Topology optimization of Stokes flow with traction boundary conditions using low-order finite elements

Carl-Johan Thore

doi:10.1016/j.cma.2021.114177

Abstract

We consider topology optimization of Stokes flow with traction boundary conditions using finite elements with low-order velocity-approximation and an element-wise constant hydrostatic pressure. The finite element formulation is stabilized using a penalty on the jump in pressure between adjacent elements. Convergence of solutions to the finite element-discretized topology optimization problem is shown, and several optimization problems are solved using a preconditioned conjugate gradient solver for the finite element matrix problem. Stable convergence to high-quality designs without an excessive number of linear solver iterations is observed, and it is seen that the finite element formulation is not particularly sensitive to the choice of the pressure jump penalty parameter, thus making it a practically useful method.

Highlights

Topology optimization (TO) in fluid flow problems is a research area which receives a lot of academic and industrial interest [1]
In this article we consider TO problems where the flow is governed by the Stokes–Brinkman model proposed by Borrvall and Petersson [2]
A drawback of traction boundary conditions is that it precludes the use of certain computationally attractive low-order finite elements (FEs) such as the Crouzeuix–Raviart P1–P0 element. (While the mass matrix in the Stokes–Brinkman system ensures, with a non-zero lower bound on the coefficient, coercivity in the discrete setting, the element can exhibit locking-type phenomena [3, Section 5].) The purpose of this work is to investigate the performance of a stabilized FE method [4,5,6,7] which enables the use of low-order velocity and element-wise constant hydrostatic pressure approximations

Summary

Introduction

Topology optimization (TO) in fluid flow problems is a research area which receives a lot of academic and industrial interest [1]. In contrast to most work on TO with this model we consider problems where traction and not velocity is prescribed on the potential inlet and/or outlet regions of the design domain. For two-dimensional problems, the nonconforming, stable P1–P0 element due to Kouhia and Stenberg [9] is an attractive alternative to the tested method: it is as simple to implement as the conforming (but useless) P1–P0 element, and introduces no additional parameters into the FE problem. It does not have a counterpart in 3D. Found any publications on TO in flow problems where stabilization is a central theme treated in detail

The continuum problem

Finite element discretization

Finite element convergence

Matrix problem

Numerical examples