# TOPOLOGY OPTIMIZATION USING A LEVEL SET PENALIZATION WITH CONSTRAINED TOPOLOGY FEATURES

8-2013

Thesis

## Degree Name

Master of Science (MS)

## Legacy Department

Mechanical Engineering

Thompson , Lonny

Qiao , Rui

Guarneri , Paolo

## Abstract

Topology optimization techniques have been applied to structural design problems in order to determine the best material distribution in a given domain. The topology optimization problem is ill-posed because optimal designs tend to have infinite number of holes. In order to regularize this problem, a geometrical constraint, for instance the perimeter of the design (i.e., the measure of the boundary of the solid region, length in 2D problems or the surface area in 3D problems) is usually imposed. In this thesis, a novel methodology to solve the topology optimization problem with a constraint on the number of holes is proposed. Case studies are performed and numerical tests evaluated as a way to establish the efficacy and reliability of the proposed method.
In the proposed topology optimization process, the material/void distribution evolves towards the optimum in an iterative process in which discretization is performed by finite elements and the material densities in each element are considered as the design variables. In this process, the material/void distribution is updated by a two-step procedure. In the first step, a temporary density function, ϕ*(x), is updated through the steepest descent direction. In the subsequent step, the temporary density function ϕ*(x) is used to model the next material/void distribution, χ*(x), by means of the level set concept. With this procedure, holes are easily created and quantified, material is conveniently added/removed.
If the design space is reduced to the elements in the boundary, the topology optimization process turns into a shape optimization procedure in which the boundaries are allowed to move towards the optimal configuration. Thus, the methodology proposed in this work controls the number of holes in the optimal design by combining both topology and shape optimization.
In order to evaluate the effectiveness of the proposed method, 2-D minimum compliance problems with volume constraints are solved and numerical tests performed. In addition, the method is capable of handling very general objective functions, and the sensitivities with respect to the design variables can be conveniently computed.

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