Structural Optimization of a Thermally Loaded Functionally Graded Pressure VesselbyJeffrey Charles YoungsA Thesis Submitted to the GraduateFaculty of Rensselaer Polytechnic Institutein Partial Fulfillment of theRequirements for the degree ofMASTER OF SCIENCEMajor Subject: Mechanical EngineeringApproved:_Ernesto Gutierrez-Miravete, Thesis AdviserRensselaer Polytechnic InstituteHartford, ConnecticutAugust 2009 Copyright 2009byJeffrey YoungsAll Rights ReservedCONTENTSCONTENTSiLIST OF TABLESvLIST OF FIGURESviLIST OF SYMBOLSxiACKNOWLEDGMENTxivABSTRACTxv1.Introduction11.1Problem Statement11.2History of Functionally Graded Materials11.3Pressure Vessel Description21.4Finite Element Theory31.5Basic Structural Optimization Theory52.Analytical Modeling72.1Exact Solution for a Thick Homogeneous Cylinder72.2Exact Solution for a Thick Homogeneous Hemisphere112.3Exact Solution for Functionally Graded Material152.4Stress-Strain Equations for a Functionally Graded Vessel162.4.1Strain Solutions for a Functionally Graded Cylinder172.4.2Stress Solutions for a Functionally Graded Cylinder182.4.3Strain Solutions for a Functionally Graded Hemisphere192.4.4Stress Solutions for a Functionally Graded Hemisphere203.Design Parameters213.1Pressure Vessel Initial Shape213.2Loading Conditions213.3Materials233.4Modeling Material Properties in Hypermesh254.Finite Element Modeling274.1Element Shape Considerations274.2Element Order Considerations274.3Axisymmetric Model284.4Solid Element Model334.5Choosing Element Type384.6Element Convergence435.Structural Optimization485.1Deciding on a Solver485.2Optimization Model in OptiStruct485.3OptiStructs Iterative Solution for Structural Optimization515.4Design Sensitivity Analysis in OptiStruct525.5Convergence in OptiStruct535.6Topology Optimization Method in OptiStruct535.7Density Method of Topology Optimization545.8Topology Optimization: Free Form555.9Defining the Optimized and Non-Optimized Regions565.10Defining the Load Sets for Structural Optimization565.11Define optimization criteria575.12Results for Free Form Topology Optimization585.13Redefining the optimized regions for topology optimization605.14Redefining the design criteria: Discreteness Parameter / Penalty Factor605.15Results for Constrained Topology Optimization605.16Added Manufacturability Constraints: Symmetry Plane and Supporting Design Parameters625.17Results for topology optimization of a 1 ” thick model655.18Shape Optimization Method in OptiStruct675.19Free Shape Optimization Method in OptiStruct675.20Defining the optimized and non-optimized regions for Free Shape Optimization685.21Defining the design criteria685.22Manufacturability Constraint: Grid Constraints705.23Control Cards for Free Shape Optimization715.24Optimization Results: Total Model Volume and Maximum Von Mises Stress, Only Hemispherical Caps Optimized715.25Optimization Results: Total Model Volume and Maximum Von Mises Stress, Whole Body Optimization745.26Results: Thickness Change for Whole Body Free Shape Optimization805.27Results: Optimization Model Shape Change825.28Free Shape Optimization Study: Whole Body, Homogeneous Material845.29Results: Optimization Model Material Distribution856.Conclusions876.1Comparison of Topology and Free Shape Optimization876.2Comparison of Analytical Solution and Finite Element Analysis for the Optimized Model886.3Future Work907.References928.Appendix948.1Excel VBA Code for Creating Material Cards for ABAQUS (12Element Layers)948.2Excel VBA Code for Creating Material Cards for OPTISTRUCT (12Element Layers)958.3Batch File Code for Input File Translation968.4Hypermesh TCL Code For Assigning Properties to Adjacent Layers to Simulate Functionally Graded Material978.5Excel VBA Code for Translating Heat Transfer Results to SPC Temperature Loads for OptiStruct Stress/Strain Analysis988.6Mesh Convergence Plots, Stresses for Functionally Graded Cylinder998.7Mesh Convergence Plots, Stresses for Functionally Graded Sphere/Hemisphere1028.8Free Form Topology Element Density Plots1058.9Topology Optimization Element Density Plots: Constrained Inner Surface, 3” Thick Vessel1088.10Topology Optimization Element Density Plots: Constrained Inner Surface, 1 ” Thick Vessel1118.11Free Shape Optimization Plots: Shape Change Compared to Non-Optimized Model (inches), Hemispherical End Cap Optimized (16Smoothing Layers, 0.125Move Factor)1148.12Free Shape Optimization Plots: Shape Change Compared to Non-Optimized Model (inches), Entire Vessel Optimized (12Smoothing Layers, 0.075Move F