Multiscale simulation of metal deformation in deep drawing using machine learning Bachelor Integration Project
Industrial Engineering & Management University of Groningen
Author: F.M. Verwijs s3090345 Supervisors: Prof. dr. A. Vakis (First supervisor) Dr. ir. A.A. Geertsema (Second supervisor) Dr. S. Solhjoo (Daily supervisor)
June 19th, 2019 Abstract Crystal plasticity modeling is a powerful and well-established computational materials science tool to investigate mechanical structure–property relations in crystalline materials. The mechanical behavior of crystalline matter is a multiscale problem, whereas the underlying deformation processes such as the slip of dislocations and their reactions and elastic interactions are microscopic problems, and the forming process itself is usually of a macroscopic nature. Information regarding the crystal plasticity of metals in deep drawing processes, can be obtained at microscale. However, for applicability, the larger scale is of importance. To transfer the crystal plasticity information to the larger scale, this information needs to be stored and arranged in a certain way. In this Integration Project, a novel method of integrating the simulations at microscale to larger scale is presented. Crystal plasticity information concerning the material behavior of deep drawing processes is obtained using the DAMASK simulation software, generating representative volume elements, based on various strain rates. An Artificial Neural Network shows the information generated from the crystal plasticity simulations in DAMASK and predicts new data points. The information that is arranged in this machine-learning neural network, will be loaded into a finite element analysis. Based on this information, simulations are performed in MSC.Marc. This ultimately yields the information at larger scale, desired in deep drawing processes. It was found that at the low strain rate used in this project, no significant differences show between the stress-strain behavior of unrolled and rolled materials. Additionally, it was concluded that the RVE representation after a certain value of strain rate does not show significant differences when a certain strain needs to be met. Furthermore, various methods of data storage were compared. It was found that for this project, the SCG-trained feedforward ANN with two hidden layers and five neurons per layer, performs best in terms of accuracy, simplicity, and computation times. Lastly, the methods of performing the integration and the selected parameters were validated. Moreover, it was found that the deep drawn sample shows mainly a strain rate deformation of 0.01s−1 and four ears are formed. This novel method significantly decreases computation times, while increasing accuracy and user simplicity. This makes this novel multiscale simulation method of the deep drawing process is very relevant in various applications.
2 Contents
List of Tables 8
List of Figures 9
1 Introduction 11
2 State of the art 12 2.1 Crystal plasticity modeling ...... 12 2.2 Artificial Neural Network ...... 12 2.3 Finite element method ...... 13 2.4 Integration of crystal plasticity and FEM ...... 14
3 Problem analysis 15 3.1 Problem description ...... 15 3.2 Problem owner analysis ...... 15 3.3 Stakeholder analysis ...... 15 3.4 System description and scope ...... 16 3.5 Cycle selection ...... 16 3.6 Resources needed ...... 17 3.6.1 DAMASK ...... 17 3.6.2 MSC.Marc ...... 18 3.6.3 MATLAB ...... 18 3.6.4 FORTRAN and Visual Studio ...... 18 3.6.5 Literature about existing body of knowledge ...... 19
4 Goal 20 4.1 Main research question ...... 20 4.2 Sub questions ...... 20
5 DAMASK 21 5.1 Comparison to previous methods ...... 21 5.2 Process ...... 21 5.3 Parameter selection ...... 22 5.4 Results ...... 24 5.4.1 General remarks ...... 24 5.4.2 Representative Volume Elements ...... 24 5.4.3 Stress-strain curves ...... 25 5.4.4 Lankford coefficient ...... 26 5.5 Discussion ...... 27
6 Data storage 28 6.1 Methods ...... 28 6.1.1 Look-up table ...... 28 6.1.2 Fitted curve ...... 28
3 6.1.3 Artificial Neural Network ...... 29 6.2 Comparison to previous methods ...... 29 6.3 Process of LUT ...... 29 6.4 Process of ANN ...... 30 6.5 Parameter selection ...... 30 6.6 Results ...... 31 6.6.1 General description of network ...... 31 6.6.2 Training functions ...... 31 6.6.3 Error estimation ...... 32 6.6.4 Number of hidden layers and number of neurons per layer ...... 33 6.6.5 Transfer to MSC.Marc ...... 34 6.6.6 Requirements to the ANN by MSC.Marc ...... 35 6.6.7 Accuracy, based on error estimation ...... 35 6.6.8 Average training time ...... 35 6.6.9 User simplicity ...... 35 6.7 Discussion ...... 35 6.7.1 Data distribution ...... 35 6.7.2 Random selection of data within set ...... 36 6.7.3 Number of neurons optimization ...... 36
7 Finite Element Analysis using the ANN 37 7.1 Comparison to previous methods ...... 37 7.2 Process ...... 37 7.3 Parameter selection ...... 37 7.4 Results ...... 42 7.4.1 Comparison of data storage methods in FEM results ...... 42 7.4.2 Interpretation and simulation using the ANN ...... 42 7.4.3 Integration of the data storage method and MSC.Marc by means of the Fortran compiler ...... 43 7.4.4 Deep drawing simulation ...... 44 7.5 Discussion ...... 44
8 Finalized methodology 45 8.1 Comparison to previous methods ...... 45 8.2 Process ...... 45 8.3 Results ...... 46 8.3.1 Accuracy ...... 46 8.3.2 Decrease in computational times ...... 46 8.3.3 Simplicity for the user ...... 46
9 Validation 47 9.1 Crystal plasticity simulations in DAMASK ...... 47 9.2 Data storage using the ANN ...... 47 9.3 FE analysis in MSC.Marc ...... 47 9.4 Integration of these methods ...... 47
10 Conclusion 48
4 11 Discussion 50
Bibliography 51
A Theoretical background 57 A.1 Crystal plasticity and anisotropy ...... 57 A.2 DAMASK ...... 57 A.3 Plastic deformation behavior ...... 58 A.4 Stress-strain curves ...... 59 A.5 Representative volume elements ...... 59 A.6 Lankford coefficient ...... 59 A.7 Hill 1948 yield criterion ...... 60 A.8 Data storage methods ...... 60 A.8.1 Artificial neural network ...... 60 A.8.2 Lookup table ...... 61 A.8.3 Fitted function ...... 62 A.9 Finite Element Method ...... 62 A.9.1 MSC.Marc ...... 62 A.10 Strain rate dependence ...... 63 A.11 Deep drawing ...... 63
B Procedures 65 B.1 DAMASK ...... 67 B.1.1 Download the DAMASK software ...... 67 B.1.2 Determine the required strain rates and simulation times ...... 67 B.1.3 Generate the general seeds file ...... 67 B.1.4 Write the preprocessing code that will perform the simulation ...... 68 B.1.5 Visualize the RVEs generated in the simulation ...... 68 B.1.6 Write the postprocessing code to conclude the simulation ...... 69 B.1.7 Perform two more simulations under rolling angles 45◦ and 90◦ ..... 69 B.1.8 Determine stress-strain curves and the Lankford coefficient ...... 69 B.2 Artificial neural network ...... 70 B.3 Integration of the ANN with FEM ...... 70 B.3.1 Generate the FORTRAN file ...... 70 B.4 Finite element analysis ...... 70 B.4.1 Download the MSC.Marc software ...... 70 B.4.2 Setting up the finite element element mesh and geometric contact bodies, material properties, boundary conditions, contact body properties . . . 70 B.4.3 Integrating the subroutine ...... 71 B.4.4 Applying the load and submitting the job ...... 71
C Representative Volume Elements for several strain rates 72
D Stress-strain curves 76
E DAMASK codes 78 E.1 DAMASK codes ...... 78 E.1.1 Generation of seeds file ...... 78
5 E.1.2 Generation of rolled seeds file ...... 78 E.1.3 Unrolled pre processing file ...... 78 E.1.4 Rolled pre processing file ...... 80 E.1.5 Post processing file ...... 81 E.1.6 Generate rotated RVEs ...... 82
F Information aluminum 83 F.1 Phenopowerlaw Aluminum ...... 83 F.2 Elaboration on parameters ...... 83
G ANN code 85
H Table input in MSC.Marc 86
I FEM figures 89
6 List of Abbreviations ANN Artificial Neural Network BP Back Propagation CODF Crystallographic Orientation Distribution Function CPFEM Crystal Plasticity Models using Finite Element Methods FCC Face-Centered Cubic FEM Finite Element Method FFT Fast Fourier Transform HL Hidden Layer LUT Loo-kup Table MSE Mean Squared Error PDE Partial Differential Equation SCG Scaled Conjugate Gradient UTT Uni-axial Tensile Test
7 List of Tables
5.1 Time required to reach the elastic part and the plastic part of the stress-strain curve under certain strain rates ...... 22 5.2 Parameters that are DAMASK’s output ...... 25 5.3 Parameters that can be calculated from DAMASK’s output ...... 25 5.4 Lankford coefficients for rotated rolling simulations ...... 26 5.5 Computational times to perform the simulations ...... 26
8 List of Figures
3.1 The relation between DAMASK and the FEM solver ...... 15 3.2 MEI-diagram for obtaining information about crystal plasticity at larger scale . 16 3.3 System description, including the software ...... 17 3.4 Hevner’s cycles ...... 18
5.1 Visualization of the operations of the MATLAB code to obtain the rotated RVE 23
6.1 The predicted (orange) and original (blue) behavior of strain σ using the LM training algorithm ...... 32 6.2 The predicted (orange) and original (blue) behavior of strain σ using the SCG training algorithm ...... 33 6.3 The predicted (orange) and original (blue) behavior of strain σ using the CGP training algorithm ...... 33 6.4 Error estimation stress delta plotted against the number of neurons using the SCG training algorithm with one HL (orange) and two HL (blue) ...... 34
A.1 Structure of the material point model ...... 58 A.2 A basic representation of a neural network [Lin et al., 2008] ...... 61 A.3 Stretching and deep drawing ...... 63 A.4 Stress-conditions in a drawn cup...... 64
B.1 System description, including the software ...... 65
C.1 The RVE of an unrolled material ...... 72 C.2 The elongated unrolled RVE ...... 72 C.3 The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 ...... 73 C.4 The RVE of a rolled material under a strain rate ˙ = 0.1 s−1 ...... 73 C.5 The RVE of a rolled material under a strain rate ˙ = 1 s−1 ...... 73 C.6 The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 and an angle of 0◦ ...... 74 C.7 The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 and an angle of 45◦ ...... 74 C.8 The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 and an angle of 90◦ ...... 75
D.1 The stress-strain curve of a rolled and unrolled material under a strain rate of ˙ = 0.01 s−1 ...... 76 D.2 The stress-strain curve of a rolled and unrolled material under a strain rate of ˙ = 0.1 s−1 ...... 77 D.3 The stress-strain curve of a rolled and unrolled material under a strain rate of ˙ = 1 s−1 ...... 77
F.1 Hard sphere unit cell representation of the Face-Centered Cubic crystal structure 84
9 I.1 The workpiece before deformation using the NN ...... 89 I.2 The workpiece after deformation using the NN ...... 89 I.3 Behavior of displacement in y-direction following the arc length of the workpiece using the NN ...... 90 I.4 Behavior of displacement in y-direction following the arc length of the workpiece (a) using table 1, (b) using table 2 ...... 90 I.5 Depiction of the reaction force in the workpiece using the NN ...... 90 I.6 Depiction of the reaction force in the workpiece (a) using table 1, (b) using table 2 90 I.7 Depiction of the equivalent plastic strain behavior using the NN ...... 91 I.8 Depiction of the equivalent plastic strain behavior (a) using table 1, (b) using table 2 ...... 91 I.9 The equivalent plastic strain behavior using the NN ...... 91 I.10 The equivalent plastic strain behavior (a) using table 1, (b) using table 2 . . . 91 I.11 Depiction of the equivalent plastic strain rate behavior using the NN ...... 92 I.12 Depiction of the equivalent plastic strain rate behavior (a) using table 1, (b) using table 2 ...... 92 I.13 The equivalent plastic strain rate behavior using the NN ...... 92 I.14 The equivalent plastic strain rate behavior (a) using table 1, (b) using table 2 . 92 I.15 The equivalent Von Mises stress behavior using the NN ...... 93 I.16 The equivalent Von Mises stress behavior using the NN (a) using table 1, (b) using table 2 ...... 93
10 Chapter 1: Introduction Crystal Plasticity modeling is a powerful and well established computational materials sci- ence tool to investigate mechanical structure–property relations in crystalline materials. [Roters et al., 2019] The mechanical behavior of crystalline matter is a multiscale problem, whereas the underlying deformation processes such as the slip of dislocations and their reactions and elastic interactions are microscopic problems, and the forming process itself is usually of a macroscopic nature. [Ma et al., 2006]. Information regarding the crystal plasticity of metals in deep drawing processes, can be obtained at microscale. This has been done in previous research. However, for applicability, the larger scale is of importance. To transfer the crystal plasticity information to the larger scale, this information needs to be stored and arranged in a certain way. In this Integration Project, crystal plasticity information concerning the material behavior of deep drawing processes will be obtained using the DAMASK simulation software, generating representative volume elements (RVEs), based on several strain rates. These simulations yield much data, that needs to be stored and arranged in a certain way. This can be done in, among other options, an artificial neural network (ANN). An ANN shows the information generated from the crystal plasticity simulations in DAMASK. In a previous Integration Project, an artificial neural network constitutive model, with inputs of strain, strain rate, and an output of stress, has been configured to maximize performance based on predictive accuracy and computational time. The information that will be arranged in the machine-learning neural network, will be loaded into a finite element analysis. Based on this information, simulations are in MSC.Marc. This ultimately yields the information at larger scale, desired in deep drawing processes. In the past, integrating the crystal plasticity simulation DAMASK and FEM analysis in MSC.Marc was time consuming. In this Integration Project, a novel method of integrating the simulations at microscale to larger scale are presented. This report shows the bachelor integration project of Industrial Engineering and Management at the University of Groningen. It is composed of a theoretical background, problem analysis, goal, the transfer from crystal plasticity from DAMASK to the finite element analysis in MSC.Marc, a conclusion and a discussion.
11 Chapter 2: State of the art
In this section, the available literature of other research related to this Integration Project is discussed, based on the several phases of this project.
2.1 Crystal plasticity modeling
Multiple methods facilitate crystal plasticity simulations. Raabe et al. [Raabe et al., 2005a] discuss several methods when comparing their DAMASK simulations. Gottstein and coworkers [Klimanek and Seefeldt, 1999] combined a Taylor model with a finite element model that does not take the influence of the grain size on the mechanical behavior of polycrystalline metals and alloys into account. Secondly, Kalidindi et al. [Kalidindi et al., 1992] designed a fully-implicit time-integration scheme. Thirdly, crystal plasticity models were developed by Mathur and Dawson [Mathur and Dawson, 1989], who present a mathematical formulation based on the response of individual grains. Fourthly, Smelser and Becker [Raabe et al., 2003] use texture components in forming simulations. Finally, Beaudoin et al. [Beaudoin et al., 1994] provide a methodology for including anisotropy in metal forming analyses, and develop a finite element formulation through massive parallel data computations. Furthermore, the Johnson-Cook plasticity model [Johnson and Cook, 1983] is widely used in engineering practice to describe the viscoplasticity of metals. Due to recent progress in the field of ductile failure, there is a growing need for highly accurate predictions of the large strain plastic response of metals. Therefore, modified Johnson-Cook models have been developed. [Li et al., 2019] Huh et al. modify the second term of the Johnson-Cook model, previously depending on equivalent plastic strain rate only, by additionally depending on strain rate. [Huh et al., 2014][Li et al., 2019] Moreover, Gruben et al.[Gruben et al., 2016] describe the strain rate dependent stress-strain response of dual phase steel in a Johnson-Cook model that adds a power law type of expression and replaces the first term, previously depending on the equivalent plastic strain, by a Voce law. [Li et al., 2019] Finally, Roters et al. [Roters et al., 2012] describe several methods in application examples to save computation time in DAMASK. One example describes the deformation of an aluminum oligo-crystal under plane strain compression. As DAMASK allows to combine several con- stitutive laws within one geometry, this modular simulation saves around 30% computation time. In another example, the elastoplastic decomposition response of deformation can be determined based on explicit or implicit time integration of the plastic deformation rate per constituent. This shortens the computation time compared to previous methods.
2.2 Artificial Neural Network
In ”The improvement on constitutive modeling of Nb-Ti micro alloyed steel by using intelligent algorithms”, Wu et al. [Wu et al., 2017] establish four constitutive models of the deformation behavior of Nb-Ti micro alloyed steel, by using intelligent algorithms such as artificial neural network and genetic algorithm. In this comparative study of the accuracy and effectiveness of the models, it shows that the ANN obtains the highest accuracy.
12 Li et al., [Li et al., 2019], include machine-learning in their temperature and rate dependent Johnson-Cook plasticity model, as an alternative to developing physics-based constitutive models. [Johnson and Cook, 1983] Li et al. perform strain rate experiments on tension speci- mens extracted from DP800 steel sheets, and develop and train a neural network based on full specimen simulations using a BP scheme. VUMAT for Abaqus is developed for use in conjunction with solid elements.
In ”Artificial neural network approach for prediction of stress-strain curve of near β titanium alloy”, Setti and Rao [Setti and Rao, 2014] describe the ANN approach to predict the stress- strain curve, based on three different NN architectures, three different transfer functions, different number of hidden layers, different number of neurons in the hidden layers, and different training algorithms. In comparison to this Integration Project, it was found that the feed-forward back-propagation network, the layer recurrent NN with a single hidden layer consisting of 11 neurons, and the Log-Sigmoid transfer function are giving the best results, with an average relative error of 1.27 ± 1.45%.
In ”Prediction of the flow stress of high-speed steel during hot deformation using a BP artificial neural network”, Liu et al. [Liu et al., 2000] predict the flow stress under hot deformation conditions using a back propagation (BP) ANN, using three input parameters of strain rate ˙, temperature T , and true strain , and an output parameter strain σ. It was found that a BP ANN performs better on efficiency and accuracy compared to the prediction method of flow stress using the Zener-Holloman parameter and hyperbolic sine stress function. In the optimized ANN, two hidden layers are adopted, the first hidden layer including nine neurons and the second ten neurons.
2.3 Finite element method
Several researchers present crystal plasticity models which incorporate both crystallographic texture and additional morphological information, using either a finite element method or a spectral method, which uses fast Fourier transformation. The former is of interest in this Integration Project.
Examples of researches in this field include Raabe et al. [Raabe et al., 2001] in ”Microme- chanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation”, who determine the microtexture of a specimen, which is mapped onto a finite element mesh. Thereafter, crystal plasticity finite element simulations are conducted. Secondly, in ”Investigation of three-dimensional aspects of grain-scale plastic surface deformation of an aluminum oligocrystal”, Zhao et al. [Zhao et al., 2008] investigate an aluminum sample which is plastically deformed under uniaxial tensile loading. Simulations are conducted using a crystal plasticity finite element model. Finally, a series of comparisons identifying aspects of good and of less good match between model predictions and experiments is presented. Lastly, in ”A full-field strategy to take texture-induced anisotropy into account during FE simulations of metal forming processes”, Van Houtte et al. [Van Houtte et al., 2011] describe a methodology to overcome the difficulty of requiring a gigantic computational effort in modeling when performing finite element simulations of the forming process.
13 2.4 Integration of crystal plasticity and FEM In ”Accurate Numerical Computation of Hot Deformation Behaviors by Integrating Finite Element Method with Artificial Neural Network”, Yu et al. [Yu et al., 2018], describe the design and optimization of hot plastic forming processes with the finite element method, where an artificial neural network was employed to learn the experimental true stress-strain data and then predict the constitutive relationships outside experimental conditions. The ANN model was written into a FEM software platform, simulating compression tests with the FE model implanted ANN model. The intercomparisons between the experimental and simulated stroke-load curves revealed that integration of FEM with ANN is a feasible approach to conduct quality numerical computation for the varied hot plastic forming processes. Furthermore, ”A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations”, Ma et al. [Ma et al., 2006], introduce and implement a dislocation density based constitutive model for face-centred cubic crystals into a crystal plasticity finite element framework. It was found that it was required to modify the assumption of a homogeneous dislocation structure which replaces the dislocation cell structure, and introduce variables to replace the dislocation cell structure, in order to be able to implement the model into commercial FEM software. Thirdly, in ”Artificial neural network modeling to evaluate and predict the deformation behavior of stainless steel type AISI 304L during hot torsion”, Mandal et al. [Mandal et al., 2009] trained an NN based on experimental torsion tests within a certain temperature range and strain range. It was found that a Resilient Propagation algorithm performs better than standard back propagation. Furthermore, sensitivity analysis showed that temperature and strain rate are the most significant parameters while strain affects the flow stress only moderately. Finally, Kessler et al. [Kessler et al., 2007] in ”Incorporating neural network material models within finite element analysis for rheological behavior prediction” describe the advantages of using an ANN in cooperation with the FEM solver ABAQUS compared to conventional means. It was found that virtual models more closely matching experimental experience as compared to conventional material modeling methods.
14 Chapter 3: Problem analysis 3.1 Problem description
Currently, it is not simple, applicable nor accurate to study the mechanical responses of metals from the microscale to a larger scale. [Alankar et al., 2009] To obtain information about the crystal plasticity at larger scale, the steps that need to be taken, are shown in the MEI-diagram in figure 3.1. However, these methods are currently not easily connected. In the past, DAMASK has been used inside the FEM solver, as shown in figure 3.1a. During this integration project, DAMASK will be run as a laboratory, and then be put into the FEM solver. The behavior of the sample is not controlled by DAMASK. Now, as shown in figure 3.1 and figure 3.1b, integrating the crystal plasticity in DAMASK and the finite element analysis in MSC.Marc by means of a multidimensional neural network will be the problem to be resolved in this Integration Project.
3.2 Problem owner analysis
The aforementioned problem has been proposed by prof. dr. Antonis Vakis, who supervises dr. Soheil Solhjoo. Dr. Solhjoo will be responsible for the execution of the research, although prof. dr. Vakis is the main problem owner. Their goal is gaining knowledge and input for publications about multiscale simulation of metal forming. Secondly, their goal is to investigate and describe case studies that develop the entire chain of processes, by means of demonstrators.
3.3 Stakeholder analysis
Besides the problem owner, more people have an interest and influence on this problem.
• Researchers interested in the possibilities of this chain of processes that are used to obtain information about crystal plasticity at a larger scale. They are interested in this for their own knowledge and research possibilities. They do not influence this research, they are only interested in it.
(a) Usage of DAMASK inside an (b) Usage of DAMASK as a labo- FEM solver, as has been done in ratory outside of the FEM solver, the past which will be the scope of this In- tegration Project
Figure 3.1: The relation between DAMASK and the FEM solver
15 Figure 3.2: MEI-diagram for obtaining information about crystal plasticity at larger scale
• The head of engineers of companies that use deep drawing, who rely on information about tensile tests to perform their operations, may be affected by the outcome of the research. Obtaining a more simple, applicable and accurate chain of processes will decrease their operational costs. They may be interested in the outcome of the research, but do not influence it.
3.4 System description and scope Many different simulations can be performed using the processes described above. The system, as described in figure 3.1, is restricted as follows. • Only simulations will be performed. • Only one material will be considered. • The material of interest will firstly be rolled using a uniaxial tensile test, and secondly be deep drawn. • The simulations will be performed for three strain rates. • A certain value of strain needs to be met to conclude the elastic and plastic region. These values for strain rates and strain times were determined considering DAMASK’s possibilities and MSC.Marc’s requirements. • The RVEs will be generated under three stress-angles: 0◦, 45◦, and 90◦. It is assumed that these three angles provide sufficient information to determine the yield surface. • The artificial neural network will be constructed using MATLAB functions. • The FEM simulation will be performed for the deep drawing process. Considering figure 3.1 and the aforementioned scope, the software supporting the chain of processes is shown in figure 3.4. It can be seen that DAMASK, Excel, MSC.Marc and an artificial neural network will be used to execute the desired functions.
3.5 Cycle selection Hevner [Hevner, 2007] recognizes three cycles in a research project, which clearly positions and differentiates design science from other research paradigms. These cycles can be found in figure 3.5 below.
16 Figure 3.3: System description, including the software
For this Integration Project, the rigor cycle was used. According to [Hevner, 2007], ”The rigor cycle provides past knowledge to the research project to ensure its innovation. It is contingent on the researchers to thoroughly research and reference the knowledge base in order to guarantee that the designs produced are research contributions and not routine designs based upon the application of well-known processes.” The rigor cycle is considered best for this Integration Project, because of the large focus of adding knowledge to adding to the already existing knowledge base (KB) of multiscale metal deformation modeling, including extensions to the original theories and including experiences gained from performing the research. The Integration Project follows the steps according to Hevner’s [Hevner, 2007] rigor cycle. Important steps in the rigor cycle include thorough literature research to the existing body of knowledge, and validation of methods and tools.
3.6 Resources needed The resources needed for the completion of this research are the following, which will be elaborated below.
• DAMASK • MATLAB • MSC.Marc • FORTRAN and Visual Studio • Literature about existing body of knowledge
3.6.1 DAMASK DAMASK is a simulation kit for the simulation of crystal plasticity within a finite-strain continuum mechanical framework. It will be used to simulate the crystal plasticity of the materials that will form the basis of the model. Preparation of simulations is done through
17 Figure 3.4: Hevner’s cycles configuration files, which are relatively straightforward to create. The code can be implemented within two popular FEM solvers: either MSC.Marc or ABAQUS, or can be used stand-alone by using the included spectral solver. In this Integration Project, MSC.Marc will be the FEM solver of interest. The virtual laboratory is conveniently implemented as a part of DAMASK in form of an automated script. This script launches virtual tests with automatically generated load cases for the considered RVEs, post-processes the results and finally performs the parameter identification for all selected yield functions until the desired fitting quality is achieved. [Zhang et al., 2016]
3.6.2 MSC.Marc
MSC.Marc is a powerful, general-purpose, nonlinear finite element analysis solution to accu- rately simulate the product behavior under static, dynamic and multi-physics loading scenarios [MSC, 2019]. While a traditional FEM simulation requires the materials characteristics as an input, MSC.Marc, when combined with DAMASK, utilizes a user subroutine. This procedure ensures that DAMASK defines the material and its behavior, instead of MSC.Marc doing this.
3.6.3 MATLAB
MATLAB is a programming platform designed specifically for engineers and scientists. The heart of MATLAB is the MATLAB language, a matrix-based language allowing the most natural expression of computational mathematics. The artificial neural network which forms the starting point of this research, is written in MATLAB code.
3.6.4 FORTRAN and Visual Studio
FORTRAN is a high-level computer programming language used especially for scientific calculations. Visual Studio is an integrated development environment, used to develop several programs and applications. FORTRAN and Visual Studio are used in this Integration Project to define the user subroutine from the ANN to MSC.Marc.
18 3.6.5 Literature about existing body of knowledge As mentioned before, in the rigor cycle, it is very important to conduct thorough literature research to the existing body of knowledge. Therefore, literature is a necessary resource to complete this research. This is done throughout the entire Integration Project, with a focus on chapter State of the Art 2.
19 Chapter 4: Goal The goal of this Integration Project is to integrate the crystal plasticity in DAMASK and the finite element analysis in MSC.Marc by means of a multidimensional neural network.
4.1 Main research question How can the chain of crystal plasticity at microscale and the finite element analysis in a larger scale be integrated?
4.2 Sub questions • How do these novel operations in DAMASK compare to previous methods? • Why and how can the DAMASK simulation be performed in this novel method? • What possibilities for data storage exist and how do they compare? • How should an Artificial Neural Network be configured to optimize the integration of microscale simulations in DAMASK and larger scale FEM simulations in MSC.Marc? Configuration in terms of the number of hidden layers, number of neurons per layer, method of error estimation, and training functions. • How can the ANN be transferred to MSC.Marc? • How does the FEM simulation of deep drawing in MSC.Marc set requirements that affect the DAMASK simulations and the ANN configuration? • How can the deep drawing process in MSC.Marc be simulated, using the input from the ANN? • How is the sample deformed after the deep drawing process? • How does the finalized methodology perform on accuracy, simplicity, and computational times compared to previous methods?
20 Chapter 5: DAMASK As mentioned before, multiple methods facilitate crystal plasticity simulations. In this In- tegration Project, DAMASK has been used to simulate the crystal plasticity of the materi- als. DAMASK has been chosen as the software to simulate crystal plasticity because it is an open, flexible, and easy to use implementation [Roters et al., 2019], its high modularity [Roters et al., 2019], and its possibility to integrate with the FEM solver MSC.Marc. Due to the large flexibility in boundary conditions applied to the RVE, it becomes then possible to mimic more complex strain paths, like biaxial tensile, compressive or shear tests, in order to extract the required parameters of the analytic material description. [Roters, 2011] This leaves possibilities for future research after this Integration Project.
5.1 Comparison to previous methods As mentioned in State of the Art in chapter 2, multiple methods facilitate crystal plasticity simulations. These methods are as follows, briefly addressed in section 2. • Taylor model [Klimanek and Seefeldt, 1999] • fully-implicit time-integration scheme [Kalidindi and Anand, 1992] • mathematical formulation [Mathur and Dawson, 1989] • texture components in forming simulations [Raabe et al., 2003] • parallel data computations [Beaudoin et al., 1994] The main difference of the DAMASK usage in this Integration Project compared to previous overviews is the internal modular structure of DAMASK, following from the hierarchy inherent to the employed continuum description. Therefore, various constitutive laws based on evolving internal state variables can be implemented to provide the response of each microstructural constituent using the responses using various laws. De facto, DAMASK has been designed to reproduce the multiscale hierarchy and multi-physics structure inherent in the underlying material physics that is associated with thermo-mechanical loading of complex materials. [Roters et al., 2019]
5.2 Process From all possibilities DAMASK offers, the information and procedures deemed useful for this Integration Project were subtracted and analyzed. This resulted into a preprocessing and a postprocessing bash file. The scripts have been analyzed and changed where necessary to provide the desired results. The initial bash files are shown under listing E.3 for preprocessing and listing for E.1.5 postprocessing. The following steps are performed in DAMASK regarding the uniaxial tensile test simulations 1. Create RVEs for rolled and unrolled metal grains, under three different strain rates 2. Perform two more simulations under rolling angles 45◦ and 90◦ for finding the anisotropic behavior of the rolled RVE (performing tensile tests on rotated RVEs
21 3. Determine stress-strain curves 4. Determine the Lankford coefficient 5. Determine yield values to determine the yield surface A more detailed description of the procedure is provided in appendix B. 5.3 Parameter selection The DAMASK code that has been used to generate the RVEs and the stress-strain curves, are shown in listing E.3. Below, all parameters implemented into the DAMASK simulation will be explained. Strain rates The strain rates for the DAMASK simulations were chosen according to the requirements of the results from the deep-drawing test in MSC.Marc, which was performed using a different method. From the deep drawing tutorial, it appears that the strain rate is almost uniform and equals 0.01s−1. However, at some spots, the sample has been deformed with a strain rate of 1s−1. Therefore, the three strain rates chosen for the DAMASK simulations in this Integration Project are 0.01s−1, 0.1s−1,and1s−1. For determining the yield surface, the strain rate of 0.01s−1 has been used, as MSC.Marc does not allow to define a rate dependent yield surface. Simulation times As mentioned before, the elastic region was decided to be restricted to a strain of 0.002. From the deep drawing tests, it was concluded that the strain needed for the MSC.Marc simulation approximates 0.6. Therefore, the simulation times were chosen to meet these strains, according to the strain rates. This is shown in table 5.3.
˙ telastic tfinal 0.01 1 60 0.1 0.1 6 1 0.01 0.6
Table 5.1: Time required to reach the elastic part and the plastic part of the stress-strain curve under certain strain rates
Geometry generation Voronoi tessellation has been selected as the method of geometry generation. In the simplest form, a Voronoi diagram can be defined as follows: Given some number of points (grains) in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor rule: Each point is associated with the region of the plane closest to it. [Burtseva et al., 2015] Constructing virtual microstructures by employing Voronoi tessellation to construct the Representative Volume Elements has been established as a powerful tool for the evaluation of field quantities. [Werner et al., 1994] [Roters et al., 2019]. Representative Volume Elements By definition, a RVE is the smallest volume over which a measurement can be made that will yield a value representative of the whole. Representative Volume Element simulations have been chosen to replace extensive experimental testing and thus saving time and costs.
22 Number of grains The number of grains was chosen to equal 40, according to the tutorials [Kumar, 2019]. It was assumed that this number of grains fulfills the requirements of an accurate representation of the grain distribution within the RVE, as well as not be computationally too expensive.
Homogenization For component-scale simulations, the volume represented by each material part is typically not composed of a single constituent but instead of an aggregate of constituents with different phases and/or crystallographic orientations. For example, sheet materials typically have a non-random Crystallographic Orientation Distribution Function (CODF), also referred to as crystallographic texture, and hence show anisotropic behavior. [Roters et al., 2019] This anisotropic behavior also occurs in the simulations in this Integration Project, and is taken into account in the DAMASK simulation. The CODF can be experimentally measured and approximated by a population of individual crystal orientations. However, the number of orientations needed for a sufficiently close approximation of the crystallographic texture is usually much larger than the number of orientations that can be computationally handled at a single material point. Therefore, DAMASK contains the HybridIA algorithm to optimally approximate a crystallographic texture with a given number of individual orientations and distribute those onto the material points. [Roters et al., 2019] Application of this homogenization algorithm in DAMASK is shown in line 15 of listing E.3.
Material In the simulations in this Integration Project, the non-existing material Phenopowerlaw Aluminum is used. This material, including the description of its properties, is available in the DAMASK software. This material was selected because it describes truly crystalline (i.e. anisotropic) behavior, and the applied material is irrelevant when performing the simulations and describing the methods. The properties of Phenopowerlaw Aluminum are described in F.1.
Rotation The rotated RVEs were generated using a MATLAB code that has been written previously. The operations of the MATLAB code are shown visually in figure 5.3.
Figure 5.1: Visualization of the operations of the MATLAB code to obtain the rotated RVE
Firstly, the rolling process is performed on one RVE. This RVE is shown by the blue square on the left side of figure 5.3. Then, the RVE is copied eight times and placed around the original RVE, as shown in the middle of the figure. This is rectified because the RVEs, by definition, represent the whole material. Then, a middle piece, rotated by the desired angle, is taken from the enlarged RVE. This RVE, finally, serves as the rotated RVE.
23 5.4 Results 5.4.1 General remarks All RVEs and the corresponding stress-strain curve are generated based on the exact same seeds configuration in DAMASK. Figure C shows the general RVE of an unrolled material. This RVE corresponds to the main *.seeds file that all rolled and/or rotated RVEs were generated from. In order to generate the rolled RVE, the dimensions of the grain were taken using a y-direction four times as large, and x and z equal to the normal RVE. This can be seen in listing E.3 and E.6 and figure C. It can be seen that the RVEs in figures C and C do not exactly match. This is because DAMASK generated a new RVE after receiving the commands to generate the long RVE. However, as this is one of the main functions of DAMASK, it is assumed that both RVEs represent the whole material correctly. Additionally, the RVEs as visual output are irrelevant in the further stages of the multiscale simulation integration. Furthermore, it appeared that DAMASK in combination with the processor used was not powerful enough to complete the simulations to the desired strain of 0.6. This was noticed by the error message after running the preprocessing part of the simulation. This also affected the post-processing file. In order to generate the desired results in the subsequent steps in the process in figure B, it was decided to extrapolate the results using Excel. From Kalpakjian [Kalpakjian and Schmid, 2008], it was determined that in the plastic part of the stress-strain curve, the behavior is linear, which rectifies the extrapolation.
5.4.2 Representative Volume Elements As mentioned in section A.5, the Representative Volume Element (RVE, also called unit cell) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. For this reason, RVEs were chosen to represent the entire material of interest. A larger deformation sample would be more computationally expensive. The RVEs created are only in 2D, as 3D would bee computationally too expensive. It was assumed that a 2D representation is sufficiently accurate in simulating the material behavior. In appendix C, the RVEs generated for this Integration Project are shown. The various grains are each represented by a different color, which is only used to distinguish the grains and does not have a physical meaning. Unrolled RVEs Figure C shows the non-rotated RVE based on the seeds file *.seeds that all rolled and/or rotated RVEs were generated from. As the unrolled RVEs under the various conditions in this Integration Project are not different visually, only one RVE is shown. It can be seen that the grains differ in size and shape. Rolled RVEs The rolled RVEs are shown in figures C, C, and C. It can be seen that the grains in the RVE of the lower strain rate of ˙ = 0.01s−1 are less squished than in the other figures. Moreover, it can be seen that the location of the grains in figures C, C, and C is roughly the same. However, it appears that C and C are very similar. Apparently, after a certain value, the strain rate does not affect the rolled RVE when reaching the same final strain. Rolled and rotated RVEs The rolled RVEs are shown in figures C, C, and C. Figure C is exactly similar to figure C,
24 because the *.geom file that generates the RVE in figure C served as the input for the rotation simulation. Furthermore, from the rolled RVEs, it can be seen that besides the angle, the grain structure does change under a rotated tensile test. The angle of 45◦ shows more granular edges, whereas the edges of 0◦ and 90◦ are smoother. Finally, it can be concluded that the rotated RVEs for 0◦ and 90◦ are exacly similar, besides the rotation. Apparently, only rolling under the angle of 45◦ affects the RVE.
5.4.3 Stress-strain curves The stress-strain curves were obtained using the data output from the post-processing file as listed in listing E.1.5. The stress-strain curves only obtained at angle zero, as the 45 ◦ and 90 ◦ angles yield insights about the yield surface. Additionally, it was mentioned that the DAMASK simulation does not reach the desired strain of 0.6 in the plastic phase, and therefore, extrapolation is applied to the data. This extrapolation is not included in the stress-strain curves in appendix D, as the results are not generated by DAMASK. The output of DAMASK can be shown in an RVE, and in a table output. The parameters that DAMASK outputs, are the following.
f The deformation gradient tensors. fe The elastic part of the deformation gradient tensors. fp The plastic part of the deformation gradient tensors. e The total strain defined as a Green-Lagrange tensor. ee The elastic strain defined as a Green-Lagrange tensor. p The first Piola-Kirchhoff stress tensors.
Table 5.2: Parameters that are DAMASK’s output
From these parameters, the following parameters shown in table 5.4.3 are calculated or derived in DAMASK:
ln(u) The left-sided Cauchy-Green spatial Hencky strain tensors derived from the deformation gradient tensors. Mises ln(v) The symmetric Cauchy stress tensors based on the deformation gradient tensors and the first Piola-Kirchhoff stress tensors. Mises e Von Mises equivalent strain based on the right-sided Cauchy-Green spatial Hencky strain tensors. Mises ee Von Mises equivalent total strain based on the Green-Lagrange strain . tensors. Mises Cauchy Von Mises equivalent stress based on the Cauchy stress tensors.
Table 5.3: Parameters that can be calculated from DAMASK’s output
[Kamps, 2018] From these values, the stress-strain curves were obtained using the data regarding strain in column 1 ln(V), which is the first entry in the Von Mises equivalent strain, and stress in column 1 Cauchy, which is the first entry in the Von Mises equivalent stress, based on the Cauchy stress tensor.
25 From the stress strain-curves as shown in appendix D, it can be concluded that the stress-strain curve is not significantly affected by the rolling process at these low strain rates, as the blue and orange line follow roughly the same curve. Kalpakjian [Kalpakjian and Schmid, 2008] describes that increasing the strain rate increases strength of a metal. However, this does not apply to these low strain rates. Any differences between the two may be due to the deviations in the various *.geom geometry files used in the simulations. No conclusions can thus be drawn from the stress-strain curves.
5.4.4 Lankford coefficient DAMASK outputs the Lankford coefficient or r-value in an Excel file for the rotated simulations. These values are shown in the table below. Strain rate Lankford coefficient 0◦ 1.08 45◦ 1.34 90◦ 0.96
Table 5.4: Lankford coefficients for rotated rolling simulations
As mentioned in chapter A.11, the Lankford coefficient indicates the variation of plastic behavior of rolled sheet metals with direction. These values are used in the Finite Element Analysis in MSC.Marc. Generally, in a cold-rolled steel, R45 is the lowest value and R90 is the highest.[Mori, 2001] However, from the DAMASK rotated UTT simulation, this appeared to be the opposite. Computation times The computations were performed on an Intel Core i5 processor. The computational times for performing the crystal plasticity simulation in DAMASK are mentioned below in table 5.5.
time old (hrs) ˙ = 0.01, unrolled 11:15 ˙ = 0.01, rolled 12:15 ˙ = 0.1, unrolled 10:30 ˙ = 0.1, rolled 11:55 ˙ = 1, unrolled 9:30 ˙ = 1, rolled 10:10 ˙ = 0.01, rotated by 45◦ 6:15 ˙ = 0.01, rotated by 90◦ 6:15
Table 5.5: Computational times to perform the simulations
From the table, it can be concluded that rolling simulations at lower strain rates need the longest computational time. This may be due to the longer required tensile simulation times, as mentioned in table 5.3. Furthermore, it appears that all simulation times are very long. This could be shortened using the method of combining several constitutive laws within one geometry, as Roters et al. [Roters et al., 2012] did. Additionally, the computation times could be shortened by using a different number of increments per simulation and by using a more powerful processor. This will be discussed in more detail in the Discussion in section 5.5.
26 5.5 Discussion During the DAMASK simulations, it appeared that DAMASK was not able to finish all simulations due to its inability to converge after a certain number of simulation steps. As the simulation has reached the plastic part to a strain of 0.47, it was decided to extrapolate the results to obtain the desired strain. This phenomenon has not been described in literature, which, however, only describes a lower final strain. This problem could be resolved by using a lower number of simulation steps. Secondly, in order to decrease the computation times in DAMASK, a more powerful processor could be used to perform the simulation. Additionally, the computation times in DAMASK depend largely on the number of increments. In this Integration Project, the number of increments for all simulations was set to 100 for the elastic phase, and 50 for the plastic phase. The frequency of outputting equaled 1 for both phases. It is expected that decreasing the number of simulation steps would decrease the simulation times significantly. However, the accuracy of the method must be closely monitored when decreasing the number of data points. If the number of data points were decreased, the common method of DAMASK simulations can still be used. Finally, it appeared that the rotated and rolled simulations in this Integration Project show irregularities regarding the Lankford coefficient and the visualization of the RVE. Although the internal code of the rotated UTT was considered a black box in this Integration Project, this could be researched in the future.
27 Chapter 6: Data storage In order to decrease the computational time when transferring the crystal plasticity simulations at microscale to FEM simulations of deep drawing at microscale, the data needs to be stored in a certain way. The data storage method must be used to define stress as a function of strain and strain rate. More specifically, the data storage method must be able to convert the two-fold input of strain, [−], and strain rate, ˙[s−1] into an output of stress, σ[MP a]. Furthermore, the various methods of data storage that exist, will be compared on • accuracy • computation times • ease of use, simplicity 6.1 Methods In this section, the methods, as well as advantages and disadvantages of using various data storage methods are described. It must be noted that, as perceived based on advantages, disadvantages, and basic assumptions, the ANN is chosen as the preferable data storage method that is finally realized in this Integration Project. Therefore, the other data storage methods are only described in detail, instead of executed with various results.
6.1.1 Look-up table In this Integration Project, the DAMASK simulations provide 100 data points for the elastic region, and 50 data points for the plastic region, each for 3 different strain rates. During the extrapolation to reach the strain of 0.6, 12 more data points were created per strain rate. This yields 486 data points in total. These data points can be arranged in a table, which provides the data input in MSC.Marc. As this method is perceived suitable considering the requirements accuracy, low computation times, and simplicity, this method is further elaborated in section 6.3.
6.1.2 Fitted curve A fitted function can be constructed based on the best fit to a number of data points. This may require interpolation or smoothing of a function. Various curve fitting methods exist, including the least square method, exponential curve fitting, or logistic curve fitting. [Arlinghaus, 1994] Furthermore, fitted function may be a function based on physical proper- ties. In case of real materials, the physical stress strain curve can be determined based on experimental data. As the stress-strain behavior in the plastic region is assumed to be linear [Kalpakjian and Schmid, 2008], a fitted curve can be determined with a high accuracy. Using a regression analysis, this accuracy can be determined. In the MSC.Marc deep drawing tutorial in its User’s Guide, the following function is used in the deep drawing simulation to determine the hardening behavior for Al2090-T3.
0.227 646 ∗ (0.025 + v1) (6.1)
28 where v1 equals the first independent variable input strain, . Based on the assumption of the linear stress-strain behavior in the plastic region, it can be assumed that a fitted function can be simply constructed based on experimental data for a certain material using various fitting methods. However, exploring these methods is a time-consuming process. The computational times for constructing the function are low, as well as the time to evaluate the linear function. The linearity makes the function simple. Its accuracy should be determined based on experimental data. Kessler et al. [Kessler et al., 2007] describe conventional material modeling and the relation- ships between various variables in the plastic region. Although this method is perceived well based on accuracy and computation times, this method is not further elaborated as the method of a LUT is more commonly used and an ANN is perceived to be a promising and novel method. 6.1.3 Artificial Neural Network The main concept of an Artificial Neural Network has been explained in chapter A.11. As mentioned before, the Artificial Neural Network should be accurate in its predictive ability and be able to convert the two-fold input of strain, [−], and strain rate, ˙ [s−1] into an output of stress, σ [MP a]. According to Lin et al., [Lin et al., 2008] neural networks can provide a fundamentally different approach to materials modeling and material processing control techniques. Multiple types of ANNs exist. Feedforward neural networks are more powerful and widely used than recurrent networks. [Nielsen, 2015] 6.2 Comparison to previous methods In a previous Integration Project, a rudimentary guide for setting up a constitutive model to determine stress under hot work in an artificial neural network is given. [Lindemann, 2019]. This Integration Project discusses the types of ANNs, and factors affecting the ANN’s performance, including error and training time. It was concluded that the number of hidden layers used should be equal to the number of input variables, and a delta error is the desired error estimation for this type of projects. Furthermore, many researchers, including [Kessler et al., 2007] and [Yu et al., 2018] describe the integration of an ANN and FEM. More information is provided in chapter 2. 6.3 Process of LUT Two tables were generated from the DAMASK simulations. Table 1 contains the data points of the variables strain and stress, whereas table 2 contains the data points of the variables strain rate, strain and stress. This was done because at first, MSC.Marc was only to read a one-input, one-output table. However, with some slight modifications in the settings of MSC.Marc, another input was allowed. To compare the requirement of two-input data, both tables were used. MSC.Marc requires the data for each variable to be rearranged in an array of four columns, which was done using MATLAB. Summarizing, the steps that were taken to submit the LUT to DAMASK were as follows
29 1. Select the desired data. 2. Rearrange the data into MSC.Marc’s desired form. 3. Adjust the necessary values in the input file. 4. Adjust the settings in MSC.Marc for it to be able to read the file. The table inputs are shown in appendix H. The exact method of submitting the table to MSC.Marc is described in appendix B.
6.4 Process of ANN The steps that were taken in performing this project are as follows. Contradictory to the process steps in DAMASK in chapter 5, the process in generating the Artificial Neural Network is an iterative process, as it concerns an optimization. 1. Determine the desired type of neural network. 2. Determine the ratio of training data vs. testing data 3. Determine the desired error estimation. 4. Determine the desired number of hidden layers and number of neurons per layer.
6.5 Parameter selection As described above, the optimization of this artificial neural network is an iterative process. Therefore, the parameters described below concern the parameters initially used. Number of inputs and outputs There are two inputs in the NN, namely strain and strain rate ˙. The output is stress σ. Type of neural network The multi-layer feed forward network with back propagation learning is the most popular of all ANN models, and is therefore chosen in this Integration Project. The feed forward back propagation neural network is actually composed of two neural network algorithms: (a) feed forward and (b) back propagation. It is not necessary to always use ’feed forward’ and ’back propagation’ together, but this is usually the case. The term ’feed forward’ refers to a method by which a neural network processes the pattern and recalls patterns, where as the term ’back propagation’ describes how this type of neural network is trained. Back propagation is a form of supervised training. [Lin et al., 2008] The backpropagation algorithm consists of two phases, referred to the forward and backward phases. In these phases, the error is computed using summations of local-gradient products over the various paths from a node to the output. [Nielsen, 2015] To overcome the inherent disadvantages of pure gradient-descent in backpropagation, RPROP, a new learning algorithm for multilayer feedforward networks, performs a local adaptation of the weight-updates according to the behaviour of the error function.[Riedmiller and Braun, 1993] All of this is incorporated in the MATLAB command feedforwardnet. Training function As researched by Lindemann [Lindemann, 2019], the training algorithm Scaled conjugate
30 gradient backpropagation ’trainscg’ was initially used. This is a network training function that updates weight and bias values according to the scaled conjugate gradient method. Lindemann compared this training function to seven other functions, based on eight 36 x 36 single-input ANNs over five iterations. Performance indicators included MSE and training time. It appeared that under these circumstances, SCG backpropagation showed the best performance compared to other functions. [Lindemann, 2019] Ratio training data vs. testing data As there are no general standards for using certain ratios in specific cases, the default settings of DAMASK were used, which was 70% training data, 15% validation and 15% testing. Error estimation MSE is a common and widely used method in error estimation. In the bachelor’s thesis by Adrian Lindemann [Lindemann, 2019], it was verified that one-input ANNs trained using stress delta perform better than ones trained using MSE. Therefore, this methodology is also used in error estimation.
6.6 Results In this section, the optimized Artificial Neural Network for the case of this Integration Project is described. The various artificial neural networks composed in this Integration Project are be compared on
• Average maximum error • Average training time • User simplicity Furthermore, the network will be described based on the number of hidden layers and the number of neurons.
6.6.1 General description of network The MATLAB code that describes the network is shown in appendix G.It was determined that the ANN is a two-input, one-output ANN, where the inputs are strain [−], and strain rate ˙[s−1], and the output is stress σ[MP a]. The finalized network is shown in appendix G.
6.6.2 Training functions Several multilayer neural network training functions are provided in MATLAB. These functions are as follows
• trainlm Levenberg-Marquardt • trainbfg BFGS Quasi-Newton • trainrp Resilient Backpropagation • trainscg Scaled Conjugate Gradient • traincgb Conjugate Gradient with Powell/BEale Restarts • traincgf Fletcher-Powell Conjugate Gradient
31 • traincgp Polak-Ribiere Conjugate Gradient • trainoss One Step Secant • traingdx Variable Learning Rate Backpropagation The MATLAB helpdesk describes which training functions are most suitable for which data sets. This Integration Project concerns a small data set, focused on generating the smallest mean squared error (MSE), without significant memory restrictions. However, computational speed is desired. Based on these requirements, the most promising training functions include trainlm because of its low mean squared error and trainscg and traincgp because of their speed. These training functions were then used in training the NN based on two hidden layers, and 20 neurons per layer, comparable to the findings in the work of [Lindemann, 2019]. Using these three training functions, the predicted and original behavior of strain σ were compared in a graph. These graphs are shown in figures 6.6.2, 6.6.2, and 6.6.2.
Figure 6.1: The predicted (orange) and original (blue) behavior of strain σ using the LM training algorithm
From these graphs, it can be concluded that the SCG training function performs best, as it deviates least from the original strain behavior. This training algorithm was then used in determining the number of hidden layers and number of neurons per layer. Finally, the performance of the network was improved by setting the performance goal of the error equal to zero (net.trainParam.goal = 0).
6.6.3 Error estimation
The mean squared error is a common error estimation used in various applications. The mean squared error approximation was used to determine the error behavior of the neural network. The mean squared error was calculated using the MATLAB function immse. In the work of [Li et al., 2019], it was shown that under certain HL, neurons per layer, and network size conditions, the mean squared error reaches a plateau. Additionally, Lindemann [Lindemann, 2019] mentions that MSE may fluctuate largely, especially during the work
32 Figure 6.2: The predicted (orange) and original (blue) behavior of strain σ using the SCG training algorithm
Figure 6.3: The predicted (orange) and original (blue) behavior of strain σ using the CGP training algorithm hardening phase. Therefore, a stress delta was introduced. This stress delta was defined as the difference between the stress value outputted by the ANN and the actual value of stress. Thus, based on research mentioning the preference of using stress delta rather than MSE, it was decided to use this in the further optimization of the NN. Furthermore, when plotting the error (either MSE or delta error) against the number of neurons per hidden layer in the stress-strain curves, it appeared that the error is extremely large in the elastic region of the stress-strain curve. As this region does not play a role in the deep drawing simulation, it is removed from the error estimation curves in order to yield a more accurate error comparison.
6.6.4 Number of hidden layers and number of neurons per layer Using the SCG training algorithm, the number of hidden layers and number of neurons per layer were compared to the stress delta error estimation in a number of iterations in MATLAB.
33 The numbers used in these tests were based on the work of [Lindemann, 2019]. The results are shown in figure 6.6.4.
Figure 6.4: Error estimation stress delta plotted against the number of neurons using the SCG training algorithm with one HL (orange) and two HL (blue)
From figure 6.6.4, it can be seen that for this data set, the error behavior of neurons shows large fluctuations. These fluctuations may indicate unstability, which is undesirable when designing a network that is, to a certain extent, based on randomness. It appears that the two hidden layer network provides reasonable results in the stress delta error. This agrees with the work of [Lindemann, 2019], who mentions that in general, the number of hidden layers should equal the number of inputs. The low error for a 2 HL network is shown at N = 5, 10, 15. Due to the large fluctuations around the N = 10 and N = 15, it was decided to select N = 5 in a 2 HL network.
It can be concluded that the research of the number of hidden layers and number of neurons per layer is not extensive enough to draw reasonable results. Therefore, literature was consulted as well. Agreeing with the work by Lindemann [Lindemann, 2019] and figure 6.6.4, the number of hidden layers is set to two and the number of neurons per hidden layer is set to 5.
6.6.5 Transfer to MSC.Marc
The neural network generated using the code in appendix G can be used as the input in another MATLAB file, after which a Fortran file modified and used. Both files are written by dr. S. Solhjoo. After the user changes the necessary lines in order to allow the code in appendix G to function, the other files can be run without any modifications. Therefore, these files will be considered as a black box. It can then be concluded that the codes in appendix G provide a complete tool set for transfer to MSC.Marc.
34 6.6.6 Requirements to the ANN by MSC.Marc As MSC.Marc is a very flexible software, it does not set any requirements to the ANN. However, in order to perform the deep drawing simulation fully, a strain rate of 0.6 needs to be reached. Obviously, this requirement depends per project. 6.6.7 Accuracy, based on error estimation The delta error value for 5 neurons in a 2HL network equals 8.2e+06. From figure 6.6.4, it can be seen that this is below average compared to other combinations of hidden layers and neurons. From research, no definitive guidelines are given about which number of neurons to use per hidden layer. Therefore, it is assumed that the number of neurons estimation based on the errors provided in figure 6.6.4 is sufficient in further optimizing the neural network. 6.6.8 Average training time The training time of NN constructed in MATLAB equals 0 seconds. After running multiple training tests, the training times most often equaled 0 seconds, although sporadically equaled 1 second. It can thus be concluded that the training time using the SCG training algorithm in MATLAB in a small data set, is negligible. 6.6.9 User simplicity When constructing an NN based on a new data set, the input is required in an Excel file called ’RawData.xslx’. The MATLAB code as shown in appendix G then provides a complete tool in generating the ANN. Obviously, the input method and file name can be changed in the MATLAB code. The subsequent steps for integrating the ANN with the FEM analysis in MSC.Marc, require two codes that are considered black boxes, as the user only needs to press Run in order to use the codes. As at most the file name of the data needs to be changed in the first MATLAB code as shown in appendix G, it can be concluded that this method is very easy to use. 6.7 Discussion 6.7.1 Data distribution From the DAMASK simulations, the number of data points in the elastic region was 100 increments over a strain from 0.0 to 0.002. In the plastic region, 50 increments were generated over a strain from 0.02 to 0.47, after which the data was extrapolated to reach a strain of 0.6, resulting in 62 data points for the plastic region. This results in 5 000 data points per strain for the elastic region, and 100 data points per strain for the elastic region. This is necessary because the slope of the stress-strain curve changes very fast in every increment. This large difference in number of data points per strain confuses the feedforward function of MATLAB. In order to resolve this, the MATLAB function linspace, which generates a linearly spaced vector was used. This decreases the number of data points used in the NN training and, presumably, decreases the accuracy of the NN. In future research, it should be looked into whether this problem can be resolved by either improving the training function of MATLAB, or generating two NNs (elastic and plastic) and coupling these, for example.
35 6.7.2 Random selection of data within set The multilayer neural network training algorithms provided by MATLAB randomly select training, testing and validation data. Therefore, the performance of varies each time the algorithm is applied to a dataset. However, it was assumed that when selecting the desired training functions, the number of hidden layers and the number of neurons per layer, the behavior of several options varied sufficiently significantly in order to be able to draw conclusions. The neural network in this Integration Project could be optimized using a larger number of analyses which rectify the random behavior of a training algorithm in a data set.
6.7.3 Number of neurons optimization Currently, the NN is designed with two hidden layers, each consisting of five neurons per layer. Performance of the NN could be improved by varying the number of neurons per layer. Although this could be looked into, it is not very promising as it is not common in similar research. Therefore, in this Integration Project, the number of neurons per layer was the same in each layer.
36 Chapter 7: Finite Element Analysis using the ANN
DAMASK provides interfaces to two commercial FE solvers (MSC.Marc and Abaqus), as well as to a spectral solver using a Fast Fourier Transform (FFT). When comparing both solvers, they both show very similar results, as the underlying grain structure is hardly visible at this low resolution. [Roters et al., 2012] MSC.Marc, however, is more powerful than Abaqus and is provided to students for free. Therefore, it was decided to use MSC.Marc.
7.1 Comparison to previous methods
In Simulation of earing of a 17% Cr stainless steel considering texture gradients, Tikhovskiy, Raabe and Roters [Tikhovskiy et al., 2008] describe using a crystal plasticity finite element method for the simulation of cup drawing of a ferritic stainless steel sheet. It predicts the development of the orientation distribution and the earing profile during cup forming considering various slip systems. Similar to this Integration Project, the earing profiles are compared to FE results obtained by use of a Hill 48 yield surface. Additionally, the results are compared to experimental data. Furthermore, contrary to this Integration Project, Tikhovskiy et al. use spherical orientation components for texture approximation, instead of yield surface approximations. Tikhovskiy et al. conclude that texture component FE simulations using the gradient texture with a certain layer texture, fits the experimental data better than that obtained from the Hill 48 yield surface approximation. However, this method requires around nine times longer than the Hill-based simulations.
7.2 Process The process of performing the finite element analysis in MSC.Marc is as follows. 1. Integrating the data storage and MSC.Marc by either
• Generating the data table • Preparing the FORTRAN file 2. Setting up the finite element element mesh and geometric contact bodies. 3. Adding the material properties. 4. Adding boundary conditions. 5. Adding contact body properties. 6. Applying the load and submitting the job.
7.3 Parameter selection In this section, the parameters chosen in or affecting the deep drawing finite element simulation are explained.
37 Anisotropy modeling Anisotropic materials show a variation of both the Lankford coefficient and the yield stress with the orientation angle α [Meuwissen, 1995]. While crystal plasticity simulations include many of the details of the underlying deformation mechanisms, in many cases they are still computationally too expensive for direct inclusion into large-scale forming simulations. Therefore, simulations at the component scale are typically conducted using the concept of yield surfaces to model the transition from elastic to elasto-plastic deformation. [Roters et al., 2019] Modeling plastic anisotropy and implementing this into finite element simulations can be complex. Wu et al.[Wu et al., 2005] make the assumption that for practical purposes, plastic strains during sheet metal forming are moderate and the change of anisotropic properties during forming is small and negligible compared to the anisotropy induced by hot forming processes. This assumption has been widely adopted in the analysis of metal forming processes, and particularly of importance for industrial applications where user-friendliness and computation times are important factors to consider [Yoon et al., 2006]. As this is also the case in this Integration Project, it has been decided to use the concepts of anisotropic yield functions and isotropic hardening in FE simulations, as the results in previous research compare well to experimental data. [Worswick and Finn, 2000] Yield criterion Various concepts exist to introduce texture-related sheet anisotropy into finite element models for sheet forming. The material anisotropy existing before sheet deformation can be incorpo- rated either through an anisotropic yield surface function or directly via the use of crystal plasticity finite element models. [Raabe et al., 2005a] The best-known and most-used yield surface description for anisotropic behavior is the Hill48 yield surface. [Roters et al., 2019] [Hill and Egon, 1948] This Hill48 yield surface description is used in this Integration Project because of its simplicity and wide applicability. The Hill48 yield surface criterion is described in more detail in appendix A.11. Yield surface models concepts are used to introduce texture related sheet anisotropy into finite element models for sheet forming. The advantage of yield surface concepts for mechanical anisotropy predictions are relatively short calculation times, when implemented into finite element models. The main disadvantage of the yield surface concept is that they do not consider that the inherited sheet starting textures may evolve further in the course of sheet forming. [Raabe et al., 2005a] [Tikhovskiy et al., 2008] In 1948, Hill [Hill and Egon, 1948] proposed an anisotropic yield criterion as the generalization of the Huber-Mises-Hencky criterion for anisotropic materials. This yield criterion may be interpreted as a surface in a six-dimensional space of the stress components. [Banabic, 2016] It can be shown that the following relation between the yield stresses and the anisotropy coefficient applies. s σ r (1 + r ) 0 = 0 90 (7.1) σ90 r90(1 + r0) where r0, r90 are the anisotropy coefficients and σ0, σ90 are the yield stresses in the directions of the principal anisotropic axes. [Banabic, 2016]
In the MSC.Marc deep drawing simulation, r0, r45 and r90 are required in determining the
38 yield surface formulation. These values were directly outputted by DAMASK. Yield surface approximation Simulations at the component scale are typically conducted using the concept of yield sur- faces to model the transition from elastic to elasto-plastic deformation.[Hill and Egon, 1948] The advantage of yield surface concepts for mechanical anisotropy predictions are relatively short calculation times, when implemented into finite element models. [Raabe et al., 2005b] Accounting for the goal of this Integration Project, namely connecting the crystal plasticity simulations and finite element analysis in an efficient manner, these advantages are the main reason for using a yield surface approximation in this case. The yield surface is usually described as a convex analytic function in the six-dimensional stress space. The best-known and probably to date most-used yield surface description for anisotropic behavior is the Hill48 yield surface model. [Hill and Egon, 1948] Instead of conducting multiple time-consuming and expensive experiments, only a small number of very basic mechanical tests to calibrate a CP constitutive law have to be performed. Then, in lieu of doing all calibrations experimentally, a number of additional virtual tests are done on the basis of RVEs using an FEM solver. This approach has the additional advantage that many tests, which experimentally can hardly be done at all, are easy to perform in a simulation. [Roters et al., 2019] The Hill48 constitutive model used in this Integration Project is based on [Hill and Egon, 1948] and [Li et al., 2019]. Below, the yield function to determine the yield surface is shown to explain the mechanism. However, the actual computation of parameters and construction of the yield surface is performed in MSC.Marc. The Cauchy stress vector is denoted as σ11 σ22 σ33 σ12 σ23 σ13 and the Hill48 equivalent stress is written as
σ¯[σ] = p(P σ)σ (7.2) where σ¯ denotes the equivalent stress measure and P denotes the symmetric matrix, given by
1 P12 −(1 + P12) 0 0 0 P12 P22 −(P12 + P 22) 0 0 0 −(1 + P12) −(P12 + P22) 1 + 2P12 + P22 0 0 0 0 0 0 P33 0 0 0 0 0 0 3 0 0 0 0 0 0 3 where P12,P22,P33 describe the anisotropy of the yield surface. [Hill and Egon, 1948]. Hill’s equivalent stress is then used to describe the yielding and plastic flow of an anisotropic material.
39 Material The material used in the MSC.Marc simulations is Phenopowerlaw Aluminum, as used in the DAMASK simulations and described in listing F.1. However, not all parameters required for the MSC.Marc simulations, are provided in the file. The material property Young’s modulus can be calculated from the geometric properties mentioned in the bash file containing Phenopowerlaw’s material properties, shown in listing F.1. The material properties concerning the entries of the stiffness matrix are as follows
c11 = 106.75e9 c12 = 60.41e9 c44 = 28.34e9 As the material has an FCC structure, it holds that
c11 = c22 = c33, c12 = c23 = c31, c44 = c55 = c66
[Hill, 1952] In the elastic part of the deformation, isotropy is assumed. This is validated because it is assumed that the final deep drawn workpiece only consists of plastically deformed material. The information about elastic behavior of the material is given by Young’s modulus and Poisson’s ratio. Hill [Hill, 1963] describes a symmetric expression in the principal components of stress and strain for an aggregate which is macroscopically isotropic, using the bulk and rigidity moduli K and G, respectively, using the Voigt and Reuss theories. Voigt [Voigt et al., 1928] proposed a method for calculating the elastic moduli expressing stress in terms of the given strain, whereas Reuss [Reuss, 1929] proposed, on the other hand, the averaging of the relations expressing the strain in terms of given stress. It follows that
KR ≤ K ≤ KV (7.3) and
GR ≤ G ≤ GV (7.4) where suffixes V and R denote values calculated in the Voigt and Reuss theories. Using Voigt [Voigt et al., 1928] and Reuss [Reuss, 1929], the Young’s modulus can be estimated using the following equation
5Gv = (c11 − c12) + 3c44 (7.5) where cij denotes the entries of the stiffness tensor, Gv denotes the rigidity modulus calculated according to the Voigt theory.
40 Additionally, it holds that 5 = 4(s11 − s12) + 3s44 (7.6) GR where GR denotes the rigidity modulus calculated according to the Reuss theory. Then, the bulk modulus is defined as follows.
C + 2C B = 11 12 (7.7) 3
Then, using the Hill empirical average, the isotropic elastic shear modulus is as follows
G + G G = V R (7.8) VRH 2
Young’s modulus E and Poisson’s ratio ν can finally be calculated from the isotropic relations
9GB E = (7.9) 3B + G and
3B − 2G ν = (7.10) 2(3B + G) [Hill, 1952] In the case of this Integration Project, this resulted in
E = 70.35GP a and µ = 0.345
As mentioned before, this information dictates the material behavior in the deep drawing simulation during the elastic phase. During the plastic phase, the data storage method dictates the material behavior. Load case and job definition The load case and job definition in this Integration Project are exactly similar to the deep drawing tutorial provided by the MSC.Marc software. The boundary conditions of interest are as follows. The blank is constrained in the global Y-direction by contact and it has to be constrained by some kinematic boundary conditions to avoid any rigid body motions. Therefore, the displacement in x- and z-direction of certain nodes is prescribed. The load case time is set to 2 and default settings in MSC.Marc are used to perform the simulation.
41 7.4 Results In this section, the results of the deep drawing simulation using the finite element analysis in MSC.Marc are described. The results can be found in appendix I. Firstly, the NN and LUT methods of data storage will be compared. Secondly, the most optimal method will be evaluated and conclusions will be drawn.
7.4.1 Comparison of data storage methods in FEM results Chapter 6 describes the usage of the NN and LUT methods of data storage. In this section, the results will be compared. It can be seen that figures of earing behavior based on displacement in y-direction (figures I and I.4), equivalent plastic strain rate behavior (figures I and I.14) show very similar results. However, the figures of the reaction force (figures I and I.6), equivalent plastic strain behavior (figures I and I.10), and the equivalent Von Mises Stress behavior (figures I and I.16) show differences between the data storage methods. The force behavior in figures I and I.6 is the reaction to the punching force applied by the punch. The differences in reaction force behavior may be addressed to inaccurate force prediction of the LUTs based on inaccurate stress-predictions. The difference is especially visible in figure I.6. This is explained by the lack of strain rate information in table 1. Therefore, table 1 is assessed inaccurate when predicting the reaction force behavior of the sample. The differences in equivalent plastic strain behavior could be addressed to the method of selecting the nodes at which to evaluate the plastic strain. This was done using the same procedure as described in MSC.Marc’s deep drawing tutorial, and was deemed an accurate representation of the whole. As the differences do not show in figures I and I.8, and only show in detail at some nodes in figures I and I.10, the difference between the NN and LUTs is assessed insignificant. The differences in equivalent Von Mises stress behavior follow the same reasoning as for equivalent plastic strain behavior. Finally, it can be concluded that the LUT using table 1 and table 2 do not differ significantly, except for the behavior of the reaction force (figure I.6).
7.4.2 Interpretation and simulation using the ANN Firstly, figure I shows the workpiece before deformation as a circular rolled plate. Secondly, figure I shows the material after deformation. It can be seen that earing has occurred, which is also shown in figure I. Earing is attributed to an inhomogeneity of radial strain along the circumferential direction in a flange of a circular blank. [Kanetake and Tozawa, 1987] Earing is the result of non-uniform flow of material into the die cavity from different anisotropy directions of the sheet. [Singh et al., 2018] From both figures, it can be seen that four ears have formed, which is according to the expectations based on the fcc-crystal structure. [Kanetake and Tozawa, 1987] Furthermore, the height and location of the ears are shown in figure I. It is visible that the ears are symmetric, and the two ears on each side show the same size and shape. Thirdly, the concentration of the reaction force is shown in figure I. The reaction force reacts to the punch force, which is the sum of the friction, compression and bending force. It is shown that the behavior of the reaction force is symmetric as well. As the punch exerts the force on the bottom of the cup in order to perform the deep drawing process, the material is stretched
42 on this side of the material near the cup bottom. Therefore, the maximum values for reaction force are located on this part of the workpiece.
Fourthly, figure I and I show the equivalent plastic strain behavior. A gradual change of deformation is observed. It is shown that the cup wall experiences the highest plain strain under the tensile stress state, as this part of the workpiece is most closely wrapped over the punch. The earforming then accounts for the lower values between the high peaks in figure I.
Fifthly, the equivalent plastic strain rate behavior is shown in figures I and I. It is shown that for the largest part of the workpiece, the strain rate ˙ stays below the value of 4.55s−1). The similarities between strain rates in the locations of the workpiece explain why the performance using LUT 1 and 2 do not differ significantly. Furthermore, figure I shows the graph of the strain rate behavior along the arc of the workpiece. It can be concluded that figure I provides more valuable results when analyzing the behavior compared to figure I, as the strain rate behavior must be analyzed over the entire workpiece rather than along the arc of the workpiece only.
Finally, figure I shows the Von Mises stress behavior along the arc of the workpiece. The Hill 1948 yield criterion was extended from the Von Mises yield criterion, and is therefore the stress included in this analysis. From figure I.16, the differences between table 1 and 2 of including strain rate in the table can be clearly seen. Furthermore, as the Von Mises stress is shown along the arc of the workpiece, the minimum and maximum values of the Von Mises stress coincide with the earforming behavior shown in figure I.
7.4.3 Integration of the data storage method and MSC.Marc by means of the Fortran compiler
ANN The constructed Artificial Neural Network can be integrated in MSC.Marc using a Fortran compiler file. This file was written by dr. S.Solhjoo and only needs to be run once, without any modifications. Therefore, this was perceived as a black box. A step-by-step guide of the integration is provided in appendix B.
In general, the integration of the ANN and MSC.Marc was very fast (within a couple of minutes) and easy, as the files are already created and do not require any modifications. This is therefore a fast and efficient method of integration.
LUT The Look-up Table can be integrated in MSC.Marc using the modification of a *.txt file. This file was shown in appendix H. A step-by-step guide of the integration is provided in appendix B.
In general, the integration of the LUT and MSC.Marc required more preparation time (around 30 minutes per data set, depending on its size) and needs to be performed for every different data set. Also, it is required that the user has insight in how the data should be organized after generation. Therefore, it can be concluded that this is not an efficient or user-friendly method.
43 7.4.4 Deep drawing simulation As mentioned before, the tutorial in MSC.Marc called Ear profiles for a Cup Drawing Simulation with Barlat Yld2004-18p model is used in a modified form. The modifications concern the following aspects, which are elaborated below. Yield criterion In the tutorial, the Barlat Yld2004-18p model was used. However, in this Integration Project, the Hill48 model is used. Therefore, the yield coefficients have changed. These values were based on the r-values from DAMASK and equations 7.3 and 7.3. Whereas the r-values are easily read from the output files in DAMASK, the equations for Young’s modulus and Poisson’s ratio need to be used in order to calculate the necessary values for the yield criterion. Material properties The materials properties consist of the elastic properties (Young’s modulus and Poisson’s ratio)) and the yield criterion parameters, and the work hardening behavior. The Young’s modulus and Poisson’s ratio provide input for the elastic region of the stress-strain curve, whereas the ANN or LUT provides input for the plastic region and work hardening behavior. Previously, the work hardening behavior was based on the equation
646(0.025 + )0.227 (7.11) where represents the equivalent plastic strain. Now, the ANN provides the table which MSC.Marc uses as input. This input is shown in appendix H. MSC.Marc transforms this information into a curve that it uses in defining the work hardening of the material. 7.5 Discussion During the deep drawing simulations, the factors of main interest are the strain behavior and formation of ears. In this Integration Project, the formation of ears has only been addressed briefly. However, many more research opportunities exist, including the relationship between the earing pattern and the r-value, the earing angle, and the influence of texture components from various layers in the sample. Furthermore, the earing behavior using a different yield criterion (for example, Von Mises), could be included in the analysis. Predicting the earing behavior based on several conditions and parameters could be the scope of an entire new Integration Project and is therfore not considered within the current scope. Furthermore, several figures have been compared in appendix I. However, MSC.Marc provides many more parameters to compare. This could be adjusted to better suit the simulation and draw better conclusions. For example, the behavior of the reaction force as shown in figure I could be researched in more detail using various parameters, in order to draw conclusions on why these maximum values do not occur on four sides of the cup. Finally, during this Integration Project, the Deep Drawing tutorial provided in MSC.Marc has been used to generate the main concept of the simulation. MSC.Marc is a very powerful tool, and has not been used to its full capacity in this project. Therefore, this Integration Project could be improved by analyzing the various options MSC.Marc offers in performing different simulations, better suitable to a desired phenomenon. However, it appeared that the MSC.Marc software is rather unknown, and the user will not receive much guidance from other sources.
44 Chapter 8: Finalized methodology In this section, the finalized methodology is described, based on chapters 5, 6, and 7.
8.1 Comparison to previous methods In previous research, many projects have been performed integrating an ANN and an FEM solver. However, no literature is available about the usage of MSC.Marc as the FEM solver. More often, ABAQUS is used. However, MSC.Marc is a very powerful and promising software and this novel integration is worth researching. Furthermore, the ANN optimization differs per research as the data set and parameters are different in each case.
8.2 Process The steps that need to be undertaken in the various phases of the integration are as follows. DAMASK 1. Create RVEs for rolled and unrolled metal grains, under three different strain rates 2. Perform two more simulations under rolling angles 45◦ and 90◦ for finding the anisotropic behavior of the rolled RVE (performing tensile tests on rotated RVEs 3. Determine stress-strain curves 4. Determine the Lankford coefficient 5. Determine yield values to determine the yield surface ANN 1. Determine the desired type of neural network. 2. Determine the ratio of training data vs. testing data 3. Determine the desired error estimation. 4. Determine the desired number of hidden layers and number of neurons per layer. MSC.Marc 1. Integrating the data storage and MSC.Marc by either preparing the FORTRAN file 2. Setting up the finite element element mesh and geometric contact bodies. 3. Adding the material properties. 4. Adding boundary conditions. 5. Adding contact body properties. 6. Applying the load and submitting the job.
45 8.3 Results The results sections of chapters 5, 6, and 7 provide the assessment on the described methods for each phase in the integration. In this section, all the aforementioned results are coupled and summarized.
8.3.1 Accuracy From section 6.6, it was concluded that an ANN with the described configurations achieves a higher accuracy than a LUT that has been done before. This is also shown in section 7.4 based on the reaction force behavior.
8.3.2 Decrease in computational times The computational times with and without the use of an ANN are mentioned in table 5.5. The DAMASK process is the largest contributor to the computation times, even though it was not able to finish all simulations due to convergence errors. In the novel method of generating the data, no DAMASK simulations need to be performed, as the NN provides the input to MSC.Marc. As the NN generation takes less than one second and the codes are already available, the only time-consuming task would be to adjust the parameters in MSC.Marc to meet the user’s requirements. Depending on the user’s knowledge about MSC.Marc, this process may range from 15 minutes to one day. However, this is also the case when performing the simulations in the previous method. Therefore, it can be concluded that the simulation times using this novel method is decreased by approximately the same amount of time required to run the DAMASK simulation, shown in figure 5.5.
8.3.3 Simplicity for the user A large part of the complexity for the user in the previous method was preparing the DAMASK simulations and providing the correct table input in MSC.Marc. As this method is eliminated, the user simplicity has increased significantly. The data is readily available for the user, and only knowledge about MSC.Marc is required, which it also was in the previous method.
46 Chapter 9: Validation The validation process of each phase in the integration process is discussed below.
9.1 Crystal plasticity simulations in DAMASK Currently, the crystal plasticity simulations have only been performed once. In order to validate the results, multiple simulations can be run. It may be the case that with a different *.seeds file, the final outcome of the crystal plasticity simulation differs. Additionally, various software is available in which crystal plasticity of metals is simulated. Some methods are described in chapter 2. These methods could be used to validate the results generated by DAMASK. Finally, experiments could validate the DAMASK simulations, although this may not be desired due to the large costs and effort.
9.2 Data storage using the ANN In this Integration Project, the ANN was optimized using input from literature as well as testing. The optimality of the ANN can be validated by conducting more experiments and testing the networks.
9.3 FE analysis in MSC.Marc Multiple software tools facilitate finite element analysis, including ABAQUS, COMSOL, and SolidWorks. These tools, as well as experiments, can be used to validate the accuracy of the MSC.Marc analysis. Furthermore, another yield criterion, for example Von Mises, can be compared to the Hill 1948 yield criterion.
9.4 Integration of these methods The accuracy of the method can be validated as described in the above sections. The user- simplicity of the integration of these methods can be validated by selecting several users who will perform this novel method of integration. This user group may, for instance, be composed of people interested in the deep drawing simulation by FEM, who have worked with similar software before and have background knowledege about the topic. These users provide feedback, which can be used validating the statements about user-simplicity. Furthermore, these users may also assess computation times.
47 Chapter 10: Conclusion In this Integration Project, the microscale deformation simulation in DAMASK and the macroscale deep drawing simulation in MSC.Marc are linked through a new integration method using an ANN. The ultimate goal of presenting this novel method was to provide an accurate method, with simple usage, at low computation times. It was concluded that this novel method presented fulfills these requirements completely. The user simplicity has increased significantly as the preparation process in DAMASK has been eliminated in the novel method. Additionally, when using the ANN, the user simplicity increases as the input in MSC.Marc is generated within two clicks using predesigned, generically- applicable files. In the LUT method, the data input in MSC.Marc needs to be rearranged a certain way, and various parameters need to be changed. Moreover, eliminating the preparation process in DAMASK decreases the computation times significantly. This was a very time-consuming process, and is now predicted using an ANN. In the LUT method, the DAMASK simulation is eliminated as well, although the preparation of the LUT input in MSC.Marc does require preparation time. Therefore, the ANN method requires the lowest computation time. Constructing the ANN and preparing the user subroutine are performed in less than five seconds. Therefore, this computation times is neglected. However, time is required to prepare the ANN and MSC.Marc configuration. The time required depends on the user’s knowledge about these software, but is assumed to be a minimum of five minutes. Therefore, it was concluded that the computation times compared to the previous methods, which require DAMASK simulations ranging from 6 to 12 hours each, decrease significantly when incorporating the new method. Finally, accuracy was shown in comparing the LUT and ANN data storage method in the deep drawing process in MSC.Marc. It was found that the results of these methods show the largest differences in the reaction force behavior. It was concluded that table 1 yields inaccurate results, whereas table 2 provides sufficiently accurate results. The ANN, however, is most accurate. In the DAMASK simulations, it was concluded that the rolled RVE at the strain rate of ˙ = 0.01s−1 consists of less-squished grains compared to the strain rates of ˙ = 0.1s−1 and ˙ = 1s−1. Moreover, it shows that the two larger strain rates show roughly the same RVE. It was concluded that after a certain value, the strain rate does not affect the rolled RVE when reaching the same final strain. Furthermore, the rolled RVEs of 0◦ and 90◦ are exactly the same, although rotated. The RVE of 45◦, on the other hand, shows more granular grain edges. It was concluded that only the 45◦ degree angle rotation affects the RVE. The stress-strain curves under various strain rates do not show significant results. It appears that the unrolled and rolled stress-strain curve follow roughly the same shape. Therefore, no conclusions can be drawn from the stress-strain curves. Finally, the Lankford coefficient under the various angles is analyzed. It was concluded that
48 although it is expected that R45 is the lowest value, and R90 is the highest value, this is the opposite. In the data storage phase, it was concluded that the ANN used in this Integration Project is optimized using the SCG-training algorithm on a feedforward backpropagation NN, consisting of two hidden layers containing five neurons in each layer. Additionally, it was concluded that the delta error should be used instead of MSE. Furthermore, in the FEM phase, the results of the deep drawing simulated were discussed. The anisotropy of the material was included using the Hill 1948 yield criterion. It was found that four ears have formed, which is as expected based on the fcc-structure. Two ears on each side show the same size and shape. The reaction force to the punch force is symmetric as well, and shows the maximum value near the cup bottom. On the other hand, the equivalent plastic strain behavior appears as a gradual change of deformation throughout the workpiece. The wall of the cup experiences highest plain strain. The formation of ears affects the graduality of the plastic strain behavior on the cup wall. Moreover, it was shown that the train rate ˙ = 0.01s−1 is most present in the deformed workpiece. This explains the ability of table 1 to predict accurate results in most cases. However, regarding the Von Mises stress behavior along the arc length, table 1 does not yield accurate results. This is explained by the formation of ears around the arc, which the table shows resemblance with. To summarize, the deep drawing process provides insights on the deformation of the workpiece. It is finally concluded that this novel method of integrating crystal plasticity simulations in DAMASK and FEM in MSC.Marc by means of an ANN, improves the simulation results based on accuracy, user simplicity, and computational times.
49 Chapter 11: Discussion As the different steps in the integration have been discussed in the corresponding chapter, the finalized methodology will be discussed in this chapter. Firstly, it must be noted that this Integration Project has a rather wide scope, and therefore, not all phases could be explored in much detail. For this reason, according to the Rigor cycle literature has been a useful tool in fastening this process by using information that was readily available. The finalized methodology is superior to the current method based on accuracy, computational times, and simplicity. However, this method could be extended in future research beyond the scope of this Integration Project using the following improvements. Firstly, the model could be extended from 2D to 3D simulations in DAMASK. This generates a larger amount of data that needs to be incorporated into the NN. However, it may be discussed whether the extension to 3D improves the accuracy of the method, as it certainly considerably increases the computation times. Secondly, the model could be extended by adding temperature conditions to the simulations. This has been done in previous research and promises interesting results. Finally, in for all phases in the Integration Project, the student versions of DAMASK, MATLAB, and MSC.Marc were used to save effort and costs. In MSC.Marc, it appeared that this resulted in some limitations. This may also be the case for DAMASK and MATLAB. Therefore, in optimization of the integration in this project, it may be considered to use the non-student versions of the software.
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56 Appendix A: Theoretical background A.1 Crystal plasticity and anisotropy Anisotropy, in Material Science, is a material’s directional dependence of a physical prop- erty. [Kocks, 2000] A consequence of crystalline anisotropy is that the associated mechanical phenomena such as shape change, crystallographic texture, strength, strain hardening, defor- mation induced surface roughening, and damage are also orientation dependent.[Roters, 2011] In such a case, anisotropy could be effectively measured directly from its stiffness tensor independently of its origin which may for instance be its texture, randomness of internal composition or defects. [Sokolowski and Kaminski, 2018] Texture patterns are often produced during manufacturing of the material. In the case of rolling, ”stringers” of texture are produced in the direction of rolling, which can lead to vastly different properties in the rolling and transverse directions. Metals tend to be more isotropic, though they can sometimes exhibit sig- nificant anisotropic behaviour. This is especially important in processes such as deep-drawing. [Doherty et al., 1997] Important properties in anisotropy are grain properties and grain orientation. It is assumed that during and after the forming and hardening of the material, the grains will recrystal- lize. During recrystallization, the grain orientations will restore to the random orientation. [Doherty et al., 1997] This means that in order to control the anisotropy of the RVE, the parameters of the grain have to be changed. During sheet metal forming, anisotropic properties of a material usually exhibit two different forms. One is concerned with the hardening behavior when measured along two different directions on the plane of the sheet, meaning the relation- ship of the stress and strain is different in different directions due to the anisotropic properties of the material. [Hu and He, 2019] During rolling of the steel, the thickness is reduced while the length is increased. This is also visible within the material; grains will deform and stretch. In order to create anisotropy within the RVE, the grains are stretched along the three main axes, xx, yy, and zz.[Doherty et al., 1997]
A.2 DAMASK DAMASK is a simulation kit for the simulation of crystal plasticity within a finite-strain continuum mechanical framework [Roters et al., 2012]. DAMASK is designed to reproduce the multi-scale structure inherent to the underlying material physics of materials deforma- tion, bridging the anisotropy introduced at the atomic scale to the global field description. [Roters et al., 2019] DAMASK is designed to provide a flexible and easy to use crystal plas- ticity implementation. Due to its highly modular character it is very flexible and can easily be extended, e.g. by adding additional constitutive laws, homogenization schemes, or solver interfaces. [Roters et al., 2012] The structure of the material point model reflects the multiscale character of continuum mechanical boundary value problems, as shown in figure A.2. In [Ma et al., 2006], Ma et al. state that ”A wide variety of currently available crystal plasticity models using finite element methods (CPFEM) can be classified into two major types. In the first type, slip resistances are used as internal state variables. In the second, dislocation density/densities are used as internal state variables. The major difference between the two is
57 Figure A.1: Structure of the material point model that the latter involves evolution of dislocation densities explicitly in the framework of the model.” In the past, DAMASK has been used inside an FEM solver, as shown in figure 3.1a. The virtual laboratory is conveniently implemented as a part of DAMASK in form of an automated script. This script launches virtual tests with automatically generated load cases for the considered RVEs, post-processes the results and finally performs the parameter identification for all selected yield functions until the desired fitting quality is achieved. [Zhang et al., 2016] In this Integration Project, however, DAMASK was used in an integrated chain with an FEM solver, as shown in figure 3.1b. The possibilities of DAMASK as a crystal plasticity simulator remain the same. Roters et al. [Roters et al., 2019] elaborately describe the possibilities of the Dusseldorf Advanced Material Simulation Kit (DAMASK), aiming to provide a highly modular and straightforward implementation of different types of constitutive laws and numerical solvers. In this Integration Project, not all of DAMASK’s features are used.
A.3 Plastic deformation behavior Plastic deformation is the permanent distortion that occurs when a material is subjected to tensile, compressive, bending, or torsion stresses that exceed its yield strength and cause it to elongate, compress, buckle, bend, or twist. [Pfeifer, 2009] High-speed tensile tests have shown that the yield stress increases with the strain rate, and this effect is more pronounced at elevated temperatures. The true strain rate in simple compression is defined as h˙ ˙ = − (A.1) h where h is the current specimen height and h˙ its rate of change. For most metals, the transition from elastic to plastic behavior is gradual, owing to successive yielding of the individual crystal grains. The location of the yield point is therefore a matter of convention. The corresponding stress Y , known as the yield stress, is generally defined as that
58 for which a specified small amount of permanent deformation is observed [Chakrabarty, 2006]. In this Integration Project, the yield stress will be determined at a strain value 0.02%. As the specimen continues to elongate under increasing load beyond the yield point Y , its cross-sectional area decreases permanently and uniformly throughout its gage length. As the load is further increased, the curve eventually reaches a maximum and then begins to decrease. the maximum stress is known as the ultimate tensile strength (UTS). Ultimate tensile strength is thus a simple and practical measure of the overall strength of a material. [Kalpakjian and Schmid, 2008]
A.4 Stress-strain curves True stress is defined as P σ = (A.2) A where P equals the force applied to the material perpendicularly, and A equals the actual area supporting the load. The tension test is the most common test for determining the strength-deformation char- acteristics of materials. With increasing load, the specimen begins to yield; undergo plastic (permanent) deformation, and the relationship between stress and strain is no longer linear. The usual practice is to define the yield stress as the point on the curve that is offset by a strain of 0.2%. As the specimen continues to elongate under increasing load beyond yield stress Y , its cross-sectional area decreases permanently and uniformly throughout its gage- length. The maximum stress is known as the ultimate tensile strength (UTS) of the material. [Kalpakjian and Schmid, 2008]
A.5 Representative volume elements The Representative Volume Element (RVE, also called unit cell) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. [Hill, 1963] This RVE will be obtained in the DAMASK simulations. Representative Volume Element simulations are of high interest for application in the so-called ”virtual laboratory” [Kraska et al., 2009], where they are used to replace extensive experimental testing. DAMASK provides interfaces to MSC.Marc, among others. [Roters et al., 2012] The RVEs obtained in this Integration Project are shown in appendix C.
A.6 Lankford coefficient The Lankford coefficient (also called Lankford value, r-value, or plastic strain ratio) is a measure of the plastic anisotropy of a rolled sheet metal. [Lankford et al., 1950] The sheet- metal industry uses Lankford coefficients as standards for characterizing a sheet’s ability to be stretched and deep drawn. [Charca Ramosa et al., 2010] In engineering applications the r-value is measured at 20% strain. [Roters, 2011] [Banabic, 2016]. The Lankford coefficient r is defined by
r = xx (A.3) zz
59 where xx is the strain in the width-direction, and zz is the strain in the thickness-direction.
However, the strain in the y direction zz can be assumed to equal practically zero, as the simulation is based on a 2D representation. Then, the strains in all directions add up to zero, by the following equation.
xx + yy + zz = 0 (A.4) where yy equals the strain in the tensile direction. Combining these two equations yields the following equation for the Lankford coefficient.
− r = xx (A.5) xx + yy
A.7 Hill 1948 yield criterion [?] Yield surface models concepts are used to introduce texture related sheet anisotropy into finite element models for sheet forming. The advantage of yield surface concepts for mechanical anisotropy predictions are relatively short calculation times, when implemented into finite element models. The main disadvantage of the yield surface concept is that they do not consider that the inherited sheet starting textures may evolve further in the course of sheet forming. [Raabe et al., 2005a] In 1948, Hill [Hill and Egon, 1948] proposed an anisotropic yield criterion as the generalization of the Huber-Mises-Hencky criterion for anisotropic materials. This yield criterion may be interpreted as a surface in a six-dimensional space of the stress components. [Banabic, 2016] It can be shown that the following relation between the yield stresses and the Lankford coefficient applies. s σ r (1 + r ) 0 = 0 90 (A.6) σ90 r90(1 + r0) where r0, r90 are the anisotropy coefficients and σ0, σ90 are the yield stresses in the directions of the principal anisotropic axes. [Banabic, 2016]
A.8 Data storage methods In this section, the various methods of data storage are described.
A.8.1 Artificial neural network In ”Foundations of Machine Learning”, Mohri defines machine learning as ”computational methods using past information to improve performance or to make accurate predictions.” [Mohri et al., 2018] [Lindemann, 2019]
The basic mechanism of an ANN is shown in figure A.8.1. The inputs xi on the left define the input nodes, parameters yi on the right define the output node. The middle layer is called the ”hidden” layers, as they have not an input nor an output. Each input is multiplied by a
60 Figure A.2: A basic representation of a neural network [Lin et al., 2008]
random weight wij or wjk and the products are summed together with a constant θ to give the output
y = Σiwixi + bi (A.7) where bi describes the preceptron’s bias, a value between 0 and 1 [Nielsen, 2015]. The summa- tion is an operation which is hidden at the hidden node. Since the weights and the bias bi were chosen at random, the value of the output will not match with experimental data. The weights are systematically changed until a best-fit description of the output is obtained as a function of the inputs; this operation is known as training the network. [Bhadeshia, 1999] In a feedforward neural network, the output from one layer is used as input to the next layer. There are no loops in the network, as shown in figure A.8.1. [Nielsen, 2015] Backpropagation involves comparing the output a network produces to the output it was meant to produce, and using the difference between them to modify the weights of the connections between the units in the network, working from the output units through the hidden units to the input units—going backward. In time, backpropagation causes the network to learn, reducing the difference between actual and intended output to the point where the two exactly coincide, making the network more accurate. [Nielsen, 2015]
A.8.2 Lookup table
A lookup table is an array or matrix of data that contains items that are searched. Lookup tables may be arranged as key-value pairs, where the keys are the data items being searched (looked up) and the values are either the actual data or pointers to where the data are located. [McNamee, 1998] A lookup table allows a computer to approximate a function or equation without performing advanced mathematical operations. [Campbell-Kelly et al., 2003] The main advantage of a lookup tables is the speed, as it may take less time to look up a value from a matrix than to calculate the number. However, the main disadvantage is a lookup
61 table’s memory usage. It is possible to be required to store numbers that may not be relevant. [O’Dwyer and Martin, 2006] It can be concluded that a lookup table is most suitable for smaller tables with less data, in methods where the values cannot easily be computed.
A.8.3 Fitted function A fitted function is a mathematical expression that describes the interrelated behavior of parameters. There may or may not be a physical meaning behind the function. The main advantage is its simplicity, although the disadvantages include the possibilities for large errors and computation times. A.9 Finite Element Method The finite element method (FEM) is a numerical method for solving problems of engineering and mathematical physics. The analytic solution of these problems generally requires the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations. To solve the problem, the method subdivides a large system into smaller, simpler parts, that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. [Logan, 2011] According to Roters et al., [Roters et al., 2010a], the finite element method (FEM) is nowadays without doubt the most popular simulation tool in structural mechanics. It allows users to incorporate various mechanisms, although it also adds complexity to the model, as the use of different competing crystallographic deformation mechanisms within a CPFE model requires the formulation of local homogenization rules. This means that at some material points, only one type of deformation mechanism may occur, whereas at others, a mix must be considered within the same material volume. Furthermore, the possibility of deformation martensite or twins may increase complexity. In a different publication, Roters et al. [Roters et al., 2010b] describe combining the CPFE method and FE method, in order to deal with macroscopicmechanical materials problems such as encountered in metal forming, tool design, or process engineering. They state that ”the particular strength of the FE method lies in studying the influence of boundary conditions on mechanical or microstructural predictions.
A.9.1 MSC.Marc Marc is a powerful, general-purpose, nonlinear finite element analysis solution to accurately simulate the product behavior under static, dynamic and multi-physics loading scenarios. Marc is ideal for product manufacturers looking for a robust nonlinear solution. It has capabilities to elegantly simulate all kinds of nonlinearities, namely geometric, material and boundary condition nonlinearity, including contact. It is also the only commercial solution that has robust manufacturing simulation and product testing simulation capabilities, with the ability to predict damage, failure and crack propagation. [MSC, 2019] Javanbakht and Ochsner¨ [Javanbakht and Ochsner,¨ 2017] describe the general-purpose finite element program MSC.Marc, and provide a comprehensive introduction to the features of the software. In this Integration Project, not all of MSC.Marc’s features will be used.
62 A.10 Strain rate dependence subsec:elasticdef According to Kalpakjian, [Kalpakjian and Schmid, 2008] ”the usual practice is to define the yield stress as the point on the curve that is offset by a strain of 0.2% or 0.002”. Therefore, the elastic region will be restricted to a strain of 0.002. The plastic region has been restricted to a strain of 0.6, due to requirements in MSC.Marc. Furthermore, Kalpakjian [Kalpakjian and Schmid, 2008] describes that increasing strain rate increases strength of the metal. However, these differences are considerably small in low strain rates.
A.11 Deep drawing In this Integration Project, the deep drawing is chosen as the forming simulation that the project will finally be based on. However, any forming process could be chosen after all. The deep drawing process is often a combination of stretch forming and deep drawing. At the beginning of the forming process, when the punch moves downwards and form out the bottom, a general stretch forming process results. When the bottom is formed, the deep drawing process follows, characterized by transfer of the drawing force from the punch through the cup wall into the flange. Within the flange results the main forming process with radial and tangential compression loads. Figure A.11 shows the whole process of deep drawing, including the stretch forming and deep drawing processes.
Figure A.3: Stretching and deep drawing
During the deep drawing process, different stress conditions result in the material. Figure A.11 shows a drawn cup with the different loading zones. The zones of interest are the flange, the side wall (cup wall), and its bottom. The stress condition in the flange area is a tensile load in radial direction and compression load in tangential direction. The stress conditions determine the forming process at deep drawing. The important stress condition in the cup wall is the point of transition of the punch radius to the cup wall area. This loading zone is characterized by a plane strain load, whereas the cup bottom has a biaxial stress load. The cup bottom and its biaxial stress load determine the stretch forming process. [Doege et al., 2001].
63 Figure A.4: Stress-conditions in a drawn cup.
64 Appendix B: Procedures
In this chapter, the procedures as described in figure B is described more elaborately.
Figure B.1: System description, including the software
As mentioned before, the steps that were followed to simulate the crystal plasticity are as follows. This will be explained in more detail in section B.1. 1. Download the DAMASK software. 2. Determine the required strain rates and simulation times. 3. Generate the general seeds file 4. Write the preprocessing code that will perform the simulation. 5. Visualize the RVEs generated in the simulation. 6. Perform two more simulations under rolling angles 45◦ and 90◦. Determine stress-strain curves and the Lankford coefficient. The steps that were followed to generate the ANN are as follows. This will be explained in more detail in section B.2. ANN 1. Determine the desired type of neural network. 2. Determine the ratio of training data vs. testing data 3. Determine the desired error estimation. 4. Determine the desired number of hidden layers and number of neurons per layer.
65 The steps that were followed to perform the finite element analysis are as follows. This will be explained in more detail in section B.4. 1. Download the MSC.Marc software 2. Setting up the finite element element mesh and geometric contact bodies, material properties, boundary conditions, contact body properties. 3. Integrating the subroutine 4. Applying the load and submitting the job. The steps that were followed to integrate the ANN with FEM are as follows. This will be explained in more detail in section B.3. 1. Generate the FORTRAN file
66 B.1 DAMASK The different steps will each be explained in one section below. As the preparation of the DAMASK files is not intuitive and requires careful operation, the steps are described in much detail. In general, the handout http://materials.iisc.ernet.in/praveenk/CrystalPlasticity/ Pe_n_DAMASK.pdf [Eisenlohr, 2015] by Eisenlohr and the tutorials by Kumar [Kumar et al., 2018] describe a code similar to this Integration Project in more detail, and can be used in most steps described below.
B.1.1 Download the DAMASK software Although it will be used in a different method compared to traditional DAMASK usage, as shown in figure A.2, the DAMASK software can be freely downloaded from their website https://damask.mpie.de/Download/WebHome. A Linux operating system is required to run the software. In general, by ”running” a code in the novel method of this Integration Project, compared to previous DAMASK usage, the following steps are considered. 1. The bash file needs to be written/modified for the desired situation. This is done by double-clicking the file, modifying it, and saving it. 2. The terminal can be opened in a desired folder by pressing CTRL + ALT + T or by clicking the right mouse button and selecting ”Open Terminal”. 3. In the terminal, the modified bash file can be made executable by typing chmod +x
B.1.2 Determine the required strain rates and simulation times The required strain rates depend on the requirements of the specific experiment. In this Integration Project, a strain of 0.2 for the elastic region, and 0.6 for the plastic region, have been used. After the strain rates have been determined, the required strains need to be determined. With this information, the simulation times can be computed using the following equation for engineering strain.
t = (B.1) ˙ where t equals the time in [s], ˙ equals the strain rate in s−1, and equals the engineering strain [−].
B.1.3 Generate the general seeds file One general seeds file will be used to generate all simulations under different strain rates and angles. This seeds file *.seeds can be generated using the code in listing E.1.1. The terminal needs to be opened in the desired folder containing all files, after which the file needs to be run.
67 B.1.4 Write the preprocessing code that will perform the simulation The DAMASK codes that can be used to perform the simulation as performed in this Integration Project, are shown in listing E.1.3. In this case, the description will be mainly based on the simulation as performed in this Integration Project. Additionally, the website https://damask.mpie.de/Documentation/WebHome describes all possibilities of DAMASK. Firstly, the so-called preprocessing bash file needs to be run. This is shown in section E.1.3 for the unrolled specimen, and section E.1.4 for the rolled specimen. All lines of code are explained on the DAMASK website, https://damask.mpie.de/Documentation/WebHome. Most importantly, the lines echo "Fdot 0.025 0 0 0 * 0 0 0 * stress * * * * 0 * * * 0 time 4incs 100 freq 1" >> tension.load and echo "Fdot 0.025 0 0 0 * 0 0 0 * stress * * * * 0 * * * 0 time 12 incs 50 freq 5 dropguessing" >> tension.load describe the elastic and plastic behavior of the specimen, respectively. The numbers indicate the deformation gradient rate tensors. A zero indicates a required strain of zero at that entry in the tensor, whereas a * denotes no restrictions for a value, which may be chosen in the convenience of the DAMASK simulation. The number after time in these lines indicates the time in sections that the simulation needs to occur for in the elastic and plastic region, respectively. The number of increments and the frequency can be changed in order to achieve faster simulation times. This is explained in detail on the DAMASK website, https://damask.mpie. de/Documentation/LoadDefinition. Furthermore, suggested changes in order to yield the desired results are provided on this website. The two different bash files for the unrolled and rolled case are provided. In the rolled case, the rescaling operations, as discussed in chapter 5 have been added. The files are run by opening the terminal in the desired folder, sourcing DAMASK, and running the bash file.
B.1.5 Visualize the RVEs generated in the simulation The RVEs can be visualized by running the following lines of code. seeds check 40 grains.seeds geom check 40grains.geom where 40grains.seeds and 40grains.geom may need to be changed into a new name for a specific simulation. The exact behavior of the codes is explained in https://damask.mpie.de/ Documentation/WebHome and the handout by Eisenlohr, https://materials.iisc.ernet. in/praveenk/CrystalPlasticity/PE_N_DAMASK.pdf [Eisenlohr, 2015]. These lines generate a *.vtr and *.vtk file, respectively. Then, the ParaView platform can be downloaded freely from https://www.paraview.org/. After installation, the resulting *.vtr file can be transfered to and opened in ParaView. The specific settings for visualization in ParaView are described in Eisenlohr’s handout [Eisenlohr, 2015].
68 B.1.6 Write the postprocessing code to conclude the simulation The postprocessing bash file is described in listing E.1.5. Nothing needs to be changed in this file when applying a different load or a rolled or unrolled case. The file is run by opening the file terminal in the desired folder, sourcing DAMASK, and running the file.
B.1.7 Perform two more simulations under rolling angles 45◦ and 90◦ The simulations to yield results under the rolling angles 45◦ and 90◦ have been performed in a different way than described above, namely following a MATLAB code, written by dr. S. Solhjoo. The bash file that needs to be run, is shown in listing E.1.6. In short, its main task is calling the MATLAB function that generates the RVEs. This MATLAB code is shown not included in this report, due to its large size and because it is treated as a black box, as nothing needs to be changed. Line 1 of E.1.6 provides the input for the MATLAB file. The input is threefold: the first input describes the geometry file that will be used, the second input is the loading file, which describes the strain rates and simulation times, and the third input describes the number of different angles, separated equally between 0◦ and 90◦. The output of the rolling simulations under different angles is posted into a folder named ”master results” and separate folders for the various angles, named ”sample x”, where x denotes a number between 1 and the number of rotations entered in the bash file E.1.6. In this bash file, the following files or functions are called. • 10grains.geom calls the geometry file that was generated from the seeds file used in all simulations. • tensionX.load calls the load file that was used to generate the load file describing the simulation time, rate, number of increments and frequency. • 10 is the parameter describing the number of rotations between 0◦ and 90◦. This can be any desired integer. When desiring to visualize the generated RVEs, the same steps as described in section B.1.5 should be followed.
B.1.8 Determine stress-strain curves and the Lankford coefficient Within the Post Proc folder, a *.txt file is generated. This file needs to be transferred to an *.xlsx file. The output that can be read from the *.xlsx file is described in subsection 5.4.3. The columns of interest differ per specific project. In this Integration Project, the parameters of interest were the strain in the xx-direction, xx, and the stress in the xx-direction, σxx. These were depicted by 1 ln(V) and 1 Cauchy in columns AA and AJ, respectively. These columns can be selected in Excel and be used to generate a scatter plot, as described in the Microsoft Office help web page, https://support.office.com/en-us/article/ present-your-data-in-a-scatter-chart-or-a-line-chart-4570a80f-599a-4d6b-a155 -104a9018b86e. Secondly, the Lankford coefficient, as determined in chapter A.11, is determined at a strain of 20%. From the Excel file that is generated from the *.txt file in the Post Proc folder, the specific row that contains the strain in tensile direction (xx) needs to be found. Then, the
69 Lankford coefficient can be calculated by equation A.6 in the specific row. In this Integration Project, this was increment 118.
B.2 Artificial neural network The methods of constructing an ANN have elaborately been described by Lindemann [Lindemann, 2019]. Therefore, this is not elaborated on in this section. Additionally, the MATLAB helpdesk provides various examples, including https://nl.mathworks.com/help/ deeplearning/ref/trainscg.html and https://nl.mathworks.com/help/deeplearning/ ref/trainscg.html.
B.3 Integration of the ANN with FEM The only step in this integration involves running the predesigned files without any modifica- tions. Therefore, this is only described briefly.
B.3.1 Generate the FORTRAN file After the NN has been constructed, it needs to be saved in the directory in which the B read netFcn ss3.m and NN Funcs.f are also saved. These files automatically recognize the NN output by MATLAB. The first file, B read netFcn ss3.m simply needs to be run. The second file, NN Funcs.f needs to be run on a pc that has the FORTRAN compiler and Microsoft Visual Studio installed on it. It is important to first install Visual Studio 2017 with the C++ compiler option checked. After that, FORTRAN can be installed. After the files have been run, the FORTRAN file NeuralNet.f is generated, which provides the subroutine input for MSC.Marc.
B.4 Finite element analysis B.4.1 Download the MSC.Marc software The MSC.Marc student version can be downloaded freely for students. Therefore, an installation request needs to be submitted through the MSC.Marc website using a Student University Card. The request is considered, which may take a couple of days. After this, a confirmation e-mail is sent containing a download link.
B.4.2 Setting up the finite element element mesh and geometric contact bodies, material properties, boundary conditions, contact body prop- erties Currently, little help about the usage of MSC.Marc is available online. Therefore, the easiest method of setting up the basic properties of the simulations are by running the tutorial provided in MSC.Marc help. This tutorial can be found under Help/User’s guide/e112. This provides a step-by-step guide of setting up the properties. On the bottom of the page, a play button is provided, which immediately applies all guidelines mentioned above. This is a safe option in ensuring that the simulation is defined properly. Then, it is important to apply the new material properties before submitting the job. This can be done by double-clicking the default material FM8. The material properties, in case of this Integration Project, can be calculated using equations 7.3 and 7.3. Furthermore, the plasticity
70 properties need to be defined according to the r-values. After inserting the experimental data, MSC.Marc computes the required data automatically.
B.4.3 Integrating the subroutine Before submitting the job, the user subroutine should be defined to be NeuralNet.f. The way to do this is elaborately described in the MSC.Marc tutorial.
B.4.4 Applying the load and submitting the job The load used in this Integration Project was the default load from the tutorial. However, this can be changed according to the requirements of the project. Before submitting the job, any errors can be checked by pressing Check. However, if any errors occur, it is unclear what the user should change exactly. This requires careful adjustments of the default tutorial. The tutorial also describes how to submit the file using the desired output parameters.
71 Appendix C: Representative Volume Elements for several strain rates In all cases, an aluminum grain with a grid of 64 x 64 x 1 and 40 random seeds was used.
Figure C.1: The RVE of an unrolled material
Figure C.2: The elongated unrolled RVE
72 Figure C.3: The RVE of a rolled material under a strain rate ˙ = 0.01 s−1
Figure C.4: The RVE of a rolled material under a strain rate ˙ = 0.1 s−1
Figure C.5: The RVE of a rolled material under a strain rate ˙ = 1 s−1
73 Figure C.6: The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 and an angle of 0◦
Figure C.7: The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 and an angle of 45◦
74 Figure C.8: The RVE of a rolled material under a strain rate ˙ = 0.01 s−1 and an angle of 90◦
75 Appendix D: Stress-strain curves
Figure D.1: The stress-strain curve of a rolled and unrolled material under a strain rate of ˙ = 0.01 s−1
76 Figure D.2: The stress-strain curve of a rolled and unrolled material under a strain rate of ˙ = 0.1 s−1
Figure D.3: The stress-strain curve of a rolled and unrolled material under a strain rate of ˙ = 1 s−1
77 Appendix E: DAMASK codes E.1 DAMASK codes E.1.1 Generation of seeds file
Listing E.1: General generation of the seeds file source /opt/netapps/DAMASK/DAMASK_env.sh seeds_fromRandom -N 40 --grid 40 40 1 > 40grains.seeds seeds_check 40grains.seeds
geom_fromVoronoiTessellation --grid 64 64 1 40grains.seeds geom_check 40grains.geom
E.1.2 Generation of rolled seeds file
Listing E.2: General generation of the rolled seeds file, based on the general file source /opt/netapps/DAMASK/DAMASK_env.sh seeds_fromRandom -N 40 --grid 40 40 1 > 40grains.seeds seeds_check 40grains.seeds
geom_fromVoronoiTessellation --grid 64 64 1 40grains.seeds geom_check 40grains.geom
E.1.3 Unrolled pre processing file
Listing E.3: Initial preprocessing bash file to generate the normal RVE using DAMASK and ParaView shopt -s extglob
rm -- !(*.sh)
source /opt/netapps/DAMASK/DAMASK_env.sh
cp 40grains.geom material.config
echo "
echo "
78 echo "{/opt/netapps/DAMASK/examples/ConfigFiles/ Phase_Phenopowerlaw_Aluminum.config}" >> material.config touch tension.load echo "Fdot 0.025 0 0 0 * 0 0 0 * stress * * * * 0 * * * 0 time 1 incs 100 freq 1" >> tension.load echo "Fdot 0.025 0 0 0 * 0 0 0 * stress * * * * 0 * * * 0 time 1 incs 50 freq 2 dropguessing" >> tension.load
DAMASK_spectral --load tension.load --geom 40grains.geom > 40 grains_tension.out & sleep 10s tail -f 40grains_tension.out
79 E.1.4 Rolled pre processing file
Listing E.4: Modified preprocessing bash file to generate the rolled RVE using DAMASK and ParaView
shopt -s extglob
rm -- !(*.sh)
source /opt/netapps/DAMASK/DAMASK_env.sh
cp 40grains_squished.geom material.config
echo "
echo "
touch tension.load echo "Fdot 0.025 0 0 0 * 0 0 0 * stress * * * * 0 * * * 0 time 4 incs 100 freq 1" >> tension.load echo "Fdot 0.025 0 0 0 * 0 0 0 * stress * * * * 0 * * * 0 time 12 incs 50 freq 5 dropguessing" >> tension.load
DAMASK_spectral --load tension.load --geom 40grains.geom > 40 grains_tension.out & sleep 10s tail -f 40grains_tension.out
80 E.1.5 Post processing file
Listing E.5: Initial postprocessing bash file to run DAMASK source /opt/netapps/DAMASK/DAMASK_env.sh
postResults 40grains_tension.spectralOut --cr f,p
cd postProc
addStrainTensors 40grains_tension.txt --left --logarithmic addCauchy 40grains_tension.txt
81 E.1.6 Generate rotated RVEs
Listing E.6: Bash file to generate the rotated RVEs
#!/bin/bash
matlab -nojvm -nodesktop -r "VL_UTT(’10grains.geom’,’tensionX. load’,10); exit;" bash lab .
# rm lab.sh
82 Appendix F: Information aluminum F.1 Phenopowerlaw Aluminum
/opt/netapps/DAMASK/examples/ConfigF iles/P hase P henopowerlaw Aluminum.config
Listing F.1: Initial postprocessing bash file to run DAMASK [ Aluminum ] elasticity hooke plasticity phenopowerlaw
(output) resistance_slip (output) shearrate_slip (output) resolvedstress_slip (output) accumulated_shear_slip (output) totalshear
lattice_structure fcc Nslip 12# per family Ntwin 0# per family
c11 106.75e9 c12 60.41 e9 c44 28.34 e9
gdot0_slip 0.001 n_slip 20 tau0_slip 31e6# per family tausat_slip 63e6# per family a_slip 2.25 h0_slipslip 75e6 interaction_slipslip 1 1 1.4 1.4 1.4 1.4 atol_resistance 1
F.2 Elaboration on parameters Hooke’s plasticity Hooke’s law describes the proportionality between stress and strain through the relationship
σ = E where σ equals stress [MPa], equals strain [-] and E equals the modulus of elasticity or Young’s modulus [MPa]. Hooke’s law holds for most metals that are stressed in tension and at relatively low levels. [Callister, 2007]
83 Metallic crystal structure The Face-Centered Cubic Crystal Structure has been chosen for phenopowerlaw aluminum in this Integration Project. This structure describes a unit cell of cubic geometry, with atoms located at each of the corners and the centers of all cube faces. Some of the familiar metals having this crystal structure are copper, aluminum, solver and gold. Figure F.2 shows the hard sphere unit cell representation of the FCC crystal structure.
Figure F.1: Hard sphere unit cell representation of the Face-Centered Cubic crystal structure
84 Appendix G: ANN code
Listing G.1: The ANN code in MATLAB clc close all;
data = xlsread(’RawData.xlsx’); eps_d = data(:,1); eps = data(:,2); sig = data(:,3);
L = length(sig)/3;
min_eps = min(eps); max_eps = max(eps); eps_space = linspace(min_eps,max_eps,L);
y_1 = interp1(eps(1:L),sig(1:L),eps_space); y_2 = interp1(eps(1:L),sig(L+1:2*L),eps_space); y_3 = interp1(eps(1:L),sig(2*L+1:3*L),eps_space);
eps = repmat(eps_space,[1,3])’; sig = [y_1 y_2 y_3]’;
x = [eps(:) eps_d(:)]’; y = sig (:) ’; nn = [20 20 20 20]; net = feedforwardnet(nn);
net.trainFcn = ’trainscg’; net.trainParam.goal = 0;
[net,tr] = train(net,x,y); genFunction(net,’netFcn’);
sig_NN = netFcn(x);
err = immse(sig,sig_NN’);
plot(sig); hold on plot(sig_NN);
85 Appendix H: Table input in MSC.Marc
Listing H.1: Table input to generate the work hardening properties of the material in MSC.Marc, based on one input # Table File Version 1 # Name hardening # Number of Independent Variables and Function Values 1 1 # First Independent Variable Label V1 # Function Label F # First Independent Variable Formula v1 # Function Formula f # Independent Variable Type(s) 15 # Independent Variable Entries 162 # Steps for Independent Variables 161 # Steps for Function Values 161 # Ranges (min and max) for Independent Variables 9.99950003333000e-05 0.602162675965254
# Ranges (min and max) for Function Values 6932126.20613000 147875385.020474
# Values for First Independent Variable 9.99950003333000e-05 0.00383951475787939 0.00757903451542548 0.0113185542729716 0.0150580740305177 0.0187975937880638 0.0225371135456099 0.0262766333031559 ## not all data points are shown here 0.568506998147339 0.572246517904885 0.575986037662431 0.579725557419977 0.583465077177524 0.587204596935070 0.590944116692616 0.594683636450162 0.598423156207708 0.602162675965254
86 # Function Values 6932126.20613000 87495244.9593108 88746238.7069800 89439380.2490505 90061281.5164046 90683182.7837587 91296109.5274040 91874741.7552901 ## not all data points are shown here 144841510.123865 145178607.334599 145515704.545334 145852801.756068 146189898.966803 146526996.177537 146864093.388271 147201190.599005 147538287.809740 147875385.020474
# Extrapolation Flag for Independent Variables 1 # Method 1
Listing H.2: Table input to generate the work hardening properties of the material in MSC.Marc, based on two inputs # Table File Version 1 # Name ss_curve_test # Number of Independent Variables and Function Values 2 1 # First Independent Variable Label V1 # Second Independent Variable Label V2 # Function Label F # First Independent Variable Formula v1 # Second Independent Variable Formula v2 # Function Formula f # Independent Variable Type(s) 15 16 # Independent Variable Entries 162 3 # Steps for Independent Variables 161 2 # Steps for Function Values 161 # Ranges (min and max) for Independent Variables
87 0.000000000000e+00 1.000000000000e+00 1.000000000000e-02 1.000000000000e+00 # Ranges (min and max) for Function Values 0.000000000000e+00 1.000000000000e+02
# Values for First Independent Variable 9.99950003333000e-05 0.00383951475787939 0.00757903451542548 0.0113185542729716 0.0150580740305177 0.0187975937880638 0.0225371135456099 0.0262766333031559 ## not all data points are shown here 0.583465077177524 0.587204596935070 0.590944116692616 0.594683636450162 0.598423156207708 0.602162675965254
# Values for Second Independent Variable 1.000000000000e-02 1.000000000000e-01 1.0
# Function Values 6932126.20613000 87495244.9593108 88746238.7069800 89439380.2490505 90061281.5164046 90683182.7837587 91296109.5274040 91874741.7552901 ## not all data points are shown here 199306530.782981 199796520.294001 200286509.805021 200776499.316040 201266488.827059 201756478.338079 202246467.849099 202736457.360119 203226446.871138 203716436.382158 # Extrapolation Flag for Independent Variables 1 1 # Method 0
88 Appendix I: FEM figures
Figure I.1: The workpiece before deformation using the NN
Figure I.2: The workpiece after deformation using the NN
89 Figure I.3: Behavior of displacement in y-direction following the arc length of the workpiece using the NN
Figure I.4: Behavior of displacement in y-direction following the arc length of the workpiece (a) using table 1, (b) using table 2
Figure I.5: Depiction of the reaction force in the workpiece using the NN
Figure I.6: Depiction of the reaction force in the workpiece (a) using table 1, (b) using table 2
90 Figure I.7: Depiction of the equivalent plastic strain behavior using the NN
Figure I.8: Depiction of the equivalent plastic strain behavior (a) using table 1, (b) using table 2
Figure I.9: The equivalent plastic strain behavior using the NN
Figure I.10: The equivalent plastic strain behavior (a) using table 1, (b) using table 2
91 Figure I.11: Depiction of the equivalent plastic strain rate behavior using the NN
Figure I.12: Depiction of the equivalent plastic strain rate behavior (a) using table 1, (b) using table 2
Figure I.13: The equivalent plastic strain rate behavior using the NN
Figure I.14: The equivalent plastic strain rate behavior (a) using table 1, (b) using table 2
92 Figure I.15: The equivalent Von Mises stress behavior using the NN
Figure I.16: The equivalent Von Mises stress behavior using the NN (a) using table 1, (b) using table 2
93