Structural Optimization of a Thermally Loaded Functionally Graded Pressure Vessel by Jeffrey Charles Youngs A Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of MASTER OF SCIENCE Major Subject: Mechanical Engineering
Approved:
______Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute Hartford, Connecticut August 2009 © Copyright 2009 by Jeffrey Youngs All Rights Reserved CONTENTS
CONTENTS...... i LIST OF TABLES...... v LIST OF FIGURES...... vi LIST OF SYMBOLS...... xi ACKNOWLEDGMENT...... xiv ABSTRACT...... xv 1. Introduction...... 1 1.1 Problem Statement...... 1 1.2 History of Functionally Graded Materials...... 1 1.3 Pressure Vessel Description...... 2 1.4 Finite Element Theory...... 3 1.5 Basic Structural Optimization Theory...... 5 2. Analytical Modeling...... 7 2.1 Exact Solution for a Thick Homogeneous Cylinder...... 7 2.2 Exact Solution for a Thick Homogeneous Hemisphere...... 11 2.3 Exact Solution for Functionally Graded Material...... 15 2.4 Stress-Strain Equations for a Functionally Graded Vessel...... 16 2.4.1 Strain Solutions for a Functionally Graded Cylinder...... 17 2.4.2 Stress Solutions for a Functionally Graded Cylinder...... 18 2.4.3 Strain Solutions for a Functionally Graded Hemisphere...... 19 2.4.4 Stress Solutions for a Functionally Graded Hemisphere...... 20 3. Design Parameters...... 21 3.1 Pressure Vessel Initial Shape...... 21 3.2 Loading Conditions...... 21 3.3 Materials...... 23 3.4 Modeling Material Properties in Hypermesh...... 25
i 4. Finite Element Modeling...... 27 4.1 Element Shape Considerations...... 27 4.2 Element Order Considerations...... 27 4.3 Axisymmetric Model...... 28 4.4 Solid Element Model...... 33 4.5 Choosing Element Type...... 38 4.6 Element Convergence...... 43 5. Structural Optimization...... 48 5.1 Deciding on a Solver...... 48 5.2 Optimization Model in OptiStruct...... 48 5.3 OptiStruct’s Iterative Solution for Structural Optimization...... 51 5.4 Design Sensitivity Analysis in OptiStruct...... 52 5.5 Convergence in OptiStruct...... 53 5.6 Topology Optimization Method in OptiStruct...... 53 5.7 Density Method of Topology Optimization...... 54 5.8 Topology Optimization: Free Form...... 55 5.9 Defining the Optimized and Non-Optimized Regions...... 56 5.10 Defining the Load Sets for Structural Optimization...... 56 5.11 Define optimization criteria...... 57 5.12 Results for Free Form Topology Optimization...... 58 5.13 Redefining the optimized regions for topology optimization...... 60 5.14 Redefining the design criteria: Discreteness Parameter / Penalty Factor...... 60 5.15 Results for Constrained Topology Optimization...... 60 5.16 Added Manufacturability Constraints: Symmetry Plane and Supporting Design Parameters...... 62 5.17 Results for topology optimization of a 1 ½” thick model...... 65 5.18 Shape Optimization Method in OptiStruct...... 67 5.19 Free Shape Optimization Method in OptiStruct...... 67
ii 5.20 Defining the optimized and non-optimized regions for Free Shape Optimization...... 68 5.21 Defining the design criteria...... 68 5.22 Manufacturability Constraint: Grid Constraints...... 70 5.23 Control Cards for Free Shape Optimization...... 71 5.24 Optimization Results: Total Model Volume and Maximum Von Mises Stress, Only Hemispherical Caps Optimized...... 71 5.25 Optimization Results: Total Model Volume and Maximum Von Mises Stress, Whole Body Optimization...... 74 5.26 Results: Thickness Change for Whole Body Free Shape Optimization...... 80 5.27 Results: Optimization Model Shape Change...... 82 5.28 Free Shape Optimization Study: Whole Body, Homogeneous Material...... 84 5.29 Results: Optimization Model Material Distribution...... 85 6. Conclusions...... 87 6.1 Comparison of Topology and Free Shape Optimization...... 87 6.2 Comparison of Analytical Solution and Finite Element Analysis for the Optimized Model...... 88 6.3 Future Work...... 90 7. References...... 92 8. Appendix...... 94 8.1 Excel VBA Code for Creating Material Cards for ABAQUS (12 Element Layers)...... 94 8.2 Excel VBA Code for Creating Material Cards for OPTISTRUCT (12 Element Layers)...... 95 8.3 Batch File Code for Input File Translation...... 96 8.4 Hypermesh TCL Code For Assigning Properties to Adjacent Layers to Simulate Functionally Graded Material...... 97 8.5 Excel VBA Code for Translating Heat Transfer Results to SPC Temperature Loads for OptiStruct Stress/Strain Analysis...... 98 8.6 Mesh Convergence Plots, Stresses for Functionally Graded Cylinder...... 99
iii 8.7 Mesh Convergence Plots, Stresses for Functionally Graded Sphere/Hemisphere ...... 102 8.8 Free Form Topology Element Density Plots...... 105 8.9 Topology Optimization Element Density Plots: Constrained Inner Surface, 3” Thick Vessel...... 108 8.10 Topology Optimization Element Density Plots: Constrained Inner Surface, 1 ½” Thick Vessel...... 111 8.11 Free Shape Optimization Plots: Shape Change Compared to Non-Optimized Model (inches), Hemispherical End Cap Optimized (16 Smoothing Layers, 0.125 Move Factor)...... 114 8.12 Free Shape Optimization Plots: Shape Change Compared to Non-Optimized Model (inches), Entire Vessel Optimized (12 Smoothing Layers, 0.075 Move Factor)117 8.13 Graphs for Material Properties of Free Shape Optimized Model...... 124
iv LIST OF TABLES
Table 1: Finite Element Model Inputs...... 29 Table 2: Number of Elements, Nodes and Degrees of Freedom for Analysis Models...37 Table 3: Run Time and Memory Usage Comparisons for Element Types...... 41 Table 4: Comparison of Element Solutions to Analytical Solution for the Hemispherical Cap...... 42 Table 5: Comparison of Element Solutions to Analytical Solution for the Cylindrical Body...... 43 Table 6: Comparisons of Von Mises Stress Results for Varying Mesh Densities...... 46 Table 7: OptiStruct/RADIOSS Finite Element Model Inputs...... 50 Table 8: RADIOSS Model Analysis Results...... 51 Table 9: Thinner Geometry for Structural Optimization Model...... 65 Table 10: Hemispherical Cap Free Shape Optimization Results...... 72 Table 11: Whole Body Free Shape Optimization Results...... 75 Table 12: Comparison of Vessel Thickness, Free Shape Optimized and Topology Optimized Models...... 81 Table 13: Functionally Graded Material Properties for the Optimized Model...... 85 Table 14: Mesh Information for Finite Element Solution of the Optimized Pressure Vessel...... 88 Table 15: Comparison of Finite Element and Analytical Solutions, Fully Optimized Hemispherical Cap...... 89 Table 16: Comparison of Finite Element and Analytical Solutions, Fully Optimized Cylindrical Body...... 89
v LIST OF FIGURES
Figure 1: Diagram of Cylinder Undergoing Pressure Loading, Front and Side Views....7 Figure 2: Diagram of Cylinder Undergoing Thermal Loading, Front and Side Views....9 Figure 3: Diagram of Hemisphere Undergoing Pressure Loading, Front and Side Views ...... 12 Figure 4: Diagram of Hemisphere Undergoing Pressure Loading, Front and Side Views ...... 13 Figure 5: Detail of Pressure Vessel Geometry...... 21 Figure 6: Pressure Loading for the Vessel...... 22 Figure 7: Thermal Loading for the Vessel...... 23 Figure 8: Hoop Stress Divided by Pressure Load for Various Material Distributions in a Functionally Graded Cylinder...... 24 Figure 9: Meshing Detail for Axisymmetric Model...... 29 Figure 10: Comparison of Axisymmetric Model to Analytical Solution, FGM Cylinder ...... 31 Figure 11: Comparison of Axisymmetric Model to Analytical Solution, FGM Sphere. 32 Figure 12: Solid Mesh Created by Spinning Elements, Top View and Isometric...... 34 Figure 13: Surface Definitions for Spherical Cap (Isometric View) and Surface Mesh (Top View)...... 35 Figure 14: Problems with High Linear Biasing with Cylinder Surface Meshes, Outer Surface of Cylindrical Body...... 36 Figure 15: Biasing for the Pressure Vessel, Outer Surface of Pressure Vessel...... 37 Figure 16: Required Alteration of Abaqus Input files for Coupled Temperature- Displacement...... 39 Figure 17: Required Addition to Static Stress/Displacement Analysis Cards for Thermal Stresses...... 40 Figure 18: Stress Sampling Regions in the Pressure Vessel Model...... 42 Figure 19: Convergence of Elements to Radial Stress Distribution for an FGM Hemisphere...... 44 Figure 20: Stress Discontinuity Plot for Thermo-Mechanically Loaded FGM Vessel (Inside Surface, Outside Surface, and Isometric View)...... 45
vi Figure 21: OptiStruct Iterative Solution...... 51 Figure 22: Element Density Plot for Free Form Topology Optimization (Side, Rear and Isometric Views)...... 58 Figure 23: Element Density Isolation Plot for Free Form Topology Optimization (Side, Rear and Isometric Views)...... 59 Figure 24: Isolated Element Density Plot for Constrained Topology Optimization (Side, Outer Surface and Isometric Views)...... 61 Figure 25: Symmetry Plane Definition in Optimization Model...... 62 Figure 26: Element Density Isolation Plots for Model with Symmetry Constraint (Outer Surface, Rear, and Isometric Views)...... 64 Figure 27: Element Density Plot, 1 ½” Thick Pressure Vessel Topology Optimization (Outer Surface, Side, and Isometric Views)...... 66 Figure 28: Final Von Mises Stress (ksi), Hemispherical Cap Free Shape Optimization Model (Inner Surface, Outer Surface and Isometric Views)...... 73 Figure 29: Shape Change Results (inches), Free Shape Optimization with Move Factor of 0.075 and 8, 12, and 16 Smoothing Layers...... 76 Figure 30: Total Volume Response (in3), Free Shape Optimization (12 Smoothing Layers)...... 77 Figure 31: Final Von Mises Stresses (ksi), Full-Body Free Shape Optimization Model (Inner Surface, Outer Surface and Isometric Views)...... 79 Figure 32: Free Shape Optimization Model Cross-Section Numbers...... 80 Figure 33: Magnitude of Final Shape Change (inches) for the Full-Body Free Shape Optimization (Top, Bottom, and Side Views)...... 82 Figure 34: Magnified Details for Shape Change (inches) of the Spherical Cap and Cylindrical Body...... 83 Figure 35: Magnified Shape Change (inches) for Homogeneous Model, Sphere and Cylinder...... 84 Figure 37: Mesh Convergence, Radial Stress for FGM Cylinder with 3x3 Top Surface Mesh...... 99 Figure 38: Mesh Convergence, Hoop Stress for FGM Cylinder with 3x3 Top Surface Mesh...... 99
vii Figure 39: Mesh Convergence, Radial Stress for FGM Cylinder with 4x4 Top Surface Mesh...... 100 Figure 40: Mesh Convergence, Hoop Stress for FGM Cylinder with 4x4 Top Surface Mesh...... 100 Figure 41: Mesh Convergence, Radial Stress for FGM Cylinder with 5x5 Top Surface Mesh...... 101 Figure 42: Mesh Convergence, Hoop Stress for FGM Cylinder with 5x5 Top Surface Mesh...... 101 Figure 43: Mesh Convergence, Radial Stress for FGM Sphere with 3x3 Top Surface Mesh...... 102 Figure 44: Mesh Convergence, Hoop Stress for FGM Sphere with 3x3 Top Surface Mesh...... 102 Figure 45: Mesh Convergence, Radial Stress for FGM Sphere with 4x4 Top Surface Mesh...... 103 Figure 46: Mesh Convergence, Hoop Stress for FGM Sphere with 4x4 Top Surface Mesh...... 103 Figure 47: Mesh Convergence, Radial Stress for FGM Sphere with 5x5 Top Surface Mesh...... 104 Figure 48: Mesh Convergence, Hoop Stress for FGM Sphere with 5x5 Top Surface Mesh...... 104 Figure 49: Free Form Topology Initial State...... 105 Figure 50: Free Form Topology, Iteration 3...... 105 Figure 51: Free Form Topology, Iteration 8...... 106 Figure 52: Free Form Topology, Iteration 16...... 106 Figure 53: Free Form Topology, Final Solution...... 107 Figure 54: Volume Response, Free Form Topology...... 107 Figure 55: 3" Constrained Topology, Initial Condition...... 108 Figure 56: 3" Constrianed Topology, 5 Iterations...... 108 Figure 57: 3" Constrained Topology, 15 Iterations...... 109 Figure 58: 3" Constrained Topology, 23 Iterations...... 109 Figure 59: 3" Constrained Topology, Final Solution...... 110
viii Figure 60: 3" Constrained Topology, Total Volume Response...... 110 Figure 61: 1 ½” Constrained Topology, Initial Condition...... 111 Figure 62: 1 ½” Constrained Topology, Iteration 3...... 111 Figure 63: 1 ½” Constrained Topology, Iteration 8...... 112 Figure 64: 1 ½” Constrained Topology, Iteration 15...... 112 Figure 65: 1 ½” Constrained Topology, Final Solution...... 113 Figure 66: 1 ½” Constrained Topology, Total Volume Response...... 113 Figure 67: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 0 (Outer Surface and Side Views) (Model Before Optimization Routine) ...... 114 Figure 68: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 1 (Outer Surface and Side Views)...... 114 Figure 69: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 2 (Outer Surface and Side Views)...... 115 Figure 70: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 3 (Outer Surface and Side Views)...... 115 Figure 71: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 4 (Outer Surface and Side Views)...... 116 Figure 72: Hemispherical Cap Free Shape Optimization, Total Volume Response.....116 Figure 73: Shape Change for Entire Vessel Free Shape Optimization, Iteration 0 (Outer Surface, Side, and Rear Views) (Model Before Optimization Routine).....117 Figure 74: Shape Change for Entire Vessel Free Shape Optimization, Iteration 1 (Outer Surface, Side, and Rear Views)...... 118 Figure 75: Shape Change for Entire Vessel Free Shape Optimization, Iteration 2 (Outer Surface, Side, and Rear Views)...... 119 Figure 76: Shape Change for Entire Vessel Free Shape Optimization, Iteration 3 (Outer Surface, Side, and Rear Views)...... 120 Figure 77: Shape Change for Entire Vessel Free Shape Optimization, Iteration 4 (Outer Surface, Side, and Rear Views)...... 121 Figure 78: Shape Change for Entire Vessel Free Shape Optimization, Iteration 5 (Outer Surface, Side, and Rear Views)...... 122
ix Figure 79: Hemispherical Cap Free Shape Optimization, Total Volume Response.....123 Figure 80: Functionally Graded Material Properties for the Optimized Cylindrical Body ...... 124 Figure 81: Functionally Graded Material Properties for the Optimized Hemispherical End Caps...... 125
x LIST OF SYMBOLS
Symbol Meaning (units) a inner radius (in) b outer radius (in) DV design variable for shape optimization (unitless) E modulus of elasticity (lbf/in2)
Eo power law equation coefficient for modulus of elasticity variation in functionally graded material (lbf/in2) f(x) objective function for optimization routine: total volume (in3) g(x) constraint function for optimization routine: von mises stress (lbf/in2) k thermal conductivity (Btu/hr-ft-ºF) ko power law equation coefficient for thermal conductivity variation in functionally graded material (Btu/hr-ft-ºF) K original stiffness matrix of an element (lbf/in) K penalized stiffness matrix of an element in density method of topology optimization (lbf/in) L axial length (in) m1 power law equation exponent for modulus of elasticity variation in functionally graded material (unitless) m2 power law equation exponent for linear coefficient of thermal expansion variation in functionally graded material (unitless) m3 power law equation exponent for thermal conductivity variation in functionally graded material (unitless) M power law equation exponent for variation in all material properties in functionally graded material (unitless) mesh_size average mesh size within the shape optimization design region (in) MVFACTOR move factor in shape optimization (unitless)
xi p penalization factor in density method of topology optimization (unitless) P Pressure (lbf/in2) PV perturbation vector for shape optimization (in) r radius (in) T temperature increase over uniform initial temperature (ºF)
Ti temperature increase at inner radius (ºF)
To temperature increase at outer radius (ºF) u radial distance (in) x vector of design variables (unitless) xL lower limit for the vector of design variables (unitless) xU upper limit for the vector of design variables (unitless) z axial distance (in) linear coefficient of thermal expansion (in/in-ºF)
power law equation coefficient for linear coefficient of thermal expansion variation in functionally graded material (in/in-ºF)
i maximum initial movement of design grids in shape optimization (in) sum of strain (in/in)
r radial strain (in/in)
hoop strain (in/in)
hoop strain (in/in)
z axial strain (in/in) angle (radians) poisson’s ratio (unitless) element density factor in density method of topology optimization (unitless)
2 r radial stress (lbf/in )
2 hoop stress (lbf/in )
2 hoop stress (lbf/in )
xii angle (radians) vector of nodal coordinates (in)
0 initial vector of nodal coordinates (in)
xiii ACKNOWLEDGMENT
I would like to thank my adviser Professor Ernesto Gutierrez-Miravete for his time and support. I would also like to thank Dan Pusey of Altair Engineering for his technical support in using Hypermesh, OptiStruct and RADIOSS. In addition, I would like to thank my co-worker Tom Roes for sharing his experience in Abaqus. Finally, I would like to thank my friends and family for their support and putting up with me for these past few months.
xiv ABSTRACT
Pressure vessels with high thermal and pressure loads require designs with large thickness to reduce the stress in the vessel. This large thickness can make the vessel very heavy and hard to manufacture. To reduce the thickness needed for the design, one can functionally grade the material to lower the stress found in the vessel and allow it to have a less thick design. However, these functionally graded materials tend to be costly, especially when bulk manufactured. Therefore, it is necessary to optimize the vessel so as to minimize the total volume while keeping it under an allowed stress. The problem posed is how to reduce the amount of material needed in a thermally loaded pressure vessel with a cylindrical body and hemispherical caps. To accomplish this, finite element models were created and optimized based on an objective of minimizing total volume while being constrained by a maximum Von Mises stress. The optimization methods used were topology and free shape optimization. The results showed that topology optimization required less design parameters and was simple to set up, but provided designs with unwanted voids unless extra controls were added. When these controls were added, the models produced were less than optimal in volume reduction. The results from the topology optimization, however, were useful as a preliminary design for the free-shape optimization to work from. The free shape method required more design parameters and was rather sensitive to the optimization controls, but provided designs with less volume. Therefore, free shape optimization is more suitable for giving a truly optimized solution.
xv 1. Introduction
1.1 Problem Statement
The goal of this project was to optimize the design of a functionally graded pressure vessel undergoing high thermal and pressure loads. This functionally graded vessel had metallic properties on the outside surface. The properties were varied through the thickness of the pressure vessel within the linear elastic regime in order to have material properties on the inner surface that are more like those of a ceramic. The objective of this optimization was to reduce the cost of the pressure vessel while keeping the vessel from surpassing a design stress constraint. The cost was dependent upon the total volume of the vessel. The greater the total volume, the more costly the design would be. To compute the stresses of the vessel, finite element models were created. The stress output of the finite element models was compared to analytical solutions for the stress distribution both before and after the optimization routine. The first method of optimization investigated was topology optimization. This optimization procedure was performed by way of effectively voiding elements in a given model. This gave guidance to where material in the model is not necessary, which in turn reduced the cost of the pressure vessel. The second method of optimization investigated was free shape optimization. In this optimization procedure, the volumes of certain elements in the model were changed by way of moving the nodes that define the shape of the elements. The smaller the elements in the model, the less material there would be and the lower the cost of the pressure vessel.
1 1.2 History of Functionally Graded Materials
Functionally graded materials (FGMs) are materials characterized by a gradual change in material properties with position throughout it. This property gradient is created by position-dependent variations in chemical composition, microstructure or atomic order. These materials allow a structure to have properties that are advantageous at various interfaces (i.e. corrosion resistance at one layer and advanced strength at another layer) while keeping the property change continuous, therefore eliminating the stress singularities due to material property mismatch or poor adhesion of layers. Japanese scientists first conceived FGMs in 1987 as a way to make a thermal shield that would be able to survive multiple cycles. The nose cone initially had a SiC protection layer on a C/C composite exposed to a supersonic gas flow at 1900 ºC that deteriorated after the first thermal cycle. When the scientists added an extra graded SiC/C graded interlayer between the SiC layer and the C/C composite, the nose cone did not show any change after 10 cycles. Since that time, the FGMs that are in production are still mostly in the form of thin layers. Thicker FGM structures, or bulk FGMs have been worked on as well, and can be preferable to thin layers on a structure due to the variation extending over a large part of the material as opposed to it being on a thin coating on the structure. While the coating may be processed to minimize the singularity effect of layer interaction, the only way to completely eliminate this effect is to go with a through-layer variation of material properties. The main problem with bulk FGMs is the cost of production. The gradual transition of materials is hard to be effectively reproduced given the current production methods. There is currently work being done to improve the production methods of bulk FGMs so that they can be commercially viable. Until an inexpensive and trustworthy method of creating these bulk FGMs arrive, though, one will have to attempt to optimize these structures so as to reduce the amount of material needed to be processed without sacrificing the structural integrity of the part being manufactured. (Neubrand, pp. 3407-3413)
2 1.3 Pressure Vessel Description
Pressure Vessels can, in theory, be any shape. However, complex shapes would make construction as well as analysis for safety too difficult. Therefore, simple shapes such as cylinders and spheres would seem to be optimal for pressure vessel design. An additional factor in pressure vessel design is the objective for the optimization. The objective for this thesis is that the design vessel be optimized so that it can reduce the amount of material used while not exceeding a stress constraint. Therefore, it would be optimal if the stress concentrations at the joints between the pressure vessel body and caps were minimal. This can be achieved by using a cylindrical tube with hemispherical end caps. This reduces the amount of sharp transitions in angularity and thickness that are the hallmarks of stress concentration were a flat cap to be used. When deciding on the conditions that the pressure vessel would have applied to it, the writer has chosen those that are similar to power plant pressure vessels. This is to say that the pressure vessel would have to hold a sizeable amount of high-temperature fluid that is at a high pressure.
1.4 Finite Element Theory
Finite Element Analysis (or FEA in short) is used to get a numerical solution to many problems in engineering that are complex in nature and therefore would make analytical solutions an encumbrance or impossible. There are two primary approaches in FEA. The first is that of using discrete elements to obtain the joint displacements and member forces of a structural framework. This method is referred to as the matrix analysis of structures. The results from this approach are identical to classical structural framework analysis. The second approach is that of using continuum elements to obtain approximate solutions for problems such as heat transfer and solid mechanics. This approach also creates discrete elements, but it uses points where the elements meet, called nodes, as reference points to give approximate values of desired parameters. While potentially less accurate, this method is preferred when working with systems with irregular-shaped bodies, several materials, nonlinear material properties, or mixed boundary conditions. This method is well suited to the proposed analysis, seeing as the pressure vessel
3 contains a discontinuity point at the intersection of the hemispherical head and the cylindrical body, as well as it having varied material properties through its thickness and mixed thermal and mechanical boundary conditions. The method involved in finite element analysis can be explained in five basic steps. The first of these steps is to discretize the region. This involves subdividing the system into elements, as well as locating and numbering the nodes that define these elements. One needs to make sure that the nodes are closer together (i.e. smaller elements) in regions where the outputs should change rapidly and further apart where the outputs are expected to be relatively constant. One should also make sure that there are nodes present in areas where output is desired. This step is rather reliant on engineering experience; otherwise the creator will need to revisit this step several times in a feat of trial and error. The next step is to specify the equations that the discrete elements will use to calculate values at the nodes. The order of the approximation is of particular interest, and most finite element software packages rely on linear and quadratic equations. These equations are calculated at the nodes by use of a shape function, which calculates the outputs based on the position of the nodes relative to the element itself. In cases where it is known that the output will be linear in form, it would be optimal to make sure all elements use linear shape functions. If it is known that the output will be of a strictly quadratic form, it may be better to use quadratic elements, because even though it has more nodes to calculate with, it will be able to fit a quadratic curve with minimal amounts of elements. Again, engineering experience is crucial in making sure that the shape functions are suitable to the model. The next step is to develop the system of equations. This, in effect, joins the nodal equations for each element together to get a system of equations that are inter-related. This can include, as in Galerkin’s method, the definition of a weighting function for each unknown nodal value and then evaluating the weighted residual integral for each node. It can also include, as in the potential energy formulation, the potential energy of the system being written in terms of nodal displacements and then minimized. Depending upon the elements used, the method of developing the system of calculations can vary.
4 Therefore, engineering experience is quite useful in making sure that the system of equations is suitable for the analysis model. After that, there is the crucial step of solving the system of equations. In order to do this, the analyst must make input material properties, as well as loading conditions and boundary conditions for the model. This is where the finite element software is most useful, seeing as it can process the system of equations much faster than a human would be able to. However, seeing as the program is not human, any mistakes in the input values for the model can become fatal and make the solution impossible without human intervention. Finally, the resulting quantities of interest must be examined. This is effectively done by finite element software in its post-processing functions. These functions make it possible for the output of the systems of equations to be seen in graphical form, making it quite easy to find points of failure or discontinuity in solution. (Cook, 2002)
1.5 Basic Structural Optimization Theory
Structural optimization is a method of design optimization in which the structure of a model is iteratively modified to accomplish an objective while being guided by design constraints. In this thesis, the objective is to minimize cost by reducing volume and the constraint is the maximum von mises stress in the model. There are many specific methods of structural optimization available, but this thesis concentrates on the topology and free shape optimization methods. Topology optimization accomplishes structural optimization by showing where material can be removed. This can be accomplished using several methods, but in this thesis uses the density method of OptiStruct. In this methodology, the optimization structure varies the stiffness of the elements by multiplying the original stiffness of the elements by a design density that is modified during each iteration. This design density can vary from zero to one. When the design density for an element is zero, it will have no stiffness and will be counted as a void. When the design density for an element is one, the stiffness will not be altered. There are several aspects that should be considered when performing topology optimization. First, there will be some elements of intermediate density present that
5 require engineering judgment as to whether or not to eliminate the elements. In addition, topology optimization has a tendency to create zero-stiffness elements in areas that would result in a design that cannot be manufactured. However, OptiStruct has controls that force the design to keep the design densities close to zero or one, reducing the amount of engineering judgment needed. In addition, OptiStruct has manufacturing constraints that greatly reduce the likelihood of having voids in undesired areas. Therefore, topology optimization should be a useful tool for reducing material in a model. Free shape optimization is a method of structural optimization in which the outer boundary of the structure is modified to accomplish the objective. With finite element models, this is accomplished by moving a grid of nodes that defines this boundary. For this thesis, the boundary will be the outer surface of the vessel and the free shape optimization will be constrained so this outer surface can only move in an inward direction. If one were to move this boundary by itself, it is quite likely that the elements that contain the boundary nodes would be changed in ways that would make them fail basic size and shape criteria. Fortunately, OptiStruct offers a number of different methods to account for the possibility of creating a bad mesh of elements through optimization. One such method of accountability is forcing nodes adjacent to the outer surface to move along with the design grid. This reduces the change in the element size and shape, reducing the chance of having failed elements. Another method of accountability is that of reducing the initial deflection of the design grid. This reduction in deflection reduces the total shape and size change and the chances of having failed elements are less likely. With these methodologies, one is able to create a feasible design that can reduce the volume of a model significantly.
6 2. Analytical Modeling
In order to find out if the models created by the finite element and optimization software are giving correct results, it is necessary to find analytical solutions for key design parameters of the optimization. As was stated before, the vessel shape chosen was that of a cylindrical pressure vessel with hemispherical end caps. The solution for the top of the hemispherical cap and the midpoint of the cylinder are sufficiently similar to the solution for those of isolated spheres and cylinders, respectively. The solution for the point where the cap meets the cylinder, however, is significantly more analytically complex. Therefore, in order to more efficiently use the time allotted in the thesis, it was deemed sufficient to use the solution for the hemisphere and the cylinder as comparisons to the model results.
2.1 Exact Solution for a Thick Homogeneous Cylinder
In order to understand the behavior of the pressure vessel, we need to figure out how the various parts of the vessel respond to the loading conditions that will be applied to it. The first effort was put into finding how the cylindrical body responds to pressure loading.
Figure 1: Diagram of Cylinder Undergoing Pressure Loading, Front and Side Views
In this diagram, the coordinate system comprises of the radius (r), the angle (), and the axial distance (z). The cylinder is of an inner radius “a”, an outer radius “b”, and axial length “L.” The inner surface has a pressure load of “P” applied to it. From Timoshenko’s work, it is known that the following are the stress and strain equations of a thick cylinder undergoing axisymmetric pressure loading.
7 u r r u r E r 1 2 r E 1 2 r
Equation 1: Stress and Strain Equations, Thick Cylinder With Axisymmetric Pressure Load
For these equations, u is the radial displacement, is poisson’s ratio, r is the radial strain, is the angular strain (also known as circumferential or hoop strain), r is the radial stress, and is the angular stress (also known as circumferential or hoop stress). If we assume that this cylinder is under an axisymmetric internal pressure load, we have the following stress equilibrium equation and the boundary conditions: r r 0 r r
r r a P
r r b 0
Equation 2: Stress Equilibrium Equation and Boundary Conditions, Thick Cylinder With Axisymmetric Pressure Load
8 From this, one can find the stress for the cylinder undergoing axisymmetric internal pressure loading (Timoshenko, pp. 205-208). a2 P b2 1 r 2 2 2 b a r a2 P b2 1 2 2 2 b a r
Equation 3: Radial and Hoop Stress, Thick Cylinder With Axisymmetric Pressure Load
The next step is to find what the behavior of a cylinder undergoing thermal loading is. As before, Timoshenko’s work was useful in finding this solution. The loading conditions are shown in Figure 2.
Figure 2: Diagram of Cylinder Undergoing Thermal Loading, Front and Side Views
In this diagram, all the coordinate and dimensional information is the same as
Figure 1. Ti is the increase in temperature of the inner wall above the initial temperature.
To is the increase in temperature of the outer wall above the initial temperature. The initial temperature is assumed to be uniform throughout the cylinder. In thermal loading, the elongations are not only in the radial and circumferential directions. Seeing as temperature loading makes a volume grow in all directions, the deflection the axis of the cylinder (the z-direction) shall also need to be taken into account. We shall refer to this deflection as w and its strain component shall be z .
9 w z z
Equation 4: Axial Strain, Thick Cylinder
In order to simplify the solution, we shall create a symbol, , for the sum of the strains. Thus, the radial and hoop stress equations for a thick cylinder with axisymmetric thermal loading are as shown below. (Timoshenko, pp.228-231)
z r E E T r r (1 ) 1 2 1 2 E E T (1 ) 1 2 1 2
Equation 5: Stress Equations, Thick Cylinder With Axisymmetric Thermal Load
Where is the linear coefficient of thermal expansion and T is the increase in temperature over a uniform initial temperature. For the loading conditions given, the temperature only varies with the radial distance (r). To find a solution for these stresses, one needs to know the stress equilibrium equation along with the following boundary conditions: r r 0 r r
r r a 0
r r b 0
T(r a) Ti
T(r b) To
Equation 6: Stress Equilibrium Equation and Boundary Conditions, Thick Cylinder With Axisymmetric Thermal Load
Now that this is known, we will need to define the temperature distribution of the temperature through the thickness of the cylinder. In this thesis, we will simplify the heat transfer through the thickness of the cylinder to that of steady state conductive heat transfer with no radiation or convection to the environment. If we assume that there is no expansion of the cylinder at the initial temperature (which would be ambient temperature
10 in the real world) and that the temperature of the outer surface is this temperature, we get the following for the temperature distribution under steady-state conditions (Incropera, p. 106). b ln r T T i b ln a
Equation 7: Temperature Distribution, Thick Cylinder With Axisymmetric Thermal Load
With these inputs, one can solve for the thermal stress. The following are the solutions for a thick cylinder undergoing steady-state thermal loading (Timoshenko, p.232):
ET b a 2 b2 b i ln 1 ln r 2 2 2 b r b a r a 21 ln a ET b a 2 b2 b i 1 ln 1 ln 2 2 2 b r b a r a 21 ln a
Equation 8: Radial and Hoop Stress, Thick Cylinder With Axisymmetric Thermal Load
Now that the stresses for a thick cylinder have been solved, one needs to find the stresses for a thick hemisphere.
2.2 Exact Solution for a Thick Homogeneous Hemisphere
To obtain the solution for the hemispherical heads, we need to figure the solution for hemispheres under similar loadings to those put upon the cylinder in the previous section. The loading conditions for the sphere are shown in Figure 3.
11 Figure 3: Diagram of Hemisphere Undergoing Pressure Loading, Front and Side Views
It can be seen from this diagram that the coordinate system has changed. The axial direction, z, is no longer used. Instead, there is another angular direction, . This angular degree of freedom has its own hoop strain, , which due to the axisymmetric nature of the load and the geometry of the sphere can be said to be equivalent to . The same can be said for the hoop stress. If we take the equivalence into account and set
and , the stress and strain equations for a thick hemisphere undergoing axisymmetric pressure loading can be said to be the following:
u r u r r E 2 r 1 2 r E (1 ) 1 2 r
Equation 9: Stress and Strain Equations, Thick Hemisphere With Axisymmetric Pressure Load
12 We will need to get the stress equilibrium equation if we wish to find the axisymmetric pressure solution for a thick hemisphere. The equilibrium equation, with its boundary conditions, is as follows: 2( ) r r 0 r r
r r a P
r r b 0
Equation 10: Stress Equilibrium Equation and Boundary Conditions, Thick Hemisphere With Axisymmetric Pressure Load
From this, one can find the stress-strain behavior of an internally loaded thick hemisphere. The equations for the radial and hoop stresses are shown below (Budynas, p.685): a3P b3 1 r 3 3 3 b a r a3P b3 1 3 3 3 2b a r
Equation 11: Radial and Hoop Stress, Thick Hemisphere With Axisymmetric Pressure Load
The next step is to find the thermal stress for the hemisphere. The loading conditions for the thick hemisphere are shown in Figure 4.
Figure 4: Diagram of Hemisphere Undergoing Pressure Loading, Front and Side Views
13 Again, we take into effect the volumetric expansion due to the thermal loading of the sphere. Therefore, the stress equations are as follows:
r E ET r r (1 ) 1 2 1 2 E ET (1 ) 1 2 1 2
Equation 12: Stress Equations, Thick Hemisphere With Axisymmetric Thermal Load
If we want to solve this system of equations, we need the stress equilibrium equation. In addition, we need to apply the same boundary conditions as we did with the cylinder. 2 r r 0 r r
r r a 0
r r b 0
T(r a) Ti
T(r b) To
Equation 13: Stress Equilibrium Equation and Boundary Conditions, Thick Hemisphere With Axisymmetric Thermal Load
We also need to find the temperature distribution through the thickness. As with the cylinder, it will be assumed that there will be steady state conductive heat transfer alone in the hemisphere. If the same assumptions that were made for the cylinder are applied to the hemisphere, we get the following for the temperature distribution at steady state. (Noda, p. 298)
a 1 r T T 1 i a 1 b
Equation 14: Temperature Distribution, Thick Hemisphere With Axisymmetric Thermal Load
14 With these inputs, we can solve for the thermal stress (Hetnarski, p. 298).
ETi ab a b ab r 3 3 1 1 b a 1 b a r r r 2 2 2 ETi ab b a a b 3 3 b a 3 1 b a 2r 2r
Equation 15: Radial and Hoop Stress, Thick Cylinder With Axisymmetric Thermal Load
Now that the stresses for both the thick hemisphere and the thick cylinder have been found, it is necessary to apply these solutions to functionally graded materials.
2.3 Exact Solution for Functionally Graded Material
All of the above equations are based on a solution in which the material is the same throughout the thickness. The pressure vessel will be functionally graded, and therefore an analytical solution that has a varied material property distribution through the thickness is needed. This variation in material properties can be modeled several ways, but in this thesis the material distribution shall be that in which the material properties vary throughout the thickness in a way describable by a power law equation. Power law equations were chosen due to their ability to conform well to various distributions with minimal inputs, allowing for a more versatile and simplistic solution. The power law equation will be limited to a single term for further simplicity. In order to simplify the solution even further, it was assumed that Poisson’s ratio would be constant throughout. To solve the equations created before, we need only to vary three material properties: the modulus of elasticity, the thermal expansion coefficient, and the thermal conductivity.
m1 Er Eo r m2 r o r m3 k(r) ko r
Equation 16: Power Law Material Property Distributions
Where the terms with “o” subscripts are the base values to be scaled and the “m” exponents are the powers that control the scaling of the base values. The exponents can be independent, but for simplicity, the thesis shall assume all the exponents are the same
15 value. The values for the material property coefficients can be found in Section 3.3, the discussion of the materials used in this study. The solution for the cylinder and the sphere is similar to that of the previous pressure and thermal loading conditions, except that in this case the thermal conductivity is needed for the steady state heat transfer solution. The variation of the thermal conductivity makes it so that the steady state temperature distribution is not exactly linear through the thickness, unlike a model with a single isotropic material. This makes the distributions given by Equations 7 and 14 no longer valid. Therefore, one needs to take into consideration the heat conduction equations for the sphere and the cylinder (Incropera, pp. 105-111). cylinder : 1 Tr r kr r r r hemisphere : 1 Tr r 2 kr r 2 r r
Equation 17: Heat Conduction Equations, Thick Cylinder and Thick Hemisphere
In order to solve for the temperature distribution and the stress-strain state, the boundary conditions for the cylinder and sphere will be as follows:
r r a P
r r b 0
T(r a) Ti
T(r b) To
Equation 18: Boundary Conditions, Functionally Graded Cylinder and Hemisphere
A more detailed solution for these systems can be found in (Eslami, 2002) and (Eslami, 2004).
2.4 Stress-Strain Equations for a Functionally Graded Vessel
With the equations set up in Sections 2.2 and 2.3, it is possible to solve for the stress and strain in a thermo-mechanically loaded vessel. However, these equations are rather complex in such as that there are too many terms in the solution to be directly input into
16 this paper. However, the equations can be made much more legible if certain values are set to numerical constants. For these equations, it was chosen to make the material properties vary linearly (as shown in Section 3.3) with a poisson’s ratio of 0.3. In addition, the geometry of the vessel is such as is described in Section 3.1. As a summary, the following are the variables set to numerical constants.
m1 m2 m3 1 0.3 a 16.75 b 19.75
2.4.1 Strain Solutions for a Functionally Graded Cylinder
The following is the radial strain for a functionally graded cylinder with linearly varied material properties that is under internal pressure and at a steady state temperature distribution through its thickness: 1 P 26968.05 T 31798.15 T 217.88 r 2.41 o i o r Eo 1 P 102.56 T 120.93T 0.33 0.59 o i o r Eo
143.35 o Ti To o r7.64 Ti 9.01To
Equation 19: Radial Strain, Functionally Graded Thick Cylinder With Axisymmetric Pressure and Thermal Loads
The hoop strain is as follows: 1 P 19176.23T 22610.78T 154.93 2.41 o i o r Eo 1 P 252.42T 297.63T 0.81 0.59 o i o r Eo
143.35 o Ti To o r3.82 Ti 4.50To
Equation 20: Hoop Strain, Functionally Graded Thick Cylinder With Axisymmetric Pressure and Thermal Loads
17 2.4.2 Stress Solutions for a Functionally Graded Cylinder
The following is the radial stress for a functionally graded cylinder with linearly varied material properties that is undergoing internal pressure and a steady state temperature distribution through its thickness: E P o 25239.94 T 29760.52 T 203.91 r 1.41 o i o r Eo P 0.41 Eo r o 283.69 Ti 334.50 To 0.91 Eo 7 2 1.7 10 Eo o rTi To Eo o r 1.47 Ti 1.73To
Equation 21: Radial Stress, Functionally Graded Thick Cylinder With Axisymmetric Pressure and Thermal Loads
The hoop stress is as follows: E P o 10255.67 T 12092.50 T 82.86 1.41 o i o r Eo P 0.41 Eo r o 398.96 Ti 470.42 To 1.29 Eo 7 2 1.7 10 Eo o rTi To Eo o r 4.41Ti 5.20 To
Equation 22: Hoop Stress, Functionally Graded Thick Cylinder With Axisymmetric Pressure and Thermal Loads
18 2.4.3 Strain Solutions for a Functionally Graded Hemisphere
The following is the radial strain for a functionally graded hemisphere with linearly varied material properties that is undergoing internal pressure and a steady state temperature distribution through its thickness: 1 P 8.34 105 T 9.38105 T 3087.07 r 3.46 o i o r Eo 1 P 105.38T 90.92 T 0.11 0.54 o i o r Eo
o r2.78 Ti 3.86 To
Equation 23: Radial Strain, Functionally Graded Thick Hemisphere With Axisymmetric Pressure and Thermal Loads
The hoop strain is as follows: 1 P 3.38105 T 3.81105 T 1252.94 3.46 o i o r Eo 1 P 227.18T 196.01T 0.25 0.54 o i o r Eo
o r1.39 Ti 1.93To
Equation 24: Hoop Strain, Functionally Graded Thick Hemisphere With Axisymmetric Pressure and Thermal Loads
19 2.4.4 Stress Solutions for a Functionally Graded Hemisphere
The following is the radial stress for a functionally graded hemisphere with linearly varied material properties that is undergoing internal pressure and a steady state temperature distribution through its thickness: E P o 7.32 105 T 8.23105 T 2709.96 r 2.46 o i o r Eo P 0.46 Eo r o 403.99 Ti 348.56 To 0.44 Eo 2 2498.56 Eo o Ti To Eo o r 1.07 Ti 1.48To
Equation 25: Radial Stress, Functionally Graded Thick Hemisphere With Axisymmetric Pressure and Thermal Loads
The hoop stress is as follows: E P o 1.70105 T 1.91105 T 628.51 2.46 o i o r Eo P 0.46 Eo r o 497.68Ti 429.39 To 0.54 Eo 2 2498.56 Eo o Ti To Eo o r 2.14 Ti 2.97 To
Equation 26: Hoop Stress, Functionally Graded Thick Hemisphere With Axisymmetric Pressure and Thermal Loads
20 3. Design Parameters
3.1 Pressure Vessel Initial Shape
As was discussed in the introduction, the pressure vessel is cylindrical with hemispherical end caps in order to reduce the discontinuity stress at the junction between said cylinder and hemisphere. The thickness of the pressure vessel was critical to the design. The inner radius was decided to be 16.75 inches and the length of the cylindrical body was decided to be 120 inches to make sure the vessel would be able to hold a large volume of fluid. In order to make sure that the pressure vessel would be able to hold high pressure and thermal loads, it was decided to make the vessel thick relative to the inner radius of the vessel. The thickness was conservatively set to 3 inches. Thus, the outer radius was set to be 19.75 inches.
Figure 5: Detail of Pressure Vessel Geometry
3.2 Loading Conditions
The loading conditions that were put upon the pressure vessel were that of a pressure 1,500 psi, an internal temperature of 500 degrees Fahrenheit above an arbitrary initial temperature, and an external temperature equal to this initial temperature, or 0 degrees Fahrenheit above the initial temperature. It is assumed that the vessel is not thermally stressed at this initial temperature.
21 To calculate the finite element analysis pressure stresses, a static stress analysis needs to be performed. The loading conditions for this analysis can be seen in Figure 6.
Symmetry Boundary Symmetry Boundary Condition at y = 0 Condition at x = 0
X-Z Symmetry Plane Internal Pressure Loading, 1500 lb/in2
Cylindrical Axis of Symmetry
Figure 6: Pressure Loading for the Vessel
The pink triangles seen on inner face of each of the components define a surface where the pressure should be applied to the vessel. The orange triangles on the sides represent the boundary conditions that are applied to those sides. Seeing as this model is one-eighth of the full vessel, the boundary conditions are those of symmetry. To get the temperature distribution needed to calculate the thermal stresses, one needs to perform a steady state heat transfer analysis. The heat transfer analysis requires different boundary conditions than the pressure model. The boundary conditions that are used in a steady state heat transfer model are shown in Figure 7.
22 External Temperature Boundary Condition, T=0
Internal Temperature Boundary Condition, T=500
Figure 7: Thermal Loading for the Vessel
The steady state analysis required that all loads shown be applied to a steady- state heat transfer solution. The outer surface had to have a temperature of 0 degrees applied to it, seeing as otherwise the steady-state solution would have the entire model be at a uniform 500 degrees.
3.3 Materials
Material choice was very important when designing a functionally graded component to be optimized. In order to properly guide the process of structural optimization, it is necessary to reasonably approximate the desired material distribution for the component. In order to help with this, theoretical solutions for radial and hoop stresses in functionally graded spheres and cylinders were obtained. The material distributions assumed that the three varied material properties each had the same exponent applied to its power law distribution function. If one solves for the thermo- mechanical problems defined in (Eslami, 2002) and (Eslami, 2004), one will find that the highest stress in a thermally loaded pressure vessel will be the hoop stress in the cylindrical portion. The value of the exponent is varied from –2 to 3, and the result for the hoop stress for the cylindrical portion is listed on the next page.
23 M = -2
M = -1
M = 0
M = 3
M = 1 M = 2
Figure 8: Hoop Stress Divided by Pressure Load for Various Material Distributions in a Functionally Graded Cylinder
From this, one can see that the material distribution that results in the least overall stress in the vessel would be that of a linear material variation (M=1). Therefore, it was decided that a linearly varied material would be used to perform the finite element analyses and the structural optimization. The outside of the vessel would be purely metallic and the material properties would linearly decrease in magnitude throughout the thickness as it approaches the inner surface. The material that was chosen for the outer surface was AISI 1050 Steel. When searching for ceramics commonly used with steels, such as Zirconia, it was found that the chosen material properties were slightly smaller for the ceramic than the steel itself. (Buschow et al., p. 3407) Therefore, this material distribution was decided to be sufficient for the stated objective of having metallic properties at the outer surface and nearly ceramic properties at the inner surface of the vessel.
24 The typical properties for AISI 1050 Steel, as found on Matweb, are as follows:
lbf E 30106 in2 in 7 106 inoF Btu k 30 hr ftoF
When the exponents are set to 1, the coefficients that need to be used for the FGM material property equations in Section 2.3 are as follows:
m1 m2 m3 M 1 lbf E 1.519106 o in2 3.4 Modeling Material 7 in o 3.5410 o Properties in in F Btu Hypermesh k 1.519 o hr ftoF Seeing as the structural optimization was performed with finite elements, it is not possible to create a smooth transition in material properties from one side to the other. Instead, the thickness of the vessel needs to be divided into several layers of elements, each layer with its own material properties. If the layers are thin enough, the discontinuities in material properties will be small enough to approximate the material distribution sufficiently. (Bhasyam, p.129) However, the thinner the layers are, the more layers will be needed, and therefore more materials. In order to make all of these materials, a spreadsheet was made to determine the material properties for each layer. Each layer is assumed to be equivalent in thickness and the material property for each layer is determined from the FGM material equations with the exponents set equal to 1 and the radius set equal to the mean radius of each layer. These material values were then exported to text files that were translated to input files for the Hypermesh preprocessor to use. The source code for the exporting and
25 translating routines are in the Appendix. The input files were imported into Hypermesh to make new materials and properties to house these materials. Finally, the material properties have to be applied to the model itself. This was accomplished by use of a macro furnished by Dan Pusey of Altair, the company that makes of Hypermesh. If one selects the innermost layer of elements and the amount of element layers that will need to have material properties assigned to them, this macro allows the user to assign it to the beginning layer and then each layer on top of it until the set amount of layers has been reached. The source for this macro is attached in the Appendix.
26 4. Finite Element Modeling
4.1 Element Shape Considerations
When considering how to mesh an object, one of the most basic decisions that have to be made is the shape of the elements to be used in the finite element model. It was decided that the mesh should use quadrilateral or brick elements, as opposed to triangular or tetrahedral elements. Triangular elements are easy to fit to most meshes and are especially useful for complex shapes. They also are not as sensitive to initial element shape as the rectangular elements are. On the other hand, the triangular and tetrahedral elements are usually overly stiff and need extremely fine meshes to obtain accurate results. In addition, it has been noted by Abaqus that triangular/tetrahedral elements exhibit slow convergence with mesh refinement. The quadrilateral elements require more effort from the mesh designer in order to keep the shape of the elements approximately rectangular. However, quadrilateral and brick elements, when meshed properly, provide better convergence when meshes are refined. Finally, the case of this material distribution gives quadrilateral and brick elements a unique advantage. The material is being varied throughout the thickness of the vessel, and seeing as finite elements require constant material properties for each element, there will need to be a simulation of the material distribution by making layers of elements with the same material properties. Within the architecture of Hypermesh, which is being used to make the mesh for these models, it is much easier to layer the materials using quadrilateral or brick elements. Therefore, quadrilateral/brick elements were decided to be optimal for the design.
4.2 Element Order Considerations
Another basic decision to be made when designing a mesh is the order of the elements being used in the analysis. The order that is being referred to is the order of the interpolating polynomial being used in the shape function for the element. For example, a first order element has linear equations defining the shape functions and a second order element has quadratic equations defining the shape functions.
27 An important aspect of choosing the proper element order is to consider the order of the equations that will define the output of the problem being looked at. After all, the shape functions are used to define the relation of the displacement of one node on an element to another. These displacements are used to calculate strain, which is used to find stress. If the element order is too low, the results may lose accuracy due to the model not being able to model the behavior properly. If the element order is too high, precious computing time is wasted. From the analytical solution, it is shown that the stress results for the pressure loading and thermal loading have non-linear terms within them. Therefore, models with elements that have higher-order output would be more suitable for conforming to the analytical solution than a model with an equivalent mesh of lower-order elements. Quadrilateral or brick elements have shape functions that make first-order elements have linear stress/strain output and second-order elements have quadrilateral stress/strain output. That is why second-order elements were chosen for the model in spite of the added nodes and degrees of freedom.
4.3 Axisymmetric Model
Now that the element shape and order have been decided upon, it must be decided how the model should be represented. The problem being solved has geometry that was decided to be too complex to be satisfied by a 1-D representation. The geometry and the loading conditions are axisymmetric, so a 2-D representation was considered for the solution. After all, using a 2-D model would drastically reduce the amount of elements needed for the solution compared to a 3-D model, thereby reducing the computing time and effort. To test whether using a 2-D mesh is viable, a sample case was used. In this case, one quarter of the model was modeled. The mesh for the model was performed using CAX8 elements. These elements are 8-noded biquadratic axisymmetric solid elements without twist. The mesh for the model was set at 32 elements through the thickness. This mesh is too fine to be seen from a full-model perspective, so a detail of the method of meshing is in Figure 9. The loading conditions for the finite element model are shown in Table 1.
28 Figure 9: Meshing Detail for Axisymmetric Model
Geometry Inner Radius, Cylinder and Sphere 16.75 inches Outer Radius, Cylinder and Sphere 19.75 inches Body Length, Cylinder 60 inches, along the Y-axis Functionally Graded Material Properties E E r E 1.519106 lbf Modulus of Elasticity o , o in2
7 in r 3.5410 o Coefficient of Thermal Expansion o , o in F Btu Thermal Conductivity k k r , ko 1.519 o hr ftoF Boundary Conditions
lbf P=1500 in 2 at inner radius Pressure lbf P=0 in2 at outer radius T=500 o F at inner radius Temperature T=0 o F at outer radius Y-Z Symmetry Plane at X=0 Symmetry Planes X-Z Symmetry Plane at Y=0 Table 1: Finite Element Model Inputs
29 With the given mesh and loading conditions, the 2D model was sent through ABAQUS to find the solution for pressure loading. In order to gauge whether the output being generated was realistic, the analytical solutions for a thick cylinder and thick sphere under pressure loading were compared to the model. The finite element data points were taken at the regions that had symmetry boundary conditions, as shown in Figure 6 in Section 3.2. These data points were chosen because they were as far away from the interaction at the sphere-cylinder junction as possible and would therefore be the best points to compare to the solutions for isolated hemispheres and cylinders. The graphs in Figures 10 and 11 compare the finite element solutions to the theoretical solutions.
30 Radial Stress for FGM Cylinder, Axisymmetric Model
0 16.75 17 17.25 17.5 17.75 18 18.25 18.5 18.75 19 19.25 19.5 19.75
-500
-1000 )
i -1500 s p (
FEA MODEL s s
e Analytical r t
S -2000
-2500
-3000
-3500 Radius (in)
Hoop Stress for FGM Cylinder, Axisymmetric Model
80000
60000
40000 )
i 20000 s p (
FEA MODEL s s
e Analytical r t
S 0 16.75 17.25 17.75 18.25 18.75 19.25 19.75
-20000
-40000
-60000 Radius (inches)
Figure 10: Comparison of Axisymmetric Model to Analytical Solution, FGM Cylinder
31 Hoop Stress for FGM Sphere, Axisymmetric Model
80000
60000
40000
20000 ) i s p (
FEA MODEL s 0 s
e Analytical r 16.75 17.25 17.75 18.25 18.75 19.25 19.75 t S
-20000
-40000
-60000
-80000 Radius (inches)
32 Radial Stress for FGM Sphere, Axisymmetric Model
4000
2000
0 16.75 17 17.25 17.5 17.75 18 18.25 18.5 18.75 19 19.25 19.5 19.75 )
i -2000 s p (
FEA MODEL s s
e Analytical r t
S -4000
-6000
-8000
-10000 Radius (in)
Figure 11: Comparison of Axisymmetric Model to Analytical Solution, FGM Sphere
From these graphs, it can be seen that the cylinder conforms very well to the analytical solution, while the hemispherical end cap does not. The most dramatic demonstration of this is the comparison of radial stresses. As was discussed in the analytical solution, the boundary conditions for the radial stress for a thick sphere should be -1500 psi at the inner radius and 0 psi at the outer radius. It can be seen that this is not the case for the data points on the hemispherical end cap, with stresses being at -5555 psi at the inner radius and 2307 psi at the outer radius. These inconsistencies show that the elements are not reliable for showing the stress at the sphere. This is likely due to the fact that these elements are located at x = 0, which is the centerline for the axisymmetric model. Seeing as the viability of the elements depends on how well they conform to the analytical solutions that were previously solved, axisymmetric elements may not sufficient for the problem at hand. Therefore, a solid element model should be created to compare these results to. If there is a significant improvement in accuracy, a solid element solution will be used for this thesis.
33 4.4 Solid Element Model
To remedy the problems encountered in the axisymmetric model, a solid element model was created. This adds a level of complexity to the meshing of the model, as well as increasing the amount of elements needed for the solution. The first attempt at modeling this was that of spinning the axisymmetric model around its centerline. The model was meshed so that it would have 12 elements made on the path of the 90-degree spin. It was decided that having 12 elements along the spin would be able to more than sufficiently model the curve that the spin will be defining. This mesh can be seen in Figure 12.
Figure 12: Solid Mesh Created by Spinning Elements, Top View and Isometric
The loading conditions shown in Figures 6 and 7 were applied to this model and the model was run through a static stress/displacement analysis in ABAQUS. Upon reviewing the results, it was seen that the results from this analysis were much like those from the axisymmetric model. This was likely due to the fact that there were many elements that shared a single node. In previous experience with finite element analysis, having many elements share one common node leads to problems with convergence on a proper answer. Therefore, this mesh was decided not to be sufficient for the problem at hand. In order to counteract this problem, it was necessary to make a mesh that doesn’t have the node-sharing problem. In order to accomplish this, a new methodology of creating a mesh was needed. The problems being encountered were on the spherical
34 cap, not the cylinder. Therefore, it was decided that the sphere should be concentrated on and that the cylinder could be looked at later. In addition, the mesh should be looked at first as a 2D skin of elements defining the inner surface of the sphere. To reduce the problems with convergence, it was decided to make it so only one element of this 1/8th skin model should be in contact with the node defining the problem area at the top of the hemispherical cap. This proved to be difficult to accomplish by simply auto-meshing an inner surface. Thus, to improve the results of this mesh, some work had to be done on the surface. To do this, an arc was made that split the surface into two equal halves. The midpoint was found for this arc, and other nodes were made at x=0 and z=0, both of which were projected from the midpoint of the arc. These nodes are shown as yellow dots in Figure 13 and the arcs that were used are defined by the red plus signs on them. The node on the mid-arc was used as a reference node for lines that were made with the two other nodes. These lines were then projected onto the surface above and were used to along with the arcs to split the surface into 3 smaller surfaces. Splitting the surface in this way made it possible to define the surface solely with quadrilaterals while eliminating the shared node problem. The mesh of the topmost of the surfaces (the surface that touches the axis of symmetry) will be used as a metric for the overall surface density for the model throughout this section. The an example of overall mesh of the cap, which has a topmost surface density of 3 x 3 elements, is shown below in Figure 13.
Topmost Surface
Figure 13: Surface Definitions for Spherical Cap (Isometric View) and Surface Mesh (Top View)
35 To create a solid mesh, it was necessary to define the outer surface as well. This was accomplished by defining the surfaces in the same manner as the inner surfaces. After these surfaces were created, the elements on the inner surface were projected to the outer surface. The solid elements were dragged from the inner surface elements to the outer surface elements with a prescribed density of elements through the thickness. This method proved to be sufficient for making a solid mesh that fit the geometry well while reducing the error in the numerical solution. It was then decided that the mesh should be modified so as to more accurately capture the stresses at the discontinuities. One of the best ways to do this in finite element analysis is to increase the amount of elements near the discontinuity. This can be accomplished by biasing the mesh so that the density of the elements gradually becomes more refined as it gets close to the discontinuity. There were several methods of biasing available in the Hypermesh preprocessor, but the only method that allowed the mesh to be biased toward the point of interest while maintaining an acceptable mesh was linear biasing. Exponential biasing tended to create abnormal meshes and bell curve biasing biased the mesh toward the center of the surface that the mesh was defined on instead of biasing toward the point of interest. Once the mesh for the sphere was found adequate, the cylinder was attempted. The first attempt was that of defining an inner surface mesh and projecting it out to an outer surface, much like the sphere. However, it was found that when one uses a large bias in this way, there is undesired curving or abnormalities in the mesh, as can be seen in Figure 14.
36 Figure 14: Problems with High Linear Biasing with Cylinder Surface Meshes, Outer Surface of Cylindrical Body
To remedy this problem, a different method of creating a solid cylinder mesh was required. This was accomplished by creating a face of 2D elements on the sphere at the surface that defines where the cylinder and sphere should meet. These 2D elements were then dragged along a line that defined the length of the cylinder to create 3D elements. This method also allowed for biasing of the mesh, and it was found that this having a high linear bias did not have any abnormalities in the creation of the mesh. An example of this biased mesh is shown in Figure 15.
Increasing Linear Bias
Increasing Linear Bias
Figure 15: Biasing for the Pressure Vessel, Outer Surface of Pressure Vessel
Several models were made to study the relationship between refining the mesh and converging to the analytical solution. These models were varied in both the density of the surface mesh and the through-thickness mesh. Table 2 shows the amount of elements, nodes, and degrees of freedom for each of the meshes used.
Top Surface Layers Through Number of Number of Number of Degrees Mesh Density the Thickness Elements Nodes of Freedom 3 Elements 12 layers 4044 18618 55854 24 layers 8088 36150 108450 by 32 layers 10784 47838 143514 3 Elements
37 4 Elements 12 layers 4704 21613 64839 24 layers 9408 41965 125895 by 32 layers 16384 72233 216699 4 Elements 5 Elements 12 layers 7764 35188 105564 24 layers 15192 66860 200580 by 32 layers 20256 88476 265428 5 Elements Table 2: Number of Elements, Nodes and Degrees of Freedom for Analysis Models
4.5 Choosing Element Type
When choosing a proper mesh, one needs to consider not only if the results match the analytical solution, but also if the solution is being done in a manner that is efficient for file size and solution time. One way to make sure that these criteria are being met is to choose the proper element type. When using Abaqus, there are many different kinds of quadratic brick elements that are available. Most are highly specialized for specific loading conditions. These are not well suited for general loading conditions, as they perform calculations that may not be applicable for the model. Therefore, elements must be selected which are suited for static stress/displacement and steady state heat transfer calculations. For heat transfer alone, there is the DC3D20 element. For static stress/displacement alone, there are the full integration C3D20 and reduced integration C3D20R. For a coupled heat transfer and stress/displacement analysis, there are the full integration C3D20T and reduced integration C3D20RT. Before choosing one type of element, it is necessary to consider the benefits for each type of element. A common way of getting acceptable results while reducing the file size and computing time is to use reduced integration elements instead of full integration elements. The reduced integration that is being referred to is the reduced amount of integration points that are used in the element, using only the vertices and not the mid-side nodes of the element. The shape function, however, is retained. The reduction in time for reduced integration elements is well documented when performing straight stress/displacement analyses. However, this case is fairly unique in as much as there are no reduced integration elements available for heat transfer alone. It will be seen if the effect of applying a non-reduced thermal solution to a reduced integration
38 stress/displacement element will have a degrading effect on the reduction of file size and solution time. When it comes to performing the heat transfer and stress/displacement calculations separately or simultaneously, one needs to weigh the ease of use with the computing power needed to perform the finite element analysis. The coupled elements (C3D20T, C3D20RT) are able to have both the heat transfer and stress/displacement loading conditions applied on one model and have a full solution after using Abaqus only once. However, if one is using Hypermesh to create the model, one needs to manually edit the input (*.inp) file of that will be used with Abaqus so that the load step couples the stress analysis with the steady state heat transfer. The specific alteration is bolded and italicized in Figure 16:
Excerpt From Sample HyperMesh Input File: **HMNAME LOADSTEP 1 pressure *STEP *STATIC *DSLOAD ID,P,1500
Required Editing of Input file: **HMNAME LOADSTEP 1 pressure *STEP *COUPLED TEMPERATURE-DISPLACEMENT, STEADY STATE *DSLOAD ID,P,1500
Figure 16: Required Alteration of Abaqus Input files for Coupled Temperature-Displacement
The alternative is to create a model for heat transfer, apply heat transfer loading to the elements (DC3D20) and solve for the temperature distribution on the model. One then needs to make an exact duplicate of that model, change the element type (C3D20, C3D20R) along with the load step and loading conditions so that they are appropriate for static stress/displacement analysis. One then needs to modify the stress/displacement input file so that Abaqus will use the output of the heat transfer analysis to calculate the thermal expansion. The modification consists of adding text to the end of the input file of the format shown in Figure 17.
39 ** READ TEMPERATURE DATA FROM THE RESULTS FILE WRITTEN DURING ** THE STEADY STATE HEAT TRANSFER ANALYSIS (FROM THE THERMAL RUN) ** *TEMPERATURE, FILE=qvM1_3s12dt, BSTEP=1, BINC=1,ESTEP=1,EINC=1 ** ** ** ** PRINT OUT THE STRESSES TO AN ASCII FILE ** ** *EL PRINT,POSITION=AVERAGED AT NODES,FREQUENCY=5 S,TRESC ** ** ** ** PRINT OUT THE STRESSES TO A BINARY FILE ** ** *EL FILE,POSITION=AVERAGED AT NODES S,TEMP *EL FILE,POSITION=NODES SINV *EL FILE,POSITION=NODES S *NODE FILE,FREQUENCY=10 U ** ** ** *END STEP ***** Figure 17: Required Addition to Static Stress/Displacement Analysis Cards for Thermal Stresses
In this example, the thermal model that is being read from is “qvM1_3s12dt.fil”. The output of the heat transfer analysis (in the form of *.fil and *.odb files) can be rather big, so this method of solution will have a larger disk size after the solution is finished. The amount of memory used in the solution and the time of the solution are not so clear- cut, however. Table 3 summarizes the findings after using the prescribed element types on the mesh shown in Figure 15, with a topmost surface mesh of 3 x 3 elements and 12 layers of elements throughout the thickness of the vessel. The inputs for the models are those listed in Table 1.
40 Total CPU Minimum Memory to Time Required Minimize I/O Output File Element Type (seconds) Memory (MB) (MB) Size (MB)1 DC3D20 5 38 116 (Note 2) C3D20 32 113 667 53.4 C3D20R 25 112 637 41.1 C3D20T 62 195 1463 23.2 C3D20RT 52 195 1433 8.8 Notes: 1.) The output files include the *.odb and *.fil files. 2.) The output files for the DC3D20 elements are accounted for in the C3D20 and C3D20R output files. Table 3: Run Time and Memory Usage Comparisons for Element Types
Several conclusions can be made from this table. First of all, one can see that there is a significant reduction in output file size when one uses coupled thermal- displacement elements. There is also a significant increase in memory required and time used while finding the solution. This is a fairly coarse mesh, so if any refinement in the mesh is required, this memory allocation will have to be even greater. This could eventually be quite a handicap when finding a solution, to the point where the memory limitations could be exceeded. One can also see that there is a definite reduction in all metrics being tracked when one compares the solution using C3D20R elements to the solution using C3D20 elements. However, the difference is not significant, save perhaps for the size of the output files. Therefore, this benefit of the reduced integration element is minimal, at least in this regard. Another important aspect of choosing the type of element used for an analysis is finding how well it converges to an existing exact solution. In this case, the elements are compared to the solution that is found using the analytical solutions outlined in Chapter 2. The regions in the model where results were taken are shown in Figure 18. The stress results can be seen in Tables 4 and 5.
41 Outer Surface
Inner Surface Sampling Region for Sampling Region for Hemispherical Cap Cylindrical Body
Figure 18: Stress Sampling Regions in the Pressure Vessel Model
Element Radial Stress (ksi) Hoop Stress (ksi) Type Inner Outer Min Max Inner Outer Min Max Used Surface Surface Stress Stress Surface Surface Stress Stress C3D20 -1.51 -0.07 -5.58 -0.07 -57.65 60.37 -57.65 60.37 C3D20R -0.98 -0.63 -5.56 -0.63 -57.92 60.70 -57.92 60.70 C3D20T -1.51 -0.07 -5.58 -0.07 -57.65 60.38 -57.65 60.38 C3D20RT -0.98 -0.63 -5.56 -0.63 -57.92 60.70 -57.92 60.70 Analytical -1.50 0 -5.53 0 -56.28 60.55 -56.28 60.55 Solution Table 4: Comparison of Element Solutions to Analytical Solution for the Hemispherical Cap
42 Element Radial Stress (ksi) Hoop Stress (ksi) Type Inner Outer Min Max Inner Outer Min Max Used Surface Surface Stress Stress Surface Surface Stress Stress C3D20 -1.51 -0.03 -3.19 -0.03 -49.80 68.27 -49.80 68.27 C3D20R -1.38 -0.2 -3.19 -0.2 -49.91 68.48 -49.91 68.48 C3D20T -1.51 -0.03 -3.19 -0.03 -49.80 68.27 -49.80 68.27 C3D20RT -1.37 -0.2 -3.18 -0.2 -49.37 68.65 -49.37 68.65 Analytical -1.50 0 -3.18 0 -48.64 68.63 -48.64 68.63 Solution Table 5: Comparison of Element Solutions to Analytical Solution for the Cylindrical Body
The results for the elements are very similar and close to the values that are predicted by the analytical solutions. The solid element finite element solutions at the hemispherical cap are significantly closer the exact solutions than the solutions for the axisymmetric model. Therefore, it was decided to use the solid elements when working with the functionally graded model. Looking further at the finite element results, there is one difference in the solid element models that stands out. The reduced integration elements do not exactly meet the radial stress boundary conditions set by the analysis. This is more dramatic in the hemispherical model than the cylindrical model, but not as dramatic as the axisymmetric model. This could be due to the fact that the reduced integration elements use less numerical integration points to find the nodal solutions, possibly making the nodal stress results less accurate. Coupled temperature-displacement elements, such as C3D20T or C3D20RT elements, will likely be sufficient for the analysis, but a more comprehensive study of the convergence of the elements to the analytical solution must be made beforehand. If it is found that a more dense mesh is needed, these elements will take too much processing time and space.
4.6 Element Convergence
As was seen in the previous section, a relatively coarse mesh seems to conform well to the analytical solution. However, the way that the functionally graded material properties are modeled in the finite element model will create stress discontinuities through the thickness, seeing as there are different material properties in each layer. The
43 stress discontinuity being referred to is derived at nodes common to two or more elements by finding the greatest difference in calculated nodal values from the contributing elements. Having large stress discontinuities can lead to less accurate results, and one common way to reduce the discontinuities is by refining the mesh of the model. One way to possibly reduce discontinuities is to refine the mesh through the thickness. Another way would be refining the surface mesh that is projected through the thickness. Both of these forms of mesh refinement were studied to see the effects they had on the stress discontinuities throughout the model. The first type of refinement was done in three models. These models had 12, 24 and 32 elements through the thickness. The surface mesh was the same as shown in Figure 15, with a topmost surface mesh of 3 x 3 elements. The results can be seen in the Appendix, but the finite element solutions for all of the meshes fit the analytical solutions very well. The most dramatic of the deviation from the analytical solution was for the radial stress in the hemispherical caps. These results can be seen in the
44 Figure 19.
Figure 19: Convergence of Elements to Radial Stress Distribution for an FGM Hemisphere
It can be seen that the only model that noticeably deviates from the analytical solution is the one with 12 elements through the thickness. Even then, it is not much of a noticeable deviation. Judging from this alone, one would think that the coarsest model would be sufficient for the solution. However, this only shows that the solution is correct at the points of interest. One should also check the discontinuities in the model to see if the mesh quality is sufficient to ensure that the model will give precise results throughout the model. Figure 20 shows the discontinuity plot for the coarsest mesh.
45
Figure 20: Stress Discontinuity Plot for Thermo-Mechanically Loaded FGM Vessel (Inside Surface, Outside Surface, and Isometric View)
As is to be expected, the point at which the spherical dome meets the cylindrical body is the point of maximum discontinuity. Specifically, the outer fiber of the vessel is the area of interest. The discontinuity is about 554 psi which, when compared to the maximum stress of 67120 psi, is not significant. However, once the structural optimization takes place, this stress discontinuity may be increased due to the morphing of the mesh. Therefore, it may be of importance to find if this discontinuity can be reduced by way of increasing the density of the mesh, also known as h-refinement. This refinement can be achieved two ways. The first has already been covered, which is by increasing the density of the mesh through the thickness. The second method is by increasing the density of the mesh on the surfaces that are projected through the thickness. For ease of description, this second method of refinement shall
46 continue to be defined by the density of elements at the topmost (y-direction) surface of the hemispherical cap. These elements are used to define the size of the other elements in the mesh, both the hemispherical head and the cylindrical body. The finite element model inputs used are those listed in Table 1. The discontinuities of the von mises stresses in the models are shown in Table 6.
Top Surface Layers Maximum Von Mises Stress, Maximum Von Mises Stress Mesh Density Through the No Averaging (ksi) Discontinuity (psi) Thickness 3 Elements 12 layers 67.12 554 24 layers 67.29 578 by 32 layers 67.33 582 3 Elements 4 Elements 12 layers 67.10 502 24 layers 67.26 527 by 32 layers 67.32 596 4 Elements 5 Elements 12 layers 67.09 456 24 layers 66.95 704 by 32 layers 67.11 533 5 Elements Table 6: Comparisons of Von Mises Stress Results for Varying Mesh Densities
It can be seen from this Table 6 that the h-refinement did not have dramatic effects on the maximum value of the von mises stress or the von mises stress discontinuities. The biggest change in the behavior was at the mesh that is 5 by 5 elements at the top surface and 24 element layers through the thickness. This is likely due to variations in element shape changing the quality of the output of the elements. While this is minimal, when coupled with the fact that the amount of elements would make it computation-heavy, the increase in discontinuity makes it less than optimal. The next biggest change is the difference in stress for the 3 by 3 and 4 by 4 surface meshes is the increase in thickness density from 12 layers to 24 layers. It is minimal for the starting mesh, and the difference is not big enough to warrant consideration when going through the steps of mesh optimization. The results are sufficiently the same when comparing the 3 by 3 and 4 by 4 surface meshes, so the less computationally heavy
47 surface mesh will be preferred. Therefore, the optimization routine shall be performed on a model with a 3x3 top surface mesh with 12 layers through the thickness to save computation time with minimal effects on the results.
48 5. Structural Optimization
5.1 Deciding on a Solver
In order to perform structural optimization, one needs a proper optimization solver. Abaqus is an effective finite element solver that has a robust collection of element types and is able to solve even computationally heavy finite element models. This makes Abaqus optimal for determining the viability of element types and mesh densities. However, the standard Abaqus package does not contain a robust structural optimization solver. There are third-party packages that can be used with Abaqus to improve the structural optimization routine, but those were not available to me. The HyperWorks package that contains Hypermesh also contains a finite element structural optimization solver called OptiStruct. This program uses RADIOSS as its finite element solver and runs an optimization routine to modify the model in order to achieve the objective given. OptiStruct is quite robust, with many different types of optimization that can be performed on 2-D and 3-D models. These optimization types include the topology optimization and free shape optimization used in this thesis. Hypermesh has the ability to convert models that are designed for one solver so that it is suited to for use in other solvers. Therefore, Hypermesh can convert the Abaqus models to be suitable with OptiStruct and RADIOSS with minimal rework required. Therefore, OptiStruct was a natural choice for performing structural optimization on the finite element model.
5.2 Optimization Model in OptiStruct
The types of elements that OptiStruct can use are rather limited in scope compared to Abaqus. For 3-D quadrilateral elements, there are only the DHEXA8 and CHEXA elements. DHEXA8 elements are only applicable for 1st order elements, and CHEXA elements are applicable for 1st order and 2nd order elements. Therefore, 2nd order CHEXA elements are needed to make a model that is similar to the ABAQUS model. It should be noted that CHEXA elements have coupled heat transfer and displacement solutions, much like the C3D20T and C3D20RT elements in ANSYS.
49 The model also needs to have a particular type of material applied to it. The type of material that can be applied to a model depends on what kind of property can be used in the model. After all, materials are assigned to a model by assigning them to properties. The CHEXA elements are only allowed to have PSOLID properties, which define properties of solid elements. In turn, PSOLID properties can only have MAT1 and MAT9 material property definitions. Both MAT1 and MAT9 define the material properties for linear, temperature-independent materials. However, MAT1 defines isotropic materials and MAT9 defines anisotropic materials. Seeing as the materials are assumed to be isotropic at each element layer, MAT1 material properties were chosen. MAT1 material properties allow the user to define properties such as the modulus of elasticity, poisson’s ratio, and the linear thermal expansion coefficient. In order to add heat transfer properties to the model, the MAT1 material property has to be use in conjunction with the MAT4 material property. MAT4 defines a temperature-independent thermal conductivity, which is sufficient for the loading case defined. A trial run was performed to see if the RADIOSS finite element solver would perform similarly to ABAQUS. The input values for the model are shown in Table 7. The results from the RADIOSS analysis are taken at the points shown previously in Figure 18 and are shown in Table 8.
50 Mesh Details Elements Used CHEXA, 20 Noded Brick Elements Top Surface Mesh Density 3 elements by 3 elements Mesh Density Through Thickness 12 elements Number of Elements 4044 Number of Degrees of Freedom 55791 Geometry Inner Radius, Cylinder and Sphere 16.75 inches Outer Radius, Cylinder and Sphere 19.75 inches Body Length, Cylinder 60 inches, along the Y-axis Functionally Graded Material Properties E E r E 1.519106 lbf Modulus of Elasticity o , o in2 3.54107 in Coefficient of Thermal Expansion o r , o ino F Btu Thermal Conductivity k k r , ko 1.519 o hr ftoF Boundary Conditions Analysis Loadcases Heat Transfer and Stress/Strain
lbf P=1500 in 2 at inner radius Pressure lbf P=0 in2 at outer radius T=500 o F at inner radius Temperature T=0 o F at outer radius Y-Z Symmetry Plane at X=0 Symmetry Planes X-Z Symmetry Plane at Y=0 Optimization Criteria Objective Function Reduce Total Volume Von Mises Stress Constraint Yield Strength of AISI 1050 Steel, 75 ksi Maximum Iterations 30
Table 7: OptiStruct/RADIOSS Finite Element Model Inputs
Radial Stress (ksi) Hoop Stress (ksi) Portion Table 8: RADIOSS Model Analysis Results Inner Outer Min Max Inner Outer Min Max Analyzed Surface Surface Stress Stress Surface Surface Stress Stress Cap -0.97 -0.63 -5.56 -0.63 -57.83 60.68 -57.83 60.68 Cylinder -1.38 -0.20 -3.20 51-0.20 -49.83 68.45 -49.83 68.45 These results closely match the results of the C3D20RT elements in the Abaqus solution, which have been shown to be sufficient for the finite element analysis in Chapter 4. Therefore, the RADIOSS finite element solver used in OptiStruct is adequate for the loading case given.
5.3 OptiStruct’s Iterative Solution for Structural Optimization
Structural optimization in OptiStruct follows an iterative procedure known as local approximation to come to a solution. The method that determines the solution is shown in Figure 21.
START
Analyze problem in Converged YES Finite Element Analysis Solution? using RADIOSS NO STO P Up to 80X Design Sensitivity Analysis
Solve for approximate optimization problem formulated using sensitivity information
Figure 21: OptiStruct Iterative Solution
The optimization routine uses an objective function, f(x), and a constraint function, g(x), to determine the suitability for a response while going through this routine. The problem for the minimization of an objective function is shown in the following equation:
min f (x) f x1 , x2 ,..., xN where :
g j (x) 0 j 1,...,m L U xi xi xi i 1,...,n
52 Equation 27: Minimization Routine for OptiStruct
The x variable is the vector of design variables while xL and xU are its lower and upper limits, respectively. The design variables applied to this vector depend on the type of optimization being performed. For topology optimization, the design variables are the element densities. For shape optimization (including free shape optimization), the design variables are the factors in a linear combination of shape perturbations. The program seeks to find a minimal f(x) while making sure that g(x) is as close to zero as possible. The program considers the constraint function active if the constraint is satisfied exactly (g=0). If g<0, the constraint is not active in determining the suitability of the solution and if g>0, the constraint is violated, forcing the solver to attempt to remedy this problem while minimally affecting the objective function. It is assumed that only small steps occur in the design with each optimization step, resulting in a local minimum. To reduce the run time, the biggest changes in the design occur in the first few optimization steps.
5.4 Design Sensitivity Analysis in OptiStruct
The update of the design in each of the optimization steps is established using sensitivity information. The three different methods by which this is accomplished are the optimality criteria method, the dual method and the primal feasible directions method. The optimality criteria method is used in common topology optimizations that use minimal compliance with a volume or a volume fraction constraint. The optimality criteria method can also be used in topology optimizations that minimally comply with a mass or mass fraction constraint, but this will not be used in this thesis. The use of dual and primal feasible directions methods depends on the number of constraints and design variables. Problems with a number of design variables that exceeds the number of constraints, as is common in topology optimization, tend to benefit from the dual method. Problems with a number of constraints that exceeds the number of design variables, as is common in shape optimization, tend to benefit from the primal method. OptiStruct, however, makes the decision between the dual and primal feasible directions methods automatically.
53 5.5 Convergence in OptiStruct
There are two ways that a solution can converge in OptiStruct, through the regular method and through the soft method. It is only necessary that one of these methods be satisfied for the solution to be deemed sufficiently converged. The regular method of convergence is dependent upon meeting the convergence criterion for two consecutive iterations. To meet this criterion, the change in the objective function must be less than the objective tolerance and any violation of the constraint is less than one percent. This requires at least three iterations, seeing as convergence is based on the comparison of values obtained from an analysis at the latest design point. The soft method is dependent upon having little or no change in the design variables for two consecutive iterations. The objective and the constraint are not used in evaluating the latest design point. Thus, soft convergence needs one less iteration than regular convergence. In addition, soft convergence does not necessarily leave the user with a feasible design. It does, however, prevent the solver from performing further iterations on an optimization solution that cannot satisfy the objective and/or the constraint.
5.6 Topology Optimization Method in OptiStruct
The stated objective of structural optimization in this thesis is to reduce the total volume of material used to create the pressure vessel. The optimization model will be constrained by making sure that the material stress limit is not violated. OptiStruct is able to accomplish this by performing topology optimization on the model. OptiStruct describes topology optimization as a mathematical technique that produces an optimized shape and material distribution for a structure within a given package space. OptiStruct performs a density method algorithm that alters the finite element model so that the discrete stiffnesses for each element will be altered through the element so as to show areas where material can be removed, thereby optimizing the user-defined objective under given constraints. There are limitations to the topology method of structural optimization in OptiStruct, however. If one decides to use a maximum stress value as a constraint for
54 the optimization model, as in this thesis, the stress constraint must be applied to the entire model and set to only one value. This is due to several phenomena that can occur, including singular topology where different materials with different permissible stresses meet in a structure and the possibility that the partial domain of a structure that has its own stress constraints could be eliminated, thus removing all stress constraints. Because of these problems, OptiStruct forces the user to use only one stress constraint that is applied to the entire model when performing topology optimization.
5.7 Density Method of Topology Optimization
Under topology optimization, there is an element density that should take a value of either 0 or 1, defining the element as either void or solid, respectively. In order to reduce the computational load required to optimize many discrete variables, OptiStruct uses a continuous variable between 0 and 1. This is accomplished in OptiStruct using either the homogenization or density method. Each of the processes has its advantages and disadvantages. The homogenization method is able to have designs form rapidly along lines of the force transmission path. However, this method also has more design variables than the density method and it is difficult to determine the orientation of the voids in a structure under multiple loading cases, such as the case proposed in this thesis. In addition, the homogenization method has limited availability for 2-D and 3-D elements. The density method is more general than the homogenization method and requires less design variables. In addition, the density method can be used for both isotropic and anisotropic materials, while the homogenization method can be used for only isotropic materials. Another important aspect to the density method is that it is the only method implemented for manufacturing constraints, such as symmetry. However, the density method tends to need more iterations to converge along the lines of the force transmission path. Judging from the advantages and disadvantages, the density method is needed for the solution for the pressure vessel. With the density method, the element density is used as a design variable. The stiffness of the material being optimized is assumed to be linearly dependent on this density. This is generally consistent with common materials.
55 An unfortunate side effect of optimization is the large areas of intermediate densities in the domain, creating large portions of elements with intermediate stiffnesses. These intermediate stiffnesses tend to make elements that have material properties that are not present in any real material. These types of solutions are not meaningful when one needs to keep the distribution of the material properties inside a domain within a certain gradient when manufacturing a FGM. To penalize the design for these fictitious materials, the density method introduces a power law representation for elasticity properties. This approach is used for both 2-D and 3-D elements in the form of the following equation: K() p K
Equation 28: Element Stiffness Penalization for Density Method of Topology Optimization
Where K and K represent the penalized and the original stiffness matrices of an element, respectively, is the element density and p is the penalization factor. The penalization factor is always greater than 1, but usually takes a value between 2 and 4. This factor is controlled by OptiStruct by default and tends to be increased as the optimization routine goes through its iteration in order to achieve a more discrete solution.
5.8 Topology Optimization: Free Form
In order to understand how OptiStruct topology optimization works, it is necessary to run a test case. In this test case, the Hypermesh model that was used in Abaqus will be modified to work in OptiStruct to see how the model would be modified to get an optimal material distribution without applying any constraints to the shape. Depending on the results, modifications may need to be made to fine-tune the optimization.
5.9 Defining the Optimized and Non-Optimized Regions
Only regions that are specified as being optimized will have topology optimization applied to them. OptiStruct accomplishes this by use of a collector called a design variable. This design variable specifies what type of structural optimization will be
56 performed. This variable also specifies which elements the structural optimization will be performed on. In this free form optimization, it was decided that all elements, and therefore the entire pressure vessel, would be optimized.
5.10 Defining the Load Sets for Structural Optimization
The method of assigning load sets for OptiStruct models is different than that of Abaqus, therefore it is necessary to create new load sets from the existing loading conditions. All of the load sets for the OptiStruct models were created using the inputs defined in Table 7. The first load set to take care of is steady state heat transfer. It should be noted that OptiStruct does not currently support iterations of heat transfer through the structural optimization process. This is not optimal for transient heat transfer, but for steady-state distributions it should be sufficient to use the same steady-state temperature distribution throughout. The heat transfer load case was solved in its own RADIOSS finite element model to find the initial temperature distribution using a non-iterative solution. This temperature distribution was translated into temperature “SPC’s”, or single point constraints, by importing the temperature distribution from the heat transfer finite element model by way using Hypermesh to import a *.fem file created using the excel VBA macro listed in the Appendix. This import allows each node of the optimization model to have its own temperature defined by a temperature “SPC”, effectively solving the steady state heat transfer for the optimization model. The second load set to take care of is thermal and pressure stress/strain. The type of load set used was linear static, which is the best suited for the type of analysis that is being performed on this model. The first load that was described was the displacement “SPC”, which was set to that of the symmetry boundary conditions at the planes of symmetry. The second load described was the distributed “LOAD”, which in this case is the pressure load applied to the inner surface of the pressure vessel. The third load to be defined was the thermal “TEMP” load, which in the case of this analysis is used to refer to the temperatures set at each node. After the creation of this load set, the optimization model was almost ready to be solved.
57 5.11 Define optimization criteria
As was stated before, the optimum design for this pressure vessel will be one in which the vessel has an objective function of minimizing material and a constraint to make sure the model does not exceed a stress limit. This is done in several steps in OptiStruct. The first step is to set the responses to be used as criteria for the structural optimization. With topology optimization, the stress response is automatically applied to all of the elements that are assigned to the design variable. The volume response, however, needs to be entered by use of the response editor in the Hypermesh preprocessor. This response needs to be set to a response type of “volume” with an additional descriptor of “total” being used so that the program will know that the volume being optimized will be that of the entire model, not just certain parts. The next step in defining the structural optimization is to assign objectives and constraints to these responses. The volume response was set in Hypermesh’s objective editor to “min.” This ensures that the optimization will strive to reduce the volume, and therefore the material, of the pressure vessel. The stress constraint for OptiStruct topology optimization is by default applied to the entire model and can only be a single von mises stress value. As was shown in Table 7, this constraint was set to the yield strength of AISI 1050 steel, 75 ksi. This is good for an initial study.
5.12 Results for Free Form Topology Optimization
The free form optimization was performed with all the required inputs and viewed using the HyperView postprocessor that comes in the HyperWorks package that Hypermesh, RADIOSS, and OptiStruct are parts of. Figure 22 shows the density plot of the elements after the optimization routine, shown using the HyperView post-processor for OptiStruct.
58 Figure 22: Element Density Plot for Free Form Topology Optimization (Side, Rear and Isometric Views)
The lower the element density, the less material is needed at that section. To better view what the final shape should be, HyperView has an option that removes all the sections of a plot that does not meet a minimum criterion for element density. In most OptiStruct examples, an isolation value of 0.3 is used for this end. Figure 23 shows this isolation plot.
59 Figure 23: Element Density Isolation Plot for Free Form Topology Optimization (Side, Rear and Isometric Views)
It can be seen from this that the free form optimization has a thin inner layer and a thin outer layer with a thick section of elements joining them at the junction between the hemispherical end cap and the cylindrical body. For the cylindrical body, there is a substantial amount of elements between the inner layer and the outer layer that do not meet the element density criteria. For the hemispherical cap, most of the elements outside of the inner layer of elements do not meet the criteria. While this design would technically hold in fluid, this is a less than optimal shape for a manufactured pressure vessel. To be specific, there must be a way to make it so there is no gap in between layers and that there must be a full inner surface at a radius of 16.75 inches.
60 5.13 Redefining the Optimized Regions for Topology Optimization
In order to maintain the amount of fluid contained in the pressure vessel as a constant, it is necessary to make it so the inner surface of the vessel is held constant at a radius of 16.75 inches. A simple way of accomplishing this is by ensuring that the elements that define the inner surface of the pressure vessel are not accounted for in the topology design variable. Fortunately, this is quite easy due with the way that this model was set up. Therefore, one can easily exempt the innermost layer of elements from the topology design variable.
5.14 Redefining the Design Criteria: Discreteness Parameter / Penalty Factor
There is an option called the discreteness parameter, which lets the user set the penalty factor (see Section 5.7) to be used in the optimization routine. The higher the value for the parameter, the closer the element densities converge to the extremities. The value was set to 3.0, which is the highest recommended value for solid models.
5.15 Results for Constrained Topology Optimization
The exclusion of the inner surface from the topology optimization ensured that the inner surface would not have its element density changed throughout all of the optimization iterations. However, there was still a slight abnormality in the solution. If one isolates the solution in HyperView so that all elements with a density less than 0.3 are excluded, the material distribution shown in Figure 24 is the result of the optimization.
61 Figure 24: Isolated Element Density Plot for Constrained Topology Optimization (Side, Outer Surface and Isometric Views)
As one can see, the inner surface has an element density of 1.0 and there are a significant amount of elements on the outer surface with an element density of 1.0. However, there are still elements with insignificant element density in between these surfaces. This is due to the solver needing to add volume to the solution after reducing the volume too far. Instead of adding volume to the inner surface of the model, it decided to add volume at the outer surface, where it seems it would be able to get the most stress relief for an equivalent amount of volume. In addition, one can see from the isolation plot that the element density is not even on one side compared to another. In order to attempt to make up for these problems with the model, manufacturing constraints need to be applied.
62 5.16 Added Manufacturability Constraints: Symmetry Plane and Supporting Design Parameters
As can be seen from Figure 24, the final design of the pressure vessel is not optimal for manufacturing. In order to improve the viability of the optimization solution, additional constraints must be made. One such manufacturing constraint is pattern grouping, which allows the user to define a single part of the domain that should be designed in a certain pattern. One such method is planar symmetry. For this design, there need only be one plane of symmetry. To define this plane of symmetry, OptiStruct needs to have an anchor node and a reference node be defined. A vector runs from the anchor node to the reference node and the plane of symmetry is normal to that vector and while passing through the anchor node. For this model, the plane of symmetry was defined as a plane that is at a 45-degree angle between the x and z directions and bisects the optimization model.
Reference Node Anchor Node
Plane of Symmetry
Figure 25: Symmetry Plane Definition in Optimization Model
63 Once the manufacturing constraint had been applied, there were extra modifications that need to be made to the model, such as adding the MINDIM parameter. This parameter sets a characteristic length that is the minimum length in any direction of any section of material or void. This eliminates common problems such as checker boarding of elements, in which there are many small sections of elements that are either voids or solids and are in close proximity to each other. In this design, it should help by forcing the model to void fewer elements near the inner and outer surfaces of the cylinder, decreasing the chance of having a void inside the cylinder wall. The MINDIM parameter enforces a solution in two iterative steps, each consisting of a number of its own iterations. The first step converges to a solution with a large number of semi-dense elements. The second step tries to refine this solution to a solution with fully dense elements. The recommended value for MINDIM is at least 3 times the average element size for all designable elements, with the average element size being defined as the average of the cubic root of the volume of the elements. This recommended value is enforced when using manufacturability constraints such as symmetry planes. To support the MINDIM parameter, additional parameters are required. First of all, the number of iterations was increased to 50 for only this model to suit the needs of the enforced iterative steps. In addition, the discreteness parameter described in Section 5.14 needs to be kept at 3.0 in order to support the second iterative step of the solution. The results of the symmetry constraint, along with all of the extra model modifications are shown in Figure 26.
64
Figure 26: Element Density Isolation Plots for Model with Symmetry Constraint (Outer Surface, Rear, and Isometric Views)
One can see that the symmetry is greatly improved. In addition, the voids inside the thickness of the model have been eliminated. Still, there is a significant amount of elements in the thickness of the cylinder that have values that are of intermediate element stiffness. This, along with the lack of elements with any notable density next to the spherical cap, seems to show that there is too much material in the model. To see if that is the case, it is necessary to create a model that is thinner to see what the behavior of this thinner model will be.
65 5.17 Results for Topology Optimization of a 1 ½” Thick Model
In order to test if the thinning of the model would indeed improve the results from the topology optimization, it is needed to make a model that is thinner than the initial model was. Judging from the element density plot from the full thickness model, it seems that a model with about ½ of the thickness of the original model should be a sufficient model. Therefore, the model will have the same diameter, but a thickness of 1 ½ inch. This model is meshed with the same density of elements at the top surface and through the thickness as was in the initial model. The material properties at each of the 12 layers are the same as those of the 12 layers of the thicker model. This would keep the material properties of the outer surface an inner surface the same, which is important for the optimization routine. Therefore, there did not need to be a change in the material properties assigned to the elements for this model. The topology inputs are the same as described in Table 7, except that the geometry was changed to the values shown in Table 9. Geometry Inner Radius, Cylinder and Sphere 16.75 inches Outer Radius, Cylinder and Sphere 19.75 inches Body Length, Cylinder 60 inches, along the Y-axis
Table 9: Thinner Geometry for Structural Optimization Model
The regions that were optimized and non-optimized were the same as described in Section 5.13. The model used the symmetry condition method stated in Section 5.16, but the maximum amount of iterations to be performed was set back to 30. On the first run, the model converged to a non-feasible design. After reviewing the iterations of the optimization, it became obvious that this problem arose due to the total volume change never converging within its tolerance until it voided all elements except for the non- design elements. With no more elements to void, it ran through iterations with only the non-design elements until it passed the soft convergence criteria. To counter this problem, it became necessary to increase the convergence tolerance to 6.5E-3. This greatly improved quality of the solution, as can be seen in Figure 27, the element density plot for this final model.
66 Figure 27: Element Density Plot, 1 ½” Thick Pressure Vessel Topology Optimization (Outer Surface, Side, and Isometric Views)
This plot shows that the element density is only lowered at the spherical dome. This makes sense, seeing as the previous models seemed to show that as a point where the most material should be removed. It should also be noted that none of the elements have a density less than 0.3, so none of the elements meet the decided criteria for voiding. This model shows where material can be reduced, but it does not do a good job of showing how the material should be eliminated. In addition, one can see from the element density plot that the outer layers of the spherical dome have a higher density than the inner layers. Judging from the behavior of the iterative solution, this occurs due to the differences in material properties. Again, more stress relief occurs from a reinforcement of the outer layers. As one can see in the Appendix, the early iterations in the optimization routine take significant portions of material off, and then later iterations add on material if the model violates the stress criteria. With this functionally graded material, the outermost material is preferred when adding material back to the model for stress relief. This makes for a model that is hard
67 to manufacture. It is apparent that there needs to be a structural optimization routine that keeps material continuity in order to prevent such problems. Free shape structural optimization is such a routine, and it will be investigated
5.18 Shape Optimization Method in OptiStruct
Shape optimization is a method of structural optimization in which the outer boundary of a structure is modified to solve the optimization problem. The shape of the structure is defined by the nodes of the FEA model, which act as grid point locations. The shape optimization acts by changing the coordinates of these nodes. In order to avoid mesh distortions due to shape changes, changes of the shape of the structural boundary must be translated into changes to the interior of the mesh. The approach that OptiStruct uses to account for these changes is the perturbation vector approach. Using the perturbation vector approach, the structural shape change is defined as a linear combination of perturbation vectors. In shape optimization, these perturbation vectors are defined by the user and are used to define the changes of nodal locations with respect to the original finite element mesh. The nodal locations are defined by the following equation: n 0 DVi PVi i1
Equation 29: Perturbation Vector Approach for Shape Optimization in OptiStruct
In this equation is the initial vector of nodal coordinates, 0 is the initial vector of nodal coordinates, and PV is the perturbation vector associated with the design variable DV. This design variable is a range of values that can be used by OptiStruct as a way of controlling the influence of the perturbation vectors when seeking to converge to a certain objective. This routine will be performed until the vector of nodal coordinates meets the convergence criteria at the nth iteration.
5.19 Free Shape Optimization Method in OptiStruct
Free-shape optimization is an optimization technique unique to OptiStruct. The main difference between free-shape and normal shape optimization is that there is no
68 need for the user to define the perturbation vectors. Instead, the allowable movement of the outer boundary is automatically determined. The vectors of nodal coordinates are perturbed in a way so as the design grid will move in a way normal to the outer boundary of the structure. For solid element structures, such as those that are used in this thesis, the design grid moves normal to the outer surface. This normal direction changes with the change in the shape of the structure and, therefore, the design grids move along the updated normals for all iterations of the solution. To reduce the effort needed in finding a solution, the free- shape optimization method will be used.
5.20 Defining the Optimized and Non-Optimized Regions for Free Shape Optimization
The method of assigning optimized and non-optimized regions is different for free shape optimization than it is for topology optimization. In free shape optimization, it is not the element density, and therefore the element stiffnesses, of the model that is manipulated, but the location of the element nodes. The nodes that will be used primarily in the optimization routine are the ones defining the outer boundary, so the nodes on the outer surface of the pressure vessel are assigned to the optimized region. The results in Sections 5.15 through 5.17 show that the spherical cap is an area that is more likely to need optimization than the cylindrical body. Therefore, there will also be studies done where only the spherical cap is being optimized. In this case, only the nodes that define the outer surface of the spherical cap will be defined as the optimized region.
5.21 Defining the Design Criteria for Free Shape Optimization
The criteria that will be set are the same that were used in the topology optimization routine, but the methods that they are enforced with are different. First, it is needed to create a volume response, which is set to total volume. This response is controlled several ways. One way that it is controlled is the same as is described in the topology optimization. That is, the volume response has an objective set to it. The other way that the volume response is controlled is by the “DSHAPE” free size optimization
69 design variable card, specifically the “DTYPE” field. This field constrains the movement of the free shape design region, which in turn constrains the movement of the elements that are confined by this outer boundary. Another criterion that must be set is the maximum stress constraint. This is first accomplished by way of creating a stress response. This response is set to a response type of “static stress”, with an additional descriptor of “von mises.” The response was applied to all the elements in the model. After that, one needs to use the “dconstraint” editor to create a constraint for the stress response. This stress constraint was applied to the static stress loadset created by the method given in Section 5.10 using inputs listed in Table 7, with modifications to geometry per Table 9. The next step is to define the objectives for the responses. For the volume response, the objective is set to “min” and the DTYPE field is set to “SHRINK” so that the grids cannot move outside of the initial part boundary. This is sufficient to tell the program that the optimization that the volume should be reduced to a minimum. The stress constraint has the same objective value as is defined in Table 7. This is accomplished by setting the constraint to an upper bound of 75,000 psi. This should be sufficient for defining the criteria for the structural optimization. In order to maintain the stability of the solution, there need to be additional criteria applied to the free shape optimization design variable card. One such criterion is the move factor (MVFACTOR). This factor sets the maximum initial movement of the grids by use of the following equation:
i MVFACTOR *mesh _ size
Equation 30: Maximum Initial Movement of Free Shape Optimization Grids Using the Move Factor
Where “mesh_size” is the average mesh size of the design region. The bigger the move factor, the faster the solution will run, but with less stability. The smaller the move factor, the slower the solution will run, but the stability will be improved. The effect of the move factor on the free shape optimization routine will be investigated. The next criterion to be applied is the number of layers for mesh smoothing (NSMOOTH). In free shape optimization, the internal node grids adjacent to the design
70 region grid have to be moved to avoid mesh distortion. The NSMOOTH card controls the amount of adjacent node grids to be moved. The larger the amount of smoothing layers, the less mesh distortion will play a factor. As a consequence, the optimization will run slower. Fewer layers will have a faster optimization, but the dangers of mesh distortion are increased.
5.22 Manufacturability Constraint: Grid Constraints
In many designs, it is necessary to make sure that certain parts of the model are not altered by the optimization routine in undesired directions. OptiStruct accounts for this by use of the GRIDCON grid constraint card. The grid constraints have three possible values: fixed, vector and planar. The fixed constraints make sure that a grid defined by nodes cannot move due to the free shape optimization. The vector constraints make sure that a nodal grid is forced to move along a certain vector, be in the positive or negative direction. The planar constraints force a nodal grid to remain on a plane that is specified as normal to a chosen vector direction. This model utilizes the fixed and planar optimization constraints. The fixed grid constraint was applied to the inside surface of the pressure vessel, making sure that the inner surface is not moved by the free shape optimization routine. This was performed so as to ensure that the volume of fluid in the pressure vessel would be kept constant throughout the optimization routine. It should be noted that this constraint is used by OptiStruct only when altering the shape of the finite element model before the model is sent to the finite element solver, RADIOSS, for each design iteration. The finite element solution does not fix the nodes on the inner surface, leaving them free to deform during thermal and pressure loads. The planar constraints were applied to the surfaces that had the symmetry boundary conditions applied to them in the finite element solution. This was performed to ensure that the symmetry boundaries would not be deformed in the optimization routine. If these surfaces were to deform, the symmetry planes would deform as well, making it so the cylindrical body and spherical dome would not longer be perfect cylinders or spheres, respectively. In order to maintain manufacturability, it is necessary to ensure that this imperfection is eliminated. The planar constraints are also used only when
71 OptiStruct alters the shape of the model before sending it to RADIOSS for each iteration. The finite element model still requires the original symmetry boundary conditions given in Table 7.
5.23 Control Cards for Free Shape Optimization
In order to aid in the process of free shape optimization, there were several control cards that were assigned to the structural optimization. One control card set was the SHAPE card. This allows the user to look at the stress results on top of the shape change in HyperView, the postprocessor. This is not necessary, but it does aid in the examination of the optimization model after the run has finished. Finally, the SCREEN control card was added. This is another card that is not necessary, but it does help by showing the progress at each optimization step while the program is running. The progress report shows, for this case, the optimization step, the value of the objective function (total volume), and the percentage by which the constrained value (static Von Mises stress) violates the maximum value constraint.
5.24 Optimization Results: Total Model Volume and Maximum Von Mises Stress, Only Hemispherical Caps Optimized
It was shown in the last topology optimization model that the only elements in the model that underwent any element density reduction were those at the hemispherical cap. Therefore, the first attempt at free shape optimization was optimizing the hemispherical cap alone. This should reduce the amount of computation work, seeing as only a small section of the model would be optimized. Several models were run using the inputs of Table 7 with its geometry modified per Table 9. Each model had one of three values for layers for mesh smoothing and one of several values for the move factor. The NSMOOTH cards were set to 8, 12 or 16. No smaller values were chosen due to problems with mesh distortion. No larger values were chosen due to the assumption that there are only 12 design grids in the model that can be affected by the NSMOOTH card. The model with 16 smoothing layers was added to test if this assumption was correct. The move factors varied from 0.05 to 0.175. No larger values were chosen due to fears of mesh distortion. No smaller values were chosen
72 because they were felt to be too small to enact any significant displacement in the design grid. There are many things that can be monitored in this analysis to determine the usefulness of the solutions of each of the models. Of primary concern is what the final volume (initial volume 3196 in3), the final stress (constraint = 75 ksi), total iterations, and total computation time was. These should give a general feel for the effects that the design criteria have on the solution as a whole. Table 10 lists the results of interest.
Smoothing Move Final Final Max Iterations Computation Time Layers Factor Volume (in3) Von Mises (hh:mm:ss) Stress (ksi) 0.05 3054 74.9 5 00:03:55 0.075 3038 75.0 4 00:03:17 0.1 3037 75.0 4 00:03:16 8 0.125 3044 74.6 3 00:02:38 0.15 3020 75.0 5 00:03:59 0.175 3014 75.1 6 00:04:36 0.05 3053 74.9 5 00:03:55 0.075 3035 74.0 4 00:03:32 0.1 3037 75.0 4 00:03:16 12 0.125 3005 75.0 4 00:03:32 0.15 2998 75.0 6 00:04:55 0.175 2975 74.9 5 00:05:00 0.05 3048 74.9 5 00:04:25 0.075 3035 74.0 4 00:03:40 0.1 3013 75.0 4 00:03:37 16 0.125 3005 75.0 4 00:03:37 0.15 2997 74.9 6 00:05:07 0.175 2975 74.9 6 00:5:07
Table 10: Hemispherical Cap Free Shape Optimization Results
It can be seen from these results that, for at least the hemispherical cap shape optimization, the assumption that having more than 12 smoothing layers will not affect the model was correct. However, having less than 12 smoothing layers resulted in less
73 optimal designs. In addition, the models with greater move factors resulted in models with smaller volumes. Specifically, the results show that the best optimization runs were those with at least 12 smoothing layers and a move factor of 0.175. This does indeed reduce the total volume (by about 7 percent) and it does not exceed the stress constraint. To get a better understanding of the optimization results, it is necessary to view a stress plot for a free shape optimization model. This plot is taken for the representative model with 16 smoothing layers and a move factor of 0.125, and is shown in Figure 28.
Figure 28: Final Von Mises Stress (ksi), Hemispherical Cap Free Shape Optimization Model (Inner Surface, Outer Surface and Isometric Views)
As one can see, the outer surface of the hemispherical cap has the nominally the same stress throughout. This shows that the optimization routine tries to make sure that there are no localized maximum stresses, to the point that the entire outer layer of the
74 design region is the about the same stress. Seeing as the cylinder was not chosen as the design area, it did not have its shape changed and may have more material than is optimal. This is evidenced by the lower stress just outside the design region of the hemispherical cap. If the thickness of this region were optimized to be less thick, it would also be close to the stress constraint. Therefore, the fact that the cylinder is not being optimized is restricting the effectiveness of the optimization routine. In order to improve the effectiveness of the free shape optimization, it is necessary to optimize the cylindrical body along with the hemispherical cap.
5.25 Optimization Results: Total Model Volume and Maximum Von Mises Stress, Whole Body Optimization
The second optimization routine was run on the entire body of the pressure vessel. Several models were run, with the same combinations of smoothing layers and move factors as the spherical cap optimization. The model inputs were the same as the hemispherical cap optimization. The values monitored were also the same as those used in the hemispherical cap optimization. Table 11 lists the results of interest.
75 Smoothing Move Final Final Max Iterations Computation Time Layers Factor Volume (in3) Von Mises (hh:mm:ss) Stress (ksi) 0.05 2843 73.8 5 00:05:39 0.075 2797 74.0 4 00:04:41 0.1 2616 78.4 30 00:29.37 8 0.125 2429 84.8 30 00:29:12 0.15 2304 86.8 30 00:29:24 0.175 2292 85.0 30 00:31:08 0.05 2838 73.7 5 00:05:54 0.075 2779 73.7 5 00:05:54 0.1 2515 78.9 30 00:31:09 12 0.125 2314 82.5 30 00:30:58 0.15 2084 92.7 30 00:30:51 0.175 1939 92.2 30 00:30:52 0.05 2838 73.7 5 00:05:54 0.075 2779 73.7 5 00:05:54 0.1 2515 78.9 4 00:31:06 16 0.125 2314 82.5 30 00:30:49 0.15 2084 92.7 30 00:30:49 0.175 1939 92.2 30 00:30:56
Table 11: Whole Body Free Shape Optimization Results
These results are notably different from those of the spherical cap. The amount of smoothing layers and the move factor seem to have notable effects on the optimization results. As one can see, the item that has the greatest effect on the results is the move factor, not the amount of smoothing layers. There is a slight difference in results between the models with 8 and 12 smoothing layers, but not between the models with 12 and 16 smoothing layers. This again seems to show that the maximum amount of smoothing layers that OptiStruct can use for this model is around 12, as was predicted. To verify this, the post-processor was used to see the shape change of models, each with a move factor of 0.075 and a smoothing layer value of 8, 12, or 16. It was seen that the
76 spherical cap had the most dramatic shape change, so it was decided that this section be used to compare the models. The comparison is shown in Figure 29.
Figure 29: Shape Change Results (inches), Free Shape Optimization with Move Factor of 0.075 and 8, 12, and 16 Smoothing Layers
It can be seen from this plot that the model with 8 smoothing layers seems to have extra layers that do not move when compared to the models with 12 and 16 smoothing layers. However, there is no difference in the shape change between the 12 and 16 smoothing layers. This shows that the amount of smoothing layers needed to get a full solution is more than 8 and less or equal to 12. Therefore, one can assume that 12 smoothing layers is the least amount needed for a full solution. It was also predicted that having more smoothing layers would reduce the likelihood of mesh distortion, but at a cost of having a slower optimization. If one looks at the number of iterations and the total run time, one can see that adding smoothing layers (up to the critical value of 12) does in fact increase the computation time for each iteration. When it comes to the move factor, it is of interest that the models that took the least amount of time to converge were the models with the smallest move factors. This is due to the fact that only the models with less than a 0.1 move factor converge to near the desired solution. This shows that there is a critical value of the move factor which, when passed, makes it impossible for the model to converge to a proper volume. However, among the models that have converged properly, it should be noted that the model with 8 smoothing layers showed a faster convergence for the model with a higher move factor. This behavior is due to the lower amount of iteration cycles that had to be
77 performed in the free shape optimization problem. This holds with the behavior that was predicted. While the behavior among the models that converged to a workable solution seems to agree with the predicted behavior, the models with move factors of 0.1 and larger were not able to converge to a proper solution. To see what the cause of this dysfunction might be, further research into the results is necessary. One possible way to glean more information would be to view the change in the optimization objective (total volume) as the optimization problems progress. Figure 30 shows the objective response, total model volume, for the models with 12 smoothing layers.
Free Shape Optimization Total Volume, 12 Smoothing Layers
3200 3150 3100 3050 3000 2950 2900 2850 2800 2750 )
3 2700
^ MV = 0.05 n
i 2650 (
MV = 0.075
e 2600
m MV = 0.1
u 2550 l
o MV = 0.125 2500 V
l MV = 0.15
a 2450 t
o MV = 0.175
T 2400 2350 2300 2250 2200 2150 2100 2050 2000 1950 1900 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Iterations
Figure 30: Total Volume Response (in3), Free Shape Optimization (12 Smoothing Layers)
This plot shows several important pieces of information. For the models with a smaller move factor, it is shown that the models converged through the regular method, as is evidenced by the fact that the model created doesn’t violate the stress constraint and the convergence required two consecutive iterations within the objective tolerance. Another key observation that can be seen in this plot is that the plots for the models with larger move factors exhibit saw tooth behavior later in the iterations. Once the model
78 got to a sufficiently small volume, the optimization software attempted to add volume to the model, but the allowed shape change in the free shape optimization had been decreased to the point that the contributions to the optimization in further iterations were insufficient to allow for a feasible design. This reduction of the allowed shape change is due to the automatic adjustments that OptiStruct performs in an effort to enhance iterative stability and convergence speed. As can be seen from the graphs, this is not always the case. The information shown so far is useful to see the methodology that OptiStruct uses when performing free-shape optimization. However, it is even more important to evaluate the suitability of the final model created by the free-shape optimization. For this evaluation, a representative model with 12 smoothing layers and a move factor of 0.075 will be used. Compared to the results for the same model undergoing optimization performed only on the spherical cap (Section 5.24), the final volume for the full-body optimization is a full 196 in3 smaller, approximately 7% smaller. Therefore, it can be shown that the full-body optimization performs better when compared to the design objective. Therefore, from a pure metrics standpoint, the full-body optimization is preferred when compared to the optimization that is performed on the spherical cap alone. In order to more completely ascertain the effectiveness of the full-body optimization, it is necessary to use the post-processor to view a physical model of the final optimization product. As was done for the spherical cap optimization, it is necessary to look at the Von Mises stress results for the final full-body optimization model. The results are shown on Figure 31.
79 Figure 31: Final Von Mises Stresses (ksi), Full-Body Free Shape Optimization Model (Inner Surface, Outer Surface and Isometric Views)
As was the case for the model in which only the hemispherical cap was optimized, the maximum stress is at the outer surface and seems to be the same throughout the outer surface of the design region. Again, this can likely be attributed to the program trying to eliminate any localized maximum stresses. One item of note is that this design did not converge as closely to the stress constraint as the hemispherical cap did. The stress results approached the stress constraint and then violated them as the move factor increased. It seems that more complex designs are more sensitive to the optimization controls and require more attention. However, the reduction in volume for the optimization of the full vessel was
80 notably more than that of optimization of the hemispherical cap alone, making the extra effort worthwhile.
5.26 Results: Thickness Change for Whole Body Free Shape Optimization
In order to more accurately portray the change in shape of the optimization model, one can show how the change in thickness for several cross-sections of the model. The cross-sections chosen for this are shown in Figure 32.
7 8 9 10 6 11 12 13 14 4 5 3
2
1
Figure 32: Free Shape Optimization Model Cross-Section Numbers
On the following page is a table of the thickness at each of the cross-sections for the pressure vessel for the final topology optimization model, whose geometry was used to make the free shape optimization model, and the fully optimized free shape model.
81 Section Number 1 2 3 4 5 6 7 Model Used Final Topology Model 1.5 1.5 1.5 1.5 1.5 1.5 1.5 Thickness (in) Free Shape Optimized 1.157 1.165 1.159 1.153 1.165 1.184 1.209 Model Thickness (in) Section Number 8 9 10 11 12 13 14 Model Used Final Topology Model 1.5 1.5 1.5 1.5 1.5 1.5 1.5 Thickness (in) Free Shape Optimized 1.260 1.309 1.349 1.361 1.354 1.354 1.351 Model Thickness (in) Table 12: Comparison of Vessel Thickness, Free Shape Optimized and Topology Optimized Models
The results can be separated into three regions. Cross-sections 1 through 5 are nominally the same and represent the hemispherical cap. Cross-sections 11 through 14 are also nominally the same and represent the cylindrical body. Cross-sections 6 through 10, however, are not nominally the same and therefore represent a transition region between the hemispherical cap and the cylindrical body. The largest change in thickness is at the hemispherical caps. The free shape optimization made a pressure vessel with and inner diameter of 16.75 inches and a nominal thickness of 1.351 inches for the cylindrical body and 1.157 inches for the hemispherical caps. This makes the cylindrical body approximately 17 percent thicker than the hemispherical caps, which is a notable difference in thickness. While there is a notable thickness change that has to occur in the transition region, this region is defined by an approximate length of 9 inches. The ratio of change in thickness and the length of the transition region denote that this is quite a gradual change. Therefore, the transition region is not prone to severe stress concentrations, as can be seen in Figure 31. In addition, the gradual transition will aid in the manufacture of the vessel. Therefore, it can be said that this model adequately meets the stress criteria as well as the design criteria. Finally, it can be said that free shape optimization significantly improved upon the topology optimization results.
82 5.27 Results: Optimization Model Shape Change
Further examination of the manufacturability of the final model must be made. One way to ascertain this is by seeing the final shape change of the optimization model. This is shown in Figure 33.
Figure 33: Magnitude of Final Shape Change (inches) for the Full-Body Free Shape Optimization (Top, Bottom, and Side Views)
83 It can be seen that the inner surface doesn’t move, as has been defined in the design constraints. This plot also shows the gradual change in thickness that was found in the previous section. Further inspection of the vessel, however, shows another interesting aspect of the shape optimization. The change in the shape seems to be more dramatic near the outer surface than at the inner surface. To verify this, a closer look at the model is necessary. Seeing as the SHAPE card was used in this analysis, it is possible to see the shape change results overlaid upon the physical shape of the final model. The magnified views for the spherical dome and the cylindrical body are shown in Figure 34.
Figure 34: Magnified Details for Shape Change (inches) of the Spherical Cap and Cylindrical Body
As one can see, the shape change for the cap is such that the outer layers are thinner than the inner layers. The cylinder, however, maintains roughly the same thickness throughout the model. This difference in shape change affects the model by making it so the material distribution is not the same throughout the model. This change in material distribution is due to the fact that the material distribution is modeled by giving individual material properties to each layer. This could be behavior in OptiStruct that, in its seeking for an optimal solution, inadvertently changes the material distribution. In order to ascertain why OptiStruct changes the thickness of the layers at the spherical cap, it is necessary to use a control model. This model will eliminate the
84 variation in material by modeling a vessel with only one material throughout. The results are shown in the next section.
5.28 Free Shape Optimization Study: Whole Body, Homogeneous Material
For this model, it was decided that there would be only one material throughout the entire model. The material applied to the whole model was the material chosen for the inner layer. The geometry, loading conditions, and optimization routine were kept the same as the functionally graded model. After the model converged, the layers of the vessel were studied. The detailed magnifications of the spherical cap and the cylindrical body are shown in Figure 35.
Figure 35: Magnified Shape Change (inches) for Homogeneous Model, Sphere and Cylinder
It can be seen that the difference in shape change between the outer and inner layers is even more dramatic for this model than the functionally graded model. In addition, it can be seen that the maximum shape change is bigger for this model than the functionally graded one. From this, it can be assumed that the shrinking behavior of the outer layers is natural for free shape optimization. The more dramatic the change in volume, the more dramatic the difference in shape change between the outer layers and the inner layers. One would assume that the program performs the optimization in a way that the elements closer to the design grid (on the outer surface for this case) move more
85 dramatically than those further away. After talking with an OptiStruct representative from Altair, this assumption was verified.
5.29 Results: Optimization Model Material Distribution
One consequence of free shape optimization is that the models were not able to keep the coefficients and exponents of the power law material distributions the same as those for the initial model. This was due to efforts to keep the material properties for the outer surface and inner surface the same. When the thickness of the model changed, the material distributions changed as well. Therefore, the new material distributions were calculated by sampling the material properties of the elements along with their radial positions. This information was then fit to power law trend lines in Excel, which can be seen in the Appendix. The new power law material property distributions can be seen in Table 13.
Geometry Inner Radius, Cylinder and Sphere 16.75 inches Outer Radius, Cylinder 18.101 inches Outer Radius, Sphere 17.906 inches Functionally Graded Material Properties E E r 2.1135 E 65793 lbf Modulus of Elasticity, Cylinder o , o in 2
E E r 2.4329 E 26660 lbf Modulus of Elasticity, Sphere o , o in 2 2.1135 1.535108 in Coefficient of Thermal Expansion, Cylinder o r , o in o F 2.4329 6.221109 in Coefficient of Thermal Expansion, Sphere o r , o in o F 2.1135 k 0.0658 Btu Thermal Conductivity, Cylinder k ko r , o hr fto F
2.4329 k 0.0267 Btu Thermal Conductivity, Sphere k ko r , o hr fto F
Table 13: Functionally Graded Material Properties for the Optimized Model
The coefficients and exponents for the power law material property distributions have changed quite a bit. Judging from the exponents, one would expect the distribution of material properties through the thickness to be nearly quadratic. However, the actual
86 distribution through the thickness tends to be more linear in form, though the distributions for the hemispherical caps are slightly less linear than those for the cylindrical body. The power law equations require that at a radius of zero, any property defined by it is equal to zero. With the AISI 1050 material properties being applied to a different outer radius, the coefficients of 1 no longer allowed the power law distributions no longer fit that criterion. Therefore, the power law distributions needed new exponents and coefficients to fit the data for the given range of radii while still making the properties equal to zero at a radius of zero. Another item of note is that all of the power law distributions for the hemispherical caps have the same exponents, as do the distributions for the cylindrical body. When coupled with the fact that the material distributions are largely linear, this shows that the final optimization model is distributed in a manner that is very close to the optimal material distribution discussed in Section 3.3. This correlation further validates the final optimization model and shows that free shape optimization is well suited to the optimization problem when the proper controls are applied.
87 6. Conclusions
6.1 Comparison of Topology and Free Shape Optimization
Both topology and free shape optimization were able to create designs that used significantly less material than the design originally used. Each optimization method had its own merits and drawbacks, but the free shape optimization was able to create the design with the least amount of material while still maintaining feasibility. The topology optimization required less design parameters and was therefore simpler to set up, but the designs tended to create unwanted voids in the model unless the proper controls were added. Once these controls were added, the optimization routine was able to make a solid structure at the cost of having a heavier design than a free shape optimization routine done on the same model. Another aspect of topology optimization that should be noted is that it does not favor material continuity when working with functionally graded materials. The early iterations of the optimization routine void significant portions of the structure and then add material on if the stress criterion is exceeded. With this model, the outermost material was best suited for stress relief, and was therefore preferred when adding material back on. This tends to make for a model that is hard to manufacture, making it undesirable for finding a final optimization solution. The topology optimization was shown to be rather useful, however, in finding an ideal vessel thickness for performing structural optimization. This is important when working with free shape optimization, which moves the nodes of elements in the optimization model. If the initial design is too thick, free shape optimization could deform the elements enough to reduce the quality of the FEA results. One can run a simple initial model in topology optimization and view the optimization results. If the model is unchanged by the optimization, the thickness is either too thin if the stress allowable is exceeded or just right if the stress allowable is met. If the model has significant portions of voids in it, the thickness is too great. In this thesis, there were significant voids in the initial topology optimization and the thickness was set to a thickness that was judged proper. One could also calculate the stiffness through the thickness for the results of the topology optimization and determine a proper thickness
88 that would have the equivalent stiffness. This way, topology optimization could be a key tool for initial design guidance. The free shape optimization required more design parameters, making it more complicated to set up. It was also fairly sensitive to the optimization controls, but free shape optimization allows for easy modification to the controls to allow for fine-tuning. When the results were tuned properly, the models converged quickly. Therefore, if a model takes an overly long time to converge, it tends to be a telltale sign that the controls need to be modified. The models analyzed in this thesis were simple and underwent simple loading conditions. Different geometries require different meshes and different loading conditions require different load collectors and element types. However, the key concepts covered should be useful for guiding structural optimization.
6.2 Comparison of Analytical Solution and Finite Element Analysis for the Optimized Model
To assess the validity of the final optimization model, it was decided to compare the finite element solutions for the model to the exact solutions for the pressure vessel. All solutions were solved using the loading conditions listed in Section 3.2. Exact analytical solutions were derived using the equations listed in Chapter 2 along with the dimensions and material properties listed in Table 13 of Section 5.29. The finite element solutions were found using the methods listed in Chapter 4 with the Abaqus and RADIOSS finite element solvers using the final optimization model with meshes defined by Table 14.
Mesh Details Elements Used, Abaqus CHEXA, 20 Noded Brick Elements Elements Used, RADIOSS C3D20T, 20 Noded Brick Elements Number of Elements 4044 Number of Degrees of Freedom 55791
Table 14: Mesh Information for Finite Element Solution of the Optimized Pressure Vessel
The sampling areas where the analytical and finite element solutions were taken are the same as shown in Figure 18 of Section 4.5. The analytical and finite element
89 solutions for the hemispherical cap are in Table 14 and Table 15 lists the solutions for the cylindrical body.
Radial Stress (ksi) Hoop Stress (ksi) Solver Inner Outer Min Max Inner Outer Min Max Used Surface Surface Stress Stress Surface Surface Stress Stress RADIOSS -1.06 -0.78 -2.71 -0.78 -51.11 71.47 -51.11 71.47 Abaqus -1.44 0.02 -2.61 0.02 -48.19 70.66 -48.19 70.66 Analytical -1.50 0 -2.74 0 -46.02 71.18 -46.02 71.18 Solution Table 15: Comparison of Finite Element and Analytical Solutions, Fully Optimized Hemispherical Cap
Radial Stress (ksi) Hoop Stress (ksi) Solver Inner Outer Min Max Inner Outer Min Max Used Surface Surface Stress Stress Surface Surface Stress Stress RADIOSS -1.44 -0.28 -1.97 -0.28 -41.47 81.87 -41.47 81.87 Abaqus -1.64 -0.16 -2.12 -0.16 -39.12 81.66 -39.12 81.66 Analytical -1.50 0 -1.92 0 -36.06 82.06 -36.06 82.06 Solution Table 16: Comparison of Finite Element and Analytical Solutions, Fully Optimized Cylindrical Body
For the radial stresses, both of the finite element solutions fit the analytical radial stress solution for the cylinder well, but the Abaqus model fit the analytical solution for the hemisphere better than the RADIOSS model did. This is likely due to the fact that the CHEXA elements that RADIOSS uses when performing finite element analysis for OptiStruct are reduced integration elements. This matches with the results obtained when comparing full integration elements to reduced integration elements in Chapter 4. However, these loads are so small compared to the hoop stresses that they are relatively inconsequential. The outer surface hoop stresses for both of the finite element solutions fit the analytical stress solutions very well for the cylinder and the hemisphere. However, the finite element hoop stresses at the inner surface were not as close to the analytical
90 stresses as the hoop stresses at the outer surface. This is likely due to the fact that the free shape optimization reduced the size of the elements on the outer surface more than it reduced the size of the elements on the inner surface. Finite elements use numerical integration in order to approximate the stresses at the nodes of the element. The elements use integration points that are a certain ratio of the distance from the center of the element to the nodes. The greater the size of the element, the greater the absolute distance from the integration point to the actual node. This can reduce the accuracy of the nodal stresses, especially if the values are calculated at a certain material property when the actual material properties at the nodes are notably different, as can be the case for the way that the functionally graded materials were modeled in this study. The elements used the average material property values for the region they were modeled in and calculated stresses based on these values. In addition, the reduced integration elements use less integration points than the full integration elements. (Cook, p. 213) This can make the results for reduced integration elements less accurate with larger elements, as can be seen in the outer surface hoop stresses. However, it should be noted that the hoop stress at the inner surface is conservatively high for the CHEXA reduced integration elements. In summary, the finite element models were able to capture the highest stresses in the sampling regions with reasonable accuracy. The surface radial stresses were better captured by the full-integration elements, but they are so small as to be not of consequence. The inner surface hoop stresses were different from the analytical solution, but conservatively so. From this, it can be decided that the use of quadratic brick elements in the optimization of the given design is sufficient, if a bit conservative.
6.3 Future Work
This project focused on a simple problem with simple loading conditions and only used a few of the routines and capabilities of OptiStruct. This optimization was run on a pressure vessel without structural reinforcement. Future studies could be performed to see if the addition of ribs to the pressure vessel would reduce the material in the vessel notably. Of particular interest would be the optimal placing of the ribs on the vessel, as well as the optimal length of said ribs.
91 Another potential study would be optimization of functionally graded pressure vessels using alternative methodologies. One possible study would be to keep the geometry constant and optimize the material distribution alone. Another alternate methodology would be to model the material distribution as a field equation instead of applying discrete material values to each element. This would open up the possibility of optimizing shell elements, in which case different optimization routines, such a size and topography optimization, would be applicable. Finally, the loading conditions were constant pressure with a steady state thermal gradient applied. Future studies could be done with transient pressure and thermal loads applied to the vessel.
92 7. References
Abaqus 6.8, help documentation.
Bhashyam, Srinivas, Shin, Ki Hoon, and Dutta, Debashish (2000). “An integrated CAD system for design of heterogeneous objects.” Rapid Prototyping Journal, 6(2), p. 119-135.
Budynas, Richard G. and Young, Warren C. (2002). Roark’s Formulas for Stress and Strain. Seventh Edition. New York: McGraw-Hill.
Buschow, K.H. Jürgen, Cahn, Robert W., Flemings, Merton C., Ilschner, Bernhard, Kramer, Edward J., and Mahajan, Subhash (2002). “The Science and Technology of Materials: An Introduction,” In: K. H. Jurgen Buschow, Robert W. Cahn, Merton C. Flemings, Bernard Ilschner (print), Edward J. Kramer, Subhash Mahajan, and Patrick Veyssiere (updates), Editor(s)-in-Chief, Encyclopedia of Materials: Science and Technology. Oxford: Elsevier. Pages xi-xiv.
Cook, Robert D., Malkus, David S., Plesha, Michael E., and Witt, Robert J. (2002). Concepts and Applications of Finite Element Analysis: Fourth Edition. Hoboken, NJ: John Wiley & Sons, Inc.
Dewitt, David P. and Incropera, Frank P. ( 2002). Fundamentals of Heat and Mass Transfer. Fifth Edition. Hoboken, NJ: John Wiley & Sons, Inc.
Eslami, M.R., Babaei, M.H., and Poultangari, R. (2005). “Thermal and mechanical stresses in a functionally graded thick sphere.” International Journal of Pressure Vessels and Piping, 82, Pages 522-527.
Eslami, M.R., Jabbari, M., and Sohrabpour, S. (2002). “Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads.” International Journal of Pressure Vessels and Piping, 79, Pages 493-497.
Hetnarski, Richard B., Noda, Naotake ,and Tanigawa, Yoshinobu (2002). Thermal Stresses: Second Edition. New York: Taylor & Francis.
Matweb. “AISI 1050 Steel, as cold drawn bar, 50-75 mm (2-3 in) round” [Data Set]. Retrieved from: http://www.matweb.com/search/DataSheet.aspx? MatGUID=9b8279fc95c043368b0d1b8589fc0cda Date Last Accessed: 08/10/2009
Neubrand, A. (2001). “Functionally Graded Materials,” In: K. H. Jurgen Buschow, Robert W. Cahn, Merton C. Flemings, Bernard Ilschner (print), Edward J. Kramer, Subhash Mahajan, and Patrick Veyssiere (updates), Editor(s)-in-Chief, Encyclopedia of Materials: Science and Technology, Elsevier, Oxford, Pages 3407-3413
93 OptiStruct 9.0, help documentation.
Timoshenko, S. (1976). Strength of Materials Part II: Advanced Theory and Problems. Malabar, Florida: Krieger Publishing Company.
94 8. Appendix
8.1 Excel VBA Code for Creating Material Cards for ABAQUS (12 Element Layers)
Private Sub cmdABAQUS_12_thick_Click() Dim fil As String 'Creates a string file used for storing the file path and name Dim filex As Variant 'Creates a variable used for storing information from an 'application Dim rngselect As Range 'Creates a variable used for storing a range of information filex = Application.GetSaveAsFilename(InitialFileName:="untitled.txt", _ FileFilter:="Input Files (*.txt), *.txt") 'Sets the filex variable to get information from the "save as" application If filex <> False Then 'If valid information can be taken from this fil = filex 'Sets the global file name to the output from the "save as" application Else 'If invalid information exists or if someone cancels out (more likely...) Exit Sub 'Get out of the subroutine End If 'End Criteria Set rngselect = ActiveSheet.Range("B8: F19") 'Select a range of values on the 'spreadsheet Open fil For Output As #1 'Open the created file in order to send text to it For i = 1 To rngselect.Rows.Count 'Go through each row of the selected range Print #1, "**HM_set_by_property " & i 'Create a property Print #1, "*SOLID SECTION, ELSET=HMprop_" & rngselect.Cells(i, 1) & _ ", MATERIAL=" & rngselect(i, 1) 'Assign a name and a material to the property Next i 'Go to the next row For i = 1 To rngselect.Rows.Count 'Go through each row of the selected range Print #1, "*MATERIAL, NAME=" & rngselect.Cells(i, 1) 'Create a material and give 'it a name Print #1, "*CONDUCTIVITY, TYPE=ISO" 'Create a conductivity property Print #1, Left(rngselect.Cells(i, 3), 10) & ",70.0" 'And give it a value Print #1, "*ELASTIC, TYPE = ISOTROPIC" 'Create a modulus of elasticity property Print #1, rngselect.Cells(i, 4) & ",0.3 ,70.0" 'And give it a value Print #1, "*EXPANSION, TYPE = ISO" 'Create a coefficient of linear thermal 'expansion property Print #1, rngselect.Cells(i, 5) & ", 70.0" 'And give it a value Next i 'Go to the next row Close #1 'Stop inputting to the file End Sub
95 8.2 Excel VBA Code for Creating Material Cards for OPTISTRUCT (12 Element Layers)
Private Sub cmdOPTI_12_thick_Click() Dim fil As String 'Creates a string file used for storing the file path and name Dim filex As Variant 'Creates a variable used for storing information from an app Dim rngselect As Range 'Creates a variable used for storing a range of information filex = Application.GetSaveAsFilename(InitialFileName:="untitled.txt", FileFilter:="Input Files (*.txt), *.txt") 'Sets the filex variable to get information from the "save as" application If filex <> False Then 'If valid information can be taken from this fil = filex 'Sets the global file name to the output from the "save as" application Else 'If invalid information exists or if someone cancels out (more likely...) Exit Sub 'Get out of the subroutine End If 'End Criteria Set rngselect = ActiveSheet.Range("B8: F19") 'Select a range of spreadsheet cells Open fil For Output As #1 ' Open file. Print #1, "$$ PSOLID Data" ' Create a place for holding property data Print #1, "$$" ' Placeholder For i = 1 To rngselect.Rows.Count 'Go through each row of the range Print #1, Left$("$HMNAME PROP ", 32 - Len(i)) & i & Chr$(34) & _ rngselect.Cells(i, 1) & Chr$(34) & " 5" 'And make a property for it Print #1, Left$("$HWCOLOR PROP ", 32 - Len(i)) & i & _ Left$(" ", 7 - Len(16 + i)) & 16 + i 'Additional property color info Print #1, Left$("PSOLID ", 16 - Len(i)) & i & Left$(" ", 8 - Len(i)) & i 'And make a note that this property is a solid element property Next i 'Go to the next row Print #1, "$$" 'Placeholder Print #1, "$$ MAT1 Data" 'Create a place for holding material data Print #1, "$$" 'Placeholder For i = 1 To rngselect.Rows.Count 'Go through each row of the range Print #1, Left$("$HMNAME MAT ", 32 - Len(i)) & i & Chr$(34) & _ rngselect.Cells(i, 1) & Chr$(34) & " " & Chr$(34) & "MAT1" & Chr$(34) 'Create the name for the material and make sure it is of the MAT1 type Print #1, Left$("$HWCOLOR MAT ", 32 - Len(i)) & i & Left$(" ", _ 7 - Len(16 + i)) & 16 + i 'Assign extra color information Print #1, Left$("MAT1 ", 16 - Len(i)) & i & Format$(rngselect.Cells(i, 4), _ "0.00E+00") & " 0.3 " & Format$(rngselect.Cells(i, 5), "0.00E+00") 'Assign MAT1 information (Modulus of Elasticity, Coefficient of Thermal Expansion) Print #1, Left$("MAT4 ", 16 - Len(i)) & i & Format$(rngselect.Cells(i, 3), _ "0.000") 'Create the MAT4 suboption and assign the proper conductivity info to it Next i 'Go to the next row in the selected range Close #1 'Stop inputting to the file End Sub
96 8.3 Batch File Code for Input File Translation
ABAQUS Batch File: cp FGM_12_DEEP_MAT.txt FGM_12_DEEP_MAT.inp
cp FGM_24_DEEP_MAT.txt FGM_24_DEEP_MAT.inp
cp FGM_32_DEEP_MAT.txt FGM_32_DEEP_MAT.inp
OPTISTRUCT Batch File: cp OPTI_FGM_12_DEEP.txt OPTI_FGM_12_DEEP.fem
cp OPTI_FGM_24_DEEP.txt OPTI_FGM_24_DEEP.fem
cp OPTI_FGM_32_DEEP.txt OPTI_FGM_32_DEEP.fem
97 8.4 Hypermesh TCL Code For Assigning Properties to Adjacent Layers to Simulate Functionally Graded Material set number_rows [hm_getint "Number of element rows" "Enter the number of element rows"] *createmarkpanel elems 1 "Select the 1st row of elements" hm_createmark elems 2 "[hm_getmark elems 1]"
*createmarkpanel prop 1 "Select the property you would like to assign to the row of elements" set prop_id [hm_getmark props 1] set prop_name [hm_getcollectorname props $prop_id] *propertyupdate elems 2 "$prop_name" hm_createmark elems 2 "[hm_getmark elems 1]" hm_appendmark elems 2 "advanced" "by adjacent" *markdifference elems 2 elems 1
*createmarkpanel prop 1 "Select the property you would like to assign to the row of elements" set prop_id [hm_getmark props 1] set prop_name [hm_getcollectorname props $prop_id] *propertyupdate elems 2 "$prop_name" set n 0 while {$n < [expr {$number_rows -2}]} { hm_appendmark elems 1 "advanced" "by adjacent" hm_createmark elems 2 "[hm_getmark elems 1]" hm_appendmark elems 2 "advanced" "by adjacent" *markdifference elems 2 elems 1 *createmarkpanel prop 1 "Select the property you would like to assign to the row of elements" set prop_id [hm_getmark props 1] set prop_name [hm_getcollectorname props $prop_id] *propertyupdate elems 2 "$prop_name" set n [expr {$n + 1}] }
98 8.5 Excel VBA Code for Translating Heat Transfer Results to SPC Temperature Loads for OptiStruct Stress/Strain Analysis
Private Sub TEMPASSIGN_Click() Dim fil As String 'Creates a string for storing a filename Dim filex As Variant 'Creates a variable used for storing information Dim rngselect As Range 'Creates a range variable for selecting a range of excel cells filex = Application.GetSaveAsFilename(InitialFileName:="untitled.txt", FileFilter:= _ "Input Files (*.txt), *.txt") 'Sets the filex variable to get information from the "save as" application If filex <> False Then 'If valid information can be taken from this fil = filex 'Sets the global file name to the output from the "save as" application Else 'If invalid information exists or if someone cancels out (more likely...) Exit Sub 'Get out of the subroutine End If 'End Criteria Set rngselect = ActiveSheet.Range("A3:B19101") ' Select a range of cells Open fil For Output As #1 ' Open file. Print #1, "$$" Print #1, "$$" Print #1, "$$------$" Print #1, "$$ HyperMesh Commands for loadcollectors name and color information $" Print #1, "$$------$" 'Adds placeholders Print #1, "$HMNAME LOADCOL 4" & Chr$(34) & "TEMPERATURE" _ & Chr$(34) 'Adds a name for the temperature load collector Print #1, "$HWCOLOR LOADCOL 4 36" 'Assigns a color to the load collector Print #1, "$$" Print #1, "$$" Print #1, "$$" Print #1, "$$" Print #1, "$$" Print #1, "$$" Print #1, "$" 'Adds placeholders For i = 1 To rngselect.Rows.Count 'Go through each row in the selected range Print #1, Left$("TEMP 4 ", 24 - Len(rngselect.Cells(i, 1))) _ & rngselect.Cells(i, 1) & Format$(rngselect.Cells(i, 2), "0.0") 'Give a temperature to the node Next i 'Iterate to the next layer Print #1, "ENDDATA" 'Finish up the iterations Close #1 'Stop inputting data to the file End Sub
99 8.6 Mesh Convergence Plots, Stresses for Functionally Graded Cylinder
Figure 36: Mesh Convergence, Radial Stress for FGM Cylinder with 3x3 Top Surface Mesh
Figure 37: Mesh Convergence, Hoop Stress for FGM Cylinder with 3x3 Top Surface Mesh
100 Figure 38: Mesh Convergence, Radial Stress for FGM Cylinder with 4x4 Top Surface Mesh
Figure 39: Mesh Convergence, Hoop Stress for FGM Cylinder with 4x4 Top Surface Mesh
101 Figure 40: Mesh Convergence, Radial Stress for FGM Cylinder with 5x5 Top Surface Mesh
Figure 41: Mesh Convergence, Hoop Stress for FGM Cylinder with 5x5 Top Surface Mesh
102 8.7 Mesh Convergence Plots, Stresses for Functionally Graded Sphere/Hemisphere
Figure 42: Mesh Convergence, Radial Stress for FGM Sphere with 3x3 Top Surface Mesh
Figure 43: Mesh Convergence, Hoop Stress for FGM Sphere with 3x3 Top Surface Mesh
103 Figure 44: Mesh Convergence, Radial Stress for FGM Sphere with 4x4 Top Surface Mesh
Figure 45: Mesh Convergence, Hoop Stress for FGM Sphere with 4x4 Top Surface Mesh
104 Figure 46: Mesh Convergence, Radial Stress for FGM Sphere with 5x5 Top Surface Mesh
Figure 47: Mesh Convergence, Hoop Stress for FGM Sphere with 5x5 Top Surface Mesh
105 8.8 Free Form Topology Element Density Plots
Figure 48: Free Form Topology Initial State
Figure 49: Free Form Topology, Iteration 3
106 Figure 50: Free Form Topology, Iteration 8
Figure 51: Free Form Topology, Iteration 16
107 Figure 52: Free Form Topology, Final Solution
Topology Objective: Total Volume
7000
6000
5000 ) 3 ^ n i 4000 ( e m u l o V l
a 3000 t o T
2000
1000
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Iteration
Figure 53: Volume Response, Free Form Topology
108 8.9 Topology Optimization Element Density Plots: Constrained Inner Surface, 3” Thick Vessel
Figure 54: 3" Constrained Topology, Initial Condition
Figure 55: 3" Constrianed Topology, 5 Iterations
109 Figure 56: 3" Constrained Topology, 15 Iterations
Figure 57: 3" Constrained Topology, 23 Iterations
110 Figure 58: 3" Constrained Topology, Final Solution
Topology Objective: Total Volume
6300 6050 5800 5550 5300 5050 4800 ) 3
^ 4550 n i ( 4300 e m
u 4050 l o
V 3800 l a t 3550 o T 3300 3050 2800 2550 2300 2050 1800 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 Iteration
Figure 59: 3" Constrained Topology, Total Volume Response
111 8.10 Topology Optimization Element Density Plots: Constrained Inner Surface, 1 ½” Thick Vessel
Figure 60: 1 ½” Constrained Topology, Initial Condition
Figure 61: 1 ½” Constrained Topology, Iteration 3
112 Figure 62: 1 ½” Constrained Topology, Iteration 8
Figure 63: 1 ½” Constrained Topology, Iteration 15
113 Figure 64: 1 ½” Constrained Topology, Final Solution
Topology Objective: Total Volume
3000
2875
2750
2625
2500
2375 e m u l 2250 o V l 2125 a t o T 2000
1875
1750
1625
1500
1375 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Iteration
Figure 65: 1 ½” Constrained Topology, Total Volume Response
114 8.11 Free Shape Optimization Plots: Shape Change Compared to Non- Optimized Model (inches), Hemispherical End Cap Optimized (16 Smoothing Layers, 0.125 Move Factor)
Figure 66: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 0 (Outer Surface and Side Views) (Model Before Optimization Routine)
Figure 67: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 1 (Outer Surface and Side Views)
115 Figure 68: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 2 (Outer Surface and Side Views)
Figure 69: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 3 (Outer Surface and Side Views)
116 Figure 70: Shape Change for Hemispherical End Cap Free Shape Optimization, Iteration 4 (Outer Surface and Side Views)
Hemispherical Cap Free Shape Optimization Objective: Total Volume
3200
3180
3160
3140 ) 3
^ 3120 n i ( e m
u 3100 l o V l a t
o 3080 T
3060
3040
3020
3000 0 1 2 3 4 Iteration
Figure 71: Hemispherical Cap Free Shape Optimization, Total Volume Response
117 8.12 Free Shape Optimization Plots: Shape Change Compared to Non- Optimized Model (inches), Entire Vessel Optimized (12 Smoothing Layers, 0.075 Move Factor)
Figure 72: Shape Change for Entire Vessel Free Shape Optimization, Iteration 0 (Outer Surface, Side, and Rear Views) (Model Before Optimization Routine)
118 Figure 73: Shape Change for Entire Vessel Free Shape Optimization, Iteration 1 (Outer Surface, Side, and Rear Views)
119 Figure 74: Shape Change for Entire Vessel Free Shape Optimization, Iteration 2 (Outer Surface, Side, and Rear Views)
120 Figure 75: Shape Change for Entire Vessel Free Shape Optimization, Iteration 3 (Outer Surface, Side, and Rear Views)
121 Figure 76: Shape Change for Entire Vessel Free Shape Optimization, Iteration 4 (Outer Surface, Side, and Rear Views)
122 Figure 77: Shape Change for Entire Vessel Free Shape Optimization, Iteration 5 (Outer Surface, Side, and Rear Views)
123 Entire Vessel Free Shape Optimization Objective: Total Volume
3200
3150
3100
3050 ) 3 ^ n i (
3000 e m u l o V
l 2950 a t o T 2900
2850
2800
2750 0 1 2 3 4 5 Iterations
Figure 78: Hemispherical Cap Free Shape Optimization, Total Volume Response
124 8.13 Graphs for Material Properties of Free Shape Optimized Model
Thermal Conductivity for the Functionally Graded Cylinder
30
29.5 ) F -
t 29 f - r h / 28.5 u
t 2.1135
B k = 0.0658r ( 28 y 2 t i R = 0.9998 v i t 27.5 c u d
n 27 o C
l 26.5 a m r
e 26 h T 25.5
25 16.6 16.8 17 17.2 17.4 17.6 17.8 18 18.2 Radius (in)
Modulus of Elasticity for the Functionally Graded Cylinder
30500000 30000000 29500000
) 2.1135 i E = 65793r s
p 29000000
( 2
y R = 0.9998 t
i 28500000 c i t s
a 28000000 l E
f 27500000 o
s u
l 27000000 u d
o 26500000 M 26000000 25500000 25000000 16.6 16.8 17 17.2 17.4 17.6 17.8 18 18.2 Radius (in)
Coefficient of Thermal Expansion for the Functionally Graded Cylinder
7.20E-06 ) F - n
i 7.00E-06 / n i (
n
o 6.80E-06 i s n
a 2.1135
p a = 1.535E-08r
x 6.60E-06
E 2
l R = 0.9998 a m
r 6.40E-06 e h T
f o
6.20E-06 t n e i c i f
f 6.00E-06 e o C 5.80E-06 16.6 16.8 17 17.2 17.4 17.6 17.8 18 18.2 Radius (in)
Figure 79: Functionally Graded Material Properties for the Optimized Cylindrical Body
125 Thermal Conductivity for the Functionally Graded Hemisphere
30
29.5 ) F -
t 29 f -
r 2.4329
h y = 0.0267x / 28.5 u
t 2
B R = 0.9972 ( 28 y t i v i t 27.5 c u d
n 27 o C
l 26.5 a m r
e 26 h T 25.5
25 16.6 16.8 17 17.2 17.4 17.6 17.8 18 Radius (in)
Modulus of Elasticity for the Functionally Graded Hemisphere
3.00E+07
2.95E+07
2.90E+07 ) i s p (
2.85E+07
y 2.4329 t
i E = 26660r c
i 2.80E+07 t 2 s R = 0.9972 a l
E 2.75E+07
f o
s 2.70E+07 u l u
d 2.65E+07 o M 2.60E+07
2.55E+07
2.50E+07 16.6 16.8 17 17.2 17.4 17.6 17.8 18 Radius (in)
Coefficient of Thermal Expansion for the Functionally Graded Hemisphere
7.20E-06 ) F - n
i 7.00E-06 / n i (
n o
i 6.80E-06 s n
a 2.4329
p a = 6.221E-9r x 6.60E-06
E 2
l R = 0.9972 a m
r 6.40E-06 e h T
f o
6.20E-06 t n e i c i f
f 6.00E-06 e o C 5.80E-06 16.6 16.8 17 17.2 17.4 17.6 17.8 18 Radius (in)
Figure 80: Functionally Graded Material Properties for the Optimized Hemispherical End Caps
126