Three-dimensional topology optimization of statically loaded porous and multi-phase structures

A novel topology optimization method called Prescribed Material Redistribution (PMR) has been under development at the University of Rhode Island for the past several years. Originally implemented through a series of Fortran subroutines used in conjunction with the commercial finite element package, Abaqus, a standalone two-dimensional Matlab code was developed and used in evaluating the method. In order to explore the capabilities of the PMR scheme for three-dimensional problems, it became necessary to develop a three-dimensional version of the PMR Matlab code. The objective of this thesis, therefore, is the development and evaluation of a three-dimensional Matlab implementation of the PMR scheme. The code allows users to analyze general topology optimization problems by defining an appropriate design domain, load conditions, support conditions, predefined fully dense or void regions, and symmetry conditions. The code also provides the capability to impose constraint conditions where coupling of displacement degrees of freedoms can be specified by the user. A primary aspect of this work is the development and implementation of hexahedral finite element equations. Since three-dimensional problems can be computationally intensive, the finite element analysis implementation includes computationally efficient algorithms where feasible. The post-processing phase of the analysis included the generation of optimized three-dimensional geometry in the standard STL file format. The STL file allows users to examine the results using standard CAD file viewers. Also, the STL file can be used as input to additive manufacturing equipment such as 3-D printing for the manufacture of physical components. The three-dimensional PMR code is evaluated by two types of optimization problems. The first set of test cases investigated are based on the identification of a known topology for a centrally loaded, simply supported beam. Although this problem can be considered using a two-dimensional analysis, performing a three-dimensional analysis allowed for the ability to consider several different symmetry cases. This is useful for evaluation of the symmetry capabilities of the three-dimensional PMR code. For the symmetry case where all three coordinate axes define symmetry place, an alternate test case was developed and used for evaluating the code. The second set of test cases were designed to identify optimal topologies for two-phase composite microstructures under general three-dimensional stress states. By defining unit cell models with appropriate loading and constraint conditions, any three-dimensional stress states can be modeled. If one of the composite phases is taken to have essentially zero stiffness, this approach can be used to determine the optimized microstructure of porous materials. Several test cases are evaluated for the identification of optimized microstructures for both porous and composite materials