Computational Optimization of Structural and Thermal Compliance Using Gradient-Based Methods

Abstract

We consider the problem of structural optimization which has many important applications in the engineering sciences. The goal is to find an optimal distribution of the material within a certain volume that will minimize the mechanical and/or thermal compliance of the structure. The physical system is governed by the standard models of elasticity and heat transfer expressed in terms of boundary-value problems for elliptic systems of partial differential equations (PDEs). The structural optimization problem is then posed as a suitably constrained PDE optimization problem, which can be solved numerically using a gradient approach. As a main contribution to the thesis, we derive expressions for gradients (sensitivities) of different objective functionals. This is done in both the continuous and discrete setting using the Riesz representation theorem and adjoint analysis. The sensitivities derived in this way are then tested computationally using simple minimization algorithms and some standard two-dimensional test problems.