Weak Material Approximation of Holes with Traction-Free Boundaries

Abstract

Consider the solution of a boundary-value problem for steady linear elasticity in which the computational domain contains one or several holes with traction-free boundaries. The presence of holes in the material can be approximated using a weak material; that is, the relative density of material $\rho$ is set to $0<\epsilon=\rho\ll1$ in the hole region. The weak material approach is a standard technique in the so-called material distribution approach to topology optimization, in which the inhomogeneous relative density of material is designated as the design variable in order to optimize the spatial distribution of material. The use of a weak material ensures that the elasticity problem is uniquely solvable for each admissible value $\rho\in[\epsilon, 1]$ of the design variable. A finite-element approximation of the boundary-value problem in which the weak material approximation is used in the hole regions can be viewed as a nonconforming but convergent approximation of a version of the original problem in which the solution is continuously and elastically extended into the holes. The error in this approximation can be bounded by two terms that depend on $\epsilon$. One term scales linearly with $\epsilon$ with a constant that is independent of the mesh size parameter $h$ but that depends on the surface traction required to fit elastic material in the deformed holes. The other term scales like $\epsilon^{1/2}$ times the finite-element approximation error inside the hole. The condition number of the weak material stiffness matrix scales like $\epsilon^{-1}$, but the use of a suitable left preconditioner yields a matrix with a condition number that is bounded independently of $\epsilon$. Moreover, the preconditioned matrix admits the limit value $\epsilon\to0$, and the solution of corresponding system of equations yields in the limit a finite-element approximation of the continuously and elastically extended problem.