# Structural optimization using the finite element method

• João Pedro Pires Mesquita (Student)

Student thesis: Master's Thesis

### Abstract

This work reports on the formulation of the conforming finite element method (FEM) and its application to structural optimization problems. The FEM is nowadays the most widely used technique for obtaining approximate solutions of complex engineering problems that cannot be solved analytically. The conforming finite element formulation can be reached using several approaches. One of the most general strategies is based on the weak-form of the equations governing the problem. However, rather than a pure mathematical approach, it is often more appealing to the structural engineer to reach the finite element formulation through more physically-meaningful strategies, such as the virtual work or the variational principle forms of the problem. All three of these strategies are briefly presented. One of the most important topics in the finite element theory is the quality of the solutions. The most commonly used method to increase the quality of a finite element solution is to increase of the number of elements used in the model (mesh refinement). In this work, two numerical examples are presented to illustrate the convergence of the finite element solutions under mesh refinement. An important application of the FEM and the focus of this work is the structural optimization. The purpose of the structural optimization is to minimize (or maximize) an objective function while respecting certain restrictions. The extremum of a continuous function on a certain interval can either correspond to a point where the gradient is null, or it may lay on the boundary of the interval. Several numerical methods for identifying null gradient points of a function are described in this work. When the extremum of a function lays on the boundary of its interval of definition, an effective minimization method should ensure that the boundaries are searched in such a way that every new iteration yields a result closer to the extremum than the one before. The basic aspects of the constrained minimization methods are briefly presented here and applied to two structural optimization problems, namely the size optimization of a bar subjected to its own weight and the topology optimization of a cantilever plate subjected to a concentrated load applied to its tip. A large scale practical example of structural optimization, consisting of the topological optimization of a wheel carrier of a motorsports car is also presented. The model is constructed using the finite element package MSC.-NASTRAN. The main goal of the optimization is to minimize the structural weight by reducing the amount of material used in order to create a new optimized design. The feasible optimized structure is analyzed and the obtained stresses and displacements are compared with the analysis results of the non-optimized structure.
Date of Award 2014 English Universidade Católica Portuguesa Ionut Dragos Moldovan (Supervisor) & Timo Prissler (Supervisor)

### Keywords

• Finite element method
• Structural optimization
• Function minimization

### Designation

• Mestrado em Engenharia Civil

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