a fundamental region. 3. Folding (deploying) process itself must be complete within the fundamental region 4. Deploying process is to be done through simple, continuous, and monotonous movement. Computational analysis of plane folding was performed to satisfy these conditions, where the solutions converged to what is called a developable double corrugation (DDC) surface, as seen in Figure 1.2, and is the basis of what is now known as the Miura-ori pattern. This design will not only fold/unfold in two mutually perpendicular directions, but also do so uniformly, meaning a one degree-of-freedom system.

Figure 1.2 The bi-axial shortening of a plane into Miura’s developable double corrugation (DDC) surface

Miura further showed the benefits of this design in [12] by introducing its use in the folding of maps. Maps were originally folded by way of orthogonal folding, resulting in a complex motion of hands and fingers to fold and unfold. Miura observed that when a map is folded again once it has been laid flat, there is a high possibility that a crease can be folded incorrectly, making the original folded state no longer possible. This phenomenon – the ease of which a crease pattern can be folded erroneously – was coined ‘the stability of folds’, and Miura stated that it was required for folding a map. A comparison of a map folded orthogonally to that of Miura’s design is shown in Figure 1.3. The figure shows the pattern of movement of two thumbs when unfolding an orthogonally folded map and Miura’s folded map. It can be seen that Miura’s design is considerably easier to unfold and fold than that of the orthogonal fold. Also, the design solves the issue of fold stability, as a crease folding in the opposite direction would require every crease to do the same in order to fold the entire structure. The main disadvantage of this method of folding is that unfolding partially, that is, having one portion unfolded while the other portion is folded, is not possible as it unfolds bi-axially.

Figure 1.3 Loci of two thumbs in unfolding a map from Miura [12] of (a) orthogonal folding and (b) Miura’s design

Despite its use primarily in map folding since its inception, the Miura-ori pattern has seen more widespread interest in recent years. Horner [13] tested a deployment method of a Miura-ori experimental solar sail model utilizing four inflatable struts. S. Lui et al. [14] created a Miura-ori patterned sheet out of Elvaloy and detailed its mechanical response in out-of-plane and in-plane direction compression tests and three-point bending tests. From the tests, it was shown that the patterned sheets mostly returned to their original dimensions, suggesting repeatable uses, and their energy absorption capacities make it potentially applicable to wearable equipment, such as cushioning for shoes or sports padding. Schenk introduced two folded metamaterials, engineered smart materials possessing properties not found within nature, which were based on the Miura-ori pattern [15]. The first, a folded shell

5 structure, was a partially folded Miura-ori sheet that was modeled kinematically both in-plane and out-of- plane. It was found that this structure had Poisson’s ratios of opposite signs but of equal magnitudes for in-plane and out-of-plane deformations. The second structure, the folded cellular metamaterial, was comprised of individual Miura-ori sheets stacked and joined on top of one another. The metamaterial was still capable of expanding and contracting uniformly by having alternating Miura-sheets of differing unit cell geometries. An example of a unit cell can be seen in Figure 1.4. Schenk also observed the Miura-ori’s ability to self-lock, halting the folding motion by altering a layer’s unit cell geometry. This ability would allow the folded cellular metamaterial to self-lock into a prescribed configuration and could provide a specific stiffening response to an applied load.

Figure 1.4 Unit cell geometry of a folded Miura-ori sheet from Schenk [15]. The parallelogram facet parameters a, b, γ, and fold angle θ or dimensions H, S, V, and L can describe the unit cell geometry. ξ and ψ are angles between fold lines and the y-axis and θ and φ are dihedral angles between facets and the xy and yz planes, respectively.

The previously mentioned works show the Miura-ori’s possible applications to space design, packaging, and force absorption. Figure 1.3 shows that fewer control points are necessary to fold/unfold the Miura-ori pattern as compared to orthogonal folds, which is advantageous in a number of engineering applications. In addition, fewer control points would suggest fewer moving parts or active elements, resulting in a more robust system. S. Lui et al. [14] showed that the Miura-ori, when created from an elastic material and subjected to deformation, was capable of returning to its original shape. This exemplifies its possible uses in reusable packaging or wearable padding where deformation of the materials is a concern. As a result, the Miura-ori is a suitable candidate for magnetic actuation, which could be implemented in future projects such as deploying the solar panels of a satellite.

The previous works also have informed the design and analysis of the active Miura-prototype design. The unit cell geometry introduced by Schenk was the basis of the design of the Miura-ori pattern in this work, seen in Section 2.1. The patterns’ 1-DOF was implemented in the calculations of the magnetic torque work applied to the structure and the criteria used to rate the magnetization orientations analyzed, detailed in Sections 2.5 and 3.3, respectively.

1.2.3 Magnetorheological/Magneto-active elastomers This work will analyze self-actuating origami by way of magnetic field actuation, employing neodymium magnets as the localized actuators while investigating the design space, but the performance of MAE patches will be additionally tested. Magnetic fields were chosen for actuation due to their bidirectionality, a trait not present among other stimuli-responsive materials such as dielectric elastomers, terpolymers, SMAs, and shape memory polymers. MAEs have a relatively fast response time (0 – 0.1s), which surpasses that of SMA and photo-thermal polymer medium response times (0.1 – 0.5s) and that of shape memory polymer and photochemical polymer slow response times (>0.5s) [16]. An added benefit of magnetic field actuation is the absence of leads and wires, unlike electric field actuation where these are required. Magnetic field actuation, however, does require structures to remain in the field to actuate them, a limitation in some instances, but one that allows for in situ applications. Furthermore, magnetic field actuation in this work will utilize unidirectional actuation stimuli, i.e. the magnetic field will maintain a constant vector orientation. Magneto-active elastomers are chosen as one of the magneto-sensitive/active elements in this study due to their ability to generate magnetic torque, be cast in nearly arbitrary planar shapes, and be magnetically aligned in orientations that provide a desired resultant magnetic torque vector. They are a recently developed offshoot of traditional magnetorheological elastomers (MREs), which are smart, composite materials comprised of an elastic substrate filled with magnetic particles. MREs have a variety of uses, including media for magnetic data storage, magnetic position sensors, flexible magnets, touch- screen displays, and electromagnet shielding [17]. Their initial widespread use can be attributed to the magnetorheological effect (MR effect) they exhibit, an increase in shear modulus when subjected to a magnetic field. This characteristic is shown to depend on the type, size, and dispersion within the matrix of the magnetic particles administered to create the MR material [18]. However, only recently have MREs been considered for actuation purposes. One study of actuation compared MREs comprised of hard magnetic powders to that of their historical composition of soft magnetic filler materials, as well as the particles’ alignment, or lack thereof, in the substrate [19]. The study used DOW Corning silicone elastomer compound combined with either 325 mesh barium hexaferrite particles as a hard-magnetic filler or 40µm iron particles as a soft-magnetic

7 filler either cured in the presence of a magnetic field or not to create four classes of materials (see Figure 1.5). The study found that MREs fabricated with soft magnetic powders had zero-remanence and no preferred magnetic orientation direction whether subjected to an external magnetic field (termed aligned) or not (termed unaligned) during curing; see Figure 1.5. By contrast, MREs containing hard magnetic powders possessed a non-zero remanent magnetization that showed either strong or no direction magnetic orientation dependence in samples subjected to (aligned) or not subjected to (unaligned) a magnetic field during curing, respectively; see Figure 1.5. Furthermore, samples with hard-magnetic fillers subjected to an external magnetic field during curing were shown to provide the greatest magnetic work in cantilever bending experiments, which is shown in Figure 1.6.

Figure 1.5 Designations of (a) soft magnetic particles and (b) hard magnetic particles. (c) Four classes of MREs based on particle alignment-magnetization permutations from von Lockette [19].

Figure 1.6 Magnetic work as a function of magnetic field strength for each class of sample from von Lockette [19].

Another study sought to further characterize hard-magnetic powder-filled MREs by testing MREs comprised of various hard magnetic particles of different particle sizes [20]. The hard magnetic particles tested were barium ferrite, strontium ferrite, samarium cobalt, and neodymium magnet at a 30% volume fraction. Cantilever beams using each powder individually, magnetically aligned through the cantilever’s thickness were placed with their lengths parallel to the direction of the magnetic field. Similar to the previous study, both the displacement response and force response was measured as the magnetic field strength was increased. Figure 1.7 shows the experimental results of the study. The neodymium magnet had the greatest displacement and blocked force compared to the other materials for the given volume fraction. The neodymium powder would also have the highest remanence. The particle size between magnetic materials varied except for barium ferrite and strontium ferrite, which both possessed particle sizes of 44μm. The samarium cobalt particle size was 595μm and the neodymium magnet particle size varied from 50μm to 250μm. Because the effects of particle size were not studied and were not known to Koo (the author), only the aforementioned materials could be compared apart from volume fraction. However, Lofland and von Lockette showed that the relative MR effect is affected by particle volume fraction and independent of particle size in MREs with iron particles [21]. Therefore, the main conclusion to be taken away is that particles with higher magnetization result in higher blocked force and displacements in MREs than other particles at the same volume fraction.

Figure 1.7 (a) Displacement change and (b) block force change of hard-magnetic powder-filled MREs with different particle types from [20].

Although the actuation of the experimental Miura-ori prototypes will be accomplished by neodymium magnets in this work, preliminary testing of MREs will be performed on a Miura-ori design close to the optimum. Von Lockette showed that aligned H – MREs would be the preferred permutation, as it produced the greatest amount of magnetic work. In terms of what type of magnetic particles should be utilized in creating the aligned H – MREs, either barium ferrite or the neodymium magnet would be suitable choices as both had significant displacement and block force changes with increasing magnetic field. This work will thus use aligned H – MREs comprised of barium ferrite particles, which will be referred to as magneto-active elastomers (MAEs). Barium ferrite particles are chosen over neodymium particles due to the safety concerns of neodymium, which becomes a flammable hazard due to internal friction that can occur during mixing. The MAEs will replace the neodymium magnets by embedding them within the substrate material.

1.2.4 Neodymium magnets As stated in Section 1.2.3, magnetic fields and neodymium magnets were chosen as the actuation elements for the Miura-ori design. Neodymium, the elemental component for which the magnets derive their name, is a rare earth element. Neodymium magnets, or more accurately neodymium-iron-boron magnets, were invented in 1983 and are typically comprised of 65% iron, 33% neodymium, and 1.2% boron, with small amounts of aluminium and niobium [22]. The magnets are either produced by powder metallurgy or by a ‘melt-spinning’ process developed by General Motors. These magnets have a large 3 remanence of 퐵푟 = 1.03 − 1.3푇 and a high energy product of 199 – 310 kJ/m . Similar to MAEs, neodymium magnets can produce magnetic torque when subjected to a magnetic field.

Prior to its use in manufacturing permanent magnets, neodymium was initially used to color glass [23]. However, neodymium magnets, and thus neodymium, have seen more widespread use in the past two decades. Large electric motors, spindles for computer hard drives, cell phones, iPods®, wind turbines, lasers, glass blowers, and welder’s goggles are just some of the uses and applications of neodymium magnets. Of particular interest is their use as actuators. Filho et. al produced a planar actuator with the use of two neodymium magnets within a truck structure that was capable of moving in the y- and x-direction [24]. Imai and Tsukioka created a microelectromechanical systems (MEMS) actuator that utilized a neodymium magnet surrounded by a magnetic fluid layer encased within a cavity between polyimide diaphragms [25]. The MEMS actuator exhibited displacements of 10 – 50 µ at the diaphragm’s center when a low magnetic flux density of 4 – 30 mT was applied, making it comparable to conventional MEMS actuators. The previously mentioned works show a neodymium magnet’s capabilities as an actuator. This work will utilize small neodymium magnets attached to the Miura-ori’s surface to fold the structure. While the strength, i.e. remanent magnetization, of these magnets is important for determining the amount of magnetic torque being applied, it is not considered in this work. The theoretical work values calculated in Sections 2.5 and 3.2 are normalized with respect to both magnetization and external field strength. Work values were normalized because magnetization directions, and thus the resultant magnetic torque directions, were considered the object of study and the fundamental part of the calculations.

1.2.5 Actuation of origami structures The designs generated in origami engineering can be actuated by external stimuli as opposed to a mechanical system, which was discussed in Section 1.2.1. These external stimuli can range from a wide variety of options, including electric current, magnetic fields, heat, and pressure. Ryu et al. used polymers with residual photoabsorbing molecules and photomechanically programmed a six-sided box shape from a rectangular sheet that folded once it was cut from the sheet [26]. Martinez et al. created and tested Ecoflex-paper composites containing a pneumatic channel of various design, exemplifying their possible use as actuators by way of pressurization [8]. Okuzaki et al. were able to create a biomorphic origami robot capable of caterpillar-like motion and an accordion-shaped origami actuator that were both actuated by an electric field [27]. One work of particular note was that of Na et al., which fabricated reversible self-folding origami based on trilayer films of photo-crosslinkable copolymers [28]. The Miura-ori was one of the patterns tested and was fabricated with dimensions of 800 x 800 µm2 and thickness of 5.64µm in the flat state. A thermal field was removed from the Miura-ori pattern and the bend angles of the creases were measured using laser scanning confocal fluorescence microscopy (LSCM). The bend angles were compared to the

11 target angles that result in the folded Miura-ori shapes shown in Figure 1.8b-d. Creases with like folding angles, whether mountain or valley folds, were grouped together for the target angles. However, Figure 1.8f shows that not all creases achieved the target angle, instead showing a distribution with an observed standard deviation of ±2%. Still, the Miura-ori was capable of returning to its flat state.

Figure 1.8 (a) The mountain (solid lines) and valley (dotted lines) crease pattern for the Miura-ori pattern with an oblique angle of s = π/4. (b-d) 3D LSCM reconstructions of self-folded Miura-ori with three different target values of |θ1| (fold angle of vertical creases) and |θ2| (fold angle of 45° angled creases). (e) LSCM reconstruction of the flat state after deswelling shows a slight degree of residual buckling due to plastic strain in the PpMS layers. (f) The target (circles) and measured folding angle distributions of θ1 and θ2 in Miura-ori (b-d). (g) Fold angles are found to be reproducible through multiple cycles of heating and cooling, as plotted for the Miura-ori in (b) from Na [28].

In addition to the target angles, the repeatability of the Miura-ori shown in Figure 1.8b through several cycles of cooling and heating was analyzed. When the Miura-ori was cooled to 22°C, it exemplified highly reproducible fold angles. Conversely, after successive heating cycles of 55°C, the fold angles did not return to their flat state, instead retaining a residual angle. These results are shown in Figure 1.8g. This phenomena was attributed by the authors to a degree of plastic deformation induced in one of the Miura-ori’s layers and can be seen in Figure 1.8e. The actuation of origami structures using smart materials and primarily by magnetic or electric fields is also being studied at the Pennsylvania State University. Von Lockette & Sheridan created and modeled a composite MAE/PDMS accordion structure that deformed in a magnetic field [29]. Three designs of an L-shaped composite capable of locomotion similar to the biomorphic origami robot in [27] were additionally tested and found to have unidirectional motion under a transverse magnetic field. A

12 compliant bistable arch of PDMS with attached MAEs studied by Crivaro showed that its bistability was dependent on the thickness of the PDMS and the initial displacement of one of its ends [30]. Work done by Bowen has sought to actuate more complex origami structures, particularly the waterbomb base and the frog’s tongue [31]. The author created a dynamic model of the waterbomb, simulating a combination of torsion springs and revolute joints as a crease. An experiment was carried out to test the model using a waterbomb fabricated from polypropylene and attached neodymium magnets subjected to increasing increments of magnetic field strength. From the experiment, Bowen refined the model by adjusting the stiffness of the torsion springs and having the magnetic torques follow the motion of their respective panels. The refined model was then expanded into an optimization tool that found the optimum magnetization poling directions for the waterbomb base and frog’s tongue to achieve the highest percent completion of the fully folded state. A representation of the optimal orientation for each structure is shown in Figure 1.9.

Figure 1.9 Schematic of the optimal orientation of magnetic material for (a) the waterbomb base and (b) the frog’s tongue for an external magnetic field coming out of the page. The arrows represent the poling directions of the magnetic material and the hatched panels represent a panel that is fixed to the ground.

This work will attempt to actuate the Miura-ori pattern, similar to what Na et. al accomplished with photo-crosslinkable polymers [28], but instead with magnetically responsive materials subject to an external applied magnetic field. The folding of the creases will be measured individually and compared to the fully folded state, the ideal folding patch, and geometric symmetries within the Miura-ori geometry, similar to what is shown in Figure 1.8f. While Na analyzed how well the creases fold to satisfy a specific target shape, this work will analyze how well the magnetic material folds the Miura-ori in achieving its completely folded state in addition to how well it follows an ideal folding path. Furthermore, a numerical optimization scheme will be used to discover the optimal magnetic orientation directions of the magnetically sensitive material to maximize the magnetic work applied to the crease per unit magnetic

13 energy density, comparable to what was done in [16, 31] for the waterbomb base and the frog’s tongue. However, the method used in this work will not require fixing a panel to the ground, thus allowing the Miura-ori to fold naturally when tested experimentally.

1.3 Research objectives This work aims to effectively actuate the Miura-ori pattern using neodymium magnets within a magnetic field. The information gathered from this work will be used to maximize the magnetic torque applied to the Miura-ori creases and improve its folding behavior. After an effective design is predicted using experimental and computational methods, a prototype Miura-ori will be constructed to further assess results. The research objectives to accomplish these goals are as follows: 1. Fabricate and test the performance of an initial finite set of magnetization orientations and observe and quantify their folding behavior, 2. Model the Miura-ori to determine the overall theoretical magnetic work applied to the structure, 3. Utilize a design space trade-off/optimization program to determine the set of magnetization orientations, from an expanded design space that includes the initial set, that maximize the overall theoretical magnetic work, 4. Develop combined metrics that assess the performance of the experimental and simulated structures to develop rankings, and 5. Fabricate and test proof-of-concept structures determined from investigation of the wider design space exploration, analyzing their folding behavior and comparing to the initial structures from objective 1.

Chapter 2. Methodology

2.1 Miura-ori design The Miura-ori fold, more commonly called the Map Fold, is the focus of this work. It consists of an alternating design of mountain and valley folds and parallelogram/trapezoidal-shaped panels whose crease pattern is actuated by grabbing opposing corners and bringing them to one another. The design of this structure can be seen in Figure 2.1. The Miura-ori was selected for reasons outlined in Section 1.2.2. The proposed structure will be actuated magnetically between the pole faces of a Walker Scientific 7H electromagnet with pole faces of approximately 13.3 cm diameter spaced 8.89 cm apart. The region of the uniform field between the pole faces necessitates structures whose bounds can be prescribed within a 13.3 cm diameter circle. Figure 2.1 shows the smallest viable Miura-ori design is a 3 x 3 panel rectangular portion with each panel numbered from one to nine. This is due to a minimum of three horizontal divisions required for the Miura-ori, as two divisions result in the Frog’s Tongue origami structure. Furthermore, to have the Miura-ori spread out while unfolding as opposed to flipping over, an odd number of divisions are required [10].

Figure 2.1 Miura-ori design with mountain folds being the dashed red lines and the valley folds being the dotted blue lines. The orange highlighted region is the smallest possible iteration of this origami structure. The bright red dashed line is a vertical line and shows the angle of the vertical creases.

The creases that run almost vertically at an alternating pattern, which will be referred to as the vertical creases, do not have a prescribed angle from the vertical. These vertical creases must be angled greater than zero degrees from the vertical and less than 90 degrees from the vertical in order for the Miura-ori to fold properly. In this work, the Miura-ori design will have vertical creases that are six degrees from the vertical as shown in Figure 2.1. A six-degree angle was chosen as it is one of the more difficult Miura-ori geometries to actuate and gives better insight into how changing magnetization orientations affects folding behavior. Having vertical creases oriented further from the vertical results in a

15 smaller usable magnetic torque region for the magnetization orientations, which will be discussed in Section 2.2. Three-dimensional models of the Miura-ori were generated utilizing the SolidWorks modeling software. This model is the basis of the experimental Miura-ori substrates that will be tested later on in this work. The first model is shown in Figure 2.2. As can be seen in the figure, the creases on the bottom surface are the mountain folds and the creases on the top surface are the valley folds. All creases go through half of the thickness of the Miura-ori and have chamfered crease edges to improve bending [5]. In addition, holes of 2.54 mm radius were introduced at the four corners of panel 5 to avoid stress concentration and bending at a point, which would be difficult for the intended material that comprised the prototypes.

Figure 2.2 Top (left) and isometric (right) views of the first Miura-ori modeled in SolidWorks with holes located at the corners of Panel 5 of 2.54 mm radii and crease thickness of 1.15 mm

The second model is similar to the first, although the creases are one-third of the Miura-ori’s thickness. Larger creases were introduced to increase the maximum possible folding angle and to allow easier folding along the creases. Consequently, the holes at the corners of panel 5 are slightly larger at 3.81 mm radii and the creases have a wider chamfer. This model can be seen in Figure 2.3. Both models have dimensions of 57.15 mm height, 53.09 mm width, and 2.30 mm thickness. In addition, both models possess the same panel dimensions, which is shown in Figure 2.4. The creases in the first model have a thickness of 1.15 mm while the creases in the second model have a thickness of 0.77 mm. The dimensions of the panels are based on Schenk’s unit cell geometry [15], with each panel having a height of 19.05 mm. Due to the rectangular shape of the Miura-ori design, the outer panels (1, 3, 4, 6, 7, and 9) are trapezoidal- shaped as opposed to the parallelogram-shaped middle panels (2, 5, and 8). Thus, an outer panel has approximately 5.25% more surface area than a middle panel. This is advantageous in the design of the Miura-ori as the smaller middle panels allow for easier actuation of the structure.

Figure 2.3 Top (left) and isometric (right) views of the second Miura-ori design in SolidWorks with hole radii of 3.81 mm and crease thickness of 0.77 mm

Figure 2.4 2D Miura-ori panel dimensions for both designs, with the creases centered on the orange highlighted lines and the holes centered on the red highlighted dots. All length dimensions are in millimeters.

2.2 Magnet orientation determination Four initial configurations were chosen after considering both the magnetization and magnetic torque directions on an arbitrary panel of the Miura-ori. First, consider a Miura-ori panel separated into four regions, i…iv, with the magnetic field (H) assumed to be coming out of the page while the magnetization (m) is assumed initially in plane. These orientations place m and H orthogonal to each other and result in magnetic torque (푻 = 풎 × 푯) residing in plane, initially, as shown in Figure 2.5. The panel has four creases, A, B, C, and D, with associated crease vectors 풆̂퐴, 풆̂퐵, 풆̂퐶, and 풆̂퐷 denoting unit vectors along the crease. Next, sweeping m through each region allows calculation of the triple product

푻푘 = (풎 × 푯) ∙ 풆̂푘, where 푻푘 is the torque acting along a crease oriented along crease vector 풆̂푘 to determine if the torque created by m is favorable. Consider 풎̂ and 푯̂, unit vectors of the magnetization and external magnetic field, respectively. The cross product between 풎̂ and 푯̂ to calculate the torque T would equal 1 as m and H are orthogonal in the flat state. Thus, torque would be equivalent to the unit vector 푻̂. The dot product between 푻̂ and 풆̂풌, an arbitrary crease unit vector, to calculate the torque along a crease 푇푘, is equivalent to

푻푘 = |푇̂||풆̂푘| cos 휃 = cos 휃. Therefore, the magnetic torque along a crease is the cosine of the angle 휃 between the torque unit vector and the crease unit vector. A favorable torque on the crease (0 < 푇푘 ≤ 1) would cause a rotation as dictated by the Miura-ori pattern, i.e. folding into a mountain or valley as prescribed. An unfavorable torque (−1 ≤ 푇푘 < 0) would work against the desired fold pattern. An example would be if crease vectors 풆̂퐴 and 풆̂퐶 pointed in the +x-direction and the torque was aligned as shown in Figure 2.5, which results in a favorable torque applied to creases A and C. However, the direction of the crease vectors are not uniform from panel to panel as a result of the crease pattern. Results are compiled in Table 2.1 with favorable orientations labelled positive (+) and unfavorable orientations labeled negative (-) for each crease. As seen in the table, only a magnetic torque vector in Region ii had all four creases on the panel folding correctly.

Figure 2.5 An arbitrary Miura-ori panel divided into four regions with example torque vector and magnetization vector orientations.

Table 2.1 Sign of 푻푘 for each crease as magnetization m is swept through each region of the panel. Region ii has all creases folding in the correct direction.

Crease

Region

Given the results in Table 2.1, the initial magnetic torque T must lie within Region ii to achieve positive rotations for all creases on a panel. The arbitrary panel divided into four regions in Figure 2.5 can be rotated to match any panel permutation within the Miura-ori pattern to determine where Region ii lies. Four possible configurations of MAEs/neodymium magnets arrangements, I, II, III, and IV, were generated actuate the structure based on this requirement, and additionally under the following conditions: 1. Minimizing the necessary number of MAEs/neodymium magnets for actuation 2. Initially maximizing the magnetic torque applied to each crease when the fold is flat These conditions were chosen as boundaries; the fewest number of MAEs/neodymium magnets means less total actuators, a cost factor, but may result in limited actuation of the Miura-ori. In comparison, having the maximum amount of magnetic torque on the creases means a greater number of actuators, but possibly resulting in increased actuation of the Miura-ori. Configurations I and II arise from condition 1. Both configurations have MAEs/neodymium magnets with magnetization m aligned with the diagonal bisecting adjacent mountain and valley folds. When only considering the magnetic torque from neodymium magnets on panels and their respective adjacent creases, each crease in these configurations has one direct contribution of magnetic torque (for example, the torque on panel 2 affects its adjacent two mountain folds and one valley fold directly). Configuration I, shown in Figure 2.6, has a total of four MAEs/neodymium magnets placed on Panels 2, 4, 6, and 8. The magnetic torque arm, i.e. 푇̂, located on the diagonal results in a value on the vertical creases and horizontal creases of 푻푘 = 0.728 (휃 = 43.3°) and 푻푘 = 0.758 (휃 = 40.7°) for Panels 2 and 8, respectively. For Panels 4 and 6, the values on the vertical creases and horizontal creases of 푻푘 = 0.766 (휃 = 40.0°) and 푻푘 = 0.719 (휃 = 44.0°), respectively. This results in a relatively equal initial distribution of torque applied to each crease.

Figure 2.6 Magnet configuration I and the magnetic field direction, H.

Configuration II, as seen in Figure 2.7, has a total of five MAEs/neodymium magnets placed on Panels 1, 3, 5, 7, and 9. Similar to configuration I, T is aligned with the diagonal separating mountain and valley folds with m poled towards the mountain folds/away from the valley folds. In addition, each crease has one direction contribution of magnetic torque. Panels 1, 3, 7, and 9 have values of 푻푘 = 0.766 (휃 =

40.0°) and 푻푘 = 0.719 (휃 = 44.0°)and Panel 5 has values of 푻푘 = 0.728 (휃 = 43.3°) and 푻푘 = 0.758 (휃 = 40.7°) for the vertical and horizontal creases, respectively.

Figure 2.7 Magnet configuration II and the magnetic field direction, H.

Configuration III is a combination of conditions 1 and 2, maximizing the magnetic torque along the creases by placing the torque vector on the boundary of region ii but also limiting the number of MAEs/neodymium magnets, which are located on Panels 1, 2, 3, 7, 8, and 9. The reasoning behind this arrangement is that the MAE/neodymium magnets on Panels 1, 3, 7, and 9 provide a torque along their adjacent vertical creases of value 푻푘 = 1, as the torque arm is parallel to the vertical creases (휃 = 0°).

The horizontal creases additionally receive a small portion of torque, equivalent to 푻푘 = 0.105 (휃 = 84°), because the torque arm is aligned 6 degrees form the vertical. The vertical creases in the middle are not directly acted on by the torques, but their folding will rely on the actuation of the aforementioned panels. As for Panels 2 and 8, the MAE/neodymium magnets provide a torque along their adjacent horizontal creases of value 푻푘 = 1, with the other horizontal creases folded indirectly from these

20 two panels. The adjacent vertical creases of Panels 2 and 8 also see a small portion of torque, equivalent to 푻푘 = cos 84° = 0.105. This configuration can be seen in Figure 2.8.

Figure 2.8 Magnet configuration III and the magnetic field direction H.

Configuration IV is based on condition 2 of maximizing the initial magnetic torque applied to each crease. As seen in Figure 2.9, each panel has a MAE/neodymium magnet attached with magnetization orientations identical to that of configuration III. The MAE/neodymium magnets on panels 1, 3, 5, 7, and 9 have magnetic torques aligned with their adjacent vertical creases, resulting in a torque along the crease value of 푻푘 = 1. These panels also contribute a small portion of magnetic torque along their adjacent horizontal creases in a favorable direction, which is 푻푘 = cos 84° = 0.105. Panels 2, 4, 6, and 8 have magnetic torques aligned with their adjacent horizontal creases, resulting in a torque along the crease value of 푻푘 = 1. Similar to the other panels, these contribute a small portion of magnetic torque along their adjacent vertical creases in a favorable direction, which is 푻푘 = cos 84° = 0.105.

Figure 2.9 Magnet configuration IV and the magnetic field direction H.

2.3 Substrate fabrication The Miura-ori substrates used in this work were created using DOW Sylgard 184 silicone rubber compound with a 10:1 base to catalyst ratio. SolidWorks models of molds were generated for the desired Miura-ori designs shown in Figures 2.2 and Figure 2.3. A representative mold is given in Figure 2.10.

The bottom mold holds the material for the prototype and creates all of the mountain folds. The top lattice mold creates all of the valley folds and allows the Sylgard to remain open to the environment for curing purposes. Due to the intricacies of the design, the molds were built with a Dimension® sst 1200es 3D printer out of ABS plastic. A total of ten mold sets were made: five of the half-thickness crease (HTC) model and five of the one-third-thickness crease (OTTC) model.

Figure 2.10 Representative bottom (left) and top lattice (right) mold for Miura substrate casting.

Three coats of Huntsman RenLease® 78-2 Aerosol Silicone Release Agent were applied to the surface of each mold to ease the removal process. The DOW Sylgard 184 compound was mixed, filling the bottom mold partially, then placing the top lattice structure onto the bottom mold. Additional compound was used to fill the mold until full and was repeated for each mold set. The molds were placed in the Model 281A Isotemp® Vacuum Oven connected to a Model TW-1A 1 Stage Vacuum Pump. The molds were subjected to a low vacuum of 54 kPa at room temperature (23°C – 26°C) for 25 – 30 minutes to remove any air pockets that may have been trapped within the compound during the filling process. The molds were removed from the vacuum oven and additional compound was added when necessary to compensate for material that leaked out of the mold during vacuuming or shrinkage due to release of entrained air. The material inside the mold was allowed to rest for 40 – 45 minutes on a level surface to achieve uniform thickness. All surface bubbles were removed by applying a low air current of compressed air and the molds were reinserted into the vacuum oven. The material was subjected to a temperature of 75°C for 45 – 50 minutes to cure. Once cured, the material was allowed to rest within the molds for 24 hours before being carefully removed from each mold. Additional repetitions of this process were carried out until a total of 20 HTC substrates and 20 OTTC substrates were created. Three substrates of each MAE/neodymium magnet configuration were tested: a total of 12 HTC substrates and 12 OTTC substrates. Each substrate was consequently analyzed by measuring the thickness of each of the nine panels using a Starrett 2720-1 Wisdom Electronic Indicator. Substrates which had all panels fall within two standard deviations of the mean panel thickness of 2.204 mm were selected. Of the substrates that satisfied this condition, 12 were chosen from each design based on the smallest individual standard deviations. The data and selection process are shown in Appendix A in Tables A.1, A.2, and A.3.

With the selection of experimental substrates to be tested completed, neodymium magnets of dimension 3.175 mm x 3.175 mm x 3.175 mm were next attached onto the Miura substrates. First, Delrin sheets of thickness 0.3 mm were cut and attached to each panel of the top surface of each substrate. The neodymium magnets were attached on top of the Delrin sheets according to each configuration’s magnetization orientations. Delrin sheets were used to increase the rigidity of the panels and promote folding at the creases instead of deformation of the panels. The shape of each Delrin sheet was based on the dimensions of the panels detailed in Section 2.1, but at a reduced size due to the width of the creases. This was to ensure that the folding behavior occurred at the prescribed creases and in the same direction. Both the sheets and magnets were attached using Loctite® Plastics Bonding System. An example image of a finalized experimental substrate can be seen in Figure 2.11.

Figure 2.11 End-product of Miura substrate fabrication with attached Delrin sheets cut to fit each panel and neodymium magnets arranged in configuration III.

2.4 Experimental setup

One of each configuration from both models was tested in a 휇0퐻 = 0 … 240 푚푇 (milliTesla) magnetic field. The OTTC prototype was observed to have relatively symmetric actuation, similar to that of the HTC prototype. However, the OTTC prototype resulted in a greater degree of actuation than the HTC prototype. This was expected due to the OTTC prototype having a wider crease and less material, i.e. a smaller thickness, resulting in more actuation. Consequently, the OTTC model was pursued for experimental data collection. For testing, each prototype was placed on a Teflon sheet suspended 3.175 mm above the bottom pole face of the Walker Scientific 7H electromagnet by an Acrylic test stand. The test stand can be seen in Figure 2.12. Both the Teflon sheet and the underside of the prototype were sprayed with a coating of Huntsman RenLease® 78-2 Aerosol Silicone Release Agent as a lubricant. For each succeeding prototype, the Teflon sheet was wiped clean and a new coating was applied. Holes cut in the Teflon sheet

23 by a 2626 JetMachining® Center water jet corresponding to the vertices of the valley vertical creases were aligned with the prototype and a designated crease was fixed to the Teflon sheet by using soldering wire. This was done to prevent the magnetic field from lifting the prototype and thus allowed the desired actuation of individual panels.

Figure 2.12 Acrylic test stand suspending the Teflon sheet. The circular base of the stand allows it to fit around the bottom pole face of the magnet.

The magnetic field strength was measured using a LakeShore 475 DSP Gaussmeter with its probe affixed to the bottom pole face of the Walker Scientific 7H electromagnet. The magnet was controlled by an HP E3615A DC power supply connected to an ELGAR 1751SX AC power amplifier. The prototypes were subjected to a magnetic field strength of 0 mT – 240 mT at 30 mT intervals. Each field strength was tested three times with an accuracy of 휇0퐻 = ±0.30 푚푇. An Extech BR350 Borescope and a Canon EOS Rebel T5i Camera (5184 x 3456 pixels) were used to take image captures of various creases’ fold angles, with the borescope probe manually aimed parallel to the crease being measured. This setup is shown in Figure 2.13. Due to the limited space within the magnet and magnetic interference with devices containing ferromagnetic metals, only six of the twelve total creases on each prototype were measured. Measured creases are shown and labeled in Figure 2.14.

Figure 2.13 Experimental setup within the big magnet. The Gauss meter probe extends underneath the Teflon Sheet (not shown) and is affixed to the top of the bottom poling face. The Acrylic Test Stand and Teflon Sheet allow the prototype to be rotated 360 degrees but remain in the center of the poling face. Paper spacers were introduced to prevent bowing of the Teflon Sheet where the Miura prototype was placed.

Figure 2.14 Designated crease and panel numbering of the Miura-ori. The creases highlighted in green (vertical creases 1, 2, 11, and 12 and horizontal creases 3 and 8) were measured in the experiment.

Fold angles were determined using digital image processing with the commercially available software, ImageJ. Lines marked on the undeformed substrate served as guides for locating creases in the deformed state. An example image of how a crease fold angle was measured is shown in Figure 2.15. A total of eight angles with three points measured for each angle were measured. The data set per configuration amounted to 432 angle measurements across each of the three prototypes’ six measurable creases.

Figure 2.15 Image capture of the folding Crease 1 of a configuration IV prototype. The black markings give a straight-edge reference for measuring the fold angle, which can be output directly by the ImageJ software.

2.5 Magnetic work analysis In addition to experiments, predictions of work performed on the structure by magnetic torques were used to assess performance across the configurations outlined in Section 2.2. Prediction of this ‘magnetic work’ required complete trajectories of all creases as they processed from fully open to fully folded states. Trajectories for each crease in the Miura-ori pattern used for this study were generated by the Rigid Origami Folder model [32-34]. The Rigid Origami Folder calculates non-intersecting folding paths of rigid panels within the pattern given a mountain-valley crease pattern constrained in a flat sheet. These simulation necessarily calculates relative points in space of all intersections in the crease pattern from which geometric data can be computed at arbitrary discrete steps along the folding path. The x-, y-, and z-coordinate locations of all vertices throughout the folding process were extracted from the model and used to calculate the work done by magnetic torque. To calculate work performed by the magnetization on a panel, it was necessary to determine the geometry of each panel computationally throughout the folding motion of the structure. Defining the magnetization on each panel with respect to adjacent crease vectors allows the calculation of both the magnetization and torque trajectories. The calculations begin by defining the crease vector:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 퐶푖 푁 = [푐푥푖 푁 , 푐푦푖 푁 , 푐푧푖 푁 ] = [푝푥푟 푁 − 푝푥푞 푁 , 푝푦푟 푁 − 푝푦푞 푁 , 푝푧푟 푁 − 푝푧푞 푁 ] (1) th where 퐶푖 is the i crease vector with components taken from step 푁 in the Rigid Origami Folder model from adjacent vertex points 푝푟 and 푝푞. Figure 2.16 shows a graphical representation of these values. Next,

26 the magnetization vector associated with a magnet on a given panel is determine using two crease vectors adjacent to the panel as its basis functions. The magnetization vector takes the form 푴 = 훼푪1 + 훽푪2 where 훼 and 훽 are unknown coefficients and 푪1 and 푪2 are crease vectors extracted from ROF. From this, the magnetization vector can be computed at every step along the folding path from ROF data. In component form: ( ) ( ) ( ) 푀푥 푁 = 훼 ∗ 푐푥1 푁 + 훽 ∗ 푐푥2 푁 (2푎) ( ) ( ) ( ) 푀푦 푛 = 훼 ∗ 푐푦1 푁 + 훽 ∗ 푐푦2 푁 (2푏) ( ) ( ) ( ) 푀푧 푛 = 훼 ∗ 푐푧1 푁 + 훽 ∗ 푐푧2 푁 (2푐)

Figure 2.16 Representation of the crease vector 퐶푖. The points 푝푟 and 푝푞 are located at adjacent crease vertex points, with crease vector 퐶푖 connecting the two points. With the values of 훼 and 훽 calculated for each magnetization vector on a given panel, the magnetization vector takes the form of ( ) ( ) ( ) ( ) ( ) ( ) 푀(푁) = [훼 ∗ 푐푥1 푁 + 훽 ∗ 푐푥2 푁 훼 ∗ 푐푦1 푁 + 훽 ∗ 푐푦2 푁 훼 ∗ 푐푧1 푁 + 훽 ∗ 푐푧2 푁 ] (3) For any step 푁 along the computed folding path. Several representative panels and their respective magnetization vectors are plotted within MATLAB at various steps of the folding process for illustration in Figure 2.17.

Figure 2.17 Panel movements during the folding process with the blue lines representing the boundaries of the panel and the red arrow representing the magnetization direction of (a) Panel 1, configuration II (b) panel 2, configuration III/IV (c) panel 4, configuration IV and (d) panel 5, configuration II. In (a), panel 1 has been translated in the x- direction to better observe its motion. In (d), panel 5 is shown to not move as the Miura-ori folds.

Magnetic torque, defined earlier as 푻 = 풎 × 푯, requires the determination of an external magnetic field vector H. However, the Rigid Origami Folder model from which vertex points are taken fixes panel 5, as a panel is required to be fixed in order to fold any origami structure, including the Miura- ori pattern. For the computational analysis to coincide with that of the experimental results where a crease was fixed, the magnetic field H cannot be fixed in the positive z-direction. From observations of folding Miura-ori structures and the folding kinematics of the Miura-ori [11,16], it was seen that the vertical valley creases (creases 1, 6, and 11) all remain in the same x-y ground plane. This can be seen in Figure 2.18 [16]. Because the Miura-ori is considered a 1-DOF system [11], symmetric folding of panels about a crease can be assumed. Therefore, the bisector of crease 6’s folding angle is collinear with the positive z- direction of the magnetic field H. The fold angles for the horizontal creases and vertical creases were calculated by determining the angle between panels 4 and 5 and the angle between panels 2 and 5, respectively. The angle between two planes is equivalent to the angle between the planes’ normal vectors,

28 which can be solved using the definition of the dot product, 푎 ∙ 푏 = |푎||푏| cos 휃. Rearranging the dot product equation to solve for 휃 results in the following equations:

−1 푢̂2(푁) ∙ 푢̂5(푁) 휃퐻(푁) = cos ( ) (4) |푢̂2(푁)| ∗ |푢̂5(푁)|

−1 푢̂4(푁) ∙ 푢̂5(푁) 휃푉(푁) = cos ( ) (5) |푢̂4(푁)| ∗ |푢̂5(푁)| where 휃퐻(푁) and 휃푉(푁) are the horizontal and vertical fold angles, respectively, and 푢̂2(푁), 푢̂4(푁), and

푢̂5(푁) are the unit normal vectors to panels 2, 4, and 5 respectively. The equation of the bisector, and consequently the magnetic field H is thus

푒̂퐻(푁) = [cos(0.5 ∗ 휃푉(푁)) ∗ cos 휑 cos(0.5 ∗ 휃푉(푁)) ∗ sin 휑 sin(0.5 ∗ 휃푉(푁))] (6) where 휑 = 6° is the angle of the vertical creases. The orientation of the magnetic field H as the Miura-ori folds from its initial flat state to its completely folded state is shown in Figure 2.19.

Figure 2.18 Folded cellular metamaterial of individual Miura-ori sheets with alternating unit cell geometries from Schenk [15]. The stacked configuration preserves the folding kinematics of the Miura-ori, and it can be observed that the valley vertical folds of layer B all remain in the same horizontal plane as the structure folds and unfolds, as outlined in blue. The same can be stated for the mountain vertical folds of layer B, which are outlined in red.

0.16

0.14

0.12

0.1

0.08

0.06

Z-coordinate 0.04 N = 1 N = 25

0.02

-0.02 0

-0.5

-1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Y-coordinate X-coordinate Figure 2.19 Directionality of the magnetic field H (bisector of crease 6’s fold angle) during the folding of the Miura-ori. The magenta lines outline the stationary panel 5.

With both the magnetizations and magnetic field defined, the magnetic torque unit vectors and the component of magnetic torques acting along creases were calculated from the triple product of equations (1), (3), and (6). The initial cross product yields the magnetic torque vector which is then normalized by dividing by its magnitude. The final dot product yields the magnetic torque applied to a crease: 푀(푁) × 푒̂ (푁) 푇̂(푁) = 퐻 (7) ‖푀(푁) × 푒̂퐻(푁)‖

푇푖(푁) = 퐶̂푖(푁) ∙ 푇̂(푁) (8) 푡ℎ where 푇̂(푁) is the magnetic torque vector, 퐶̂푖(푁) is the 푖 crease unit vector, and 푇푖(푁) is the applied magnetic torque on a crease at an arbitrary step 푛 in the fold path. With the magnetic torque and crease magnetic torque defined, the magnetic work applied to each crease and the total normalized work applied to the Miura-ori can be calculated. The incremental magnetic work applied to each crease is defined as ( ) ̅ ( ) ( ) ̅ [̅ ̅ ̅ ̅ ̅ ̅ ] 푊퐻휆 푁 = 푇휆 푁 ∗ ∆휃퐻 푁 푇휆 = 푇3, 푇4, 푇5, 푇8, 푇9, 푇10 (9) ( ) ̅ ( ) ( ) ̅ [̅ ̅ ̅ ̅ ̅ ̅ ] 푊푉휁 푁 = 푇휁 푁 ∗ ∆휃푉 푁 푇휁 = 푇1, 푇2, 푇6, 푇7, 푇11, 푇12 (10) ( ) ( ) where 푊퐻휆 푁 is the magnetic work on a horizontal crease, 푊푉휁 푁 is the magnetic work on a vertical crease, 푇̅휆(푁) and 푇̅휁(푁) are the average crease magnetic torque between steps 푁 and 푁 + 1 for horizontal and vertical creases, respectively, ∆휃퐻(푁) is the fold angle change (in radians) of the horizontal creases, and ∆휃푉(푁) is the average fold angle change (in radians) of the vertical creases. Both

∆휃퐻(푁) and ∆휃푉(푁) are both calculated under the assumption of the small-angle approximation, as the

30 change in angle for both the vertical creases and horizontal creases did not exceed a tenth of a radian. The total normalized work applied to the Miura-ori is the summation of the magnetic work applied to each crease over 푛 steps, i.e. the summation of equations (9) and (10):

128 6 6 1 ( ) ( ) 푊푙푡표푡 = ∑ [∑ 푊퐻휆 푁 + ∑ 푊푉휁 푁 ] (11) 훤푙 푁=1 휆=1 휁=1 where 훤푙 is the number of MAEs/neodymium magnets on configuration l. The total magnetic work was normalized as each configuration possesses a different number of neodymium magnets. Therefore determining the amount of work each neodymium magnet performs throughout the folding process of the Miura-ori allows comparison between configurations. The results of these calculations will be discussed in further detail in Chapter 3. To view the full MATLAB code, see Appendix B.

Chapter 3. Results and Discussion 3.1 Experimental data analysis As discussed in Section 2.5, each Miura prototype was subjected to an increasing external magnetic field, with each measurable crease’s fold angle taken at intervals of magnetic field strength. For example, Figure 3.1 shows a configuration II prototype folding at increasing external magnetic field strengths. Images of the other configurations’ actuation in the magnetic field can be seen in Appendix C. Each set of prototypes within each configuration came from a different batch of substrate material which is shown in Table 3.1.

Figure 3.1 Configuration II prototype within the Walker Scientific 7H electromagnetic subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

Table 3.1 List of prototypes and their respective batch within each configuration

Substrate Batch A B C D Prototype 1 X Prototype 2 X

Config. I Prototype 3 X

Prototype 1 X Prototype 2 X

Config. II Prototype 3 X

Prototype 1 X Prototype 2 X X

Config. III Prototype 3

Prototype 1 X Prototype 2 X X Config. IV Prototype 3

The results of the experiment can be seen in Figure 3.2 which shows the fold angle of a given crease versus external field strength averaged over three prototypes, each of which were tested three times for repeatability, per configuration. The error bars within each figure represent the standard deviation of the fold angle data collected at each external field strength between the 9 sets of data collected for each configuration. Consequently, the error bars combine variance due to fabrication with that due to experimental procedure. The experimental data can be seen in Appendix C. Creases 1, 2, 11, and 12 had maximum average fold angles, at an external field strength of 240 mT, ranging from 52 – 110 degrees across all configurations. As for creases 3 and 8, their maximum average fold angle response at the maximum external field strength range from 0 – 27 degrees. The maximum average fold angle of the horizontal creases across all configurations was 17.06 degrees with a standard deviation of 10.18 degrees. The maximum average fold angle of the vertical creases across all configurations was 79.05 degrees with a standard deviation of 17.54 degrees.

130 130

110 110

90 90

70 70 Angle (deg) Angle

50 50

30 30 Average FoldAverage 10 (deg) Angle Average Fold 10

-10 0 50 100 150 200 250 -10 0 50 100 150 200 250 Field Strength (mT) Field Strength (mT)

(b) (a) 130 130

110

110 90 90 70 70 50 50 30

30 Average (deg) Angle Average Fold

Average (deg) Angle FoldAverage 10 10 -10 0 50 100 150 200 250 -10 0 50 100 150 200 250 Field Strength (mT) Field Strength (mT) (d) (c)

Crease 1 Crease 2 Crease 3 Crease 8 Crease 11 Crease 12

Figure 3.2 Average Fold Angle vs. Applied Field Strength results for all prototypes of (a) configuration I, (b) configuration II, (c) configuration III, and (d) configuration IV. The error bars show the standard deviation from the three measurements taken per field strength per prototype, resulting in a total of nine fold angle measurements per field strength setting. The negative average fold angle values in (a) and (c) represent a prescribed mountain and/or valley fold folding into a valley and/or mountain fold, respectively. The square data points represent vertical creases 1, 2, 11, and 12 and the diamond data points represent the horizontal creases 3 and 8.

For most crease responses, it can be observed that as the field strength increases, the average fold angle increases, but at a decreasing rate, e.g. the response saturates. However, some cases do remain linear, notably creases 3 and 8 in configurations I, II, and IV (Figures 3.2a, b, and d). Saturation is attributed to the neodymium magnets’ magnetizations becoming more aligned with that of the applied magnetic field direction, resulting in less magnetic torque applied to folding the creases. It is assumed that the neodymium magnets’ magnetizations remain at their remanence values and are unaffected by the field strengths used; ideal hard-magnetic behavior is assumed. Comparing the configurations, configuration IV

(Figure 3.2d) yielded the greatest average fold angles, ranging from 75 – 110 degrees for the vertical creases and 22 – 27 degrees for the horizontal creases, and thus the greatest amount of actuation among the Miura prototypes. Configuration III (Figure 3.2c) exhibited maximum average fold angles ranging from 50 – 92 degrees among its vertical creases, but very little actuation in its horizontal creases, which had maximum average fold angles less than 10 degrees. The maximum average fold angle values for each configuration can be seen in more detail in Figure 3.3 and Table 3.2. Table 3.2 Maximum average fold angles (240 mT external field strength) in degrees for each configuration

Maximum Average Fold Angle (deg)

Crease 1 Crease 2 Crease 3 Crease 8 Crease 11 Crease 12 Config. I 52.1 70.9 8.4 23.0 75.7 76.5 Config. II 89.7 84.0 24.7 23.2 75.4 61.2 Config. III 86.2 52.1 7.2 0.4 92.4 66.7 Config. IV 93.6 75.5 27.1 22.4 110.3 102.6

120

100

0 Crease 1 Crease 2 Crease 3 Crease 8 Crease 11 Crease 12 Average Average FoldAngle 240 at mT (deg) -20

Configuration I Configuration II Configuration III Configuration IV

Figure 3.3 Bar chart representation of the maximum average fold angles for each crease and configuration shown in Table 3.1. The error bars represent one standard deviation.

It should be noted that crease 11 for configurations III and IV and crease 1 for configuration II had the greatest average fold angles among other creases within each configuration. This can be explained by the fact that the aforementioned creases were those chosen to be fixed for the experiment. While the fixed creases experienced purely folding behavior, the other unfixed creases experienced both folding and translational motion from the magnetic torques applied by the neodymium magnets. On average, the values of the second highest average fold angles for the aforementioned configurations were 93.3% of the maximum average fold angles. As for configuration I, which was fixed at crease 11, had a value for its

35 second highest average fold angle maximum average fold angle of 99.0% of the maximum average fold angle, which occurred at crease 12. This suggests that any effects due to boundary conditions on the structure are small with respect to the total actuation. The one standard deviation error bars in Figure 3.2 and Figure 3.3 represent the error from both the fabrication variance among the three prototypes within each configuration and the variance in the measured crease fold angles across the three cycles within each prototype. Another metric utilized to compare configurations was the ability to follow the ideal behavior of a folding Miura-ori pattern. In the ideal case, all horizontal creases fold the same amount and likewise all vertical creases fold the same amount for a given external field strength; horizontal and vertical creases however fold different amounts for a given external field. Figure 3.4 plots average mountain fold angle vs. average valley fold angle, showing visually how effective each configuration was in achieving ideal folding behavior. The data were calculated by averaging the fold angles of creases with mountain folds and plotting against the average fold angle of creases with valley folds. In other words, the vertical crease data are the average fold angles of crease 2 and crease 12 (mountain folds) vs. the average fold angles of crease 1 and 11 (valley folds). The horizontal crease data are the average fold angles of crease 8 (a mountain fold) vs. the average fold angles of crease 3 (a valley fold). The vertical error bars represent one standard deviation between creases 2 and 12 and that of crease 8 for the vertical and horizontal creases, respectively. The horizontal error bars represent one standard deviation between creases 1 and 11 and that of crease 3 for the vertical and horizontal creases, respectively.

110 110

90 90

70 70

50 50

30 30

10 10

Average Mountain Fold Angle (deg) Angle Fold MountainAverage Average (deg) Angle FoldAverage Mountain

-10-10 10 30 50 70 90 110 -10-10 10 30 50 70 90 110 Average Valley Fold Angle (deg) Average Valley Fold Angle (deg)

(a) (b)

110 110

90 90

70 70

50 50

30 30

10 10

Average (deg) Angle FoldAverage Mountain Average Mountain Fold Angle (deg) Angle Fold MountainAverage

-10-10 10 30 50 70 90 110 -10-10 10 30 50 70 90 110 Average Valley Fold Angle (deg) Average Valley Fold Angle (deg)

Ideal Behavior Vertical Creases Horizontal Creases

Figure 3.4 Mountain Fold Angle vs. Valley Fold Angle compared to the ideal, unit slope, behavior at increasing field strength. Results shown for vertical and horizontal creases in (a) configuration I, (b) configuration II, (c) configuration III, and (d) configuration IV. Increasing applied magnetic field strength is in the +x/+y direction.

From Figure 3.4, it appears that configurations II and IV possess both vertical and horizontal creases that followed the ideal behavior most closely. Configuration I’s vertical creases similarly followed the ideal behavior, but the horizontal creases deviate as crease 8 folded to a greater degree than that of crease 3, which can be seen in Figure 3.2a. As for configuration III, the figure shows the slight actuation of the horizontal creases and the valley fold (creases 1 and 11) favored actuation of the vertical creases.

A calculated metric of the data shown in Figure 3.4 was constructed to quantify observed behavior. The experimental deviation from ideal behavior was calculated from:

푁 2 1 휃푒 퐷̅ = 1 − √ ∑ (1 − 푖) (12) 푁 휃푓 푖=1 푖 where 퐷̅ is the average deviation of the vertical and horizontal creases, 푁 is the number of points 휃 collected, and 푒푖 is the ratio between creases 푒 and 푓 at the 푖푡ℎ data point (which corresponds to a given 휃 푓푖 external field strength). The ratios compared using equation (12) are: 휃 휃 1,11 푎푛푑 3 휃2,12 휃8 휃 where 1,11 is the ratio of the average fold angles between creases 1 and 11 (valley folds) and the average 휃2,12 휃 fold angles between creases 2 and 12 (mountain folds) and 3 is the ratio of average fold angles between 휃8 crease 3 (valley fold) and crease 8 (mountain fold). A value of 1 would signify ideal folding behavior between all mountain and valley folds of the Miura-ori pattern at each field strength. This calculation was done for each prototype test, a total of three tests per prototype, for each of the three prototypes in a configuration for the vertical and horizontal creases (푁 = 18) and the results are shown below in Table 3.3.

Table 3.3 Average deviation values 퐷̅ for each configuration when comparing mountain vs. valley fold behavior across all measured creases.

Field Strength (mT) Config. I Config. II Config. III Config. IV 0 1.000 1.000 1.000 1.000 30 0.965 0.978 0.918 0.954 60 0.943 0.958 0.875 0.926 90 0.925 0.946 0.848 0.917 120 0.907 0.941 0.833 0.907 150 0.899 0.941 0.823 0.897 180 0.892 0.937 0.828 0.896 210 0.887 0.939 0.824 0.893 240 0.883 0.930 0.820 0.894 Average 0.922 0.952 0.863 0.920 Standard Deviation 0.040 0.023 0.061 0.036

The analysis shows that with respect to mountain vs. valley fold behavior, configuration II is the best as it is closest to a value of 1, i.e. perfectly ideal behavior, at all field strengths, as well as its average

퐷̅ values over all field strengths. Additionally, configuration II has the smallest standard deviation of 퐷̅ values, meaning it has the most consistent ideal behavior. As expected, configuration III was the furthest from ideal behavior due to significantly more folding along the valley folds than that of the mountain folds across both vertical and horizontal creases. While the deviation of experimental behavior from that of ideal behavior and its expression across configurations is primarily driven by arrangement of the neodymium magnets, some degree of the response and its variation within and across prototypes can be attributed to the fabrication process of the prototypes and measurement of creases across cycles within a prototype. This variance is quantified by standard deviation error bars shown in Figure 3.4. As discussed in Section 2.5, the total magnetic work potential for each configuration was calculated from the simulated Miura-ori structure to provide estimates for the magnetic work potential experienced by experimental prototypes of the four configurations. The results of these calculations can be seen in Figure 3.5, which shows magnetic work done per total magnetic energy potential as a function of horizontal fold angle and vertical fold angle. Configuration I was found to have the highest amount of normalized work performed on its creases with a value of 3.40. Configurations II, III, and IV had normalized work values of 2.76, 2.66, and 2.83, respectively. It should be noted that the vertical creases of the Miura-ori structure dominate the horizontal creases with regards to folding in the beginning portions of the Miura-ori actuation. Considering that the magnitude of the magnetic torque applied to the creases is greatest at the Miura-ori’s flat state (when m and H are orthogonal), the bulk of the magnetic work therefore comes from the folding of the vertical creases. This is evident in Figure 3.5, where the magnetic work and the vertical angle increase steeply relative to the increase in horizontal angle during the Miura-ori’s initial folding.

Configuration I Configuration II Configuration III Configuration IV 3.5

2.5

1.5

0.5

0 200 180 160 140 200

Magnetic Work Done/Total Magnetic Energy Potential [] Potential Energy Magnetic Work Done/Total Magnetic 180 120 160 140 100 120 80 100 60 80 60 40 40 20 20 0 0 Horizontal Fold Angle (deg) Vertical Fold Angle (deg)

Figure 3.5 Simulated magnetic work done per total magnetic energy potential as a function of Horizontal angle (fold angle of horizontal creases) and Vertical Angle (fold angle of vertical creases). At zero degrees for both Horizontal and Vertical angle, the Miura-ori structure is in its initial flat state; it is completely folded at 180 degrees for both angles.

3.2 Design space exploration In order to observe how the magnetic torque work changes with respect to the orientation of each neodymium magnet/MAE patch for each configuration, the MATLAB function ‘Fminsearch’ was initially used to find each configuration’s magnetic orientation that results in the maximum amount of magnetic work. Magnetic orientation symmetry, as shown in Figure 3.6, was assumed in the script and results in four independent magnetizations. Note that each configuration uses a specific subset of the available independent orientation variables, 휓푖; in Table 3.4, the blacked out columns for a given configuration are not required for that configuration. The assumption corresponds to ideal behavior, which allows for comparison across best case scenarios even though experimentally, a given configuration may not follow the ideal response. Each configuration was run a total of 100 times, with each run having a new, randomized starting point. The resultant normalized work values and corresponding independent magnetization values can be seen in Table 3.4. The runs for configurations I, II, and III provided similar results for both theoretical work and magnetization orientations, all of which had variances that did not exceed 5.11 x 10-23. Configuration IV had a significantly higher variance among its 100 runs, resulting in a value of 0.0146, which is still small relative to the normalized work value. Configuration I generated the

40 most theoretical normalized magnetic work yielding a value of 5.20, 12.5% above the second highest value of configuration IV. The full code for each configuration of Fminsearch is detailed in Appendix D.

Figure 3.6 Panel (yellow) and magnetization orientation numbering for the Miura-ori. Simplification by symmetry yields the independent magnetizations ψ1 (red colored magnetization angles), ψ2 (black colored magnetization angles), ψ3 (green colored magnetization angles), and ψ4 (purple colored magnetization angle).

Table 3.4 MATLAB’s Fminsearch results for maximizing the theoretical normalized magnetic work.

Config. Norm. Work ψ1 (deg) ψ2 (deg) ψ3 (deg) ψ4 (deg) I 5.20 -6.1 187.3 II 4.16 173.8 6.1 III 4.51 173.8 -6.1 IV 4.62 173.8 -6.1 187.3 6.1

However, several runs for configurations IV either exceeded the maximum number of function evaluations and stopped at a lower theoretical normalized magnetic work value (15 runs) or converged at a lower or higher value, and thus different magnetization orientations, than the values shown in the table. The runs that exceeded the maximum amount of function evaluations had work values ranging from 92.9% – 99.9% of 4.62. The lowest theoretical normalized work values converged to be only 76.7% of the work value in Table 3.4. The higher theoretical work value was only slightly larger, being only 0.048% greater than the work value shown for configuration IV. This is most likely due to either the limitations of Fminsearch, which may only give local solutions, or the default termination tolerances that stop the iterations of Fminsearch. Consequently, the ARL Trade Space Visualizer (ATSV) created by ARL at The Pennsylvania State University was utilized to gain a better understanding of the design space. ATSV allows the user to visualize the entire trade space, a point of interest, or a region of high preference defined by the user of an objective function comprised of multiple combinations of design parameters [35-37]. This enables the user to steer the focus of the trade space exploration and locate the best design.

Trade space exploration was performed by linking the existing MATLAB code used to calculate normalized magnetic work to the ATSV program. See Appendix D for the MATLAB code for each configuration. The orientations of the magnets were altered and the resultant normalized magnetic work was calculated on each run. Two cases were considered when using ATSV. For the first case of ATSV, the magnetization orientations were simplified by way of symmetry in the MATLAB function. Figure 3.6 shows the labeling of the resulting reduced set of independent magnetizations angles 휓1 … 휓4. The Basic Sampler was run 500 times to populate the design space. Each configurations simplified design space can be seen in Figure 3.5. Magnetizations 휓1 and 휓3 were allowed to range from 0 to 360 degrees, but magnetizations 휓2 and 휓4 range from -180 to 180 degrees to obtain a better view of the design space. Configuration IV in Figure 3.7d varies in appearance due to it utilizing four independent magnetizations angles, as opposed to the other configurations that only have two independent magnetizations, which requires additional iconography to distinguish variation in the results of the additional variables. However, it can be seen that it still possesses the same peak shape as what is shown in Figure 3.7a, b, and c.

Figure 3.7 Trade spaces for (a) configuration I, (b) configuration II, (c) configuration III, and (d) configuration IV. Each trade space assumes magnetization orientation symmetry. For (d), the peak shape shown in (a), (b), and (c) is still visible, but not restricted to one contour. 휓3 is defined by the size of the point, small to large mirroring 0° to 360° and 휓4 is defined by the color of the point, blue to red mirroring -180° to +180°. As can be observed from Figure 3.7, each design space appears to converge to a point of maximum normalized work. Subsequently, the Preference Sampler of ATSV was then utilized to find the design for each configuration that resulted in the highest amount of normalized work. Using the Brush/Preference Controls, the variable ‘Norm. Work’ was set to be maximized, and the design space was modified to only show points that had a positive normalized work value. The Preference Sampler was run 250 times, with the resulting design spaces, essentially more detailed depictions of the maximal regions in

43 each case, shown in Figure 3.8. The values of the magnetization orientations that generated the most normalized work can be seen in Table 3.5.

Figure 3.8 Preference Sampler trade space results for (a) configuration I, (b) configuration II, (c) configuration III, and (d) configuration IV. For (d), 휓3 is defined by the size of the point, small to large mirroring 0° to 360° and 휓4 is defined by the color of the point, blue to red mirroring -180° to +180°.

Table 3.5 Preference Sampler results for the symmetry case for the maximum normalized work of each configuration and the respective input magnetization values.

Config. Norm. Work ψ1 (deg) ψ2 (deg) ψ3 (deg) ψ4 (deg) I 5.19 -6.4 187.6 II 4.15 173.9 6.8 III 4.50 173.5 -6.9 IV 4.38 169.4 -6.6 210.6 25.0

A comparison was performed between the results of the symmetry cases for both Fminsearch and ATSV. The percent difference between Fminsearch and ATSV for the normalized work and the magnetization orientations was calculated and the results are shown in Table 3.6. ATSV showed good correlation with the results of Fminsearch for configurations I, II, and III, having maximum theoretical normalized work values within 0.25%. ATSV’s results for configuration IV showed much higher deviation from that of Fminsearch, with magnetization orientation 휓3 and 휓4 values being 11.1% and 75.4% greater than what was seen in Fminsearch. As a result, the ATSV magnitude of the normalized work value for configuration IV is much lower than the Fminsearch result, but the percent difference between ATSV and Fminsearch remains relatively small. Table 3.6 Percent difference comparison between the symmetry cases of Fminsearch and ATSV.

Percent Difference (Fminsearch / ATSV)

Config. Norm. Work (%) ψ1 (%) ψ2 (%) ψ3 (%) ψ4 (%) I 0.16 -4.84 -0.16 II 0.18 -0.07 -10.07 III 0.21 0.19 -12.24 IV 5.48 2.59 -7.58 -11.10 -75.39

Another set of simulations were performed within ATSV where the magnetization orientations for each configuration were all independent, resulting in a total of nine possible magnetizations (one for each panel noting that some configuration only use a subset of all nine panels). See Appendix D for the MATLAB code for the independent magnetization orientations case. The Basic Sampler was run 600 times to populate the design space for each configuration and the Preference Sampler was run 400 times to determine the point of highest normalized work. The results of the Preference Sampler below in Table 3.7 shows that ATSV was not able to generate normalized work values in the independent case comparable to what was seen in the symmetry case. Within each configuration, ATSV found one or more magnetization orientations that were significantly different than other panels which, by symmetry, would be equivalent to one another. An example would be configuration II having a θM1 value of 173.4 degrees, similar to the Fminsearch and ATSV symmetry cases for 휓1, but having its θM3, θM7, and θM9 values ranging from 14 – 24 degrees lower; all four angles would be equal in the symmetry case. The difference was more pronounced for configurations III and IV as more magnetization orientations, and thus input variables, were present. Overall, the normalized work values in this case were 2.31%, 6.51%, 9.11%, and 11.41% lower for configurations I, II, III, and IV, respectively, when compared to their respective values in the ATSV symmetry case.

Table 3.7 Preference Sampler results for independent case for the maximum normalized work of each configuration and the respective input magnetization values.

θM1 θM2 θM3 θM4 θM5 θM6 θM7 θM8 θM9 Config. Norm. Work (deg) (deg) (deg) (deg) (deg) (deg) (deg) (deg) (deg) I 5.08 -7.7 207.5 193.3 -5.8 II 3.88 173.4 158.6 5.8 149.4 159.2 III 4.09 172.3 -6.3 171.4 133.1 -23.8 163.9 IV 3.88 171.5 -24.7 103.8 256.3 5.4 212.2 158.8 -6.4 180.3

3.3 Configuration optimization One of the research objectives of this work is to optimize the neodymium magnet/MAE patch magnetization directions in order to maximize the magnetic torque work applied to the Miura-ori. To accomplish this, it was first necessary to select a configuration to optimize. In the previous section, it was shown that while configuration I was the most efficient in terms of magnetic work done per neodymium magnet, it neither actuated the most nor followed ideal behavior the best. This highlights the inherent trade-offs between configurations across the calculated performance metrics. Therefore, a weighted sum model was utilized to compare each configuration combining the criteria covered in Sections 3.1 and 3.2: actuation, ideal behavior, and theoretical work. Before using the weighted sum model to determine the best suited configuration for optimization, the three criteria were normalized such that no single criteria would overly bias results due to the scale of its particular metric. All criteria are therefore normalized to unity. For the actuation, the following equation normalized the fold angles across each configuration at the maximum tested field strength of 240 mT: 6 1 휃휉 퐴 = ∑ (1 − ) 휃 = [휃 , 휃 , 휃 , 휃 , 휃 , 휃 ] (13) 6 180 휉 1 2 3 8 11 12 휉=1 where 휃휉 is the fold angle of a crease. This form was used because the crease fold angles of the Miura-ori prototypes at its initial flat state were denoted to be at 180 degrees while folding approached 0 degrees. For the ideal behavior, a more in-depth analysis was needed than what was performed in Section 3.1. The previous method’s analysis only compared average mountain fold angles vs. average valley fold angles for vertical creases and horizontal creases separately. In contrast, the ideal behavior of the Miura- ori has all vertical creases folding equally and all horizontal creases equally. The previous method did not compare vertical creases or horizontal creases of the same fold (mountain or valley). Furthermore, the ideal behavior criteria must be condensed into one metric that applies to each entire Miura-ori configuration, not to specific subsets of creases. Therefore, this analysis will focus on averaging the ratio

46 between all vertical creases and the ratio between the two horizontal creases. The following ratios were compared among each configuration using equation (12): 휃 휃 휃 휃 휃 휃 휃 1 , 1 , 1 , 2 , 11 , 11 , 3 휃11 휃2 휃12 휃12 휃2 휃12 휃8 These ratios were analyzed for each test of each prototype (a total of 3 tests), resulting in 푁 = 21 for each configuration’s three prototypes, and then averaged for each configuration. The theoretical normalized work, which was calculated in Section 3.1 and 3.2, was the final criterion. This value was separated into two categories: the total theoretical normalized work based on the initial configurations with fixed magnetization orientations, and the total maximum theoretical normalized work based on the initial configurations with varying magnetization orientations. The work values based on the fixed magnetization orientations were normalized by dividing each work value by the highest theoretical work value, which belonged to configuration I, in order to scale each value from 0 to 1. The values of each criterion for the configurations, their respective weights, and the results of the weighted sum models can be seen in Table 3.8. The weighted sum model used calculated a combined score for each configuration from the following equation: 푛

휌푖 = ∑ 푤푗푎푖푗 푓표푟 푖 = 1,2, … , 푚 (14) 푗=1 where 휌푖 is the total performance value of configuration 푖, 푚 is the total number of configurations, 푤푗 is the relative weight of importance of decision criteria 푗, 푛 is the total number of decision criteria, and 푎푖푗 is the performance value of alternative 푖 of decision criteria 푗. The values of each criteria shown in Table 3.8 are multiplied by their respective weights and summed across each configuration, resulting in total performance values for each configuration. For the actuation and ideal behavior, its value for each configuration at the external field strength of 240 mT is listed in the table. A graphical representation of the weighted sum performance values are shown in Figure 3.9. The error bars show the propagation of error from both the actuation and ideal behavior criteria values. Table 3.8 Actuation, Ideal behavior, and Theoretical Normalized Work from fixed magnetization orientations criteria values and the weighted sum model. The weights are highlighted in red.

Ideal Theoretical Norm. Weighted Actuation behavior Work (fixed) Sum Config. 0.333 0.333 0.333 I 0.284 0.835 1.000 0.706 II 0.332 0.854 0.812 0.666 III 0.282 0.799 0.781 0.621 IV 0.400 0.776 0.833 0.669

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Figure 3.9 Weighted sum performance values for the fixed magnetization orientations. The error bars represent the highest and lowest possible weighted performance values for each configuration based on the propagation of error from the actuation and ideal behavior criterion.

Another weighted sum model was performed for the work values based on the varying magnetization orientations, which consisted of the following cases: Fminsearch using symmetry, ATSV using symmetry, and ATSV using independent magnetization orientations. In each case, the work values were normalized further by dividing by the highest theoretical work value, which belonged to configuration I, in order to scale each value from 0 to 1. Weighted sum performance values were calculated individually for each case, the results of which are shown in Table 3.9 and Figure 3.10. The error bars in Figure 3.10 represent the propagation of error from the actuation and ideal behavior criterion. Table 3.9 Actuation, Ideal behavior, and Theoretical Normalized Work from varying magnetization orientations criterion values and the weighted sum model. The weights are highlighted in red.

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Figure 3.10 Weighted sum performance values for each configuration and case of the varied magnetization orientations. The error bars represent the highest and lowest possible weighted performance values for each configuration based on the propagation of error from the actuation and ideal behavior criterion.

The weighting was determined to be equal across actuation, ideal behavior, and theoretical normalized work, as each criterion is an important aspect in the folding of the Miura-ori pattern by way of magnetic fields. As shown in Figures 3.9 and Figure 3.10, configuration I had the highest weighted sum performance value among all configurations and both theoretical normalized work categories. Therefore, configuration I, with magnetic material on Panels 2, 4, 6, and 8, was the chosen configuration to experimentally test the maximum theoretical work magnetization orientations.

3.4 Maximum work configuration fabrication and analysis The weighted sum models in Section 3.3 showed that configuration I, for fixed or optimized magnetization orientations, was the ideal configuration for testing of the magnetic orientations corresponding to the maximum theoretical normalized work found by Fminsearch and ATSV. The new configuration that arose from optimizing magnetization orientations based on Fminsearch was designated as configuration I*; it generated the highest theoretical normalized work value at orientations of ψ2 =

−6.1° and ψ3 = 187.3°, shown in Table 3.2. A magnetization orientation comparison between configuration I and configuration I* can be seen in Figure 3.11. Another magnetization orientation of

ψ2 = −15.3° and ψ3 = 198.1°, resulting in a theoretical normalized work value 6.2% lower than that of the highest theoretical normalized work value in the Fminsearch case, was also tested and was designated as configuration I**. The values of configuration I** stem from a small correction to the calculation of the bisector/magnetic field vector, which does not affect previous weighted sum rankings. Configuration I* was tested with attached neodymium magnets, while configuration I** was tested by both using

49 attached neodymium magnets and embedded MAE patches within the substrate. For the neodymium magnet prototypes, the selection and fabrication processes covered in Section 2.3 were repeated using three prototypes from an additional batch F for configuration I* and two prototypes from the original four batches of substrates. Another prototype from an additional batch E for configuration I** was also used. Each prototype satisfied the minimum/maximum individual panel thickness check. See Appendix E for more details.

Figure 3.11 Magnetization orientation directions and corresponding initial magnetic torque directions for (a) configuration I and (b) configuration I*. Note that in (b) the magnetization directions found by Fminsearch are oriented to apply substantially more magnetic torque to the vertical creases than the horizontal creases.

The MAE patches for the test set of configuration I** with embedded MAE patches were created from molds designed in SolidWorks. The designed MAE patches possess the shape of their respective panels, with each side’s length reduced to 55% of that of the panel to ensure there was no interference with the creases. A total of four molds, one mold per panel, with three MAE patches per mold were created and cut from 1.5mm Acrylic using an Epilog Helix 60 Watt laser cutter. The MAE patch mold dimensions are shown in Figure 3.12 and the full mold for Panel 4 is shown in Figure 3.13. To create MAEs, barium hexaferrite (325 mesh) from ESPI Metals was mixed with the Sylgard 184 at a 30% volume fraction and filled each patch layer of the four molds. Each mold was oriented vertically within a 0.7 mT magnetic field for approximately 11 minutes in order to pole the MAE patches in-plane. The MAE patches were left to cure for two days before removal from the molds. Similar procedures have been and are detailed elsewhere [29].

Figure 3.12 MAE patch layer dimensions of (a) Panels 2 and 8, (b) Panel 4, and (c) Panel 6. All length measurements have units of millimeters. The fillet on the edge signifies the top of the mold, used for orienting the mold during the curing process.

Figure 3.13 3-layer mold for Panel 4. The acrylic layers are connected by plastic screws and nuts through the circular holes. The bottom and patch layers are attached first using the outer circular holes, and the top layer is attached using the inner circular holes after the patch layer is filled.

A thin layer of Sylgard was applied to Panels 2, 4, 6, and 8 of the Miura-ori molds. The top lattice molds were attached and the molds were subjected to a vacuum of 68 kPa at room temperature for 10 minutes to remove any air pockets. The molds were then subjected to a temperature of 75°C for 20 minutes to cure. The MAE patches were situated at the center of their respective panels to the thin layer of

Sylgard and adhered on using Loctite® Plastics Bonding System. The Miura-ori molds were filled with Sylgard and the vacuuming/curing process outlined in Section 2.3 was repeated. Delrin sheets of thickness 0.3 mm were attached to panels not containing an MAE patch in order to improve their rigidity. Figure 3.14 shows one of the finalized embedded MAE patch prototypes.

Figure 3.14 An embedded MAE prototype of configuration I**.

Both sets of configuration I* and configuration I** prototypes were experimentally tested and analyzed using the setup discussed in Section 2.4. Figure 3.15 shows one of the configuration I* prototype’s folding behavior at increasing external magnetic field strengths. Images of the neodymium configuration I**’s and MAE configuration I**’s actuation in the magnetic field can be seen in Appendix E. The results of the experiment are shown in Figure 3.16. For the complete set of data, see Appendix E. It can be observed from Figure 3.16 that creases 1 and 11 actuate to a lesser extent for both the neodymium and MAE prototypes. This behavior is more significant for the MAE prototypes, which had crease 1 and 11 average folding angles approximately 67.8% that of the crease 2 and 12 at the maximum field strength of 240 mT. For the neodymium prototypes, the fraction between crease 1 and 11’s and crease 2 and 12’s average folding angles was 80.6% at the field strength of 240 mT for both configuration I* and configuration I**.

Figure 3.15 Neodymium configuration I* prototype within the Walker Scientific 7H electromagnetic subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

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(b) Figure 3.16 Average Fold Angle vs. Applied Field Strength results for all prototypes of (a) configuration I* with neodymium magnets, (b) configuration I** with neodymium magnets, and (c) configuration I** with MAE patches. The error bars show the standard deviation from the three measurements taken per field strength per prototype, resulting in a total of nine fold angle measurements per field strength setting. The square data points represent vertical creases 1, 2, 11, and 12 and the diamond data points represent the horizontal creases 3 and 8.

The discrepancy between the vertical mountain (crease 2 and 12) and vertical valley (crease 1 and 11) average fold angles can be attributed to the nonuniform crease thickness discussed in Section 3.1 and the magnetization orientation of configuration I* and configuration I**. Looking back at Figure 3.11 shows that the magnetizations on Panels 2, 6, and 8 act on the mountain vertical creases, lifting them within the magnetic field. If the left edge of the Miura-ori is imagined to be another set of mountain vertical creases, only the magnetization on Panel 4 contributed in lifting that edge. However, it is believed that in a larger Miura-ori pattern, this phenomena would only be localized at the edges of the Miura-ori.

Normalized actuation and idealness factor values were additionally calculated for both configuration I* prototypes and configuration I** prototypes and compared to the original four configurations. This comparison can be seen in Figure 3.17. The normalized actuation at the maximum external field strength of 240 mT for the neodymium configuration I*, neodymium configuration I**, and the MAE configuration I* were 0.352, 0.382 and 0.321, respectively. The neodymium configuration I* had a normalized actuation 7.94% lower than that of the neodymium configuration I**. Both neodymium configuration I* and neodymium configuration I** prototypes’ maximum actuation factors exceeded that of configuration I’s maximum actuation factor of 0.284 by 23.9% and 34.6%, respectively. However, neither outperformed configuration IV’s maximum actuation factor of 0.400, both of which were 88.0% and 95.6% of configuration IV’s value, respectively. The idealness factors at the field strength of 240 mT for the neodymium configuration I*, neodymium configuration I**, and MAE configuration I** prototypes were 0.783, 0.756 and 0.687, respectively. When compared to configuration I’s idealness factor of 0.835, both sets of neodymium prototypes were 93.8% and 90.5% of this value, respectively. With respect to the highest average idealness factor observed of 0.854 achieved by configuration II, both sets of neodymium prototypes were 91.7% and 88.5% of this value, respectively.

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Config. I Config. II Config. III Config. IV Config. I* - Neo Config. I** - Neo Config. I** - MAE

Figure 3.17 (a) Normalized actuation and (b) idealness factor as a function of field strength for all tested configurations. Both normalized actuation and idealness factor are dimensionless, with the maximum value for each being 1. For (a), the error bars represent one standard deviation. For (b), the error bars represent the propagation of error, i.e. the lowest and highest possible idealness factor values for each configuration at each field strength.

The neodymium configuration I* prototypes performed better in idealness, but worse in actuation, than its neodymium configuration I** counterparts. This was contrary to the expectation that configuration I* would have the greater amount of actuation when compared to configuration I** with neodymium magnets. However, configuration I**’s average fold angles, and thus actuation factor, have higher standard deviations at each field strength than configuration I*. This is most likely due to one of its prototypes exhibiting more folding behavior than the other prototypes within the neodymium configuration I** set, which would account for this discrepancy. The greater average fold angles for the configuration I* prototype sets comparative to configuration I, and by extension the actuation factor, can be attributed to the magnetization orientations applying more magnetic work to the vertical creases than the horizontal creases, as was discussed in Section 3.1. In terms of idealness factor, both the neodymium configuration I* and configuration I** prototypes remained relatively close to the other configurations, but the neodymium configuration I** prototypes had one of the lowest idealness factor. This discrepancy is explained by the asymmetric folding of the vertical mountain and vertical valley creases, which can be seen in Figure 3.16. While the MAE configuration I** prototypes performed well in actuation and idealness, no direct comparisons can be made with respect to the other configurations with neodymium magnets; the MAE configuration I** was a proof-of-concept design meant to test feasibility. Furthermore, the magnetization, from which the torque was developed, was not normalized between the two sets of neodymium and MAE. A much better understanding of MAEs, specifically the structure of the barium hexaferrite particles within its matrix and how they interact with one another during the curing process, is needed before more substantial comparisons can be made. Additionally, an extended weighted sum model based on the varied theoretical normalized work shown in Table 3.9 was calculated to include both configuration I* and configuration I**, utilizing the same process presented in Section 3.3. The results of the weighted sum performance values of configuration I* with neodymium magnets, configuration I** with neodymium magnets, configuration I** with MAE patches, and the initial four configurations are shown in Table 3.10 and Figure 3.18. The error bars in Figure 3.18 represent the propagation of error from the actuation and ideal behavior criterion. The difference in theoretical normalized work value is more significant for both configuration I** configurations for the symmetry and independent ATSV cases, as both cases had lower theoretical normalized work values for configuration I than that of the symmetry Fminsearch case, as was observed in Section 3.2. Configuration I* and configuration I have the same theoretical work values for all cases as it is based on configuration I. Configuration I* with neodymium magnets produced the greatest weighted sum performance value of 0.712.

Table 3.10 Weighted sum model and the respective criterion of Actuation, Ideal behavior, and Theoretical Norm. Work from varying magnetization orientations for all experimentally tested configurations. The weights are highlighted in red.

Theoretical Norm. Work (varied) Weighted Sum Actuation Ideal behavior Fmin - S ATSV - S ATSV - I Fmin - S ATSV - S ATSV - I Config. 0.333 0.333 0.333 0.333 0.333 I 0.284 0.835 1.000 1.000 1.000 0.706 0.706 0.706 II 0.332 0.854 0.806 0.806 0.787 0.664 0.664 0.658 III 0.282 0.799 0.817 0.819 0.827 0.633 0.633 0.636 IV 0.400 0.776 0.892 0.872 0.668 0.689 0.683 0.615 I* - Neo 0.352 0.783 1.000 1.000 1.000 0.712 0.712 0.712 I** - Neo 0.382 0.756 0.938 0.940 0.960 0.692 0.692 0.699 I** - MAE 0.321 0.687 0.938 0.940 0.960 0.648 0.649 0.656

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Figure 3.18 Weighted sum performance values for all configurations and theoretical normalized work cases of the varied magnetization orientations. The error bars represent the highest and lowest possible weighted performance values for each configuration based on the propagation of error from the actuation and ideal behavior criterion.

Chapter 4. Conclusion This work shows the capabilities of magnetically sensitive components interacting with external magnetic fields in actuating a Miura-ori pattern on a silicone elastomer substrate. Four configurations of the Miura-ori with varying number of magnetic material and magnetization orientations were tested and compared based on the criterion of actuation, following the ideal folding behavior of the Miura-ori, and the total theoretical normalized magnetic work applied to the Miura-ori. The actuation and ideal folding behavior were determined experimentally by measuring the fold angles of selected creases when the Miura-ori prototypes were subjected to an increasing external magnetic field. The theoretical normalized work was calculated within MATLAB by utilizing the triple scalar product to determine the amount of torque applied to each of the Miura-ori’s creases. Additionally, the theoretical normalized work was maximized via varying magnetization orientations within a configuration by the use of MATLAB’s ‘Fminsearch’ function and the design space visualizer ATSV. Configuration IV had the greatest amount of actuation and configuration II exhibited behavior closest to the ideal folding of the Miura-ori. Configuration I generated the most theoretical normalized work, both in the fixed and varied magnetization orientation categories. Configuration I was determined to be the best based on the three criteria by a weighted sum model using equal weightings. Three sets of Miura-ori prototypes were created with magnetization orientations found by Fminsearch and ATSV, configuration I* using the magnetization orientations from the maximum theoretical work value and configuration I** using magnetization orientations that resulted in a theoretical work value 2% lower than the maximum. One set of configuration I* prototypes were generated using neodymium magnets and two sets of configuration I** prototypes were generated, one with attached neodymium magnets and the other with embedded MAE patches. Both the neodymium configuration I* and neodymium configuration I** prototypes produced better actuation than that of configuration I but reduced values in idealness. Weighted sum performance averages for I, I*, I** neodymium configurations all lie within overlapping error ranges. The reduction in the idealness factor for the neodymium configuration I* and neodymium configuration I** can be attributed to the vertical mountain creases folding more than the vertical valley creases as a result of the magnetization location and orientation. However, it is believed that this could be remedied by having Miura-ori patterns with vertical mountain creases at both ends of the Miura-ori and an even number of panels running horizontally as can be seen in Figure 4.1. Using this pattern, an entire edge would not need to be actuated by one neodymium magnet/MAE patch, which was the case for Panel 4 of configuration I* and configuration I**.

Figure 4.1 Miura-ori design with mountain folds being the dashed red lines and the valley folds being the dotted blue lines. The orange highlighted region is the Miura-ori used in this work. The green highlighted region is the suggested design to prevent an entire edge being actuated by one neodymium magnet/MAE patch.

In addition, the theoretical work model used to estimate the magnetic work applied to the Miura- ori assumes the complete folding of the Miura-ori. Experimentally, this is difficult, if not impossible, to realize. The magnetization will generate decreasing torque as the pattern approaches a fully folded state and the neodymium magnets/MAEs become more aligned with fixed, external field. This is of special concern for the particular Miura-ori geometry used in this work with vertical creases oriented 6 degrees from the vertical. As a result of this geometry, the vertical creases fold significantly more than the horizontal creases when starting from an initially flat state, but the trend reverses when nearing the completely folded state. Because the magnetic torque is greatest at the initially flat state, the Fminsearch and ATSV results for magnetization orientations with the maximum theoretical magnetic work prioritize folding the vertical creases over the horizontal creases. While this is favorable in the beginning stages of actuation, it may result in significant deviation when horizontal crease folding overtakes vertical crease folding in the ideal behavior. Despite the theoretical normalized work model assuming a completely and ideally folded Miura- ori pattern, it can be the basis of a more comprehensive model. Introducing concepts such as magnetic torque saturation, stiffness of the substrate, thickness and stiffness of the creases, and the weight of each respective panel would result in a more accurate representation of the theoretical normalized work applied to the Miura-ori structure. The method employed in Section 2.2 to determine the magnetization orientations that result in favorable crease folding could be expanded and applied to a variety of origami structures’ patterns for magnetic actuation purposes.

As shown in this work, the three criterion of actuation, ideal behavior, and theoretical normalized work were assumed to have equal weighting within the weighted sum model that compared configurations. However, the theoretical magnetic work may not necessarily be of equal importance to actuation and ideal behavior. As evidenced by the results, configuration IV had the greatest actuation while configuration II was the closest to ideal behavior. While fabrication of the Miura-ori prototypes certainly has an effect on its performance in these criterion, it’s possible that there is decoupling between actuation and theoretical normalized work. With the use of ATSV and more complex models, a design space of the criterion of actuation, ideal behavior, and theoretical normalized work could be explored to find new configurations and magnetization orientations that optimizes the three in a fully coupled manner. Future work with the Miura-ori will analyze how altering its geometry effects the ability of MAE patches in actuating the pattern. Miura-ori patterns containing vertical creases with larger angles from the vertical may result in more equal folding progression of vertical and horizontal creases, allowing for easier actuation. Future work will also aim to generate Miura-ori substrates from one piece of MAE material. This could be accomplished by photo-curing and magnetically poling in prescribed directions for panels individually. This new process would allow for thinner Miura-ori substrates, resulting in better actuation of the structure in order to achieve its completely folded state.

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Appendix A Prototype selection data for initial four configurations

Table A.1. Excel Statistical Analysis output of panel thickness data and calculation of the upper and lower bounds of the 2 standard deviation condition

Mean 2.204022 Standard Error 0.012673 Median 2.2 Mode 2.066 Standard Deviation 0.24046 Lower bounds Sample Variance 0.057821 2.204022 mm - 2*0.24046 mm = 1.723102 mm Kurtosis -0.42038 Skewness 0.252366 Upper bounds Range 1.096 2.204022 mm + 2*0.24046 mm = 2.684943 mm Minimum 1.728 Maximum 2.824 Sum 793.448 Count 360

Table A.2 Crease panel thickness data and the selection of suitable experimental half-thickness crease (HTC) prototypes. Each prototype is labeled by a mold number and a batch letter.

Prototype Panel 1A 2A 3A 4A 5A 1B 2B 3B 4B 5B 1C 2C 3C 4C 5C 1D 2D 3D 4D 5D 1 2.292 2.396 2.358 2.114 2.366 2.374 2.534 2.302 2.574 2.390 2.290 2.244 2.158 2.346 2.430 2.136 2.132 2.214 2.076 2.218 2 2.198 2.392 2.310 2.072 2.216 2.334 2.522 2.330 2.532 2.358 2.226 2.240 2.178 2.328 2.494 2.104 2.138 2.226 2.094 2.534

3 1.982 2.310 2.082 2.004 1.942 2.136 2.326 2.254 2.306 2.272 2.006 2.176 2.070 2.148 2.434 1.944 1.964 2.170 2.026 2.816 4 2.544 2.554 2.594 2.310 2.762 2.768 2.778 2.524 2.824 2.594 2.618 2.496 2.456 2.542 2.578 2.428 2.432 2.392 2.278 2.188 5 2.406 2.536 2.410 2.212 2.592 2.472 2.742 2.352 2.740 2.484 2.440 2.422 2.338 2.516 2.536 2.306 2.406 2.280 2.208 2.494

6 2.148 2.356 2.208 2.092 2.296 2.420 2.460 2.234 2.536 2.390 2.208 2.352 2.230 2.312 2.400 2.150 2.282 2.196 2.146 2.760 Thickness (mm) Thickness 7 2.366 2.438 2.514 2.358 2.804 2.806 2.618 2.414 2.684 2.502 2.512 2.490 2.434 2.346 2.566 2.312 2.346 2.336 2.202 2.064 8 2.290 2.396 2.352 2.272 2.648 2.646 2.544 2.324 2.646 2.382 2.382 2.394 2.340 2.316 2.490 2.268 2.308 2.306 2.168 2.378

9 1.998 2.184 2.090 2.042 2.232 2.418 2.302 2.118 2.308 2.192 2.108 2.232 2.168 2.066 2.292 2.064 2.110 2.076 2.026 2.566

AVG 2.247 2.396 2.324 2.164 2.429 2.486 2.536 2.317 2.572 2.396 2.310 2.338 2.264 2.324 2.469 2.190 2.235 2.244 2.136 2.446

SD 0.186 0.112 0.175 0.127 0.289 0.216 0.163 0.114 0.178 0.121 0.196 0.120 0.134 0.151 0.091 0.150 0.157 0.095 0.087 0.257

pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass fail pass pass pass pass fail fail fail pass fail pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass fail pass fail pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass fail pass pass pass pass fail fail pass pass pass pass pass pass pass pass pass pass pass pass pass pass

2 Standard Deviation 2Check Deviation Standard pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

= accepted for use as an experimental substrate

= rejected for having one or more panels outside 2 standard deviations from the mean

Table A.3 Crease panel thickness data and the selection of suitable experimental one-third-thickness crease OTTC prototypes. Each prototype is labeled by a mold number and a batch letter.

Prototype Panel 1A 2A 3A 4A 5A 1B 2B 3B 4B 5B 1C 2C 3C 4C 5C 1D 2D 3D 4D 5D 1 1.828 2.024 1.846 1.968 2.158 1.790 1.800 1.922 1.990 2.272 1.954 2.130 1.992 1.948 2.042 2.236 1.938 2.090 2.066 2.100 2 1.944 1.958 1.902 2.080 2.048 1.756 1.762 1.772 1.950 2.090 1.884 2.006 1.780 1.780 1.876 2.240 1.784 1.888 2.004 1.990

3 2.382 1.940 2.310 2.336 2.094 1.960 2.032 1.914 2.066 2.000 1.980 2.150 1.766 1.840 1.782 2.316 1.772 1.848 2.152 2.164 4 1.922 2.084 1.860 2.012 2.154 1.940 1.738 2.138 2.114 2.434 1.952 1.970 2.056 1.972 2.176 2.266 2.216 2.120 2.196 1.896 5 2.186 2.092 1.988 2.246 2.074 2.070 1.728 2.114 2.086 2.290 1.986 1.840 1.892 1.836 2.018 2.292 2.136 1.894 2.140 1.808

6 2.572 1.982 2.376 2.468 2.090 2.262 2.024 2.232 2.198 2.108 2.058 1.970 1.820 1.846 1.848 2.272 2.072 1.836 2.234 1.940 Thickness (mm) Thickness 7 2.066 2.078 2.102 2.008 2.066 2.100 1.858 2.408 2.154 2.468 1.886 1.926 2.016 1.926 2.204 2.168 2.412 2.010 2.214 1.822 8 2.338 2.044 2.280 2.232 2.056 2.252 1.910 2.410 2.230 2.326 1.920 1.850 1.940 1.812 2.062 2.222 2.338 1.910 2.194 1.778 9 2.740 2.076 2.686 2.428 2.140 2.488 2.242 2.532 2.376 2.242 2.036 2.024 1.990 1.826 1.980 2.334 2.236 1.874 2.292 1.958 AVG 2.220 2.031 2.150 2.198 2.098 2.069 1.899 2.160 2.129 2.248 1.962 1.985 1.917 1.865 1.999 2.261 2.100 1.941 2.166 1.940 SD 0.312 0.058 0.284 0.189 0.042 0.238 0.172 0.259 0.129 0.157 0.061 0.108 0.107 0.067 0.144 0.051 0.230 0.106 0.088 0.131

pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

2 Standard Deviation 2Check Deviation Standard pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

fail pass fail pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

= accepted for use as an experimental substrate

= rejected for having one or more panels outside 2 standard deviations from the mean

Appendix B MATLAB code for experimental theoretical normalized work clear all close all

% Loads Zhonghua's miura fold point data data = load('Point_test.txt');

% Point coordinates (x,y,z) of each folding step (row) P1 = [data(:,1) data(:,2) data(:,3)]; P2 = [data(:,4) data(:,5) data(:,6)]; P3 = [data(:,7) data(:,8) data(:,9)]; P4 = [data(:,10) data(:,11) data(:,12)]; P5 = [data(:,13) data(:,14) data(:,15)]; P6 = [data(:,16) data(:,17) data(:,18)]; P7 = [data(:,19) data(:,20) data(:,21)]; P8 = [data(:,22) data(:,23) data(:,24)]; P9 = [data(:,25) data(:,26) data(:,27)]; P10 = [data(:,28) data(:,29) data(:,30)]; P11 = [data(:,31) data(:,32) data(:,33)]; P12 = [data(:,34) data(:,35) data(:,36)]; P13 = [data(:,37) data(:,38) data(:,39)]; P14 = [data(:,40) data(:,41) data(:,42)]; P15 = [data(:,43) data(:,44) data(:,45)]; P16 = [data(:,46) data(:,47) data(:,48)];

% Rotation around Y-axis to match experimental orientation RM = [cos(pi) 0 sin(pi); 0 1 0; -sin(pi) 0 cos(pi)]; P4c = transpose(RM*transpose(P1)); P3c = transpose(RM*transpose(P2)); P2c = transpose(RM*transpose(P3)); P1c = transpose(RM*transpose(P4)); P5c = transpose(RM*transpose(P8)); P6c = transpose(RM*transpose(P7)); P7c = transpose(RM*transpose(P6)); P8c = transpose(RM*transpose(P5)); P9c = transpose(RM*transpose(P12)); P10c = transpose(RM*transpose(P11)); P11c = transpose(RM*transpose(P10)); P12c = transpose(RM*transpose(P9)); P13c = transpose(RM*transpose(P16)); P14c = transpose(RM*transpose(P15)); P15c = transpose(RM*transpose(P14)); P16c = transpose(RM*transpose(P13));

% Crease vectors C1 = P2c - P6c; C2 = P3c - P7c; C3 = P5c - P6c; C4 = P7c - P6c; C5 = P8c - P7c; C6 = P6c - P10c; C7 = P7c - P11c; C8 = P9c - P10c; C9 = P10c - P11c; C10 = P12c - P11c; C11 = P14c - P10c; C12 = P15c - P11c;

% Normalized crease vectors C1_norm = C1; C2_norm = C2; C3_norm = C3; C4_norm = C4; C5_norm = C5; 67

C6_norm = C6; C7_norm = C7; C8_norm = C8; C9_norm = C9; C10_norm = C10; C11_norm = C11; C12_norm = C12; for i = 1:129 C1_norm(i,:) = C1(i,:)/norm(C1(i,:)); C2_norm(i,:) = C2(i,:)/norm(C2(i,:)); C3_norm(i,:) = C3(i,:)/norm(C3(i,:)); C4_norm(i,:) = C4(i,:)/norm(C4(i,:)); C5_norm(i,:) = C5(i,:)/norm(C5(i,:)); C6_norm(i,:) = C6(i,:)/norm(C6(i,:)); C7_norm(i,:) = C7(i,:)/norm(C7(i,:)); C8_norm(i,:) = C8(i,:)/norm(C8(i,:)); C9_norm(i,:) = C9(i,:)/norm(C9(i,:)); C10_norm(i,:) = C10(i,:)/norm(C10(i,:)); C11_norm(i,:) = C11(i,:)/norm(C11(i,:)); C12_norm(i,:) = C12(i,:)/norm(C12(i,:)); end

% Calculation of unit normal vectors to panels for i = 1:129 Pan1norm(i,:) = cross(C1(i,:),C3(i,:)); Pan2norm(i,:) = cross(C1(i,:),C4(1,:)); Pan3norm(i,:) = -cross(C2(i,:),C5(i,:)); Pan4norm(i,:) = -cross(C6(1,:),C8(i,:)); Pan5norm(i,:) = cross(C6(1,:),C9(1,:)); Pan6norm(i,:) = cross(C7(1,:),C10(i,:)); Pan7norm(i,:) = cross(C8(i,:),C11(i,:)); Pan8norm(i,:) = cross(C11(i,:),C9(i,:)); Pan9norm(i,:) = -cross(C10(i,:),C12(i,:)); end

% Calculation of horizontal crease angles for i = 1:129 AngleH(i,1) = acos(dot(Pan2norm(i,:),Pan5norm(1,:))/(norm(... % Used in further calculations Pan2norm(i,:))*norm(Pan5norm(1,:)))); AngleH2_5(i,1) = AngleH(i,1); AngleH1_4(i,1) = acos(dot(Pan4norm(i,:),Pan1norm(i,:))/(norm(... Pan4norm(i,:))*norm(Pan1norm(i,:)))); AngleH3_6(i,1) = acos(dot(Pan3norm(i,:),Pan6norm(i,:))/(norm(... Pan3norm(i,:))*norm(Pan6norm(i,:)))); AngleH4_7(i,1) = acos(dot(Pan4norm(i,:),Pan7norm(i,:))/(norm(... Pan4norm(i,:))*norm(Pan7norm(i,:)))); AngleH5_8(i,1) = acos(dot(Pan8norm(i,:),Pan5norm(1,:))/(norm(... Pan8norm(i,:))*norm(Pan5norm(1,:)))); AngleH6_9(i,1) = acos(dot(Pan6norm(i,:),Pan9norm(i,:))/(norm(... Pan6norm(i,:))*norm(Pan9norm(i,:)))); end

% Calculation of vertical crease angles for i = 1:129 AngleV(i,1) = acos(dot(Pan4norm(i,:),Pan5norm(i,:))/(norm(... Pan4norm(i,:))*norm(Pan5norm(i,:)))); AngleV4_5(i,1) = AngleV(i,1); AngleV1_2(i,1) = acos(dot(Pan1norm(i,:),Pan2norm(i,:))/(norm(... Pan1norm(i,:))*norm(Pan2norm(i,:)))); AngleV2_3(i,1) = acos(dot(Pan2norm(i,:),Pan3norm(i,:))/(norm(... Pan2norm(i,:))*norm(Pan3norm(i,:)))); AngleV5_6(i,1) = acos(dot(Pan5norm(i,:),Pan6norm(i,:))/(norm(... Pan5norm(i,:))*norm(Pan6norm(i,:)))); AngleV7_8(i,1) = acos(dot(Pan7norm(i,:),Pan8norm(i,:))/(norm(... Pan7norm(i,:))*norm(Pan8norm(i,:)))); AngleV8_9(i,1) = acos(dot(Pan8norm(i,:),Pan9norm(i,:))/(norm(... Pan8norm(i,:))*norm(Pan9norm(i,:)))); 68 end

% Defining the bisector/magnetic field unit vector phi = 6 * pi/180; % Angle of the vertical creases w.r.t. the vertical for i = 1:129 bisector(i,:) = [cos(0.5*AngleV(i,1))*cos(phi) cos(0.5*AngleV(i,1))... *sin(phi) sin(0.5*AngleV(i,1))]; H_dir(i,:) = bisector(i,:); norm_H(i,:) = H_dir(i,:)/norm(H_dir(i,:)); end

% Calculation of the change in horizontal fold angle and the change in % vertical fold angle between steps N for i = 1:128 delVangle(i,:) = AngleV(i,1) - AngleV(i+1,1); delHangle(i,:) = AngleH(i,1) - AngleH(i+1,1); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Configuration I Work Calculations V7_2(1,:) = P7c(1,:) - P2c(1,:); MagV7_2 = sqrt(V7_2(1,1)^2 + V7_2(1,2)^2 + V7_2(1,3)^2); Mag4a = sqrt(C4(1, 1)^2 + C4(1, 2)^2 + C4(1, 3)^2); M12angle = acos(dot(V7_2(1,:),C4(1,:))/(MagV7_2*Mag4a)); M12dangle = M12angle*180/pi; M18angle = M12angle; V9_6(1,:) = P9c(1,:) - P6c(1,:); MagV9_6 = sqrt(V9_6(1,1)^2 + V9_6(1,2)^2 + V9_6(1,3)^2); Mag8a = sqrt(C8(1, 1)^2 + C8(1, 2)^2 + C8(1, 3)^2); M14angle = acos(dot(V9_6(1,:),C8(1,:))/(MagV9_6*Mag8a)); M14dangle = M14angle*180/pi; V11_8(1,:) = P11c(1,:) - P8c(1,:); MagV11_8 = sqrt(V11_8(1,1)^2 + V11_8(1,2)^2 + V11_8(1,3)^2); Mag10a = sqrt(C10(1, 1)^2 + C10(1, 2)^2 + C10(1, 3)^2); M16angle = pi - acos(dot(V11_8(1,:),C10(1,:))/(MagV11_8*Mag10a)); M16dangle = M16angle*180/pi; alpha14 = -0.7235645029; beta14 = 0.7700215803; alpha16 = alpha14; beta16 = -beta14;

% Magnetization orientations for i = 1:129 M12(i,:) = [cos(M12angle) sin(M12angle)*cos(AngleH(i,1)) ... sin(M12angle)*sin(AngleH(i,1))]; M14(i,:) = [alpha14*C6_norm(i,1) + beta14*C8_norm(i,1) alpha14*... C6_norm(i,2) + beta14*C8_norm(i,2) alpha14*C6_norm(i,3) + ... beta14*C8_norm(i,3)]; M16(i,:) = [alpha16*C7_norm(i,1) + beta16*C10_norm(i,1) alpha16*... C7_norm(i,2) + beta16*C10_norm(i,2) alpha16*C7_norm(i,3) + ... beta16*C10_norm(i,3)]; M18(i,:) = [cos(M18angle) sin(M18angle)*cos(AngleH(i,1)) ... sin(M18angle)*sin(AngleH(i,1))]; end

% Torques for i = 1:129 T12(i,:) = cross(M12(i,:), norm_H(i,:)); MagT12 = sqrt(T12(i, 1)^2 + T12(i, 2)^2 + T12(i, 3)^2); nT12(i,:) = T12(i,:) / MagT12; T14(i,:) = cross(M14(i,:), norm_H(i,:)); MagT14 = sqrt(T14(i, 1)^2 + T14(i, 2)^2 + T14(i, 3)^2); nT14(i,:) = T14(i,:) / MagT14; T16(i,:) = cross(M16(i,:), norm_H(i,:)); MagT16 = sqrt(T16(i, 1)^2 + T16(i, 2)^2 + T16(i, 3)^2); nT16(i,:) = T16(i,:) / MagT16; 69

T18(i,:) = cross(M18(i,:), norm_H(i,:)); MagT18 = sqrt(T18(i, 1)^2 + T18(i, 2)^2 + T18(i, 3)^2); nT18(i,:) = T18(i,:) / MagT18; end

% Torques along creases for i = 1:129 T121(i,:) = dot(nT12(i,:), C1_norm(i,:)); T122(i,:) = dot(nT12(i,:), C2_norm(i,:)); T124(i,:) = dot(nT12(i,:), C4_norm(i,:)); T143(i,:) = dot(nT14(i,:), C3_norm(i,:)); T146(i,:) = dot(nT14(i,:), C6_norm(i,:)); T148(i,:) = dot(nT14(i,:), C8_norm(i,:)); T165(i,:) = dot(nT16(i,:), C5_norm(i,:)); T167(i,:) = dot(nT16(i,:), C7_norm(i,:)); T1610(i,:) = dot(nT16(i,:), C10_norm(i,:)); T1811(i,:) = dot(nT18(i,:), C11_norm(i,:)); T189(i,:) = dot(nT18(i,:), C9_norm(i,:)); T1812(i,:) = dot(nT18(i,:), C12_norm(i,:)); end

% Torque between each step for i = 1:128 AvgT121(i,:) = (T121(i,:) + T121(i+1,:))/2; AvgT122(i,:) = (T122(i,:) + T122(i+1,:))/2; AvgT124(i,:) = (T124(i,:) + T124(i+1,:))/2; AvgT143(i,:) = (T143(i,:) + T143(i+1,:))/2; AvgT146(i,:) = (T146(i,:) + T146(i+1,:))/2; AvgT148(i,:) = (T148(i,:) + T148(i+1,:))/2; AvgT165(i,:) = (T165(i,:) + T165(i+1,:))/2; AvgT167(i,:) = (T167(i,:) + T167(i+1,:))/2; AvgT1610(i,:) = (T1610(i,:) + T1610(i+1,:))/2; AvgT1811(i,:) = (T1811(i,:) + T1811(i+1,:))/2; AvgT189(i,:) = (T189(i,:) + T189(i+1,:))/2; AvgT1812(i,:) = (T1812(i,:) + T1812(i+1,:))/2; end

% Torque Work on each crease l = 128; W121 = 0; W122 = 0; W124 = 0; W143 = 0; W146 = 0; W148 = 0; W165 = 0; W167 = 0; W1610 = 0; W1811 = 0; W189 = 0; W1812 = 0; for i = 1:l W121 = W121 + AvgT121(i,:)*delVangle(i,:); W121s(i,:) = W121; W122 = W122 + AvgT122(i,:)*delVangle(i,:); W122s(i,:) = W122; W124 = W124 + AvgT124(i,:)*delHangle(i,:); W124s(i,:) = W124; W143 = W143 + AvgT143(i,:)*delHangle(i,:); W143s(i,:) = W143; W146 = W146 + AvgT146(i,:)*delVangle(i,:); W146s(i,:) = W146; W148 = W148 + AvgT148(i,:)*delHangle(i,:); W148s(i,:) = W148; W165 = W165 + AvgT165(i,:)*delHangle(i,:); W165s(i,:) = W165; W167 = W167 + AvgT167(i,:)*delVangle(i,:); W167s(i,:) = W167; 70

W1610 = W1610 + AvgT1610(i,:)*delHangle(i,:); W1610s(i,:) = W1610; W1811 = W1811 + AvgT1811(i,:)*delVangle(i,:); W1811s(i,:) = W1811; W189 = W189 + AvgT189(i,:)*delHangle(i,:); W189s(i,:) = W189; W1812 = W1812 + AvgT1812(i,:)*delVangle(i,:); W1812s(i,:) = W1812; end for i = 1:l W1VSUM(i,:) = -W121s(i,:) - W122s(i,:) + W146s(i,:) + W167s(i,:) + ... W1811s(i,:) + W1812s(i,:); W1HSUM(i,:) = -W124s(i,:) + W143s(i,:) - W165s(i,:) + W148s(i,:) + ... W189s(i,:) - W1610s(i,:); W1TOT(i,:) = -W121s(i,:) - W122s(i,:) - W124s(i,:) + W143s(i,:) + ... W146s(i,:) + W148s(i,:) - W1610s(i,:) - W165s(i,:) + W167s(i,:) ... + W1811s(i,:) + W1812s(i,:) + W189s(i,:); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Configuration II Work Calculations V6_1(1,:) = P6c(1,:) - P1c(1,:); MagV6_1 = sqrt(V6_1(1,1)^2 + V6_1(1,2)^2 + V6_1(1,3)^2); Mag3a = sqrt(C3(1, 1)^2 + C3(1, 2)^2 + C3(1, 3)^2); M21angle = pi - acos(dot(V6_1(1,:),C3(1,:))/(MagV6_1*Mag3a)); M21dangle = M21angle*180/pi;

V8_3(1,:) = P8c(1,:) - P3c(1,:); MagV8_3 = sqrt(V8_3(1,1)^2 + V8_3(1,2)^2 + V8_3(1,3)^2); Mag5a = sqrt(C5(1, 1)^2 + C5(1, 2)^2 + C5(1, 3)^2); M23angle = acos(dot(V8_3(1,:),C5(1,:))/(MagV8_3*Mag5a)); M23dangle = M23angle*180/pi;

V7_10(1,:) = P7c(1,:) - P10c(1,:); MagV7_10 = sqrt(V7_10(1,1)^2 + V7_10(1,2)^2 + V7_10(1,3)^2); Mag9a = sqrt(C9(1, 1)^2 + C9(1, 2)^2 + C9(1, 3)^2); M25angle = pi - acos(dot(V7_10(1,:),C9(1,:))/(MagV7_10*Mag9a)); M25dangle = M25angle*180/pi;

V9_14(1,:) = P9c(1,:) - P14c(1,:); MagV9_14 = sqrt(V9_14(1,1)^2 + V9_14(1,2)^2 + V9_14(1,3)^2); M27angle = acos(dot(V9_14(1,:),C8(1,:))/(MagV9_14*Mag8a)); M27dangle = M27angle*180/pi;

V11_16(1,:) = P11c(1,:) - P16c(1,:); MagV11_16 = sqrt(V11_16(1,1)^2 + V11_16(1,2)^2 + V11_16(1,3)^2); M29angle = pi - acos(dot(V11_16(1,:),C10(1,:))/(MagV11_16*Mag10a)); M29dangle = M29angle*180/pi; alpha21 = 0.7235634799; beta21 = 0.7700227765; alpha23 = alpha21; beta23 = -beta21; alpha27 = -alpha21; beta27 = beta21; alpha29 = -alpha21; beta29 = -beta21;

% Magnetization orientations for i = 1:129 M21(i,:) = [alpha21*C1_norm(i,1) + beta21*C3_norm(i,1) alpha21*... C1_norm(i,2) + beta21*C3_norm(i,2) alpha21*C1_norm(i,3) + ... beta21*C3_norm(i,3)]; M23(i,:) = [alpha23*C2_norm(i,1) + beta23*C5_norm(i,1) alpha23*... C2_norm(i,2) + beta23*C5_norm(i,2) alpha23*C2_norm(i,3) + ... beta23*C5_norm(i,3)]; M25(i,:) = [cos(M25angle) sin(M25angle) 0]; 71

M27(i,:) = [alpha27*C11_norm(i,1) + beta27*C8_norm(i,1) alpha27*... C11_norm(i,2) + beta27*C8_norm(i,2) alpha27*C11_norm(i,3) + ... beta27*C8_norm(i,3)]; M29(i,:) = [alpha29*C12_norm(i,1) + beta29*C10_norm(i,1) alpha29*... C12_norm(i,2) + beta29*C10_norm(i,2) alpha29*C12_norm(i,3) + ... beta29*C10_norm(i,3)]; end

% Torques for i = 1:129 T21(i,:) = cross(M21(i,:), norm_H(i,:)); MagT21 = sqrt(T21(i, 1)^2 + T21(i, 2)^2 + T21(i, 3)^2); nT21(i,:) = T21(i,:) / MagT21; T23(i,:) = cross(M23(i,:), norm_H(i,:)); MagT23 = sqrt(T23(i, 1)^2 + T23(i, 2)^2 + T23(i, 3)^2); nT23(i,:) = T23(i,:) / MagT23; T25(i,:) = cross(M25(i,:), norm_H(i,:)); MagT25 = sqrt(T25(i, 1)^2 + T25(i, 2)^2 + T25(i, 3)^2); nT25(i,:) = T25(i,:) / MagT25; T27(i,:) = cross(M27(i,:), norm_H(i,:)); MagT27 = sqrt(T27(i, 1)^2 + T27(i, 2)^2 + T27(i, 3)^2); nT27(i,:) = T27(i,:) / MagT27; T29(i,:) = cross(M29(i,:), norm_H(i,:)); MagT29 = sqrt(T29(i, 1)^2 + T29(i, 2)^2 + T29(i, 3)^2); nT29(i,:) = T29(i,:) / MagT29; end

% Torques along creases for i = 1:129 T211(i,:) = dot(nT21(i,:), C1_norm(i,:)); T213(i,:) = dot(nT21(i,:), C3_norm(i,:)); T232(i,:) = dot(nT23(i,:), C2_norm(i,:)); T235(i,:) = dot(nT23(i,:), C5_norm(i,:)); T254(i,:) = dot(nT25(i,:), C4_norm(i,:)); T256(i,:) = dot(nT25(i,:), C6_norm(i,:)); T257(i,:) = dot(nT25(i,:), C7_norm(i,:)); T259(i,:) = dot(nT25(i,:), C9_norm(i,:)); T278(i,:) = dot(nT27(i,:), C8_norm(i,:)); T2711(i,:) = dot(nT27(i,:), C11_norm(i,:)); T2910(i,:) = dot(nT29(i,:), C10_norm(i,:)); T2912(i,:) = dot(nT29(i,:), C12_norm(i,:)); end

% Torque between each step for i = 1:128 AvgT211(i,:) = (T211(i,:) + T211(i+1,:))/2; AvgT213(i,:) = (T213(i,:) + T213(i+1,:))/2; AvgT232(i,:) = (T232(i,:) + T232(i+1,:))/2; AvgT235(i,:) = (T235(i,:) + T235(i+1,:))/2; AvgT254(i,:) = (T254(i,:) + T254(i+1,:))/2; AvgT256(i,:) = (T256(i,:) + T256(i+1,:))/2; AvgT257(i,:) = (T257(i,:) + T257(i+1,:))/2; AvgT259(i,:) = (T259(i,:) + T259(i+1,:))/2; AvgT278(i,:) = (T278(i,:) + T278(i+1,:))/2; AvgT2711(i,:) = (T2711(i,:) + T2711(i+1,:))/2; AvgT2910(i,:) = (T2910(i,:) + T2910(i+1,:))/2; AvgT2912(i,:) = (T2912(i,:) + T2912(i+1,:))/2; end

% Torque Work on each crease W211 = 0; W213 = 0; W232 = 0; W235 = 0; W254 = 0; W256 = 0; W257 = 0; W259 = 0; 72

W278 = 0; W2711 = 0; W2910 = 0; W2912 = 0; for i = 1:l W211 = W211 + AvgT211(i,:)*delVangle(i,:); W211s(i,:) = W211; W213 = W213 + AvgT213(i,:)*delHangle(i,:); W213s(i,:) = W213; W232 = W232 + AvgT232(i,:)*delVangle(i,:); W232s(i,:) = W232; W235 = W235 + AvgT235(i,:)*delHangle(i,:); W235s(i,:) = W235; W254 = W254 + AvgT254(i,:)*delHangle(i,:); W254s(i,:) = W254; W256 = W256 + AvgT256(i,:)*delVangle(i,:); W256s(i,:) = W256; W257 = W257 + AvgT257(i,:)*delVangle(i,:); W257s(i,:) = W257; W259 = W259 + AvgT259(i,:)*delHangle(i,:); W259s(i,:) = W259; W278 = W278 + AvgT278(i,:)*delHangle(i,:); W278s(i,:) = W278; W2711 = W2711 + AvgT2711(i,:)*delVangle(i,:); W2711s(i,:) = W2711; W2910 = W2910 + AvgT2910(i,:)*delHangle(i,:); W2910s(i,:) = W2910; W2912 = W2912 + AvgT2912(i,:)*delVangle(i,:); W2912s(i,:) = W2912; end for i = 1:l W2VSUM(i,:) = W211s(i,:) + W232s(i,:) - W256s(i,:) - W257s(i,:) - ... W2711s(i,:) - W2912s(i,:); W2HSUM(i,:) = -W213s(i,:) + W254s(i,:) + W235s(i,:) - W278s(i,:) - ... W259s(i,:) + W2910s(i,:); W2TOT(i,:) = W211s(i,:) - W213s(i,:) + W232s(i,:) + W235s(i,:) + W254s(i,:) ... - W256s(i,:) - W257s(i,:) - W259s(i,:) - W2711s(i,:) - W278s(i,:) ... + W2910s(i,:) - W2912s(i,:); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Configuration III Work Calculations alpha31 = 0.1051042448; beta31 = 1.00550837; alpha33 = alpha31; beta33 = -beta31; alpha37 = -alpha31; beta37 = beta31; alpha39 = -alpha31; beta39 = -beta31;

% Magnetization orientations for i = 1:129 M31(i,:) = [alpha31*C1_norm(i,1) + beta31*C3_norm(i,1) alpha31*... C1_norm(i,2) + beta31*C3_norm(i,2) alpha31*C1_norm(i,3) + ... beta31*C3_norm(i,3)]; M32(i,:) = [0 cos(AngleH(i,1)) sin(AngleH(i,1))]; M33(i,:) = [alpha33*C2_norm(i,1) + beta33*C5_norm(i,1) alpha33*... C2_norm(i,2) + beta33*C5_norm(i,2) alpha33*C2_norm(i,3) + ... beta33*C5_norm(i,3)]; M37(i,:) = [alpha37*C11_norm(i,1) + beta37*C8_norm(i,1) alpha37*... C11_norm(i,2) + beta37*C8_norm(i,2) alpha37*C11_norm(i,3) + ... beta37*C8_norm(i,3)]; M38(i,:) = [0 cos(AngleH(i,1)) sin(AngleH(i,1))]; M39(i,:) = [alpha39*C12_norm(i,1) + beta39*C10_norm(i,1) alpha39*... C12_norm(i,2) + beta39*C10_norm(i,2) alpha39*C12_norm(i,3) + ... beta39*C10_norm(i,3)]; 73 end

% Torques for i = 1:129 T31(i,:) = cross(M31(i,:), norm_H(i,:)); MagT31 = sqrt(T31(i, 1)^2 + T31(i, 2)^2 + T31(i, 3)^2); nT31(i,:) = T31(i,:) / MagT31; T32(i,:) = cross(M32(i,:), norm_H(i,:)); MagT32 = sqrt(T32(i, 1)^2 + T32(i, 2)^2 + T32(i, 3)^2); nT32(i,:) = T32(i,:) / MagT32; T33(i,:) = cross(M33(i,:), norm_H(i,:)); MagT33 = sqrt(T33(i, 1)^2 + T33(i, 2)^2 + T33(i, 3)^2); nT33(i,:) = T33(i,:) / MagT33; T37(i,:) = cross(M37(i,:), norm_H(i,:)); MagT37 = sqrt(T37(i, 1)^2 + T37(i, 2)^2 + T37(i, 3)^2); nT37(i,:) = T37(i,:) / MagT37; T38(i,:) = cross(M38(i,:), norm_H(i,:)); MagT38 = sqrt(T38(i, 1)^2 + T38(i, 2)^2 + T38(i, 3)^2); nT38(i,:) = T38(i,:) / MagT38; T39(i,:) = cross(M39(i,:), norm_H(i,:)); MagT39 = sqrt(T39(i, 1)^2 + T39(i, 2)^2 + T39(i, 3)^2); nT39(i,:) = T39(i,:) / MagT39; end

% Torques along the creases for i = 1:129 T311(i,:) = dot(nT31(i,:), C1_norm(i,:)); T313(i,:) = dot(nT31(i,:), C3_norm(i,:)); T321(i,:) = dot(nT32(i,:), C1_norm(i,:)); T324(i,:) = dot(nT32(i,:), C4_norm(i,:)); T322(i,:) = dot(nT32(i,:), C2_norm(i,:)); T332(i,:) = dot(nT33(i,:), C2_norm(i,:)); T335(i,:) = dot(nT33(i,:), C5_norm(i,:)); T378(i,:) = dot(nT37(i,:), C8_norm(i,:)); T3711(i,:) = dot(nT37(i,:), C11_norm(i,:)); T389(i,:) = dot(nT38(i,:), C9_norm(i,:)); T3811(i,:) = dot(nT38(i,:), C11_norm(i,:)); T3812(i,:) = dot(nT38(i,:), C12_norm(i,:)); T3910(i,:) = dot(nT39(i,:), C10_norm(i,:)); T3912(i,:) = dot(nT39(i,:), C12_norm(i,:)); end

% Torque between each step for i = 1:128 AvgT311(i,:) = (T311(i,:) + T311(i+1,:))/2; AvgT313(i,:) = (T313(i,:) + T313(i+1,:))/2; AvgT321(i,:) = (T321(i,:) + T321(i+1,:))/2; AvgT324(i,:) = (T324(i,:) + T324(i+1,:))/2; AvgT322(i,:) = (T322(i,:) + T322(i+1,:))/2; AvgT332(i,:) = (T332(i,:) + T332(i+1,:))/2; AvgT335(i,:) = (T335(i,:) + T335(i+1,:))/2; AvgT378(i,:) = (T378(i,:) + T378(i+1,:))/2; AvgT3711(i,:) = (T3711(i,:) + T3711(i+1,:))/2; AvgT389(i,:) = (T389(i,:) + T389(i+1,:))/2; AvgT3811(i,:) = (T3811(i,:) + T3811(i+1,:))/2; AvgT3812(i,:) = (T3812(i,:) + T3812(i+1,:))/2; AvgT3910(i,:) = (T3910(i,:) + T3910(i+1,:))/2; AvgT3912(i,:) = (T3912(i,:) + T3912(i+1,:))/2; end

% Torque Work on each crease W311 = 0; W313 = 0; W321 = 0; W324 = 0; W322 = 0; W332 = 0; W335 = 0; 74

W378 = 0; W3711 = 0; W389 = 0; W3811 = 0; W3812 = 0; W3910 = 0; W3912 = 0; for i = 1:l W311 = W311 + AvgT311(i,:)*delVangle(i,:); W311s(i,:) = W311; W313 = W313 + AvgT313(i,:)*delHangle(i,:); W313s(i,:) = W313; W321 = W321 + AvgT321(i,:)*delVangle(i,:); W321s(i,:) = W321; W324 = W324 + AvgT324(i,:)*delHangle(i,:); W324s(i,:) = W324; W322 = W322 + AvgT322(i,:)*delVangle(i,:); W322s(i,:) = W322; W332 = W332 + AvgT332(i,:)*delVangle(i,:); W332s(i,:) = W332; W335 = W335 + AvgT335(i,:)*delHangle(i,:); W335s(i,:) = W335; W378 = W378 + AvgT378(i,:)*delHangle(i,:); W378s(i,:) = W378; W3711 = W3711 + AvgT3711(i,:)*delVangle(i,:); W3711s(i,:) = W3711; W389 = W389 + AvgT389(i,:)*delHangle(i,:); W389s(i,:) = W389; W3811 = W3811 + AvgT3811(i,:)*delVangle(i,:); W3811s(i,:) = W3811; W3812 = W3812 + AvgT3812(i,:)*delVangle(i,:); W3812s(i,:) = W3812; W3910 = W3910 + AvgT3910(i,:)*delHangle(i,:); W3910s(i,:) = W3910; W3912 = W3912 + AvgT3912(i,:)*delVangle(i,:); W3912s(i,:) = W3912; end for i = 1:l W31C(i,:) = W311s(i,:) - W321s(i,:); W32C(i,:) = W332s(i,:) - W322s(i,:); W311C(i,:) = -W3711s(i,:) + W3811s(i,:); W312C(i,:) = -W3912s(i,:) + W3812s(i,:); W3VSUM(i,:) = W31C(i,:) + W32C(i,:) + W311C(i,:) + W312C(i,:); W3HSUM(i,:) = W313s(i,:) - W324s(i,:) + W335s(i,:) - W378s(i,:) + ... W389s(i,:) + W3910s(i,:); W3TOT(i,:) = W31C(i,:) + W32C(i,:) + W311C(i,:) + W312C(i,:) - ... W313s(i,:) - W324s(i,:) + W335s(i,:) - W378s(i,:) + W389s(i,:) ... + W3910s(i,:); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Configuration IV Work Calculations alpha44 = -1.005508338; beta44 = 0.1051047898; alpha46 = alpha44; beta46 = -beta44;

M41 = M31; M42 = M32; M43 = M33; M47 = M37; M48 = M38; M49 = M39;

% Magnetization orientations for i = 1:129 M44(i,:) = [alpha44*C6_norm(i,1) + beta44*C8_norm(i,1) alpha44*... 75

C6_norm(i,2) + beta44*C8_norm(i,2) alpha44*C6_norm(i,3) + ... beta44*C8_norm(i,3)]; M45(i,:) = [cos(phi) sin(phi) 0]; M46(i,:) = [alpha46*C7_norm(i,1) + beta46*C10_norm(i,1) alpha46*... C7_norm(i,2) + beta46*C10_norm(i,2) alpha46*C7_norm(i,3) + ... beta46*C10_norm(i,3)]; end

% Torques nT41 = nT31; nT42 = nT32; nT43 = nT33; nT47 = nT37; nT48 = nT38; nT49 = nT39; for i = 1:129 T44(i,:) = cross(M44(i,:), norm_H(i,:)); MagT44 = sqrt(T44(i, 1)^2 + T44(i, 2)^2 + T44(i, 3)^2); nT44(i,:) = T44(i,:) / MagT44; T45(i,:) = cross(M45(i,:), norm_H(i,:)); MagT45 = sqrt(T45(i, 1)^2 + T45(i, 2)^2 + T45(i, 3)^2); nT45(i,:) = T45(i,:) / MagT45; T46(i,:) = cross(M46(i,:), norm_H(i,:)); MagT46 = sqrt(T46(i, 1)^2 + T46(i, 2)^2 + T46(i, 3)^2); nT46(i,:) = T46(i,:) / MagT46; end

% Torques along the creases T411 = T311; T413 = T313; T421 = T321; T422 = T322; T424 = T324; T432 = T332; T435 = T335; T478 = T378; T4711 = T3711; T489 = T389; T4811 = T3811; T4812 = T3812; T4910 = T3910; T4912 = T3912; for i = 1:129 T443(i,:) = dot(nT44(i,:), C3_norm(i,:)); T446(i,:) = dot(nT44(i,:), C6_norm(i,:)); T448(i,:) = dot(nT44(i,:), C8_norm(i,:)); T454(i,:) = dot(nT45(i,:), C4_norm(i,:)); T456(i,:) = dot(nT45(i,:), C6_norm(i,:)); T457(i,:) = dot(nT45(i,:), C7_norm(i,:)); T459(i,:) = dot(nT45(i,:), C9_norm(i,:)); T465(i,:) = dot(nT46(i,:), C5_norm(i,:)); T467(i,:) = dot(nT46(i,:), C7_norm(i,:)); T4610(i,:) = dot(nT46(i,:), C10_norm(i,:)); end

% Torque between each step for i = 1:128 AvgT443(i,:) = (T443(i,:) + T443(i+1,:))/2; AvgT446(i,:) = (T446(i,:) + T446(i+1,:))/2; AvgT448(i,:) = (T448(i,:) + T448(i+1,:))/2; AvgT454(i,:) = (T454(i,:) + T454(i+1,:))/2; AvgT456(i,:) = (T456(i,:) + T456(i+1,:))/2; AvgT457(i,:) = (T457(i,:) + T457(i+1,:))/2; AvgT459(i,:) = (T459(i,:) + T459(i+1,:))/2; AvgT465(i,:) = (T465(i,:) + T465(i+1,:))/2; AvgT467(i,:) = (T467(i,:) + T467(i+1,:))/2; 76

AvgT4610(i,:) = (T4610(i,:) + T4610(i+1,:))/2; end

% Torque Work on each crease W411 = W311; W413 = W313; W421 = W321; W424 = W324; W422 = W322; W432 = W332; W435 = W335; W478 = W378; W4711 = W3711; W489 = W389; W4811 = W3811; W4812 = W3812; W4910 = W3910; W4912 = W3912; W443 = 0; W446 = 0; W448 = 0; W454 = 0; W456 = 0; W457 = 0; W459 = 0; W465 = 0; W467 = 0; W4610 = 0; W413s = W313s; W424s = W324s; W4910s = W3910s; W435s = W335s; W478s = W378s; W489s = W389s; W41C = W31C; W42C = W32C; W411C = W311C; W412C = W312C; for i = 1:l W443 = W443 + AvgT443(i,:)*delHangle(i,:); W443s(i,:) = W443; W446 = W446 + AvgT446(i,:)*delVangle(i,:); W446s(i,:) = W446; W448 = W448 + AvgT448(i,:)*delHangle(i,:); W448s(i,:) = W448; W454 = W454 + AvgT454(i,:)*delHangle(i,:); W454s(i,:) = W454; W456 = W456 + AvgT456(i,:)*delVangle(i,:); W456s(i,:) = W456; W457 = W457 + AvgT457(i,:)*delVangle(i,:); W457s(i,:) = W457; W459 = W459 + AvgT459(i,:)*delHangle(i,:); W459s(i,:) = W459; W465 = W465 + AvgT465(i,:)*delHangle(i,:); W465s(i,:) = W459; W467 = W467 + AvgT467(i,:)*delVangle(i,:); W467s(i,:) = W467; W4610 = W4610 + AvgT4610(i,:)*delHangle(i,:); W4610s(i,:) = W4610; end for i = 1:l W43C(i,:) = -W413s(i,:) + W443s(i,:); W44C(i,:) = -W424s(i,:) + W454s(i,:); W45C(i,:) = W435s(i,:) - W465s(i,:); W46C(i,:) = W446s(i,:) - W456s(i,:); W47C(i,:) = -W457s(i,:) + W467s(i,:); 77

W48C(i,:) = -W478s(i,:) + W448s(i,:); W49C(i,:) = -W459s(i,:) + W489s(i,:); W410C(i,:) = W4910s(i,:) - W4610s(i,:); W4VSUM(i,:) = W41C(i,:) + W42C(i,:) + W46C(i,:) + W47C(i,:) + ... W411C(i,:) + W412C(i,:); W4HSUM(i,:) = W43C(i,:) + W44C(i,:) + W45C(i,:) + W48C(i,:) + ... W49C(i,:) + W410C(i,:); W4TOT(i,:) = W41C(i,:) + W42C(i,:) + W43C(i,:) + W44C(i,:) + ... W45C(i,:) + W46C(i,:) + W47C(i,:) + W48C(i,:) + W49C(i,:) + ... W410C(i,:) + W411C(i,:) + W412C(i,:); end

%% Sanity Check for panel movement and magnetization directions % Panel 1

C1_2 = P1c - P2c; C1_5 = P1c - P5c; x1m_1 = C1(1,1)/2; y1m_1 = C1(1,2)/2; z1m_1 = C1(1,3)/2; x1m_25 = C1(25,1)/2; y1m_25 = C1(25,2)/2; z1m_25 = C1(25,3)/2; x1m_49 = C1(49,1)/2; y1m_49 = C1(49,2)/2; z1m_49 = C1(49,3)/2; x1m_73 = C1(73,1)/2; y1m_73 = C1(73,2)/2; z1m_73 = C1(73,3)/2; x1m_97 = C1(97,1)/2; y1m_97 = C1(97,2)/2; z1m_97 = C1(97,3)/2; x1m_110 = C1(110,1)/2; y1m_110 = C1(110,2)/2; z1m_110 = C1(110,3)/2; x1m_120 = C1(120,1)/2; y1m_120 = C1(120,2)/2; z1m_120 = C1(120,3)/2; x1m_129 = C1(129,1)/2; y1m_129 = C1(129,2)/2; z1m_129 = C1(129,3)/2; x2_1 = P2c(1,1); y2_1 = P2c(1,2); z2_1 = P2c(1,3); x2_25 = P2c(25,1); y2_25 = P2c(25,2); z2_25 = P2c(25,3); x2_49 = P2c(49,1); y2_49 = P2c(49,2); z2_49 = P2c(49,3); x2_73 = P2c(73,1); y2_73 = P2c(73,2); z2_73 = P2c(73,3); x2_97 = P2c(97,1); y2_97 = P2c(97,2); z2_97 = P2c(97,3); x2_110 = P2c(110,1); y2_110 = P2c(110,2); z2_110 = P2c(110,3); x2_120 = P2c(120,1); y2_120 = P2c(120,2); z2_120 = P2c(120,3); x2_129 = P2c(129,1); y2_129 = P2c(129,2); z2_129 = P2c(129,3); x5_1 = P5c(1,1); y5_1 = P5c(1,2); 78 z5_1 = P5c(1,3); x5_25 = P5c(25,1); y5_25 = P5c(25,2); z5_25 = P5c(25,3); x5_49 = P5c(49,1); y5_49 = P5c(49,2); z5_49 = P5c(49,3); x5_73 = P5c(73,1); y5_73 = P5c(73,2); z5_73 = P5c(73,3); x5_97 = P5c(97,1); y5_97 = P5c(97,2); z5_97 = P5c(97,3); x5_110 = P5c(110,1); y5_110 = P5c(110,2); z5_110 = P5c(110,3); x5_120 = P5c(120,1); y5_120 = P5c(120,2); z5_120 = P5c(120,3); x5_129 = P5c(129,1); y5_129 = P5c(129,2); z5_129 = P5c(129,3); x6 = P6c(1,1); y6 = P6c(1,2); z6 = P6c(1,3); scale = 0; scale2 = 0.4; scale3 = 0.3; scale4 = 0.15; figure quiver3(x6,y6,z6,C1(1,1),C1(1,2),C1(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.1,y6,z6,C1(25,1),C1(25,2),C1(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.2,y6,z6,C1(49,1),C1(49,2),C1(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.3,y6,z6,C1(73,1),C1(73,2),C1(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.4,y6,z6,C1(97,1),C1(97,2),C1(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.5,y6,z6,C1(110,1),C1(110,2),C1(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.6,y6,z6,C1(120,1),C1(120,2),C1(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.7,y6,z6,C1(129,1),C1(129,2),C1(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(1,1),C3(1,2),C3(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.1,y6,z6,C3(25,1),C3(25,2),C3(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.2,y6,z6,C3(49,1),C3(49,2),C3(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.3,y6,z6,C3(73,1),C3(73,2),C3(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.4,y6,z6,C3(97,1),C3(97,2),C3(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.5,y6,z6,C3(110,1),C3(110,2),C3(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.6,y6,z6,C3(120,1),C3(120,2),C3(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.7,y6,z6,C3(129,1),C3(129,2),C3(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_1,y2_1,z2_1,C1_2(1,1),C1_2(1,2),C1_2(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_25+0.1,y2_25,z2_25,C1_2(25,1),C1_2(25,2),C1_2(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) 79 hold on quiver3(x2_49+0.2,y2_49,z2_49,C1_2(49,1),C1_2(49,2),C1_2(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_73+0.3,y2_73,z2_73,C1_2(73,1),C1_2(73,2),C1_2(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_97+0.4,y2_97,z2_97,C1_2(97,1),C1_2(97,2),C1_2(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_110+0.5,y2_110,z2_110,C1_2(110,1),C1_2(110,2),C1_2(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_120+0.6,y2_120,z2_120,C1_2(120,1),C1_2(120,2),C1_2(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_129+0.7,y2_129,z2_129,C1_2(129,1),C1_2(129,2),C1_2(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_1,y5_1,z5_1,C1_5(1,1),C1_5(1,2),C1_5(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_25+0.1,y5_25,z5_25,C1_5(25,1),C1_5(25,2),C1_5(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_49+0.2,y5_49,z5_49,C1_5(49,1),C1_5(49,2),C1_5(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_73+0.3,y5_73,z5_73,C1_5(73,1),C1_5(73,2),C1_5(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_97+0.4,y5_97,z5_97,C1_5(97,1),C1_5(97,2),C1_5(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_110+0.5,y5_110,z5_110,C1_5(110,1),C1_5(110,2),C1_5(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_120+0.6,y5_120,z5_120,C1_5(120,1),C1_5(120,2),C1_5(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_129+0.7,y5_129,z5_129,C1_5(129,1),C1_5(129,2),C1_5(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on

% Configuration II Magnetization quiver3(x6,y6,z6,M21(1,1),M21(1,2),M21(1,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.1,y6,z6,M21(25,1),M21(25,2),M21(25,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.2,y6,z6,M21(49,1),M21(49,2),M21(49,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.3,y6,z6,M21(73,1),M21(73,2),M21(73,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.4,y6,z6,M21(97,1),M21(97,2),M21(97,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.5,y6,z6,M21(110,1),M21(110,2),M21(110,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.6,y6,z6,M21(120,1),M21(120,2),M21(120,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+0.7,y6,z6,M21(129,1),M21(129,2),M21(129,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4)

% % Configuration III & IV Magnetizations % quiver3(x6+x1m_1,y6+y1m_1,z6+z1m_1,M31(1,1),M31(1,2),M31(1,3),scale3,'r', 'MaxHeadSize',0.3) % hold on

% quiver3(x6+x1m_25+0.1,y6+y1m_25,z6+z1m_25,M31(25,1),M31(25,2),M31(25,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6+x1m_49+0.2,y6+y1m_49,z6+z1m_49,M31(49,1),M31(49,2),M31(49,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6+x1m_73+0.3,y6+y1m_73,z6+z1m_73,M31(73,1),M31(73,2),M31(73,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6+x1m_97+0.4,y6+y1m_97,z6+z1m_97,M31(97,1),M31(97,2),M31(97,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6+x1m_110+0.5,y6+y1m_110,z6+z1m_110,M31(110,1),M31(110,2),M31(110,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6+x1m_120+0.6,y6+y1m_120,z6+z1m_120,M31(120,1),M31(120,2),M31(120,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6+x1m_129+0.7,y6+y1m_129,z6+z1m_129,M31(129,1),M31(129,2),M31(129,3),scale3, 'r', 'MaxHeadSize',0.3) % hold on

% Panel 2 C2_3 = P3c - P2c; x7 = P7c(1,1); y7 = P7c(1,2); z7 = P7c(1,3); x2_1 = P2c(1,1); y2_1 = P2c(1,2); z2_1 = P2c(1,3); x2_25 = P2c(25,1); y2_25 = P2c(25,2); z2_25 = P2c(25,3); x2_49 = P2c(49,1); y2_49 = P2c(49,2); z2_49 = P2c(49,3); x2_73 = P2c(73,1); y2_73 = P2c(73,2); z2_73 = P2c(73,3); x2_97 = P2c(97,1); y2_97 = P2c(97,2); z2_97 = P2c(97,3); x2_110 = P2c(110,1); y2_110 = P2c(110,2); z2_110 = P2c(110,3); x2_120 = P2c(120,1); y2_120 = P2c(120,2); z2_120 = P2c(120,3); x2_129 = P2c(129,1); y2_129 = P2c(129,2); z2_129 = P2c(129,3); x2m_1 = C2_3(1,1)/2; y2m_1 = C2_3(1,2)/2; z2m_1 = C2_3(1,3)/2; x2m_25 = C2_3(25,1)/2; y2m_25 = C2_3(25,2)/2; z2m_25 = C2_3(25,3)/2; x2m_49 = C2_3(49,1)/2; y2m_49 = C2_3(49,2)/2; z2m_49 = C2_3(49,3)/2; x2m_73 = C2_3(73,1)/2; y2m_73 = C2_3(73,2)/2; z2m_73 = C2_3(73,3)/2; x2m_97 = C2_3(97,1)/2; y2m_97 = C2_3(97,2)/2; z2m_97 = C2_3(97,3)/2; x2m_110 = C2_3(110,1)/2; 81 y2m_110 = C2_3(110,2)/2; z2m_110 = C2_3(110,3)/2; x2m_120 = C2_3(120,1)/2; y2m_120 = C2_3(120,2)/2; z2m_120 = C2_3(120,3)/2; x2m_129 = C2_3(129,1)/2; y2m_129 = C2_3(129,2)/2; z2m_129 = C2_3(129,3)/2; figure quiver3(x6,y6,z6,C4(1,1),C4(1,2),C4(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(1,1),C1(1,2),C1(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(25,1),C1(25,2),C1(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(49,1),C1(49,2),C1(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(73,1),C1(73,2),C1(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(97,1),C1(97,2),C1(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(110,1),C1(110,2),C1(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(120,1),C1(120,2),C1(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C1(129,1),C1(129,2),C1(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(1,1),C2(1,2),C2(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(25,1),C2(25,2),C2(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(49,1),C2(49,2),C2(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(73,1),C2(73,2),C2(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(97,1),C2(97,2),C2(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(110,1),C2(110,2),C2(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(120,1),C2(120,2),C2(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(129,1),C2(129,2),C2(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_1,y2_1,z2_1,C2_3(1,1),C2_3(1,2),C2_3(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_25,y2_25,z2_25,C2_3(25,1),C2_3(25,2),C2_3(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_49,y2_49,z2_49,C2_3(49,1),C2_3(49,2),C2_3(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_73,y2_73,z2_73,C2_3(73,1),C2_3(73,2),C2_3(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_97,y2_97,z2_97,C2_3(97,1),C2_3(97,2),C2_3(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_110,y2_110,z2_110,C2_3(110,1),C2_3(110,2),C2_3(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_120,y2_120,z2_120,C2_3(120,1),C2_3(120,2),C2_3(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_129,y2_129,z2_129,C2_3(129,1),C2_3(129,2),C2_3(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on 82

% Configuration I Magnetization % quiver3(x2_1,y2_1,z2_1,M12(1,1),M12(1,2),M12(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_25,y2_25,z2_25,M12(25,1),M12(25,2),M12(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_49,y2_49,z2_49,M12(49,1),M12(49,2),M12(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_73,y2_73,z2_73,M12(73,1),M12(73,2),M12(73,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_97,y2_97,z2_97,M12(97,1),M12(97,2),M12(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_110,y2_110,z2_110,M12(110,1),M12(110,2),M12(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_120,y2_120,z2_120,M12(120,1),M12(120,2),M12(120,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x2_129,y2_129,z2_129,M12(129,1),M12(129,2),M12(129,3),scale2, 'r', 'MaxHeadSize',0.3)

% Configuration III & IV Magnetizations quiver3(x2_1+x2m_1,y2_1+y2m_1,z2_1+z2m_1,M32(1,1),M32(1,2),M32(1,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_25+x2m_25,y2_25+y2m_25,z2_25+z2m_25,M32(25,1),M32(25,2),M32(25,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_49+x2m_49,y2_49+y2m_49,z2_49+z2m_49,M32(49,1),M32(49,2),M32(49,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_73+x2m_73,y2_73+y2m_73,z2_73+z2m_73,M32(73,1),M32(73,2),M32(73,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_97+x2m_97,y2_97+y2m_97,z2_97+z2m_97,M32(97,1),M32(97,2),M32(97,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_110+x2m_110,y2_110+y2m_110,z2_110+z2m_110,M32(110,1),M32(110,2),M32(110,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_120+x2m_120,y2_120+y2m_120,z2_120+z2m_120,M32(120,1),M32(120,2),M32(120,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x2_129+x2m_129,y2_129+y2m_129,z2_129+z2m_129,M32(129,1),M32(129,2),M32(129,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3)

% Panel 3 C4_3 = P4c - P3c; C4_8 = P4c - P8c; x3_1 = P3c(1,1); y3_1 = P3c(1,2); z3_1 = P3c(1,3); x3_25 = P3c(25,1); y3_25 = P3c(25,2); z3_25 = P3c(25,3); x3_49 = P3c(49,1); y3_49 = P3c(49,2); z3_49 = P3c(49,3); x3_73 = P3c(73,1); y3_73 = P3c(73,2); z3_73 = P3c(73,3); x3_97 = P3c(97,1); y3_97 = P3c(97,2); z3_97 = P3c(97,3); x3_110 = P3c(110,1); y3_110 = P3c(110,2); z3_110 = P3c(110,3); x3_120 = P3c(120,1); y3_120 = P3c(120,2); z3_120 = P3c(120,3); 83 x3_129 = P3c(129,1); y3_129 = P3c(129,2); z3_129 = P3c(129,3); x8_1 = P8c(1,1); y8_1 = P8c(1,2); z8_1 = P8c(1,3); x8_25 = P8c(25,1); y8_25 = P8c(25,2); z8_25 = P8c(25,3); x8_49 = P8c(49,1); y8_49 = P8c(49,2); z8_49 = P8c(49,3); x8_73 = P8c(73,1); y8_73 = P8c(73,2); z8_73 = P8c(73,3); x8_97 = P8c(97,1); y8_97 = P8c(97,2); z8_97 = P8c(97,3); x8_110 = P8c(110,1); y8_110 = P8c(110,2); z8_110 = P8c(110,3); x8_120 = P8c(120,1); y8_120 = P8c(120,2); z8_120 = P8c(120,3); x8_129 = P8c(129,1); y8_129 = P8c(129,2); z8_129 = P8c(129,3); figure quiver3(x7,y7,z7,C2(1,1),C2(1,2),C2(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.1,y7,z7,C2(25,1),C2(25,2),C2(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.2,y7,z7,C2(49,1),C2(49,2),C2(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.3,y7,z7,C2(73,1),C2(73,2),C2(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.4,y7,z7,C2(97,1),C2(97,2),C2(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.5,y7,z7,C2(110,1),C2(110,2),C2(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.6,y7,z7,C2(120,1),C2(120,2),C2(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.7,y7,z7,C2(129,1),C2(129,2),C2(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(1,1),C5(1,2),C5(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.1,y7,z7,C5(25,1),C5(25,2),C5(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.2,y7,z7,C5(49,1),C5(49,2),C5(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.3,y7,z7,C5(73,1),C5(73,2),C5(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.4,y7,z7,C5(97,1),C5(97,2),C5(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.5,y7,z7,C5(110,1),C5(110,2),C5(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.6,y7,z7,C5(120,1),C5(120,2),C5(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x7-0.7,y7,z7,C5(129,1),C5(129,2),C5(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_1,y3_1,z3_1,C4_3(1,1),C4_3(1,2),C4_3(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_25-0.1,y3_25,z3_25,C4_3(25,1),C4_3(25,2),C4_3(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_49-0.2,y3_49,z3_49,C4_3(49,1),C4_3(49,2),C4_3(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_73-0.3,y3_73,z3_73,C4_3(73,1),C4_3(73,2),C4_3(73,3),scale, 'b', 'MaxHeadSize',0.01) 84 hold on quiver3(x3_97-0.4,y3_97,z3_97,C4_3(97,1),C4_3(97,2),C4_3(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_110-0.5,y3_110,z3_110,C4_3(110,1),C4_3(110,2),C4_3(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_120-0.6,y3_120,z3_120,C4_3(120,1),C4_3(120,2),C4_3(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x3_129-0.7,y3_129,z3_129,C4_3(129,1),C4_3(129,2),C4_3(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_1,y8_1,z8_1,C4_8(1,1),C4_8(1,2),C4_8(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_25-0.1,y8_25,z8_25,C4_8(25,1),C4_8(25,2),C4_8(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_49-0.2,y8_49,z8_49,C4_8(49,1),C4_8(49,2),C4_8(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_73-0.3,y8_73,z8_73,C4_8(73,1),C4_8(73,2),C4_8(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_97-0.4,y8_97,z8_97,C4_8(97,1),C4_8(97,2),C4_8(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_110-0.5,y8_110,z8_110,C4_8(110,1),C4_8(110,2),C4_8(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_120-0.6,y8_120,z8_120,C4_8(120,1),C4_8(120,2),C4_8(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x8_129-0.7,y8_129,z8_129,C4_8(129,1),C4_8(129,2),C4_8(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on

% Configuration II Magnetization % quiver3(x8_1,y8_1,z8_1,M23(1,1),M23(1,2),M23(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_25-0.1,y8_25,z8_25,M23(25,1),M23(25,2),M23(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_49-0.2,y8_49,z8_49,M23(49,1),M23(49,2),M23(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_73-0.3,y8_73,z8_73,M23(73,1),M23(73,2),M23(73,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_97-0.4,y8_97,z8_97,M23(97,1),M23(97,2),M23(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_110-0.5,y8_110,z8_110,M23(110,1),M23(110,2),M23(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_120-0.6,y8_120,z8_120,M23(120,1),M23(120,2),M23(120,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_129-0.7,y8_129,z8_129,M23(129,1),M23(129,2),M23(129,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% Configurations III & IV Magnetization quiver3(x8_1+C4_8(1,1)/2,y8_1+C4_8(1,2)/2,z8_1+C4_8(1,3)/2,M33(1,1),M33(1,2),M33(1,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x8_25+C4_8(25,1)/2- 0.1,y8_25+C4_8(25,2)/2,z8_25+C4_8(25,3)/2,M33(25,1),M33(25,2),M33(25,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x8_49+C4_8(49,1)/2- 0.2,y8_49+C4_8(49,2)/2,z8_49+C4_8(49,3)/2,M33(49,1),M33(49,2),M33(49,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x8_73+C4_8(73,1)/2- 0.3,y8_73+C4_8(73,2)/2,z8_73+C4_8(73,3)/2,M33(73,1),M33(73,2),M33(73,3),scale3, 'r', 'MaxHeadSize',0.3) hold on

85 quiver3(x8_97+C4_8(97,1)/2- 0.4,y8_97+C4_8(97,2)/2,z8_97+C4_8(97,3)/2,M33(97,1),M33(97,2),M33(97,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x8_110+C4_8(110,1)/2- 0.5,y8_110+C4_8(110,2)/2,z8_110+C4_8(110,3)/2,M33(110,1),M33(110,2),M33(110,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x8_120+C4_8(120,1)/2- 0.6,y8_120+C4_8(120,2)/2,z8_120+C4_8(120,3)/2,M33(120,1),M33(120,2),M33(120,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x8_129+C4_8(129,1)/2- 0.7,y8_129+C4_8(129,2)/2,z8_129+C4_8(129,3)/2,M33(129,1),M33(129,2),M33(129,3),scale3, 'r', 'MaxHeadSize',0.3) hold on

% Panel 4 C9_5 = P9c - P5c; x10 = P10c(1,1); y10 = P10c(1,2); z10 = P10c(1,3); x3m_1 = C3(1,1)/2; y3m_1 = C3(1,2)/2; z3m_1 = C3(1,3)/2; x3m_25 = C3(25,1)/2; y3m_25 = C3(25,2)/2; z3m_25 = C3(25,3)/2; x3m_49 = C3(49,1)/2; y3m_49 = C3(49,2)/2; z3m_49 = C3(49,3)/2; x3m_73 = C3(73,1)/2; y3m_73 = C3(73,2)/2; z3m_73 = C3(73,3)/2; x3m_97 = C3(97,1)/2; y3m_97 = C3(97,2)/2; z3m_97 = C3(97,3)/2; x3m_110 = C3(110,1)/2; y3m_110 = C3(110,2)/2; z3m_110 = C3(110,3)/2; x3m_120 = C3(120,1)/2; y3m_120 = C3(120,2)/2; z3m_120 = C3(120,3)/2; x3m_129 = C3(129,1)/2; y3m_129 = C3(129,2)/2; z3m_129 = C3(129,3)/2; figure quiver3(x10,y10,z10,C6(1,1),C6(1,2),C6(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(25,1),C6(25,2),C6(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(49,1),C6(49,2),C6(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(73,1),C6(73,2),C6(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(97,1),C6(97,2),C6(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(110,1),C6(110,2),C6(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(120,1),C6(120,2),C6(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(129,1),C6(129,2),C6(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(1,1),C8(1,2),C8(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(25,1),C8(25,2),C8(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) 86 hold on quiver3(x10,y10,z10,C8(49,1),C8(49,2),C8(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(73,1),C8(73,2),C8(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(97,1),C8(97,2),C8(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(110,1),C8(110,2),C8(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(120,1),C8(120,2),C8(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(129,1),C8(129,2),C8(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(1,1),C3(1,2),C3(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(25,1),C3(25,2),C3(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(49,1),C3(49,2),C3(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(73,1),C3(73,2),C3(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(97,1),C3(97,2),C3(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(110,1),C3(110,2),C3(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(120,1),C3(120,2),C3(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(129,1),C3(129,2),C3(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_1,y5_1,z5_1,C9_5(1,1),C9_5(1,2),C9_5(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_25,y5_25,z5_25,C9_5(25,1),C9_5(25,2),C9_5(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_49,y5_49,z5_49,C9_5(49,1),C9_5(49,2),C9_5(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_73,y5_73,z5_73,C9_5(73,1),C9_5(73,2),C9_5(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_97,y5_97,z5_97,C9_5(97,1),C9_5(97,2),C9_5(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_110,y5_110,z5_110,C9_5(110,1),C9_5(110,2),C9_5(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_120,y5_120,z5_120,C9_5(120,1),C9_5(120,2),C9_5(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_129,y5_129,z5_129,C9_5(129,1),C9_5(129,2),C9_5(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on

% Configuration I Magnetization % quiver3(x6,y6,z6,M14(1,1),M14(1,2),M14(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6,y6,z6,M14(25,1),M14(25,2),M14(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6,y6,z6,M14(49,1),M14(49,2),M14(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6,y6,z6,M14(73,1),M14(73,2),M14(73,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6,y6,z6,M14(97,1),M14(97,2),M14(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6,y6,z6,M14(110,1),M14(110,2),M14(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x6,y6,z6,M14(120,1),M14(120,2),M14(120,3),scale2, 'r', 'MaxHeadSize',0.3) 87

% hold on % quiver3(x6,y6,z6,M14(129,1),M14(129,2),M14(129,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% Configuration IV Magnetization quiver3(x6+x3m_1,y6+y3m_1,z6+z3m_1,M44(1,1),M44(1,2),M44(1,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_25,y6+y3m_25,z6+z3m_25,M44(25,1),M44(25,2),M44(25,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_49,y6+y3m_49,z6+z3m_49,M44(49,1),M44(49,2),M44(49,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_73,y6+y3m_73,z6+z3m_73,M44(73,1),M44(73,2),M44(73,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_97,y6+y3m_97,z6+z3m_97,M44(97,1),M44(97,2),M44(97,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_110,y6+y3m_110,z6+z3m_110,M44(110,1),M44(110,2),M44(110,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_120,y6+y3m_120,z6+z3m_120,M44(120,1),M44(120,2),M44(120,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on quiver3(x6+x3m_129,y6+y3m_129,z6+z3m_129,M44(129,1),M44(129,2),M44(129,3),scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.4) hold on

% Panel 5 x11 = P11c(1,1); y11 = P11c(1,2); z11 = P11c(1,3); x6m_1 = C6(1,1)/2; y6m_1 = C6(1,2)/2; z6m_1 = C6(1,3)/2; figure quiver3(x10,y10,0,C6(1,1),C6(1,2),0,scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,0,C4(1,1),C4(1,2),0,scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,0,C7(1,1),C7(1,2),0,scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,0,C9(1,1),C9(1,2),0,scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on

% Configuration II Magnetization quiver3(x10,y10,0,M25(1,1),M25(1,2),0,scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on

% Configuration IV Magnetization % quiver3(x10+x6m_1,y10+y6m_1,0,M45(1,1),M45(1,2),0,scale3, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) % hold on

% Panel 6 C12_8 = P12c - P8c; x5m_1 = C5(1,1)/2; y5m_1 = C5(1,2)/2; z5m_1 = C5(1,3)/2; x5m_25 = C5(25,1)/2; y5m_25 = C5(25,2)/2; z5m_25 = C5(25,3)/2; x5m_49 = C5(49,1)/2; y5m_49 = C5(49,2)/2; z5m_49 = C5(49,3)/2; 88 x5m_73 = C5(73,1)/2; y5m_73 = C5(73,2)/2; z5m_73 = C5(73,3)/2; x5m_97 = C5(97,1)/2; y5m_97 = C5(97,2)/2; z5m_97 = C5(97,3)/2; x5m_110 = C5(110,1)/2; y5m_110 = C5(110,2)/2; z5m_110 = C5(110,3)/2; x5m_120 = C5(120,1)/2; y5m_120 = C5(120,2)/2; z5m_120 = C5(120,3)/2; x5m_129 = C5(129,1)/2; y5m_129 = C5(129,2)/2; z5m_129 = C5(129,3)/2; figure quiver3(x11,y11,z11,C7(1,1),C7(1,2),C7(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(1,1),C10(1,2),C10(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(25,1),C10(25,2),C10(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(49,1),C10(49,2),C10(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(73,1),C10(73,2),C10(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(97,1),C10(97,2),C10(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(110,1),C10(110,2),C10(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(120,1),C10(120,2),C10(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(129,1),C10(129,2),C10(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(1,1),C5(1,2),C5(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(25,1),C5(25,2),C5(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(49,1),C5(49,2),C5(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(73,1),C5(73,2),C5(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(97,1),C5(97,2),C5(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(110,1),C5(110,2),C5(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(120,1),C5(120,2),C5(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(129,1),C5(129,2),C5(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_1,y8_1,z8_1,C12_8(1,1),C12_8(1,2),C12_8(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_25,y8_25,z8_25,C12_8(25,1),C12_8(25,2),C12_8(25,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_49,y8_49,z8_49,C12_8(49,1),C12_8(49,2),C12_8(49,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_73,y8_73,z8_73,C12_8(73,1),C12_8(73,2),C12_8(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_97,y8_97,z8_97,C12_8(97,1),C12_8(97,2),C12_8(97,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) 89 hold on quiver3(x8_110,y8_110,z8_110,C12_8(110,1),C12_8(110,2),C12_8(110,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_120,y8_120,z8_120,C12_8(120,1),C12_8(120,2),C12_8(120,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_129,y8_129,z8_129,C12_8(129,1),C12_8(129,2),C12_8(129,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on

% Configuration I Magnetization % quiver3(x8_1,y8_1,z8_1,M16(1,1),M16(1,2),M16(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_25,y8_25,z8_25,M16(25,1),M16(25,2),M16(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_49,y8_49,z8_49,M16(49,1),M16(49,2),M16(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_73,y8_73,z8_73,M16(73,1),M16(73,2),M16(73,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_97,y8_97,z8_97,M16(97,1),M16(97,2),M16(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_110,y8_110,z8_110,M16(110,1),M16(110,2),M16(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_120,y8_120,z8_120,M16(120,1),M16(120,2),M16(120,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x8_129,y8_129,z8_129,M16(129,1),M16(129,2),M16(129,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% Configuration IV Magnetization quiver3(x7+x5m_1,y7+y5m_1,z7+z5m_1,M46(1,1),M46(1,2),M46(1,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_25,y7+y5m_25,z7+z5m_25,M46(25,1),M46(25,2),M46(25,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_49,y7+y5m_49,z7+z5m_49,M46(49,1),M46(49,2),M46(49,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_73,y7+y5m_73,z7+z5m_73,M46(73,1),M46(73,2),M46(73,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_97,y7+y5m_97,z7+z5m_97,M46(97,1),M46(97,2),M46(97,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_110,y7+y5m_110,z7+z5m_110,M46(110,1),M46(110,2),M46(110,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_120,y7+y5m_120,z7+z5m_120,M46(120,1),M46(120,2),M46(120,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x7+x5m_129,y7+y5m_129,z7+z5m_129,M46(129,1),M46(129,2),M46(129,3),scale3, 'r', 'MaxHeadSize',0.3) hold on

% Panel 7 C13_9 = P13c - P9c; C13_14 = P13c - P14c; x9_1 = P9c(1,1); y9_1 = P9c(1,2); z9_1 = P9c(1,3); x9_25 = P9c(25,1); y9_25 = P9c(25,2); z9_25 = P9c(25,3); x9_49 = P9c(49,1); y9_49 = P9c(49,2); z9_49 = P9c(49,3); x9_73 = P9c(73,1); y9_73 = P9c(73,2); z9_73 = P9c(73,3); x9_97 = P9c(97,1); y9_97 = P9c(97,2); 90 z9_97 = P9c(97,3); x9_110 = P9c(110,1); y9_110 = P9c(110,2); z9_110 = P9c(110,3); x9_120 = P9c(120,1); y9_120 = P9c(120,2); z9_120 = P9c(120,3); x9_129 = P9c(129,1); y9_129 = P9c(129,2); z9_129 = P9c(129,3); x14_1 = P14c(1,1); y14_1 = P14c(1,2); z14_1 = P14c(1,3); x14_25 = P14c(25,1); y14_25 = P14c(25,2); z14_25 = P14c(25,3); x14_49 = P14c(49,1); y14_49 = P14c(49,2); z14_49 = P14c(49,3); x14_73 = P14c(73,1); y14_73 = P14c(73,2); z14_73 = P14c(73,3); x14_97 = P14c(97,1); y14_97 = P14c(97,2); z14_97 = P14c(97,3); x14_110 = P14c(110,1); y14_110 = P14c(110,2); z14_110 = P14c(110,3); x14_120 = P14c(120,1); y14_120 = P14c(120,2); z14_120 = P14c(120,3); x14_129 = P14c(129,1); y14_129 = P14c(129,2); z14_129 = P14c(129,3); x11m_1 = C11(1,1)/2; y11m_1 = C11(1,2)/2; z11m_1 = C11(1,3)/2; x11m_25 = C11(25,1)/2; y11m_25 = C11(25,2)/2; z11m_25 = C11(25,3)/2; x11m_49 = C11(49,1)/2; y11m_49 = C11(49,2)/2; z11m_49 = C11(49,3)/2; x11m_73 = C11(73,1)/2; y11m_73 = C11(73,2)/2; z11m_73 = C11(73,3)/2; x11m_97 = C11(97,1)/2; y11m_97 = C11(97,2)/2; z11m_97 = C11(97,3)/2; x11m_110 = C11(110,1)/2; y11m_110 = C11(110,2)/2; z11m_110 = C11(110,3)/2; x11m_120 = C11(120,1)/2; y11m_120 = C11(120,2)/2; z11m_120 = C11(120,3)/2; x11m_129 = C11(129,1)/2; y11m_129 = C11(129,2)/2; z11m_129 = C11(129,3)/2; figure quiver3(x10,y10,z10,C8(1,1),C8(1,2),C8(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.1,y10,z10,C8(25,1),C8(25,2),C8(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.2,y10,z10,C8(49,1),C8(49,2),C8(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.3,y10,z10,C8(73,1),C8(73,2),C8(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on 91 quiver3(x10+0.4,y10,z10,C8(97,1),C8(97,2),C8(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.5,y10,z10,C8(110,1),C8(110,2),C8(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.6,y10,z10,C8(120,1),C8(120,2),C8(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.7,y10,z10,C8(129,1),C8(129,2),C8(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(1,1),C11(1,2),C11(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.1,y10,z10,C11(25,1),C11(25,2),C11(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.2,y10,z10,C11(49,1),C11(49,2),C11(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.3,y10,z10,C11(73,1),C11(73,2),C11(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.4,y10,z10,C11(97,1),C11(97,2),C11(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.5,y10,z10,C11(110,1),C11(110,2),C11(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.6,y10,z10,C11(120,1),C11(120,2),C11(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10+0.7,y10,z10,C11(129,1),C11(129,2),C11(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_1,y9_1,z9_1,C13_9(1,1),C13_9(1,2),C13_9(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_25+0.1,y9_25,z9_25,C13_9(25,1),C13_9(25,2),C13_9(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_49+0.2,y9_49,z9_49,C13_9(49,1),C13_9(49,2),C13_9(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_73+0.3,y9_73,z9_73,C13_9(73,1),C13_9(73,2),C13_9(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_97+0.4,y9_97,z9_97,C13_9(97,1),C13_9(97,2),C13_9(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_110+0.5,y9_110,z9_110,C13_9(110,1),C13_9(110,2),C13_9(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_120+0.6,y9_120,z9_120,C13_9(120,1),C13_9(120,2),C13_9(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x9_129+0.7,y9_129,z9_129,C13_9(129,1),C13_9(129,2),C13_9(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_1,y14_1,z14_1,C13_14(1,1),C13_14(1,2),C13_14(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_25+0.1,y14_25,z14_25,C13_14(25,1),C13_14(25,2),C13_14(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_49+0.2,y14_49,z14_49,C13_14(49,1),C13_14(49,2),C13_14(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_73+0.3,y14_73,z14_73,C13_14(73,1),C13_14(73,2),C13_14(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_97+0.4,y14_97,z14_97,C13_14(97,1),C13_14(97,2),C13_14(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_110+0.5,y14_110,z14_110,C13_14(110,1),C13_14(110,2),C13_14(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_120+0.6,y14_120,z14_120,C13_14(120,1),C13_14(120,2),C13_14(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_129+0.7,y14_129,z14_129,C13_14(129,1),C13_14(129,2),C13_14(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on

% Configuration II Magnetization 92

% quiver3(x14_1,y14_1,z14_1,M27(1,1),M27(1,2),M27(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_25+0.1,y14_25,z14_25,M27(25,1),M27(25,2),M27(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_49+0.2,y14_49,z14_49,M27(49,1),M27(49,2),M27(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_73+0.3,y14_73,z14_73,M27(73,1),M27(73,2),M27(73,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_97+0.4,y14_97,z14_97,M27(97,1),M27(97,2),M27(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_110+0.5,y14_110,z14_110,M27(110,1),M27(110,2),M27(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_120+0.6,y14_120,z14_120,M27(120,1),M27(120,2),M27(120,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x14_129+0.7,y14_129,z14_129,M27(129,1),M27(129,2),M27(129,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% Configuration III & IV Magnetizations quiver3(x10+x11m_1,y10+y11m_1,z10+z11m_1,M37(1,1),M37(1,2),M37(1,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_25+0.1,y10+y11m_25,z10+z11m_25,M37(25,1),M37(25,2),M37(25,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_49+0.2,y10+y11m_49,z10+z11m_49,M37(49,1),M37(49,2),M37(49,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_73+0.3,y10+y11m_73,z10+z11m_73,M37(73,1),M37(73,2),M37(73,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_97+0.4,y10+y11m_97,z10+z11m_97,M37(97,1),M37(97,2),M37(97,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_110+0.5,y10+y11m_110,z10+z11m_110,M37(110,1),M37(110,2),M37(110,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_120+0.6,y10+y11m_120,z10+z11m_120,M37(120,1),M37(120,2),M37(120,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x10+x11m_129+0.7,y10+y11m_129,z10+z11m_129,M37(129,1),M37(129,2),M37(129,3),scale3, 'r', 'MaxHeadSize',0.3) hold on

% Panel 8 C15_14 = P15c - P14c; x9m_1 = C9(1,1)/2; y9m_1 = C9(1,2)/2; z9m_1 = C9(1,3)/2; x9m_25 = C9(25,1)/2; y9m_25 = C9(25,2)/2; z9m_25 = C9(25,3)/2; x9m_49 = C9(49,1)/2; y9m_49 = C9(49,2)/2; z9m_49 = C9(49,3)/2; x9m_73 = C9(73,1)/2; y9m_73 = C9(73,2)/2; z9m_73 = C9(73,3)/2; x9m_97 = C9(97,1)/2; y9m_97 = C9(97,2)/2; z9m_97 = C9(97,3)/2; x9m_110 = C9(110,1)/2; y9m_110 = C9(110,2)/2; z9m_110 = C9(110,3)/2; x9m_120 = C9(120,1)/2; y9m_120 = C9(120,2)/2; 93 z9m_120 = C9(120,3)/2; x9m_129 = C9(129,1)/2; y9m_129 = C9(129,2)/2; z9m_129 = C9(129,3)/2; figure quiver3(x11,y11,z11,C9(1,1),C9(1,2),C9(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(1,1),C11(1,2),C11(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(25,1),C11(25,2),C11(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(49,1),C11(49,2),C11(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(73,1),C11(73,2),C11(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(97,1),C11(97,2),C11(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(110,1),C11(110,2),C11(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(120,1),C11(120,2),C11(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(129,1),C11(129,2),C11(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(1,1),C12(1,2),C12(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(25,1),C12(25,2),C12(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(49,1),C12(49,2),C12(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(73,1),C12(73,2),C12(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(97,1),C12(97,2),C12(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(110,1),C12(110,2),C12(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(120,1),C12(120,2),C12(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(129,1),C12(129,2),C12(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_1,y14_1,z14_1,C15_14(1,1),C15_14(1,2),C15_14(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_25,y14_25,z14_25,C15_14(25,1),C15_14(25,2),C15_14(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_49,y14_49,z14_49,C15_14(49,1),C15_14(49,2),C15_14(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_73,y14_73,z14_73,C15_14(73,1),C15_14(73,2),C15_14(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_97,y14_97,z14_97,C15_14(97,1),C15_14(97,2),C15_14(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_110,y14_110,z14_110,C15_14(110,1),C15_14(110,2),C15_14(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_120,y14_120,z14_120,C15_14(120,1),C15_14(120,2),C15_14(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x14_129,y14_129,z14_129,C15_14(129,1),C15_14(129,2),C15_14(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on

% Configuration I Magnetization % quiver3(x10,y10,z10,M18(1,1),M18(1,2),M18(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x10,y10,z10,M18(25,1),M18(25,2),M18(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x10,y10,z10,M18(49,1),M18(49,2),M18(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x10,y10,z10,M18(73,1),M18(73,2),M18(73,3),scale2, 'r', 'MaxHeadSize',0.3) 94

% hold on % quiver3(x10,y10,z10,M18(97,1),M18(97,2),M18(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x10,y10,z10,M18(110,1),M18(110,2),M18(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x10,y10,z10,M18(120,1),M18(120,2),M18(120,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x10,y10,z10,M18(129,1),M18(129,2),M18(129,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% Configuration III & IV Magnetizations quiver3(x11+x9m_1,y11+y9m_1,z11+z9m_1,M38(1,1),M38(1,2),M38(1,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_25,y11+y9m_25,z11+z9m_25,M38(25,1),M38(25,2),M38(25,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_49,y11+y9m_49,z11+z9m_49,M38(49,1),M38(49,2),M38(49,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_73,y11+y9m_73,z11+z9m_73,M38(73,1),M38(73,2),M38(73,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_97,y11+y9m_97,z11+z9m_97,M38(97,1),M38(97,2),M38(97,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_110,y11+y9m_110,z11+z9m_110,M38(110,1),M38(110,2),M38(110,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_120,y11+y9m_120,z11+z9m_120,M38(120,1),M38(120,2),M38(120,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x11+x9m_129,y11+y9m_129,z11+z9m_129,M38(129,1),M38(129,2),M38(129,3),scale3, 'r', 'MaxHeadSize',0.3) hold on

% Panel 9 C16_12 = P16c - P12c; C16_15 = P16c - P15c; x15_1 = P15c(1,1); y15_1 = P15c(1,2); z15_1 = P15c(1,3); x15_25 = P15c(25,1); y15_25 = P15c(25,2); z15_25 = P15c(25,3); x15_49 = P15c(49,1); y15_49 = P15c(49,2); z15_49 = P15c(49,3); x15_73 = P15c(73,1); y15_73 = P15c(73,2); z15_73 = P15c(73,3); x15_97 = P15c(97,1); y15_97 = P15c(97,2); z15_97 = P15c(97,3); x15_110 = P15c(110,1); y15_110 = P15c(110,2); z15_110 = P15c(110,3); x15_120 = P15c(120,1); y15_120 = P15c(120,2); z15_120 = P15c(120,3); x15_129 = P15c(129,1); y15_129 = P15c(129,2); z15_129 = P15c(129,3); x12_1 = P12c(1,1); y12_1 = P12c(1,2); z12_1 = P12c(1,3); x12_25 = P12c(25,1); y12_25 = P12c(25,2); 95 z12_25 = P12c(25,3); x12_49 = P12c(49,1); y12_49 = P12c(49,2); z12_49 = P12c(49,3); x12_73 = P12c(73,1); y12_73 = P12c(73,2); z12_73 = P12c(73,3); x12_97 = P12c(97,1); y12_97 = P12c(97,2); z12_97 = P12c(97,3); x12_110 = P12c(110,1); y12_110 = P12c(110,2); z12_110 = P12c(110,3); x12_120 = P12c(120,1); y12_120 = P12c(120,2); z12_120 = P12c(120,3); x12_129 = P12c(129,1); y12_129 = P12c(129,2); z12_129 = P12c(129,3); x16_1 = P16c(1,1); y16_1 = P16c(1,2); z16_1 = P16c(1,3); x16_25 = P16c(25,1); y16_25 = P16c(25,2); z16_25 = P16c(25,3); x16_49 = P16c(49,1); y16_49 = P16c(49,2); z16_49 = P16c(49,3); x16_73 = P16c(73,1); y16_73 = P16c(73,2); z16_73 = P16c(73,3); x16_97 = P16c(97,1); y16_97 = P16c(97,2); z16_97 = P16c(97,3); x16_110 = P16c(110,1); y16_110 = P16c(110,2); z16_110 = P16c(110,3); x16_120 = P16c(120,1); y16_120 = P16c(120,2); z16_120 = P16c(120,3); x16_129 = P16c(129,1); y16_129 = P16c(129,2); z16_129 = P16c(129,3); figure quiver3(x11,y11,z11,C12(1,1),C12(1,2),C12(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.1,y11,z11,C12(25,1),C12(25,2),C12(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.2,y11,z11,C12(49,1),C12(49,2),C12(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.3,y11,z11,C12(73,1),C12(73,2),C12(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.4,y11,z11,C12(97,1),C12(97,2),C12(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.5,y11,z11,C12(110,1),C12(110,2),C12(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.6,y11,z11,C12(120,1),C12(120,2),C12(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.7,y11,z11,C12(129,1),C12(129,2),C12(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(1,1),C10(1,2),C10(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.1,y11,z11,C10(25,1),C10(25,2),C10(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.2,y11,z11,C10(49,1),C10(49,2),C10(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.3,y11,z11,C10(73,1),C10(73,2),C10(73,3),scale, 'b', 'MaxHeadSize',0.01) 96 hold on quiver3(x11-0.4,y11,z11,C10(97,1),C10(97,2),C10(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.5,y11,z11,C10(110,1),C10(110,2),C10(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.6,y11,z11,C10(120,1),C10(120,2),C10(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x11-0.7,y11,z11,C10(129,1),C10(129,2),C10(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_1,y15_1,z15_1,C16_15(1,1),C16_15(1,2),C16_15(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_25-0.1,y15_25,z15_25,C16_15(25,1),C16_15(25,2),C16_15(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_49-0.2,y15_49,z15_49,C16_15(49,1),C16_15(49,2),C16_15(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_73-0.3,y15_73,z15_73,C16_15(73,1),C16_15(73,2),C16_15(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_97-0.4,y15_97,z15_97,C16_15(97,1),C16_15(97,2),C16_15(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_110-0.5,y15_110,z15_110,C16_15(110,1),C16_15(110,2),C16_15(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_120-0.6,y15_120,z15_120,C16_15(120,1),C16_15(120,2),C16_15(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x15_129-0.7,y15_129,z15_129,C16_15(129,1),C16_15(129,2),C16_15(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_1,y12_1,z12_1,C16_12(1,1),C16_12(1,2),C16_12(1,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_25-0.1,y12_25,z12_25,C16_12(25,1),C16_12(25,2),C16_12(25,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_49-0.2,y12_49,z12_49,C16_12(49,1),C16_12(49,2),C16_12(49,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_73-0.3,y12_73,z12_73,C16_12(73,1),C16_12(73,2),C16_12(73,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_97-0.4,y12_97,z12_97,C16_12(97,1),C16_12(97,2),C16_12(97,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_110-0.5,y12_110,z12_110,C16_12(110,1),C16_12(110,2),C16_12(110,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_120-0.6,y12_120,z12_120,C16_12(120,1),C16_12(120,2),C16_12(120,3),scale, 'b', 'MaxHeadSize',0.01) hold on quiver3(x12_129-0.7,y12_129,z12_129,C16_12(129,1),C16_12(129,2),C16_12(129,3),scale, 'b', 'MaxHeadSize',0.01) hold on

% Configuration II Magnetization % quiver3(x16_1,y16_1,z16_1,M29(1,1),M29(1,2),M29(1,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x16_25-0.1,y16_25,z16_25,M29(25,1),M29(25,2),M29(25,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x16_49-0.2,y16_49,z16_49,M29(49,1),M29(49,2),M29(49,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x16_73-0.3,y16_73,z16_73,M29(73,1),M29(73,2),M29(73,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x16_97-0.4,y16_97,z16_97,M29(97,1),M29(97,2),M29(97,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% quiver3(x16_110-0.5,y16_110,z16_110,M29(110,1),M29(110,2),M29(110,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x16_120-0.6,y16_120,z16_120,M29(120,1),M29(120,2),M29(120,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on % quiver3(x16_129-0.7,y16_129,z16_129,M29(129,1),M29(129,2),M29(129,3),scale2, 'r', 'MaxHeadSize',0.3) % hold on

% Configuration III & IV Magnetizations quiver3(x12_1+C16_12(1,1)/2,y12_1+C16_12(1,2)/2,z12_1+C16_12(1,3)/2,M39(1,1),M39(1,2),M39(1,3),scal e3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_25+C16_12(25,1)/2- 0.1,y12_25+C16_12(25,2)/2,z12_25+C16_12(25,3)/2,M39(25,1),M39(25,2),M39(25,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_49+C16_12(49,1)/2- 0.2,y12_49+C16_12(49,2)/2,z12_49+C16_12(49,3)/2,M39(49,1),M39(49,2),M39(49,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_73+C16_12(73,1)/2- 0.3,y12_73+C16_12(73,2)/2,z12_73+C16_12(73,3)/2,M39(73,1),M39(73,2),M39(73,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_97+C16_12(97,1)/2- 0.4,y12_97+C16_12(97,2)/2,z12_97+C16_12(97,3)/2,M39(97,1),M39(97,2),M39(97,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_110+C16_12(110,1)/2- 0.5,y12_110+C16_12(110,2)/2,z12_110+C16_12(110,3)/2,M39(110,1),M39(110,2),M39(110,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_120+C16_12(120,1)/2- 0.6,y12_120+C16_12(120,2)/2,z12_120+C16_12(120,3)/2,M39(120,1),M39(120,2),M39(120,3),scale3, 'r', 'MaxHeadSize',0.3) hold on quiver3(x12_129+C16_12(129,1)/2- 0.7,y12_129+C16_12(129,2)/2,z12_129+C16_12(129,3)/2,M39(129,1),M39(129,2),M39(129,3),scale3, 'r', 'MaxHeadSize',0.3) hold on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Plot of the Normalized Total Magnetic Work as a function of Vertical Fold Angle and Horizontal Fold Angle for i = 1:128 W1TOT_c(1,:) = 0; W1TOT_c(i+1,:) = W1TOT(i,:)/4; W2TOT_c(1,:) = 0; W2TOT_c(i+1,:) = W2TOT(i,:)/5; W3TOT_c(1,:) = 0; W3TOT_c(i+1,:) = W3TOT(i,:)/6; W4TOT_c(1,:) = 0; W4TOT_c(i+1,:) = W4TOT(i,:)/9; end

AngleV_c = (pi - AngleV)*(180/pi); AngleH_c = (pi - AngleH)*(180/pi); figure plot3(AngleV_c, AngleH_c, W1TOT_c, 'r', 'LineWidth', 3) hold on plot3(AngleV_c, AngleH_c, W2TOT_c, 'b', 'LineWidth', 3) hold on plot3(AngleV_c, AngleH_c, W3TOT_c, 'k', 'LineWidth', 3) hold on plot3(AngleV_c, AngleH_c, W4TOT_c, 'g', 'LineWidth', 3) 98 hold on grid on xlabel('Vertical Angle (deg)') ylabel('Horizontal Angle (deg)') zlabel('Magnetic Work Done/Total Magnetic Energy Potential []') legend('Configuration I', 'Configuration II', 'Configuration III', 'Configuration IV')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Miura-ori plots of Configurations I, II, III, and IV with magnetization and torque vectors at the initial flat state

% Miura-ori pattern figure quiver3(x6,y6,z6,C1(1,1),C1(1,2),C1(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(1,1),C3(1,2),C3(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_1,y2_1,z2_1,C1_2(1,1),C1_2(1,2),C1_2(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_1,y5_1,z5_1,C1_5(1,1),C1_5(1,2),C1_5(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C4(1,1),C4(1,2),C4(1,3),scale, 'r', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C2(1,1),C2(1,2),C2(1,3),scale, 'r', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x2_1,y2_1,z2_1,C2_3(1,1),C2_3(1,2),C2_3(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(1,1),C5(1,2),C5(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x3_1,y3_1,z3_1,C4_3(1,1),C4_3(1,2),C4_3(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_1,y8_1,z8_1,C4_8(1,1),C4_8(1,2),C4_8(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C6(1,1),C6(1,2),C6(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(1,1),C8(1,2),C8(1,3),scale, 'r', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_1,y5_1,z5_1,C9_5(1,1),C9_5(1,2),C9_5(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,0,C7(1,1),C7(1,2),0,scale, 'r', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,0,C9(1,1),C9(1,2),0,scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C10(1,1),C10(1,2),C10(1,3),scale, 'r', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x7,y7,z7,C5(1,1),C5(1,2),C5(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x8_1,y8_1,z8_1,C12_8(1,1),C12_8(1,2),C12_8(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C11(1,1),C11(1,2),C11(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x9_1,y9_1,z9_1,C13_9(1,1),C13_9(1,2),C13_9(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x14_1,y14_1,z14_1,C13_14(1,1),C13_14(1,2),C13_14(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,z11,C12(1,1),C12(1,2),C12(1,3),scale, 'r', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x14_1,y14_1,z14_1,C15_14(1,1),C15_14(1,2),C15_14(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) 99 hold on quiver3(x15_1,y15_1,z15_1,C16_15(1,1),C16_15(1,2),C16_15(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x12_1,y12_1,z12_1,C16_12(1,1),C16_12(1,2),C16_12(1,3),scale, 'k', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on axis([-1.2 0.2 -1.2 0.2 -1 1])

% Configuration I MIP2 = [cosd(-M12dangle) sind(-M12dangle) 0]; MIP8 = MIP2; MIP4 = [cosd(M14dangle+180) sind(M14dangle+180) 0]; MIP6 = MIP4; TIP2 = cross(MIP2, H_dir(1,:)); TIP8 = TIP2; TIP4 = cross(MIP4, H_dir(1,:)); TIP6 = TIP4; quiver3((x3_1 + x2_1)/2,y6/2,0,MIP2(1,1),MIP2(1,2),MIP2(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3((x3_1 + x2_1)/2,y6/2,0,TIP2(1,1),TIP2(1,2),TIP2(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3((x9_1 + x10)/2,(y9_1 + y6)/2,0,MIP4(1,1),MIP4(1,2),MIP4(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3((x9_1 + x10)/2,(y9_1 + y6)/2,0,TIP4(1,1),TIP4(1,2),TIP4(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3(x7/2,(y9_1 + y6)/2,0,MIP6(1,1),MIP6(1,2),MIP6(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3(x7/2,(y9_1 + y6)/2,0,TIP6(1,1),TIP6(1,2),TIP6(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3((x3_1 + x2_1)/2,(y14_1 + y11)/2,0,MIP8(1,1),MIP8(1,2),MIP8(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on quiver3((x3_1 + x2_1)/2,(y14_1 + y11)/2,0,TIP8(1,1),TIP8(1,2),TIP8(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) hold on

% Configuration II MIIP1 = [cosd(162.0001) sind(162.0001) 0]; MIIP3 = MIIP1; MIIP7 = MIIP1; MIIP9 = MIIP1; MIIP5 = [cosd(3.8403) sind(3.8403) 0]; TIIP1 = cross(MIIP1, H_dir(1,:)); TIIP3 = TIIP1; TIIP7 = TIIP1; TIIP9 = TIIP1; TIIP5 = cross(MIIP5, H_dir(1,:));

% quiver3((-1 + x2_1)/2,y6/2,0,MIIP1(1,1),MIIP1(1,2),MIIP1(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,y6/2,0,TIIP1(1,1),TIIP1(1,2),TIIP1(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,y6/2,0,MIIP3(1,1),MIIP3(1,2),MIIP3(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,y6/2,0,TIIP3(1,1),TIIP3(1,2),TIIP3(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on 100

% quiver3((x6 + x7)/2,(y6 + y10)/2,0,MIIP5(1,1),MIIP5(1,2),MIIP5(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x6 + x7)/2,(y6 + y10)/2,0,TIIP5(1,1),TIIP5(1,2),TIIP5(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,(y14_1 + y11)/2,0,MIIP7(1,1),MIIP7(1,2),MIIP7(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,(y14_1 + y11)/2,0,TIIP7(1,1),TIIP7(1,2),TIIP7(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y14_1 + y11)/2,0,MIIP9(1,1),MIIP9(1,2),MIIP9(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y14_1 + y11)/2,0,TIIP9(1,1),TIIP9(1,2),TIIP9(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on

% Configuration III MIIIP1 = [cosd(162.1169) sind(162.1169) 0]; MIIIP3 = MIIIP1; MIIIP7 = MIIIP1; MIIIP9 = MIIIP1; MIIIP2 = [cosd(344.6915) sind(344.6915) 0]; MIIIP8 = MIIIP2; TIIIP1 = cross(MIIIP1, H_dir(1,:)); TIIIP3 = TIIIP1; TIIIP7 = TIIIP1; TIIIP9 = TIIIP1; TIIIP2 = cross(MIIIP2, H_dir(1,:)); TIIIP8 = TIIIP2;

% quiver3((-1 + x2_1)/2,y6/2,0,MIIIP1(1,1),MIIIP1(1,2),MIIIP1(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,y6/2,0,TIIIP1(1,1),TIIIP1(1,2),TIIIP1(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,y6/2,0,MIIIP2(1,1),MIIIP2(1,2),MIIIP2(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,y6/2,0,TIIIP2(1,1),TIIIP2(1,2),TIIIP2(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,y6/2,0,MIIIP3(1,1),MIIIP3(1,2),MIIIP3(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,y6/2,0,TIIIP3(1,1),TIIIP3(1,2),TIIIP3(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,(y14_1 + y11)/2,0,MIIIP7(1,1),MIIIP7(1,2),MIIIP7(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,(y14_1 + y11)/2,0,TIIIP7(1,1),TIIIP7(1,2),TIIIP7(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,(y14_1 + y11)/2,0,MIIIP8(1,1),MIIIP8(1,2),MIIIP8(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,(y14_1 + y11)/2,0,TIIIP8(1,1),TIIIP8(1,2),TIIIP8(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y14_1 + y11)/2,0,MIIIP9(1,1),MIIIP9(1,2),MIIIP9(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y14_1 + y11)/2,0,TIIIP9(1,1),TIIIP9(1,2),TIIIP9(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) 101

% hold on

% Configuration IV MIVP1 = [cosd(162.1169) sind(162.1169) 0]; MIVP3 = MIVP1; MIVP7 = MIVP1; MIVP9 = MIVP1; MIVP2 = [cosd(344.6915) sind(344.6915) 0]; MIVP8 = MIVP2; MIVP4 = [cosd(198.0589) sind(198.0589) 0]; MIVP6 = MIVP4; MIVP5 = [cosd(3.5188) sind(3.5188) 0]; TIVP1 = cross(MIVP1, H_dir(1,:)); TIVP3 = TIVP1; TIVP7 = TIVP1; TIVP9 = TIVP1; TIVP2 = cross(MIVP2, H_dir(1,:)); TIVP8 = TIVP2; TIVP4 = cross(MIVP4, H_dir(1,:)); TIVP6 = TIVP4; TIVP5 = cross(MIVP5, H_dir(1,:));

% quiver3((-1 + x2_1)/2,y6/2,0,MIVP1(1,1),MIVP1(1,2),MIVP1(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,y6/2,0,TIVP1(1,1),TIVP1(1,2),TIVP1(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,y6/2,0,MIVP2(1,1),MIVP2(1,2),MIVP2(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,y6/2,0,TIVP2(1,1),TIVP2(1,2),TIVP2(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,y6/2,0,MIVP3(1,1),MIVP3(1,2),MIVP3(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,y6/2,0,TIVP3(1,1),TIVP3(1,2),TIVP3(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x9_1 + x10)/2,(y9_1 + y6)/2,0,MIVP4(1,1),MIVP4(1,2),MIVP4(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x9_1 + x10)/2,(y9_1 + y6)/2,0,TIVP4(1,1),TIVP4(1,2),TIVP4(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x6 + x7)/2,(y6 + y10)/2,0,MIVP5(1,1),MIVP5(1,2),MIVP5(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x6 + x7)/2,(y6 + y10)/2,0,TIVP5(1,1),TIVP5(1,2),TIVP5(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y9_1 + y6)/2,0,MIVP6(1,1),MIVP6(1,2),MIVP6(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y9_1 + y6)/2,0,TIVP6(1,1),TIVP6(1,2),TIVP6(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,(y14_1 + y11)/2,0,MIVP7(1,1),MIVP7(1,2),MIVP7(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((-1 + x2_1)/2,(y14_1 + y11)/2,0,TIVP7(1,1),TIVP7(1,2),TIVP7(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3((x3_1 + x2_1)/2,(y14_1 + y11)/2,0,MIVP8(1,1),MIVP8(1,2),MIVP8(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on

102

% quiver3((x3_1 + x2_1)/2,(y14_1 + y11)/2,0,TIVP8(1,1),TIVP8(1,2),TIVP8(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y14_1 + y11)/2,0,MIVP9(1,1),MIVP9(1,2),MIVP9(1,3),scale4, 'color', [0 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on % quiver3(x7/2,(y14_1 + y11)/2,0,TIVP9(1,1),TIVP9(1,2),TIVP9(1,3),scale4, 'color', [1 0.5 0], 'LineWidth', 2, 'MaxHeadSize',0.5) % hold on

% Panels 4 and 5 for Bisector figure figure quiver3(x10,y10,z10,C6(1,1),C6(1,2),C6(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10+0.8,y10,z10,C6(73,1),C6(73,2),C6(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,z10,C8(1,1),C8(1,2),C8(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10+0.8,y10,z10,C8(73,1),C8(73,2),C8(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,z6,C3(1,1),C3(1,2),C3(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.8,y6,z6,C3(73,1),C3(73,2),C3(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_1,y5_1,z5_1,C9_5(1,1),C9_5(1,2),C9_5(1,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x5_73+0.8,y5_73,z5_73,C9_5(73,1),C9_5(73,2),C9_5(73,3),scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x10,y10,0,C6(1,1),C6(1,2),0,scale, 'b', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6,y6,0,C4(1,1),C4(1,2),0,scale, 'm', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,0,C7(1,1),C7(1,2),0,scale, 'm', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11,y11,0,C9(1,1),C9(1,2),0,scale, 'm', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x6+0.8,y6,0,C4(1,1),C4(1,2),0,scale, 'm', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11+0.8,y11,0,C7(1,1),C7(1,2),0,scale, 'm', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on quiver3(x11+0.8,y11,0,C9(1,1),C9(1,2),0,scale, 'm', 'LineWidth', 2, 'MaxHeadSize',0.01) hold on

% Bisector quiver3(x10+x6m_1,y10+y6m_1,0,bisector(1,1),bisector(1,2),bisector(1,3),scale4, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on quiver3(x10+x6m_1+0.8,y10+y6m_1,0,bisector(73,1),bisector(73,2),bisector(73,3),scale2, 'r', 'LineWidth', 2, 'MaxHeadSize',0.3) hold on

103

MuPAD calculations for coefficients α and β: Configuration I Calculations Defining M14 M14x := `α`*-0.104529008728949 + `β`*-1 + cos(46.021523270560360*PI/180);

M14y := `α`*0.994521838037830 + `β`*0 + sin(46.021523270560360*PI/180);

Equations14 := simplify(matrix([[M14x],[M14y]]));

Solset14 := solve(Equations14, [`α`,`β`]);

Defining M16 M16x := `α`*-0.104529008728949 + `β`*1 + cos(46.021523270560350*PI/180);

M16y := `α`*0.994521838037830 + `β`*0 + sin(46.021523270560350*PI/180);

Equations16 := simplify(matrix([[M16x],[M16y]]));

Solset41 :=solve(Equations61, [`α`,`β`]);

Configuration II Calculations Defining M21 M21x := `α`*0.104529318889919 + `β`*-1 + cos(46.021437381561670*PI/180);

M21y := `α`*0.994521805438377 + `β`*0 - sin(46.021437381561670*PI/180);

Equations21 := simplify(matrix([[M21x],[M21y]]));

Solset21 := solve(Equations21, [`α`,`β`]);

Defining M23 M23x := `α`*0.104529318889920 + `β`*1 + cos(46.021437381561690*PI/180);

M23y := `α`*0.994521805438377 + `β`*0 - sin(46.021437381561690*PI/180);

Equations23 := simplify(matrix([[M23x],[M23y]]));

Solset23 := solve(Equations23, [`α`,`β`]);

Defining M27 M27x := `α`*-0.104529318889919 + `β`*-1 + cos(46.021437381561704*PI/180);

104

M27y := `α`*-0.994521805438377 + `β`*0 - sin(46.021437381561704*PI/180);

Equations27 := simplify(matrix([[M27x],[M27y]]));

Solset27 := solve(Equations27, [`α`,`β`]);

Defining M29 M29x := `α`*-0.104529318889919 + `β`*1 + cos(46.021437381561670*PI/180);

M29y := `α`*-0.994521805438377 + `β`*0 - sin(46.021437381561670*PI/180);

Equations29 := simplify(matrix([[M29x],[M29y]]));

Solset29 := solve(Equations29, [`α`,`β`]);

Configuration III Calculations Defining M31 M31x := `α`*0.104529318889919 + `β`*-1 + cos(6*PI/180);

M31y := `α`*0.994521805438377 + `β`*0 - sin(6*PI/180);

Equations31 := simplify(matrix([[M31x],[M31y]]));

Solset31 := solve(Equations31, [`α`,`β`]);

Defining M33 M33x := `α`*0.104529318889920 + `β`*1 + cos(6*PI/180);

M33y := `α`*0.994521805438377 + `β`*0 - sin(6*PI/180);

Equations33 := simplify(matrix([[M33x],[M33y]]));

Solset33 := solve(Equations33, [`α`,`β`]);

Defining M37 M37x := `α`*-0.104529318889919 + `β`*-1 + cos(6*PI/180);

105

M37y := `α`*-0.994521805438377 + `β`*0 - sin(6*PI/180);

Equations37 := simplify(matrix([[M37x],[M37y]]));

Solset37 := solve(Equations37, [`α`,`β`]);

Defining M93 M39x := `α`*-0.104529318889919 + `β`*1 + cos(6*PI/180);

M39y := `α`*-0.994521805438377 + `β`*0 - sin(6*PI/180);

Equations39 := simplify(matrix([[M39x],[M39y]]));

Solset39 := solve(Equations39, [`α`,`β`]);

Configuration IV Calculations Defining M44 M44x := `α`*-0.104529008728948 + `β`*-1 - 0;

M44y := `α`*0.994521838037830 + `β`*0 + 1;

Equations44 := simplify(matrix([[M44x],[M44y]]));

Solset44 := solve(Equations44, [`α`,`β`]);

Defining M64 M46x := `α`*-0.104529008728949 + `β`*1 - 0;

M46y := `α`*0.994521838037830 + `β`*0 + 1;

Equations46 := simplify(matrix([[M46x],[M46y]]));

Solset46 := solve(Equations46, [`α`,`β`]);

106

Appendix C Experimental data of initial four configurations

Figure A.1 Configuration I prototype within the Walker Scientific 7H electromagnet subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

107

Figure A.2 Configuration III prototype within the Walker Scientific 7H electromagnet subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

Figure A.3 Configuration IV prototype within the Walker Scientific 7H electromagnet subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

108

Table A.4 Experimental data of the configuration I prototypes Configuration I OTTC_2A OTTC_2C OTTC_1D Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 175.222 173.096 174.394 174.268 175.306 174.122 173.220 172.372 173.983 173.998 6.002 0.969 60 165.530 165.294 166.203 165.391 165.255 164.071 163.617 162.401 162.241 164.445 15.555 1.432 90 157.730 158.871 158.153 158.933 157.728 158.396 156.199 154.705 154.296 157.223 22.777 1.745 120 151.982 150.308 148.929 153.095 153.700 153.166 147.422 148.949 147.503 150.562 29.438 2.490

Crease 1 150 146.107 145.069 144.331 148.455 147.261 146.157 139.172 139.675 140.066 144.033 35.967 3.505 180 139.514 138.993 139.309 140.806 141.044 139.998 135.740 134.928 135.998 138.481 41.519 2.307 210 132.921 133.397 132.744 135.901 134.844 135.972 129.545 128.496 127.631 132.383 47.617 3.133 240 129.156 128.578 127.763 130.380 130.656 129.007 124.715 125.360 125.458 127.897 52.103 2.224

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 171.432 172.399 173.050 169.684 170.226 170.477 160.841 162.108 159.036 167.695 12.305 5.432 60 162.007 163.311 160.311 154.280 153.241 155.004 148.358 144.824 147.401 154.304 25.696 6.624 90 151.438 151.763 152.950 142.409 142.426 140.264 133.335 133.524 133.129 142.360 37.640 8.156 120 144.445 142.871 143.276 132.186 133.790 134.143 120.752 122.967 120.668 132.789 47.211 9.611

Crease 2 150 138.180 139.093 137.604 123.699 126.730 125.435 112.319 114.503 114.893 125.828 54.172 10.624 180 134.540 133.007 131.555 121.018 119.012 117.374 108.065 103.847 105.444 119.318 60.682 11.906 210 125.767 126.354 125.958 113.421 112.966 109.943 98.860 100.859 100.718 112.761 67.239 11.272 240 123.094 123.709 122.292 106.560 107.940 107.220 95.906 96.740 98.632 109.121 70.879 11.353

60 182.462 183.020 181.034 176.037 178.628 179.019 179.207 179.889 179.521 179.869 0.131 2.107 90 180.609 181.145 180.514 175.825 176.348 176.079 179.643 180.319 179.023 178.834 1.166 2.152

120 177.905 177.550 178.852 174.008 172.291 175.075 180.315 177.786 178.220 176.889 3.111 2.559 Crease 3 109

150 177.236 178.153 177.674 172.167 170.343 170.959 176.725 177.589 179.045 175.543 4.457 3.382 180 174.759 179.172 176.139 169.539 171.068 168.088 177.173 176.275 177.241 174.384 5.616 3.869 210 175.682 176.933 175.971 165.540 168.636 166.369 173.973 175.262 175.661 172.670 7.330 4.504 240 174.398 175.273 173.215 166.616 166.253 165.361 174.262 173.434 175.545 171.595 8.405 4.217

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 176.424 176.858 176.748 174.002 172.426 173.092 177.077 176.633 176.487 175.527 4.473 1.820 60 172.758 173.265 173.173 173.207 171.391 171.731 175.416 174.930 173.774 173.294 6.706 1.313 90 169.964 169.233 169.706 169.740 167.197 167.602 172.972 171.053 171.325 169.866 10.134 1.799 120 165.338 170.726 165.877 167.281 164.986 167.643 169.919 167.919 169.089 167.642 12.358 2.015

Crease 8 150 162.590 163.848 164.351 166.739 164.431 165.035 165.620 165.255 164.671 164.727 15.273 1.161 180 164.002 163.302 165.441 160.673 161.501 162.458 162.587 162.797 163.113 162.875 17.125 1.374 210 160.252 160.376 159.175 159.700 157.756 158.795 162.073 162.383 163.791 160.478 19.522 1.929 240 157.197 158.876 157.141 157.407 157.255 156.328 156.104 156.441 156.149 156.989 23.011 0.873

60 150.404 149.450 147.057 152.186 151.195 152.202 154.506 154.244 153.148 151.599 28.401 2.382

90 135.563 134.436 134.445 138.855 137.251 137.245 141.799 139.055 138.612 137.473 42.527 2.413 120 126.603 125.216 126.790 128.751 129.346 131.405 132.281 132.376 129.748 129.168 50.832 2.583

Crease 11 150 120.761 119.151 119.219 122.925 122.350 124.000 123.482 122.210 121.046 121.683 58.317 1.754 180 110.552 112.215 112.449 117.890 116.663 115.776 113.583 117.068 113.295 114.388 65.612 2.539 210 106.008 106.211 106.701 110.119 109.819 109.398 111.644 110.585 110.351 108.982 71.018 2.104 240 103.423 100.698 101.303 103.927 105.736 105.200 105.604 106.151 106.482 104.280 75.720 2.110

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 166.074 166.177 165.547 161.143 163.652 163.583 158.515 160.045 160.726 162.829 17.171 2.828 60 152.194 153.244 153.300 149.271 148.033 147.808 138.220 138.790 137.585 146.494 33.506 6.551 Crease 12 90 139.101 141.508 140.310 134.735 134.507 135.568 125.405 124.592 124.565 133.366 46.634 6.830 110

120 131.260 132.171 130.141 126.261 125.934 125.472 116.305 114.749 113.575 123.985 56.015 7.260 150 126.080 126.940 126.417 118.689 117.240 118.105 106.850 107.454 106.627 117.156 62.844 8.482 180 121.540 119.053 121.211 112.418 111.304 110.609 104.346 103.121 101.449 111.672 68.328 7.713 210 116.815 116.706 115.472 109.164 107.351 109.842 95.943 98.924 97.155 107.486 72.514 8.348 240 111.642 111.038 111.237 103.438 105.295 105.426 94.570 95.508 93.723 103.542 76.458 7.324

111

Table A.5 Experimental data of the configuration II prototypes Configuration II OTTC_4A OTTC_2D OTTC_3C Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 164.326 162.770 160.810 164.667 164.404 165.196 166.482 166.232 166.581 164.608 15.392 1.881 60 138.884 139.162 141.996 148.728 147.679 148.193 149.505 149.172 149.680 145.889 34.111 4.531 90 125.464 124.894 124.990 135.612 133.746 133.992 137.394 136.456 134.272 131.869 48.131 5.201 120 112.009 111.040 111.032 123.932 124.032 124.104 125.125 125.487 123.415 120.020 59.980 6.530

Crease 1 150 102.028 100.742 102.090 116.106 113.393 114.459 114.186 115.644 114.366 110.335 69.665 6.595 180 94.469 94.592 93.550 107.660 106.979 108.597 104.725 106.207 105.734 102.501 77.499 6.326 210 88.302 88.616 88.680 100.159 100.327 99.644 99.913 97.720 97.955 95.702 84.298 5.454 240 82.745 84.448 83.264 94.407 95.103 93.852 93.440 92.291 92.907 90.273 89.727 5.172

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 168.042 169.949 167.100 163.841 162.440 163.614 166.752 168.750 167.422 166.434 13.566 2.561 60 148.913 148.684 150.560 150.379 149.590 150.047 155.587 157.039 156.665 151.940 28.060 3.443 90 136.315 134.317 138.133 135.834 134.845 135.630 142.348 142.762 141.686 137.986 42.014 3.388 120 124.539 124.591 124.072 123.170 122.860 123.115 131.617 129.957 131.553 126.164 53.836 3.739

Crease 2 150 114.751 112.256 113.622 113.048 113.313 114.293 120.378 119.458 121.425 115.838 64.162 3.543 180 107.198 105.267 107.309 105.009 106.944 107.577 114.338 113.476 112.615 108.859 71.141 3.599 210 98.720 99.050 98.161 99.156 100.473 100.641 106.690 104.677 106.223 101.532 78.468 3.382 240 94.123 94.380 93.385 94.770 93.103 93.511 100.375 101.092 99.478 96.024 83.976 3.283

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 177.333 175.799 176.881 177.165 176.396 176.496 179.854 178.580 178.310 177.424 2.576 1.273 60 171.074 171.767 171.464 173.028 171.336 172.541 174.855 175.057 174.721 172.871 7.129 1.622

Crease 3 90 165.899 165.961 166.366 168.404 167.336 169.237 170.858 172.085 171.409 168.617 11.383 2.409 120 162.160 162.193 162.000 165.137 166.459 165.849 166.792 166.785 167.079 164.939 15.061 2.194 112

150 157.147 158.656 159.865 162.470 163.797 165.156 164.700 164.390 165.073 162.362 17.638 3.041 180 157.320 156.263 156.361 159.122 159.113 158.916 163.488 162.705 163.868 159.684 20.316 2.972 210 154.955 153.845 155.289 157.765 157.156 159.207 160.384 161.328 161.173 157.900 22.100 2.803 240 150.650 150.540 150.213 158.138 157.536 156.469 159.081 156.913 157.952 155.277 24.723 3.683

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 176.333 177.643 178.560 178.694 178.313 178.351 174.670 177.561 176.129 177.362 2.638 1.371 60 170.924 174.293 172.364 174.767 174.180 175.489 174.537 174.842 175.058 174.050 5.950 1.464 90 167.633 168.210 165.783 172.057 172.777 171.413 172.441 172.411 173.936 170.740 9.260 2.803 120 164.593 166.112 163.709 169.303 169.744 168.243 170.449 172.686 172.813 168.628 11.372 3.278

Crease 8 150 160.084 159.930 157.893 165.577 166.507 164.589 168.047 168.333 168.688 164.405 15.595 4.092 180 156.698 158.725 159.605 162.614 160.668 162.795 165.634 162.488 162.590 161.313 18.687 2.667 210 154.321 152.477 153.215 157.673 159.086 157.523 162.002 163.131 161.791 157.913 22.087 3.954 240 152.216 153.181 152.077 156.295 157.085 156.220 161.519 162.313 160.334 156.804 23.196 3.909

60 148.280 149.311 150.618 160.226 159.344 158.110 155.993 154.965 156.049 154.766 25.234 4.390

90 136.151 136.075 136.217 145.749 147.771 146.486 141.953 143.510 142.597 141.834 38.166 4.645 120 125.647 127.006 125.957 136.910 138.635 138.961 132.440 131.024 131.443 132.003 47.997 5.242

Crease 11 150 119.562 120.619 119.711 129.194 129.581 129.298 122.515 123.553 122.310 124.038 55.962 4.198 180 114.609 113.193 115.017 122.515 122.204 121.605 112.269 114.865 114.974 116.806 63.194 4.086 210 108.889 106.840 108.713 115.684 115.937 114.418 109.169 107.693 108.502 110.649 69.351 3.613 240 102.505 102.357 101.750 110.858 108.558 110.413 101.854 100.864 101.930 104.565 75.435 4.105

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 176.441 175.711 174.536 172.184 173.787 174.976 177.719 177.329 176.889 175.508 4.492 1.812 60 163.719 163.411 163.517 166.795 165.873 164.011 168.737 170.117 170.128 166.256 13.744 2.819 Crease 12 90 150.625 155.068 152.324 157.177 156.083 156.935 159.467 160.289 161.589 156.617 23.383 3.601 113

120 139.952 140.244 140.474 148.596 148.625 150.954 149.389 149.606 149.234 146.342 33.658 4.641 150 131.903 131.918 134.527 141.275 142.366 141.383 140.345 140.122 140.364 138.245 41.755 4.221 180 126.181 122.506 125.602 134.924 133.370 133.869 132.510 131.831 132.520 130.368 49.632 4.408 210 115.318 116.546 116.624 128.692 125.629 128.830 126.168 126.822 126.647 123.475 56.525 5.596 240 113.472 113.568 114.008 123.919 121.229 121.948 120.399 120.668 119.769 118.776 61.224 3.996

114

Table A.6 Experimental data of the configuration III prototypes Configuration III OTTC_3D OTTC_4C OTTC_5B Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 154.086 151.197 153.303 163.856 163.774 163.827 153.626 152.194 150.959 156.314 23.686 5.724 60 135.932 132.710 131.321 153.011 150.833 152.211 132.970 132.610 129.987 139.065 40.935 9.856 90 112.811 114.095 113.892 142.467 141.550 140.990 120.649 121.247 119.218 125.213 54.787 12.706 120 101.905 103.907 103.982 133.867 135.423 133.030 108.158 107.920 108.158 115.150 64.850 14.395

Crease 1 150 95.554 95.920 93.706 126.624 126.104 125.800 97.517 99.784 99.219 106.692 73.308 14.730 180 92.155 91.205 92.230 120.783 119.425 119.299 94.226 94.171 93.231 101.858 78.142 13.523 210 86.636 87.982 89.307 115.222 113.685 114.166 87.382 87.974 86.862 96.580 83.420 13.361 240 86.314 87.306 86.055 110.331 108.104 109.149 86.854 86.438 84.095 93.850 86.150 11.556

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 173.828 173.923 175.722 177.984 178.242 177.518 177.795 179.821 178.590 177.047 2.953 2.093 60 162.690 163.737 162.984 174.989 174.310 174.947 169.431 168.211 169.395 168.966 11.034 5.051 90 153.035 152.461 150.393 171.769 171.918 171.666 160.677 160.365 159.604 161.321 18.679 8.655 120 142.542 141.605 140.672 164.910 165.605 164.883 152.866 152.581 152.279 153.105 26.895 10.209

Crease 2 150 132.121 130.270 129.843 159.530 158.776 158.132 143.988 144.338 143.936 144.548 35.452 12.179 180 127.606 129.261 132.001 149.013 151.253 148.353 134.683 133.504 133.189 137.651 42.349 9.204 210 118.685 120.131 119.847 144.800 145.079 145.762 127.609 127.577 129.871 131.040 48.960 11.321 240 119.296 118.612 119.235 140.191 141.291 141.309 123.514 123.845 124.100 127.933 52.067 9.973

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 183.626 181.242 182.828 186.186 185.902 186.342 181.813 182.465 183.200 183.734 -3.734 1.941 60 181.352 180.436 181.976 186.964 187.524 187.436 181.645 180.230 180.163 183.081 -3.081 3.234

Crease 3 90 180.157 182.341 180.899 187.949 188.620 187.409 181.087 180.835 180.283 183.287 -3.287 3.596 120 181.906 182.477 181.301 185.025 186.840 187.121 180.857 177.218 176.407 182.128 -2.128 3.789 115

150 178.836 178.904 177.172 182.846 183.064 183.455 176.546 173.969 174.110 178.767 1.233 3.696 180 175.427 175.796 176.267 183.563 181.914 181.553 172.831 174.818 175.947 177.568 2.432 3.754 210 170.942 171.068 173.502 181.836 183.380 183.545 173.045 174.387 173.704 176.157 3.843 5.220 240 171.199 171.846 172.495 180.452 178.929 179.039 167.572 165.990 167.383 172.767 7.233 5.496

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 177.969 178.448 178.887 181.373 181.630 182.498 179.438 179.285 179.597 179.903 0.097 1.560 60 178.738 177.351 176.452 184.130 183.437 185.764 180.349 181.327 180.618 180.907 -0.907 3.125 90 176.168 177.301 179.471 186.223 187.427 186.671 181.950 180.118 180.445 181.753 -1.753 4.138 120 177.053 177.385 178.923 186.481 187.668 186.187 178.447 179.329 177.888 181.040 -1.040 4.378

Crease 8 150 178.259 178.833 177.115 187.041 186.770 187.281 178.445 177.748 178.454 181.105 -1.105 4.472 180 178.215 177.706 179.187 185.316 184.832 183.011 179.324 179.574 178.697 180.651 -0.651 2.922 210 179.369 179.790 178.757 182.653 185.089 182.617 177.520 178.446 178.482 180.303 -0.303 2.545 240 178.687 179.313 178.935 182.154 180.752 181.497 178.036 178.325 178.918 179.624 0.376 1.472

60 132.136 131.959 132.741 147.264 146.658 147.094 130.962 131.203 130.369 136.710 43.290 7.754

90 116.272 117.910 118.614 135.231 136.689 135.272 117.551 117.625 116.105 123.474 56.526 9.234 120 106.873 107.510 105.713 125.347 125.251 126.133 107.419 107.740 107.801 113.310 66.690 9.225

Crease 11 150 98.268 95.428 96.837 116.584 115.680 116.383 95.691 96.483 95.555 102.990 77.010 9.959 180 92.970 90.699 89.807 109.611 106.400 109.491 89.523 89.051 89.916 96.385 83.615 9.199 210 87.874 85.301 88.071 101.887 101.250 101.167 83.723 83.462 82.683 90.602 89.398 8.331 240 87.189 85.918 84.027 99.813 98.439 97.610 77.846 78.638 79.159 87.627 92.373 8.869

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 171.481 171.727 172.685 172.832 173.856 173.592 174.660 174.774 175.141 173.416 6.584 1.326 60 160.513 161.465 154.957 166.656 164.296 165.178 165.736 166.872 166.321 163.555 16.445 3.933 Crease 12 90 149.991 147.470 149.064 158.427 160.245 159.083 151.716 152.310 154.021 153.592 26.408 4.665 116

120 138.040 138.757 137.722 152.581 151.582 153.264 141.251 140.963 141.449 143.957 36.043 6.544 150 121.984 126.059 125.761 145.832 146.388 145.317 130.902 128.587 129.452 133.365 46.635 9.701 180 113.609 117.655 112.562 136.747 137.297 136.919 117.655 117.056 117.660 123.018 56.982 10.637 210 110.650 112.257 113.652 126.982 127.163 126.912 110.149 110.555 109.468 116.421 63.579 8.042 240 110.621 111.671 111.263 125.666 125.164 125.488 102.369 102.521 104.709 113.275 66.725 9.783

117

Table A.7 Experimental data of the configuration IV prototypes Configuration IV OTTC_3B OTTC_4D OTTC_5C Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 153.862 154.036 152.852 158.103 158.982 160.305 162.665 160.226 162.272 158.145 21.855 3.715 60 136.234 134.189 133.892 138.965 141.863 140.822 139.904 138.527 138.481 138.097 41.903 2.793 90 118.321 118.814 117.600 127.767 126.587 125.878 126.072 127.625 127.079 123.971 56.029 4.351 120 103.730 103.308 104.320 112.788 111.527 111.833 113.218 111.618 114.516 109.651 70.349 4.501

Crease 1 150 94.071 94.697 93.893 104.444 105.224 104.345 100.500 101.734 100.293 99.911 80.089 4.607 180 89.819 88.779 88.744 94.241 93.235 93.080 96.714 97.845 97.309 93.307 86.693 3.582 210 83.892 83.423 83.762 89.361 90.171 90.725 93.410 94.414 93.359 89.169 90.831 4.425 240 79.358 79.113 78.483 87.702 87.257 89.912 90.978 91.797 93.221 86.425 93.575 5.882

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 166.836 165.552 168.799 163.944 163.954 164.765 173.246 173.786 173.001 168.209 11.791 4.137 60 152.334 152.432 152.411 148.948 149.844 149.794 162.050 160.652 158.236 154.078 25.922 4.939 90 136.291 135.593 136.987 139.708 136.700 136.759 146.019 145.464 147.746 140.141 39.859 4.869 120 121.179 119.216 122.014 122.212 120.573 122.682 136.041 136.859 136.101 126.320 53.680 7.581

Crease 2 150 114.465 112.446 114.088 111.862 110.973 110.698 128.454 129.152 130.527 118.074 61.926 8.585 180 104.486 106.224 108.017 109.960 111.575 110.598 121.003 120.636 119.316 112.424 67.576 6.320 210 102.463 101.299 102.147 102.862 102.093 102.762 118.895 115.868 114.788 107.020 72.980 7.216 240 99.129 99.065 97.396 100.717 99.994 97.260 116.193 115.278 115.235 104.474 75.526 8.397

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 176.679 176.260 178.137 177.214 177.232 179.410 178.532 176.196 178.830 177.610 2.390 1.164 60 169.625 173.775 169.565 170.636 173.696 174.190 169.799 174.344 171.211 171.871 8.129 2.094

Crease 3 90 166.154 167.989 166.557 168.168 166.115 167.446 168.969 171.263 169.054 167.968 12.032 1.661 120 159.766 164.751 160.401 164.919 165.966 166.356 165.251 167.324 167.362 164.677 15.323 2.772 118

150 156.908 154.844 157.894 160.844 159.910 158.649 165.168 163.232 165.027 160.275 19.725 3.621 180 152.810 152.961 151.857 158.081 156.281 157.845 161.041 160.076 159.170 156.680 23.320 3.397 210 152.660 151.206 150.493 152.681 154.906 160.007 158.591 157.913 159.544 155.333 24.667 3.736 240 148.520 150.881 146.553 151.422 152.249 151.481 158.387 158.162 158.194 152.872 27.128 4.387

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 173.712 172.715 175.806 175.186 176.652 174.541 173.941 173.354 172.777 174.298 5.702 1.362 60 168.901 165.767 166.905 170.885 167.866 169.835 172.148 170.308 169.375 169.110 10.890 2.001 90 167.860 166.809 167.452 165.200 164.134 166.198 167.953 165.716 167.959 166.587 13.413 1.370 120 169.733 168.919 169.137 168.207 163.947 165.806 165.740 166.697 165.332 167.058 12.942 2.011

Crease 8 150 164.208 165.362 165.894 164.565 162.308 164.727 164.580 164.797 164.629 164.563 15.437 0.982 180 163.847 163.629 166.127 163.073 162.724 162.604 162.219 162.094 161.452 163.085 16.915 1.364 210 163.228 163.226 162.510 160.843 160.682 161.135 161.013 160.216 161.093 161.550 18.450 1.132 240 154.087 150.769 151.015 160.372 161.702 160.852 157.719 160.325 161.264 157.567 22.433 4.448

60 123.056 120.427 119.646 131.174 131.550 130.911 128.078 127.784 129.152 126.864 53.136 4.644

90 102.758 103.191 101.830 110.639 115.861 114.222 112.884 110.720 109.856 109.107 70.893 5.242 120 89.743 88.883 91.872 99.805 100.194 97.717 98.966 99.307 95.811 95.811 84.189 4.490

Crease 11 150 81.900 82.962 83.763 92.329 89.386 90.545 86.512 87.685 88.764 87.094 92.906 3.592 180 76.230 75.049 73.774 82.896 82.910 83.112 83.103 81.996 81.907 80.109 99.891 3.892 210 69.085 68.083 67.783 77.378 76.713 77.994 77.072 76.639 74.957 73.967 106.033 4.328 240 64.916 63.206 63.944 70.536 72.940 72.360 72.545 72.652 74.186 69.698 110.302 4.379

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 157.886 160.539 159.853 160.159 161.604 158.089 162.997 163.849 162.355 160.815 19.185 2.074 60 134.871 132.440 134.308 132.821 133.167 132.891 149.063 148.120 148.361 138.449 41.551 7.591 Crease 12 90 116.433 115.994 119.560 115.977 115.401 115.678 133.785 131.718 131.957 121.834 58.166 8.101 119

120 103.784 102.843 99.937 101.333 99.220 102.264 118.567 118.613 117.470 107.115 72.885 8.446 150 91.649 89.052 91.567 88.056 89.264 90.535 106.987 107.062 109.019 95.910 84.090 8.928 180 85.001 83.136 80.984 82.407 83.379 83.020 99.723 98.935 98.380 88.329 91.671 8.086 210 77.394 75.396 74.719 81.361 79.228 80.750 91.465 90.797 92.012 82.569 97.431 6.998 240 71.004 70.004 69.293 75.416 79.328 75.163 86.489 84.950 84.904 77.395 102.605 6.800

120

Appendix D Fminsearch/ATSV MATLAB code

MATLAB code for function ‘getsome’ used in symmetry cases: function [norm_energy] = getsome(orientation, config)

% Loads Zhonghua's miura fold point data data = load('Point_data.txt');

% Normalized crease vectors C1_norm = C1; C2_norm = C2; C3_norm = C3; C4_norm = C4; C5_norm = C5; 121

% Magnetization 1 (Panels 1, 3, 7, and 9) syms A1 B1 M1x = A1*C1_norm(1,1) + B1*C3_norm(1,1) - cosd(orientation(1,1)); M1y = A1*C1_norm(1,2) + B1*C3_norm(1,2) - sind(orientation(1,1)); [solA1, solB1] = solve(M1x == 0, M1y == 0, A1, B1); solA1 = double(solA1); % Determining values for magnetization vector M1 solB1 = double(solB1); for i = 1:129 M1(i,:) = [solA1*C1_norm(i,1) + solB1*C3_norm(i,1) solA1*... C1_norm(i,2) + solB1*C3_norm(i,2) solA1*C1_norm(i,3) + ... solB1*C3_norm(i,3)]; % Vector components of M1 T1(i,:) = cross(M1(i,:),norm_H(i,:)) / norm(cross(M1(i,:),norm_H(i,:))); % Normalized Torque T1 T1C1(i,:) = dot(T1(i,:),C1_norm(i,:)); % Torque on crease 1 (C1) T1C3(i,:) = dot(T1(i,:),-C3_norm(i,:)); % Torque on crease 3 (C3) end

WM1C1 = 0; WM1C3 = 0; for i = 1:128 WM1C1 = WM1C1 + ((T1C1(i,:) + T1C1(i+1,:))/2)*delVangle(i,:); WM1C1s(i,:) = WM1C1; % Work on C1 WM1C3 = WM1C3 + ((T1C3(i,:) + T1C3(i+1,:))/2)*delHangle(i,:); WM1C3s(i,:) = WM1C3; % Work on C3 end

WM1 = WM1C1s(128,1) + WM1C3s(128,1); % Summation of work due to M1

% Magnetization 2 (Panels 2 and 8) for i = 1:129 M2(i,:) = [cosd(orientation(1,2)) sind(orientation(1,2))*cos(pi - ... AngleH(i,1)) sind(orientation(1,2))*sin(-AngleH(i,1))]; % Vector components of M2 T2(i,:) = cross(M2(i,:),norm_H(i,:)) / norm(cross(M2(i,:),norm_H(i,:))); % Normalized Torque T2 T2C1(i,:) = dot(T2(i,:),-C1_norm(i,:)); % Torque on crease 1 (C1) T2C2(i,:) = dot(T2(i,:),-C2_norm(i,:)); % Torque on crease 2 (C2) T2C4(i,:) = dot(T2(i,:),-C4_norm(i,:)); % Torque on crease 4 (C4) end

WM2C1 = 0; WM2C2 = 0; WM2C4 = 0; for i = 1:128 WM2C1 = WM2C1 + ((T2C1(i,:) + T2C1(i+1,:))/2)*delVangle(i,:); WM2C1s(i,:) = WM2C1; % Work on C1 123

WM2C2 = WM2C2 + ((T2C2(i,:) + T2C2(i+1,:))/2)*delVangle(i,:); WM2C2s(i,:) = WM2C2; % Work on C2 WM2C4 = WM2C4 + ((T2C4(i,:) + T2C4(i+1,:))/2)*delHangle(i,:); WM2C4s(i,:) = WM2C4; % Work on C4 end

WM2 = WM2C1s(128,1) + WM2C2s(128,1) + WM2C4s(128,1); % Summation of work due to M2

% Magnetization 3 (Panels 4 and 6) syms A3 B3 M3x = A3*C6_norm(1,1) + B3*C8_norm(1,1) - cosd(orientation(1,3)); M3y = A3*C6_norm(1,2) + B3*C8_norm(1,2) - sind(orientation(1,3)); [solA3, solB3] = solve(M3x == 0, M3y == 0, A3, B3); solA3 = double(solA3); % Determining values for magnetization vector M3 solB3 = double(solB3); for i = 1:129 M3(i,:) = [solA3*C6_norm(i,1) + solB3*C8_norm(i,1) solA3*... C6_norm(i,2) + solB3*C8_norm(i,2) solA3*C6_norm(i,3) + ... solB3*C8_norm(i,3)]; % Vector components of M3 T3(i,:) = cross(M3(i,:),norm_H(i,:)) / norm(cross(M3(i,:),norm_H(i,:))); % Normalized Torque T3 T3C3(i,:) = dot(T3(i,:),C3_norm(i,:)); % Torque on crease 3 (C3) T3C6(i,:) = dot(T3(i,:),C6_norm(i,:)); % Torque on crease 6 (C6) T3C8(i,:) = dot(T3(i,:),C8_norm(i,:)); % Torque on crease 8 (C8) end

WM3C3 = 0; WM3C6 = 0; WM3C8 = 0; for i = 1:128 WM3C3 = WM3C3 + ((T3C3(i,:) + T3C3(i+1,:))/2)*delHangle(i,:); WM3C3s(i,:) = WM3C3; % Work on C3 WM3C6 = WM3C6 + ((T3C6(i,:) + T3C6(i+1,:))/2)*delVangle(i,:); WM3C6s(i,:) = WM3C6; % Work on C6 WM3C8 = WM3C8 + ((T3C8(i,:) + T3C8(i+1,:))/2)*delHangle(i,:); WM3C8s(i,:) = WM3C8; % Work on C8 end

WM3 = WM3C3s(128,1) + WM3C6s(128,1) + WM3C8s(128,1); % Summation of work due to M3

% Magnetization 4 (Panel 5) for i = 1:129 M4(i,:) = [cosd(orientation(1,4)) sind(orientation(1,4)) 0]; % Vector components of M4 T4(i,:) = cross(M4(i,:),norm_H(i,:)) / norm(cross(M4(i,:),norm_H(i,:))); % Normalized Torque T4 T4C4(i,:) = dot(T4(i,:),C4_norm(i,:)); % Torque on crease 4 (C4) T4C6(i,:) = dot(T4(i,:),-C6_norm(i,:)); % Torque on crease 6 (C6) T4C7(i,:) = dot(T4(i,:),-C7_norm(i,:)); % Torque on crease 7 (C7) T4C9(i,:) = dot(T4(i,:),-C9_norm(i,:)); % Torque on crease 9 (C9) end

WM4C4 = 0; WM4C6 = 0; WM4C7 = 0; WM4C9 = 0; for i = 1:128 WM4C4 = WM4C4 + ((T4C4(i,:) + T4C4(i+1,:))/2)*delHangle(i,:); WM4C4s(i,:) = WM4C4; % Work on C4 WM4C6 = WM4C6 + ((T4C6(i,:) + T4C6(i+1,:))/2)*delVangle(i,:); WM4C6s(i,:) = WM4C6; % Work on C6 WM4C7 = WM4C7 + ((T4C7(i,:) + T4C7(i+1,:))/2)*delVangle(i,:); WM4C7s(i,:) = WM4C7; % Work on C7 WM4C9 = WM4C9 + ((T4C9(i,:) + T4C9(i+1,:))/2)*delHangle(i,:); WM4C9s(i,:) = WM4C9; % Work on C9 end 124

WM4 = WM4C4s(128,1) + WM4C6s(128,1) + WM4C7s(128,1) + WM4C9s(128,1); % Summation of work due to M4

Work = [4*WM1; 2*WM2; 2*WM3; WM4]; norm_energy = config*Work; % Total Normalized Work on a configuration end

Configuration I ‘Fminsearch’ MATLAB script: clear all close all

% Configuration I config1 = (1/4)*[0 1 1 0]; % Configuration I magnetization array (includes normalization) orientationx = zeros(100,4); X = zeros(100,4); FVAL = zeros(100,1); for i = 1:100 orientationx(i,:) = randi([0 360],1,4); funk = @(con) getsome(con,config1); [X(i,:),FVAL(i,:)] = fminsearch(funk,orientationx(i,:)); end

Configuration II ‘Fminsearch’ MATLAB script: clear all close all

% Configuration II config2 = (1/5)*[1 0 0 1]; % Configuration II magnetization array (includes normalization) orientationx = zeros(100,4); X = zeros(100,4); FVAL = zeros(100,1); for i = 1:100 orientationx(i,:) = randi([0 360],1,4); funk = @(con) getsome(con,config2); [X(i,:),FVAL(i,:)] = fminsearch(funk,orientationx(i,:)); end

Configuration III ‘Fminsearch’ MATLAB script: clear all close all

% Configuration III config3 = (1/6)*[1 1 0 0]; % Configuration III magnetization array (includes normalization) orientationx = zeros(100,4); X = zeros(100,4); FVAL = zeros(100,1); for i = 1:100 orientationx(i,:) = randi([0 360],1,4); funk = @(con) getsome(con,config3); [X(i,:),FVAL(i,:)] = fminsearch(funk,orientationx(i,:)); end

Configuration IV ‘Fminsearch’ MATLAB script: clear all close all

% Configuration IV config4 = (1/9)*[1 1 1 1]; % Configuration IV magnetization array (includes normalization) orientationx = zeros(100,4); X = zeros(100,4); 125

FVAL = zeros(100,1); options = optimset('MaxFunEvals',1600); for i = 1:100 orientationx(i,:) = randi([0 360],1,4); funk = @(con) getsome(con,config4); [X(i,:),FVAL(i,:)] = fminsearch(funk,orientationx(i,:),options); end

Configuration I ATSV (symmetry)/MATLAB link script: clear all close all

% Configuration I config1 = (1/4)*[0 1 1 0]; % Configuration I symmetry magnetization array (includes normalization) fileID = fopen('CI_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = 0; Ang2 = str2double(input{2}{1}); Ang3 = str2double(input{2}{2}); Ang4 = 0; fclose(fileID); orientation1 = [Ang1 Ang2 Ang3 Ang4]; norm_energy = getsome(orientation1,config1);

ANS = sprintf('Norm. Work, %1.5f',norm_energy); fileID2 = fopen('CI_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

Configuration II ATSV (symmetry)/MATLAB link script: clear all close all

% Configuration II config2 = (1/5)*[1 0 0 1]; % Configuration II symmetry magnetization array (includes normalization) fileID = fopen('CII_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = str2double(input{2}{1}); Ang2 = 0; Ang3 = 0; Ang4 = str2double(input{2}{2}); fclose(fileID); orientation2 = [Ang1 Ang2 Ang3 Ang4]; norm_energy = getsome(orientation2,config2);

ANS = sprintf('Norm. Work, %1.5f',norm_energy); fileID2 = fopen('CII_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

Configuration III ATSV (symmetry)/MATLAB link script: 126 clear all close all

% Configuration III config3 = (1/6)*[1 1 0 0]; % Configuration III symmetry magnetizaiton array (includes normalization) fileID = fopen('CIII_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = str2double(input{2}{1}); Ang2 = str2double(input{2}{2}); Ang3 = 0; Ang4 = 0; fclose(fileID); orientation3 = [Ang1 Ang2 Ang3 Ang4]; norm_energy = getsome(orientation3,config3);

ANS = sprintf('Norm. Work, %1.5f',norm_energy); fileID2 = fopen('CIII_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

Configuration IV ATSV (symmetry)/MATLAB link script: clear all close all

% Configuration IV config4 = (1/9)*[1 1 1 1]; % Configuration IV symmetry magnetization array (includes normalization) fileID = fopen('CIV_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = str2double(input{2}{1}); Ang2 = str2double(input{2}{2}); Ang3 = str2double(input{2}{3}); Ang4 = str2double(input{2}{4}); fclose(fileID); orientation4 = [Ang1 Ang2 Ang3 Ang4]; norm_energy = getsome(orientation4,config4);

ANS = sprintf('Norm. Work, %1.5f',norm_energy); fileID2 = fopen('CIV_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

MATLAB code for function ‘getmore’ used in independent magnetization case: function [norm_work] = getmore(orientation,config)

% Loads Zhonghua's miura fold point data data = load('Point_data.txt');

127

P5 = [data(:,13) data(:,14) data(:,15)]; P6 = [data(:,16) data(:,17) data(:,18)]; P7 = [data(:,19) data(:,20) data(:,21)]; P8 = [data(:,22) data(:,23) data(:,24)]; P9 = [data(:,25) data(:,26) data(:,27)]; P10 = [data(:,28) data(:,29) data(:,30)]; P11 = [data(:,31) data(:,32) data(:,33)]; P12 = [data(:,34) data(:,35) data(:,36)]; P13 = [data(:,37) data(:,38) data(:,39)]; P14 = [data(:,40) data(:,41) data(:,42)]; P15 = [data(:,43) data(:,44) data(:,45)]; P16 = [data(:,46) data(:,47) data(:,48)];

% Normalized crease vectors C1_norm = C1; C2_norm = C2; C3_norm = C3; C4_norm = C4; C5_norm = C5; C6_norm = C6; C7_norm = C7; C8_norm = C8; C9_norm = C9; C10_norm = C10; C11_norm = C11; C12_norm = C12; for i = 1:129 C1_norm(i,:) = C1(i,:)/norm(C1(i,:)); C2_norm(i,:) = C2(i,:)/norm(C2(i,:)); C3_norm(i,:) = C3(i,:)/norm(C3(i,:)); C4_norm(i,:) = C4(i,:)/norm(C4(i,:)); C5_norm(i,:) = C5(i,:)/norm(C5(i,:)); C6_norm(i,:) = C6(i,:)/norm(C6(i,:)); C7_norm(i,:) = C7(i,:)/norm(C7(i,:)); 128

C8_norm(i,:) = C8(i,:)/norm(C8(i,:)); C9_norm(i,:) = C9(i,:)/norm(C9(i,:)); C10_norm(i,:) = C10(i,:)/norm(C10(i,:)); C11_norm(i,:) = C11(i,:)/norm(C11(i,:)); C12_norm(i,:) = C12(i,:)/norm(C12(i,:)); end

% Calculation of the change in horizontal fold angle and the change in % vertical fold angle between steps N for i = 1:128 delVangle(i,:) = AngleV(i,1) - AngleV(i+1,1); 129

delHangle(i,:) = AngleH(i,1) - AngleH(i+1,1); end

% Magnetization 1 (Panel 1) syms A1 B1 M1x = A1*C1_norm(1,1) + B1*C3_norm(1,1) - cosd(orientation(1,1)); M1y = A1*C1_norm(1,2) + B1*C3_norm(1,2) - sind(orientation(1,1)); [solA1, solB1] = solve(M1x == 0, M1y == 0, A1, B1); solA1 = double(solA1); % Determining values for magnetization vector M1 solB1 = double(solB1);

M1 = zeros(129,3); T1 = zeros(129,3); T1C1 = zeros(129,1); T1C3 = zeros(129,1); for i = 1:129 M1(i,:) = [solA1*C1_norm(i,1) + solB1*C3_norm(i,1) solA1*... C1_norm(i,2) + solB1*C3_norm(i,2) solA1*C1_norm(i,3) + ... solB1*C3_norm(i,3)]; % Vector components of M1 T1(i,:) = cross(M1(i,:),norm_H(i,:)) / norm(cross(M1(i,:),norm_H(i,:))); % Normalized Torque T1 T1C1(i,:) = dot(T1(i,:),C1_norm(i,:)); % Torque on crease 1 (C1) T1C3(i,:) = dot(T1(i,:),-C3_norm(i,:)); % Torque on crease 3 (C3) end

WM1C1 = 0; WM1C3 = 0; WM1C1s = zeros(128,1); WM1C3s = zeros(128,1); for i = 1:128 WM1C1 = WM1C1 + ((T1C1(i,:) + T1C1(i+1,:))/2)*delVangle(i,:); WM1C1s(i,:) = WM1C1; % Work on C1 WM1C3 = WM1C3 + ((T1C3(i,:) + T1C3(i+1,:))/2)*delHangle(i,:); WM1C3s(i,:) = WM1C3; % Work on C3 end

WM1 = WM1C1s(128,1) + WM1C3s(128,1); % Summation of work due to M1

% Magnetization 2 (Panel 2)

M2 = zeros(129,3); T2 = zeros(129,3); T2C1 = zeros(129,1); T2C2 = zeros(129,1); T2C4 = zeros(129,1); for i = 1:129 M2(i,:) = [cosd(orientation(1,2)) sind(orientation(1,2))*cos(pi - ... AngleH(i,1)) sind(orientation(1,2))*sin(-AngleH(i,1))]; % Vector components of M2 T2(i,:) = cross(M2(i,:),norm_H(i,:)) / norm(cross(M2(i,:),norm_H(i,:))); % Normalized Torque T2 T2C1(i,:) = dot(T2(i,:),-C1_norm(i,:)); % Torque on crease 1 (C1) T2C2(i,:) = dot(T2(i,:),-C2_norm(i,:)); % Torque on crease 2 (C2) T2C4(i,:) = dot(T2(i,:),-C4_norm(i,:)); % Torque on crease 4 (C4) end

WM2C1 = 0; WM2C2 = 0; WM2C4 = 0; WM2C1s = zeros(128,1); WM2C2s = zeros(128,1); WM2C4s = zeros(128,1); for i = 1:128 WM2C1 = WM2C1 + ((T2C1(i,:) + T2C1(i+1,:))/2)*delVangle(i,:); WM2C1s(i,:) = WM2C1; % Work on C1 130

WM2C2 = WM2C2 + ((T2C2(i,:) + T2C2(i+1,:))/2)*delVangle(i,:); WM2C2s(i,:) = WM2C2; % Work on C2 WM2C4 = WM2C4 + ((T2C4(i,:) + T2C4(i+1,:))/2)*delHangle(i,:); WM2C4s(i,:) = WM2C4; % Work on C4 end

WM2 = WM2C1s(128,1) + WM2C2s(128,1) + WM2C4s(128,1); % Summation of work due to M2

% Magnetization 3 (Panel 3) syms A3 B3 M3x = A3*C2_norm(1,1) + B3*C5_norm(1,1) - cosd(orientation(1,3)); M3y = A3*C2_norm(1,2) + B3*C5_norm(1,2) - sind(orientation(1,3)); [solA3, solB3] = solve(M3x == 0, M3y == 0, A3, B3); solA3 = double(solA3); % Determining values for magnetization vector M1 solB3 = double(solB3);

M3 = zeros(129,3); T3 = zeros(129,3); T3C2 = zeros(129,1); T3C5 = zeros(129,1); for i = 1:129 M3(i,:) = [solA3*C2_norm(i,1) + solB3*C5_norm(i,1) solA3*... C2_norm(i,2) + solB3*C5_norm(i,2) solA3*C2_norm(i,3) + ... solB3*C5_norm(i,3)]; % Vector components of M1 T3(i,:) = cross(M3(i,:),norm_H(i,:)) / norm(cross(M3(i,:),norm_H(i,:))); % Normalized Torque T1 T3C2(i,:) = dot(T3(i,:),C2_norm(i,:)); % Torque on crease 2 (C2) T3C5(i,:) = dot(T3(i,:),C5_norm(i,:)); % Torque on crease 5 (C5) end

WM3C2 = 0; WM3C5 = 0; WM3C2s = zeros(128,1); WM3C5s = zeros(128,1); for i = 1:128 WM3C2 = WM3C2 + ((T3C2(i,:) + T3C2(i+1,:))/2)*delVangle(i,:); WM3C2s(i,:) = WM3C2; % Work on C2 WM3C5 = WM3C5 + ((T3C5(i,:) + T3C5(i+1,:))/2)*delHangle(i,:); WM3C5s(i,:) = WM3C5; % Work on C3 end

WM3 = WM3C2s(128,1) + WM3C5s(128,1); % Summation of work due to M3

% Magnetization 4 (Panel 4) syms A4 B4 M4x = A4*C6_norm(1,1) + B4*C8_norm(1,1) - cosd(orientation(1,4)); M4y = A4*C6_norm(1,2) + B4*C8_norm(1,2) - sind(orientation(1,4)); [solA4, solB4] = solve(M4x == 0, M4y == 0, A4, B4); solA4 = double(solA4); % Determining values for magnetization vector M4 solB4 = double(solB4);

M4 = zeros(129,3); T4 = zeros(129,3); T4C3 = zeros(129,1); T4C6 = zeros(129,1); T4C8 = zeros(129,1); for i = 1:129 M4(i,:) = [solA4*C6_norm(i,1) + solB4*C8_norm(i,1) solA4*... C6_norm(i,2) + solB4*C8_norm(i,2) solA4*C6_norm(i,3) + ... solB4*C8_norm(i,3)]; % Vector components of M4 T4(i,:) = cross(M4(i,:),norm_H(i,:)) / norm(cross(M4(i,:),norm_H(i,:))); % Normalized Torque T4 T4C3(i,:) = dot(T4(i,:),C3_norm(i,:)); % Torque on crease 3 (C3) T4C6(i,:) = dot(T4(i,:),C6_norm(i,:)); % Torque on crease 6 (C6) T4C8(i,:) = dot(T4(i,:),C8_norm(i,:)); % Torque on crease 8 (C8) end 131

WM4C3 = 0; WM4C6 = 0; WM4C8 = 0; WM4C3s = zeros(128,1); WM4C6s = zeros(128,1); WM4C8s = zeros(128,1); for i = 1:128 WM4C3 = WM4C3 + ((T4C3(i,:) + T4C3(i+1,:))/2)*delHangle(i,:); WM4C3s(i,:) = WM4C3; % Work on C3 WM4C6 = WM4C6 + ((T4C6(i,:) + T4C6(i+1,:))/2)*delVangle(i,:); WM4C6s(i,:) = WM4C6; % Work on C6 WM4C8 = WM4C8 + ((T4C8(i,:) + T4C8(i+1,:))/2)*delHangle(i,:); WM4C8s(i,:) = WM4C8; % Work on C8 end

WM4 = WM4C3s(128,1) + WM4C6s(128,1) + WM4C8s(128,1); % Summation of work due to M4

% Magnetization 5 (Panel 5)

M5 = zeros(129,3); T5 = zeros(129,3); T5C4 = zeros(129,1); T5C6 = zeros(129,1); T5C7 = zeros(129,1); T5C9 = zeros(129,1); for i = 1:129 M5(i,:) = [cosd(orientation(1,5)) sind(orientation(1,5)) 0]; % Vector components of M5 T5(i,:) = cross(M5(i,:),norm_H(i,:)) / norm(cross(M5(i,:),norm_H(i,:))); % Normalized Torque T5 T5C4(i,:) = dot(T5(i,:),C4_norm(i,:)); % Torque on crease 4 (C4) T5C6(i,:) = dot(T5(i,:),-C6_norm(i,:)); % Torque on crease 6 (C6) T5C7(i,:) = dot(T5(i,:),-C7_norm(i,:)); % Torque on crease 7 (C7) T5C9(i,:) = dot(T5(i,:),-C9_norm(i,:)); % Torque on crease 9 (C9) end

WM5C4 = 0; WM5C6 = 0; WM5C7 = 0; WM5C9 = 0; WM5C4s = zeros(128,1); WM5C6s = zeros(128,1); WM5C7s = zeros(128,1); WM5C9s = zeros(128,1); for i = 1:128 WM5C4 = WM5C4 + ((T5C4(i,:) + T5C4(i+1,:))/2)*delHangle(i,:); WM5C4s(i,:) = WM5C4; % Work on C4 WM5C6 = WM5C6 + ((T5C6(i,:) + T5C6(i+1,:))/2)*delVangle(i,:); WM5C6s(i,:) = WM5C6; % Work on C6 WM5C7 = WM5C7 + ((T5C7(i,:) + T5C7(i+1,:))/2)*delVangle(i,:); WM5C7s(i,:) = WM5C7; % Work on C7 WM5C9 = WM5C9 + ((T5C9(i,:) + T5C9(i+1,:))/2)*delHangle(i,:); WM5C9s(i,:) = WM5C9; % Work on C9 end

WM5 = WM5C4s(128,1) + WM5C6s(128,1) + WM5C7s(128,1) + WM5C9s(128,1); % Summation of work due to M5

% Magnetization 6 (Panel 6) syms A6 B6 M6x = A6*C7_norm(1,1) + B6*C10_norm(1,1) - cosd(orientation(1,6)); M6y = A6*C7_norm(1,2) + B6*C10_norm(1,2) - sind(orientation(1,6)); [solA6, solB6] = solve(M6x == 0, M6y == 0, A6, B6); solA6 = double(solA6); % Determining values for magnetization vector M6 solB6 = double(solB6); 132

M6 = zeros(129,3); T6 = zeros(129,3); T6C5 = zeros(129,1); T6C7 = zeros(129,1); T6C10 = zeros(129,1); for i = 1:129 M6(i,:) = [solA6*C7_norm(i,1) + solB6*C10_norm(i,1) solA6*... C7_norm(i,2) + solB6*C10_norm(i,2) solA6*C7_norm(i,3) + ... solB6*C10_norm(i,3)]; % Vector components of M4 T6(i,:) = cross(M6(i,:),norm_H(i,:)) / norm(cross(M6(i,:),norm_H(i,:))); % Normalized Torque T6 T6C5(i,:) = dot(T6(i,:),-C5_norm(i,:)); % Torque on crease 5 (C5) T6C7(i,:) = dot(T6(i,:),C7_norm(i,:)); % Torque on crease 7 (C7) T6C10(i,:) = dot(T6(i,:),-C10_norm(i,:)); % Torque on crease 10 (C10) end

WM6C5 = 0; WM6C7 = 0; WM6C10 = 0; WM6C5s = zeros(128,1); WM6C7s = zeros(128,1); WM6C10s = zeros(128,1); for i = 1:128 WM6C5 = WM6C5 + ((T6C5(i,:) + T6C5(i+1,:))/2)*delHangle(i,:); WM6C5s(i,:) = WM6C5; % Work on C3 WM6C7 = WM6C7 + ((T6C7(i,:) + T6C7(i+1,:))/2)*delVangle(i,:); WM6C7s(i,:) = WM6C7; % Work on C6 WM6C10 = WM6C10 + ((T6C10(i,:) + T6C10(i+1,:))/2)*delHangle(i,:); WM6C10s(i,:) = WM6C10; % Work on C8 end

WM6 = WM6C5s(128,1) + WM6C7s(128,1) + WM6C10s(128,1); % Summation of work due to M6

% Magnetization 7 (Panel 7) syms A7 B7 M7x = A7*C11_norm(1,1) + B7*C8_norm(1,1) - cosd(orientation(1,7)); M7y = A7*C11_norm(1,2) + B7*C8_norm(1,2) - sind(orientation(1,7)); [solA7, solB7] = solve(M7x == 0, M7y == 0, A7, B7); solA7 = double(solA7); % Determining values for magnetization vector M7 solB7 = double(solB7);

M7 = zeros(129,3); T7 = zeros(129,3); T7C8 = zeros(129,1); T7C11 = zeros(129,1); for i = 1:129 M7(i,:) = [solA7*C11_norm(i,1) + solB7*C8_norm(i,1) solA7*... C11_norm(i,2) + solB7*C8_norm(i,2) solA7*C11_norm(i,3) + ... solB7*C8_norm(i,3)]; % Vector components of M7 T7(i,:) = cross(M7(i,:),norm_H(i,:)) / norm(cross(M7(i,:),norm_H(i,:))); % Normalized Torque T7 T7C11(i,:) = dot(T7(i,:),-C11_norm(i,:)); % Torque on crease 11 (C11) T7C8(i,:) = dot(T7(i,:),-C8_norm(i,:)); % Torque on crease 8 (C8) end

WM7C11 = 0; WM7C8 = 0; WM7C11s = zeros(128,1); WM7C8s = zeros(128,1); for i = 1:128 WM7C11 = WM7C11 + ((T7C11(i,:) + T7C11(i+1,:))/2)*delVangle(i,:); WM7C11s(i,:) = WM7C11; % Work on C1 WM7C8 = WM7C8 + ((T7C8(i,:) + T7C8(i+1,:))/2)*delHangle(i,:); WM7C8s(i,:) = WM7C8; % Work on C3 133 end

WM7 = WM7C11s(128,1) + WM7C8s(128,1); % Summation of work due to M7

% Magnetization 8 (Panel 8)

M8 = zeros(129,3); T8 = zeros(129,3); T8C9 = zeros(129,1); T8C11 = zeros(129,1); T8C12 = zeros(129,1); for i = 1:129 M8(i,:) = [cosd(orientation(1,8)) sind(orientation(1,8))*cos(pi - ... AngleH(i,1)) sind(orientation(1,8))*sin(-AngleH(i,1))]; % Vector components of M8 T8(i,:) = cross(M8(i,:),norm_H(i,:)) / norm(cross(M8(i,:),norm_H(i,:))); % Normalized Torque T8 T8C11(i,:) = dot(T8(i,:),C11_norm(i,:)); % Torque on crease 11 (C11) T8C12(i,:) = dot(T8(i,:),C12_norm(i,:)); % Torque on crease 12 (C12) T8C9(i,:) = dot(T8(i,:),C9_norm(i,:)); % Torque on crease 9 (C9) end

WM8C11 = 0; WM8C12 = 0; WM8C9 = 0; WM8C11s = zeros(128,1); WM8C12s = zeros(128,1); WM8C9s = zeros(128,1); for i = 1:128 WM8C11 = WM8C11 + ((T8C11(i,:) + T8C11(i+1,:))/2)*delVangle(i,:); WM8C11s(i,:) = WM8C11; % Work on C1 WM8C12 = WM8C12 + ((T8C12(i,:) + T8C12(i+1,:))/2)*delVangle(i,:); WM8C12s(i,:) = WM8C12; % Work on C2 WM8C9 = WM8C9 + ((T8C9(i,:) + T8C9(i+1,:))/2)*delHangle(i,:); WM8C9s(i,:) = WM8C9; % Work on C4 end

WM8 = WM8C11s(128,1) + WM8C12s(128,1) + WM8C9s(128,1); % Summation of work due to M8

% Magnetization 9 (Panel 9) syms A9 B9 M9x = A9*C12_norm(1,1) + B9*C10_norm(1,1) - cosd(orientation(1,9)); M9y = A9*C12_norm(1,2) + B9*C10_norm(1,2) - sind(orientation(1,9)); [solA9, solB9] = solve(M9x == 0, M9y == 0, A9, B9); solA9 = double(solA9); % Determining values for magnetization vector M9 solB9 = double(solB9);

M9 = zeros(129,3); T9 = zeros(129,3); T9C10 = zeros(129,1); T9C12 = zeros(129,1); for i = 1:129 M9(i,:) = [solA9*C12_norm(i,1) + solB9*C10_norm(i,1) solA9*... C12_norm(i,2) + solB9*C10_norm(i,2) solA9*C12_norm(i,3) + ... solB9*C10_norm(i,3)]; % Vector components of M9 T9(i,:) = cross(M9(i,:),norm_H(i,:)) / norm(cross(M9(i,:),norm_H(i,:))); % Normalized Torque T9 T9C12(i,:) = dot(T9(i,:),-C12_norm(i,:)); % Torque on crease 12 (C12) T9C10(i,:) = dot(T9(i,:),C10_norm(i,:)); % Torque on crease 10 (C10) end

WM9C12 = 0; WM9C10 = 0; WM9C12s = zeros(128,1); WM9C10s = zeros(128,1);

134 for i = 1:128 WM9C12 = WM9C12 + ((T9C12(i,:) + T9C12(i+1,:))/2)*delVangle(i,:); WM9C12s(i,:) = WM9C12; % Work on C12 WM9C10 = WM9C10 + ((T9C10(i,:) + T9C10(i+1,:))/2)*delHangle(i,:); WM9C10s(i,:) = WM9C10; % Work on C10 end

WM9 = WM9C12s(128,1) + WM9C10s(128,1); % Summation of work due to M9

Work = [WM1; WM2; WM3; WM4; WM5; WM6; WM7; WM8; WM9]; norm_work = config*Work; % Total Normalized Work on a configuration end

Configuration I ATSV (independent)/MATLAB link script: clear all close all

% Configuration I config1 = (1/4)*[0 1 0 1 0 1 0 1 0]; % Configuration I magnetization array (includes normalization) fileID = fopen('IM_CI_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = 0; Ang2 = str2double(input{2}{1}); Ang3 = 0; Ang4 = str2double(input{2}{2}); Ang5 = 0; Ang6 = str2double(input{2}{3}); Ang7 = 0; Ang8 = str2double(input{2}{4}); Ang9 = 0; fclose(fileID); orientation1 = [Ang1 Ang2 Ang3 Ang4 Ang5 Ang6 Ang7 Ang8 Ang9]; norm_work = getmore(orientation1,config1);

ANS = sprintf('Norm. Work, %1.5f',norm_work); fileID2 = fopen('IM_CI_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

Configuration II ATSV (independent)/MATLAB link script: clear all close all

% Configuration II config2 = (1/5)*[1 0 1 0 1 0 1 0 1]; % Configuration II magnetization array (includes normalization) fileID = fopen('IM_CII_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = str2double(input{2}{1}); Ang2 = 0; Ang3 = str2double(input{2}{2}); Ang4 = 0; Ang5 = str2double(input{2}{3}); Ang6 = 0; Ang7 = str2double(input{2}{4}); Ang8 = 0; 135

Ang9 = str2double(input{2}{5}); fclose(fileID); orientation2 = [Ang1 Ang2 Ang3 Ang4 Ang5 Ang6 Ang7 Ang8 Ang9]; norm_work = getmore(orientation2,config2);

ANS = sprintf('Norm. Work, %1.5f',norm_work); fileID2 = fopen('IM_CII_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

Configuration III ATSV (independent)/MATLAB link script: clear all close all

% Configuration III config3 = (1/6)*[1 1 1 0 0 0 1 1 1]; % Configuration III magnetization array (includes normalization) fileID = fopen('IM_CIII_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = str2double(input{2}{1}); Ang2 = str2double(input{2}{2}); Ang3 = str2double(input{2}{3}); Ang4 = 0; Ang5 = 0; Ang6 = 0; Ang7 = str2double(input{2}{4}); Ang8 = str2double(input{2}{5}); Ang9 = str2double(input{2}{6}); fclose(fileID); orientation3 = [Ang1 Ang2 Ang3 Ang4 Ang5 Ang6 Ang7 Ang8 Ang9]; norm_work = getmore(orientation3,config3);

ANS = sprintf('Norm. Work, %1.5f',norm_work); fileID2 = fopen('IM_CIII_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

Configuration IV ATSV (independent)/MATLAB link script: clear all close all

% Configuration IV config4 = (1/9)*[1 1 1 1 1 1 1 1 1]; % Configuration IV magnetization array (includes normalization) fileID = fopen('IM_CIV_input.txt'); input = textscan(fileID,'%7s %s'); Ang1 = str2double(input{2}{1}); Ang2 = str2double(input{2}{2}); Ang3 = str2double(input{2}{3}); Ang4 = str2double(input{2}{4}); Ang5 = str2double(input{2}{5}); Ang6 = str2double(input{2}{6}); Ang7 = str2double(input{2}{7}); Ang8 = str2double(input{2}{8}); 136

Ang9 = str2double(input{2}{9}); fclose(fileID); orientation4 = [Ang1 Ang2 Ang3 Ang4 Ang5 Ang6 Ang7 Ang8 Ang9]; norm_work = getmore(orientation4,config4);

ANS = sprintf('Norm. Work, %1.5f',norm_work); fileID2 = fopen('IM_CIV_Work.txt', 'w'); fprintf(fileID2,'%s', ANS); fclose(fileID2); quit

137

Appendix E Configuration I* and I** prototype selection and experimental data

Table A.8 Miura-ori substrate selection data for the configuration I* and configuration I** prototypes

Prototype Panel 1A 3A 1E 2E 3E 4E 5E 1F 2F 3F 4F 5F 2M 3M 5M 1 2.292 2.358 1.980 1.738 1.874 1.864 2.258 2.224 2.018 2.006 2.034 2.240 1.954 1.998 2.016 2 2.198 2.310 1.888 1.672 1.848 1.732 2.288 2.274 2.140 2.118 2.054 2.376 2.062 2.036 2.148

3 1.982 2.082 1.668 1.668 1.766 1.652 2.198 2.214 2.222 2.100 2.048 2.334 2.178 2.038 2.296 4 2.544 2.594 2.160 1.950 2.082 2.132 2.236 2.404 2.160 2.270 2.334 2.544 2.244 2.114 2.160 5 2.406 2.410 2.058 1.840 2.054 1.964 2.242 2.472 2.226 2.290 2.318 2.564 2.118 2.138 2.312

6 2.148 2.208 1.790 1.714 1.894 1.836 2.046 2.534 2.216 2.290 2.286 2.506 2.192 2.052 2.216 Thickness (mm) Thickness 7 2.366 2.514 2.010 2.054 2.062 2.186 1.910 2.320 2.222 2.312 2.424 2.458 2.082 2.042 2.100 8 2.290 2.352 1.886 1.922 2.004 1.996 1.860 2.302 2.232 2.308 2.372 2.490 2.094 2.194 2.146

9 1.998 2.090 1.590 1.706 1.830 1.774 1.728 2.160 2.194 2.254 2.290 2.430 2.082 2.070 2.192

AVG 2.247 2.324 1.892 1.807 1.935 1.904 2.085 2.323 2.181 2.216 2.240 2.438 2.112 2.076 2.176

SD 0.186 0.175 0.185 0.140 0.117 0.180 0.207 0.125 0.069 0.112 0.152 0.105 0.085 0.061 0.092

pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

pass pass pass fail pass pass pass pass pass pass pass pass pass pass pass pass pass fail fail pass fail pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass fail pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

2 Standard Deviation 2Check Deviation Standard pass pass pass pass pass pass pass pass pass pass pass pass pass pass pass

pass pass fail fail pass pass pass pass pass pass pass pass pass pass pass

= accepted for use as an experimental substrate for neodymium configuration I** Lower bounds:

= accepted for use as an experimental substrate for MAE configuration I** 2.204 mm - 2*0.240 mm = 1.723 mm Upper = accepted for use as an experimental substrate for neodymium configuration I* bounds: = rejected for having one or more panels outside 2 standard deviations from the mean 2.204 mm + 2*0.240 mm = 2.685 mm

138

Figure A.4 Neodymium configuration I** prototype within the Walker Scientific 7H electromagnet subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

Figure A.5 MAE configuration I** prototype within the Walker Scientific 7H electromagnet subjected to a (a) 0 mT, (b) 80 mT, (c) 160 mT, and (d) 240 mT magnetic field.

139

Table A.9 Experimental data of the configuration I* - Neodymium prototypes Neodymium configuration I* OTTC_2F OTTC_3F OTTC_5F Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 162.463 161.524 163.088 166.960 168.961 168.041 165.270 166.282 165.020 165.290 14.710 2.547 60 148.480 148.989 149.745 155.182 153.487 153.900 153.583 150.769 151.743 151.764 28.236 2.400 90 133.737 132.602 132.121 143.829 143.133 142.810 137.336 136.791 135.764 137.569 42.431 4.617 120 119.993 122.134 119.796 134.374 133.188 132.551 126.599 122.835 125.449 126.324 53.676 5.744

Crease 1 150 107.050 108.938 106.956 123.724 120.590 122.998 114.103 113.649 112.970 114.553 65.447 6.528 180 99.245 98.832 98.143 114.316 115.626 116.046 102.839 103.369 103.603 105.780 74.220 7.447 210 91.492 92.223 92.998 108.529 107.837 109.041 94.197 95.270 96.315 98.656 81.344 7.510 240 87.734 88.766 89.102 103.875 103.683 101.787 89.319 90.542 90.820 93.959 86.041 6.951

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 148.802 150.947 149.016 155.966 155.789 156.895 160.835 163.767 162.894 156.101 23.899 5.680 60 124.507 123.931 124.765 133.161 132.849 132.617 140.854 139.642 141.048 132.597 47.403 6.996 90 107.708 107.024 106.529 115.337 114.758 114.380 125.375 124.372 125.108 115.621 64.379 7.772 120 92.664 92.474 93.037 99.910 99.612 99.298 110.077 110.853 110.073 100.889 79.111 7.692

Crease 2 150 84.355 83.801 84.491 88.061 87.679 87.175 99.666 100.586 103.193 91.001 88.999 7.813 180 78.714 79.446 78.333 80.463 79.814 80.384 89.877 92.062 90.725 83.313 96.687 5.749 210 72.831 73.291 72.757 74.701 74.316 74.909 85.752 84.053 85.256 77.541 102.459 5.678 240 70.778 69.541 69.506 72.061 72.328 70.336 79.591 80.702 79.547 73.821 106.179 4.705

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 180.138 180.278 180.055 179.233 181.479 181.854 180.247 181.871 180.305 180.607 -0.607 0.912 60 180.270 180.973 182.381 180.809 182.734 181.436 181.773 181.094 181.850 181.480 -1.480 0.785

Crease 3 90 181.100 180.029 180.446 181.750 182.032 180.165 182.848 183.169 182.556 181.566 -1.566 1.185 120 179.643 180.579 182.740 180.924 181.001 181.149 183.152 183.091 182.976 181.695 -1.695 1.306 140

150 181.033 181.788 180.867 182.255 181.567 180.625 182.023 181.406 182.897 181.607 -1.607 0.722 180 180.740 180.928 182.560 181.062 181.404 181.457 185.222 184.922 183.979 182.475 -2.475 1.781 210 181.319 182.362 182.555 182.969 182.823 182.147 182.441 182.540 183.110 182.474 -2.474 0.529 240 178.576 181.097 182.462 184.502 182.242 184.494 181.151 182.018 182.073 182.068 -2.068 1.802

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 180.020 180.646 180.486 182.218 180.992 181.473 181.439 179.886 179.741 180.767 -0.767 0.836 60 180.075 180.010 178.986 179.012 179.928 179.693 178.570 178.639 178.965 179.320 0.680 0.603 90 178.755 177.833 177.009 179.122 179.953 179.747 176.538 177.100 176.479 178.060 1.940 1.366 120 176.605 177.336 177.681 177.142 176.241 175.593 174.840 172.246 175.101 175.865 4.135 1.680

Crease 8 150 172.698 170.382 172.840 174.735 174.311 175.037 173.569 174.222 172.770 173.396 6.604 1.428 180 172.245 170.386 169.512 175.484 173.854 173.430 174.644 174.226 174.007 173.088 6.912 1.994 210 170.157 169.427 169.577 171.837 172.511 171.984 170.764 170.988 172.085 171.037 8.963 1.139 240 167.438 168.391 168.451 171.487 171.430 170.332 170.675 168.207 171.371 169.754 10.246 1.617

60 146.320 145.749 146.493 155.195 153.481 153.540 146.142 145.479 147.366 148.863 31.137 3.971

90 133.379 134.057 133.233 143.795 142.758 141.996 132.153 135.430 135.138 136.882 43.118 4.604 120 122.718 123.129 121.174 134.644 134.715 132.972 124.767 124.650 123.459 126.914 53.086 5.521

Crease 11 150 114.298 115.338 113.391 127.584 126.928 123.453 115.824 117.473 117.216 119.056 60.944 5.465 180 107.308 106.958 107.487 118.887 116.624 116.959 109.012 109.853 108.526 111.290 68.710 4.773 210 101.737 99.345 100.225 111.606 112.044 111.376 102.636 101.065 102.931 104.774 75.226 5.294 240 96.305 96.608 97.041 107.556 105.669 105.344 98.699 97.119 97.202 100.171 79.829 4.600

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 150.006 149.648 149.697 160.660 161.201 162.094 162.314 164.806 164.905 158.370 21.630 6.596 60 124.340 123.578 123.436 137.757 137.222 136.629 144.526 145.477 144.102 135.230 44.770 9.192 Crease 12 90 107.754 104.382 106.089 118.338 119.599 120.034 129.833 130.254 130.037 118.480 61.520 10.441 141

120 93.835 93.767 93.979 104.025 104.787 104.665 118.829 115.777 119.017 105.409 74.591 10.463 150 86.850 86.530 87.541 93.491 93.010 93.815 108.182 108.734 109.529 96.409 83.591 9.728 180 82.062 80.171 79.539 87.007 87.988 89.286 103.305 99.971 102.218 90.172 89.828 9.408 210 76.251 75.522 76.709 84.795 83.880 82.931 93.838 95.009 93.879 84.757 95.243 7.884 240 74.111 73.797 74.330 79.257 79.467 79.624 88.220 87.225 88.362 80.488 99.512 6.060

142

Table A.10 Experimental data of the configuration I** - Neodymium prototypes Neodymium configuration I**

TC_1A TC_3A TC_5E Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 163.703 166.421 164.823 172.210 172.209 172.042 161.849 159.172 160.542 165.886 14.114 5.172 60 149.823 149.184 149.366 159.075 158.720 157.590 142.512 141.765 140.156 149.799 30.201 7.395 90 137.491 137.092 136.915 145.981 146.482 145.427 127.915 125.777 125.415 136.499 43.501 8.531 120 126.703 127.039 127.816 134.930 132.402 134.308 115.471 115.047 113.021 125.193 54.807 8.574

Crease 1 150 118.041 118.059 119.201 126.789 124.166 122.212 103.842 101.033 101.034 114.931 65.069 10.160 180 111.839 110.870 111.100 115.544 115.728 117.416 93.569 92.512 93.887 106.941 73.059 10.459 210 102.549 104.150 102.642 108.227 106.850 106.338 89.502 88.586 86.761 99.512 80.488 8.654 240 98.447 99.330 100.545 103.318 102.354 102.138 85.298 82.634 85.169 95.470 84.530 8.495

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 155.312 156.665 157.038 162.355 161.889 163.842 144.125 143.489 144.452 154.352 25.648 8.254 60 133.998 134.779 135.472 140.140 139.918 139.725 115.897 115.153 115.710 130.088 49.912 11.113 90 110.841 114.786 114.304 124.404 124.840 122.813 98.172 97.284 98.414 111.762 68.238 11.412 120 102.491 102.597 101.624 109.106 111.141 110.363 87.982 84.078 85.826 99.468 80.532 10.761

Crease 2 150 93.807 92.375 92.764 98.878 100.689 99.157 79.939 79.372 79.225 90.690 89.310 8.879 180 84.305 83.875 84.142 93.513 92.162 90.826 72.344 71.131 72.400 82.744 97.256 8.844 210 76.328 75.894 76.746 88.622 86.056 88.620 70.350 69.511 68.413 77.838 102.162 8.074 240 71.004 71.763 70.622 83.266 82.574 83.897 66.967 64.745 65.232 73.341 106.659 7.833

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 181.820 180.832 181.562 180.776 181.167 180.260 182.116 179.282 180.353 180.908 -0.908 0.879 60 181.118 180.858 179.150 180.741 180.575 179.838 179.437 179.208 179.359 180.032 -0.032 0.787

Crease 3 90 178.245 179.518 178.814 180.489 180.478 179.556 177.477 178.876 177.572 179.003 0.997 1.116 120 177.285 176.655 177.570 178.516 179.104 179.476 177.950 179.040 177.651 178.139 1.861 0.949 143

150 176.273 177.818 175.676 177.663 178.041 177.352 177.344 177.441 177.151 177.195 2.805 0.757 180 175.057 174.725 175.087 178.338 178.332 179.184 175.796 175.572 175.967 176.451 3.549 1.688 210 177.890 179.707 179.051 176.978 175.990 176.085 174.766 173.385 175.803 176.628 3.372 2.013 240 177.188 174.853 176.429 175.602 178.436 177.800 170.517 171.441 173.965 175.137 4.863 2.751

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 177.261 176.984 177.035 179.301 178.590 179.435 179.405 178.907 179.300 178.469 1.531 1.068 60 175.747 175.238 175.794 177.684 178.806 178.395 178.106 178.045 178.140 177.328 2.672 1.344 90 176.420 173.725 174.708 176.194 175.205 177.272 173.205 173.294 175.149 175.019 4.981 1.437 120 171.331 172.689 174.747 172.342 172.527 171.466 171.771 172.448 171.595 172.324 7.676 1.038

Crease 8 150 170.973 170.729 168.773 169.268 170.078 168.308 169.383 168.101 168.413 169.336 10.664 1.056 180 168.523 166.243 167.181 168.491 169.906 167.379 166.342 164.365 164.678 167.012 12.988 1.820 210 163.912 161.837 162.339 164.982 163.241 163.954 163.732 162.954 163.141 163.344 16.656 0.936 240 164.898 164.759 161.010 163.801 162.667 162.392 160.128 160.427 159.679 162.196 17.804 1.995

60 145.613 146.319 144.487 145.331 145.234 145.513 129.820 129.187 129.821 140.147 39.853 7.919

90 133.598 135.864 133.551 131.625 133.267 132.078 116.575 115.676 118.972 127.912 52.088 8.256 120 124.477 124.685 125.438 121.905 120.773 121.708 105.576 107.873 108.586 117.891 62.109 8.094

Crease 11 150 118.705 116.357 117.847 112.602 113.220 112.497 96.112 95.489 98.940 109.085 70.915 9.483 180 110.832 111.160 109.758 106.994 106.302 105.405 90.239 90.453 90.581 102.414 77.586 9.204 210 104.519 104.461 104.058 98.563 99.397 99.836 85.206 85.327 86.439 96.423 83.577 8.383 240 98.243 100.302 97.112 94.381 95.549 95.831 78.591 76.792 78.634 90.604 89.396 9.612

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 154.168 152.888 153.109 155.224 154.215 157.624 134.424 134.031 133.735 147.713 32.287 10.330 60 129.233 128.964 129.771 130.353 130.378 130.186 102.006 103.365 101.864 120.680 59.320 13.716 Crease 12 90 112.883 113.225 111.879 114.747 111.521 113.381 85.626 85.537 87.318 104.013 75.987 13.430 144

120 101.677 101.793 103.589 100.770 100.553 100.625 76.083 76.577 75.104 92.975 87.025 12.828 150 93.066 92.808 93.215 93.530 93.043 92.088 69.318 68.852 69.570 85.054 94.946 11.864 180 88.329 85.571 87.444 83.940 84.997 86.334 64.960 64.286 62.654 78.724 101.276 11.157 210 84.518 82.580 83.480 78.543 80.296 80.391 61.535 58.906 58.609 74.318 105.682 11.149 240 77.131 80.979 80.829 75.700 76.412 75.874 56.543 56.571 56.655 70.744 109.256 10.789

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Table A.11 Experimental data of the configuration I** - MAE prototypes MAE configuration I** MAE2 MAE3 MAE5 Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 165.797 165.425 166.058 167.779 166.909 165.729 171.768 172.134 172.019 168.180 11.820 2.933 60 151.569 151.295 151.222 158.337 156.683 157.109 157.883 158.140 158.555 155.644 24.356 3.265 90 139.237 138.742 138.899 146.034 146.449 147.114 147.694 148.005 147.619 144.421 35.579 4.144 120 129.319 130.627 130.389 137.870 138.279 136.557 137.406 137.483 137.177 135.012 44.988 3.721

Crease 1 150 124.697 122.456 123.954 133.572 133.267 132.225 127.846 128.357 127.985 128.262 51.738 4.096 180 115.917 117.136 117.218 128.554 128.466 128.276 122.136 122.084 120.729 122.280 57.720 5.107 210 107.746 106.847 109.481 123.489 124.042 124.290 116.083 115.261 115.584 115.869 64.131 6.932 240 104.053 102.676 102.650 119.874 121.714 121.090 109.832 109.232 109.057 111.131 68.869 7.832

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 156.246 156.457 155.746 160.570 162.857 161.541 163.719 165.823 165.182 160.905 19.095 3.918 60 129.648 128.665 129.625 140.365 139.718 139.163 143.778 143.427 143.785 137.575 42.425 6.438 90 112.134 109.847 109.972 124.649 121.790 122.966 127.538 123.008 125.670 119.730 60.270 7.040 120 96.501 96.507 95.093 109.533 109.076 108.032 111.803 113.118 110.955 105.624 74.376 7.357

Crease 2 150 87.996 87.145 87.704 99.047 98.600 96.443 104.370 101.953 101.768 96.114 83.886 6.766 180 79.780 80.446 78.737 91.259 90.408 89.087 94.819 94.720 95.093 88.261 91.739 6.795 210 75.240 75.987 75.224 83.381 84.731 83.912 88.845 88.908 88.564 82.755 97.245 5.845 240 73.743 73.154 72.048 79.960 79.317 81.178 84.618 83.526 84.541 79.121 100.879 4.982

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 182.415 180.677 180.851 178.087 178.796 178.826 181.544 184.389 181.375 180.773 -0.773 1.989 60 179.688 180.618 179.588 177.566 178.596 177.855 181.604 181.285 181.612 179.824 0.176 1.569

Crease 3 90 178.143 180.847 178.451 176.765 177.353 176.759 180.430 181.710 182.233 179.188 0.812 2.142 120 178.730 178.656 178.626 175.779 176.835 176.365 180.258 181.387 182.192 178.759 1.241 2.216 146

150 178.940 179.195 181.376 175.077 175.500 176.343 181.997 181.209 180.868 178.945 1.055 2.686 180 178.047 179.068 178.433 175.388 175.611 176.289 180.717 180.546 179.763 178.207 1.793 2.043 210 178.266 178.570 179.694 176.508 176.290 175.411 179.214 179.913 179.970 178.204 1.796 1.723 240 178.472 179.671 181.008 175.145 175.521 176.098 179.962 181.594 181.749 178.802 1.198 2.622

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD 0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 179.006 179.958 180.589 178.939 179.842 178.510 179.398 178.246 180.000 179.388 0.612 0.772 60 177.325 178.823 179.705 179.307 178.629 178.754 179.332 179.402 179.803 179.009 0.991 0.754 90 179.153 178.627 178.266 178.015 178.266 178.837 176.938 177.160 177.314 178.064 1.936 0.777 120 175.915 176.184 175.492 176.388 177.058 176.521 173.516 177.114 177.793 176.220 3.780 1.226

Crease 8 150 174.869 171.142 173.131 177.193 178.303 176.439 174.548 173.783 173.744 174.795 5.205 2.210 180 173.337 172.781 171.569 176.693 176.648 177.218 173.064 174.748 172.911 174.330 5.670 2.064 210 171.025 169.624 171.545 176.293 176.404 176.023 172.499 169.720 170.328 172.607 7.393 2.867 240 169.736 168.989 170.787 174.618 176.216 175.289 172.059 171.573 170.601 172.208 7.792 2.571

60 149.036 149.248 147.991 153.504 152.893 153.075 152.179 151.289 152.166 151.265 28.735 2.011

90 139.083 138.711 139.229 143.344 142.353 142.330 142.271 141.493 141.542 141.151 38.849 1.699 120 132.387 131.605 130.716 137.017 136.504 136.279 132.485 132.755 132.266 133.557 46.443 2.366

Crease 11 150 126.051 125.300 125.326 130.722 130.910 131.281 126.720 126.714 126.330 127.706 52.294 2.505 180 121.168 120.863 120.544 125.247 125.429 125.593 120.229 118.365 119.704 121.905 58.095 2.758 210 115.466 115.171 114.830 122.316 121.944 120.681 115.355 114.956 115.326 117.338 62.662 3.266 240 110.957 109.873 110.387 117.668 116.336 117.690 109.159 110.000 111.353 112.603 67.397 3.549

Field Strength (mT) Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 Cycle 1 Cycle 2 Cycle 3 AVG AVG_C SD

0 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 180.000 0.000 0.000 30 158.149 155.947 156.431 157.758 159.477 159.775 162.906 161.563 160.746 159.195 20.805 2.333 60 128.112 127.192 127.969 138.343 135.902 137.223 139.166 140.789 138.822 134.835 45.165 5.477 Crease 12 90 108.655 108.918 105.391 119.761 119.195 120.331 120.956 122.549 122.951 116.523 63.477 6.829 147

120 94.441 95.211 94.208 107.146 109.036 107.319 111.139 110.088 111.322 104.434 75.566 7.506 150 88.652 87.375 85.419 98.507 98.014 97.329 101.710 103.134 102.518 95.851 84.149 6.876 180 82.825 81.141 91.975 91.279 90.741 89.114 95.588 94.314 95.559 90.282 89.718 5.206 210 77.541 77.225 76.800 83.044 84.889 84.508 90.222 89.604 88.381 83.579 96.421 5.352 240 74.478 74.125 73.826 80.007 80.387 79.116 84.691 83.860 87.300 79.754 100.246 4.921

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