Available online at www.sciencedirect.com

ScienceDirect

Comput. Methods Appl. Mech. Engrg. 278 (2014) 640–663 www.elsevier.com/locate/cma

An integrated approach to shape optimization and mesh adaptivity based on material residual forces

Stefan Riehl, Paul Steinmann∗

Chair of Applied Mechanics, Friedrich–Alexander University Erlangen-Nu¨rnberg, Egerlandstr. 5, 91058 Erlangen, Germany

Received 7 March 2014; received in revised form 19 May 2014; accepted 6 June 2014

Available online 18 June 2014

Abstract

This contribution is concerned with the coupling of finite element based shape optimization to methods of mesh adaptivity.

Therein, the nodal points of a finite element mesh serve as design variables in an optimization problem that aims to minimize a cost functional with respect to different constraints. In order to avoid the occurrence of oscillating boundaries in the optimal design trials, we generate the respective design updates through solving a series of fictitious boundary value problems on an updated Lagrangian configuration. The design update process is accompanied by adaptive mesh refinement to obtain more accurate results for the structural analysis and the design sensitivity analysis. For the mesh adaptation process, we consider r -adaptive node relocation, h-adaptive mesh refinement and a combination of both approaches. In each case, the derivations rest upon the notion of material residual forces induced by finite element discretization and the coupling of shape optimization and mesh adaptivity is of intermittent type, i.e. mesh adaptivity is invoked once a certain number of design updates has been generated. We examine the benefits of the proposed method on the basis of some numerical examples in comparison to the same shape optimization method not involving adaptive mesh refinement where we evaluate the corresponding numerical costs, the gain in accuracy and the effects on the optimal shapes being obtained. c⃝ 2014 Elsevier B.V. All rights reserved.

Keywords: Shape optimization; Material force method; Adaptive finite elements; Traction method 1. Introduction

The mechanics of material forces is a branch of continuum mechanics that is concerned with the energetic changes upon a material variation of a continuum point with respect to its ambient material. These considerations date back to at least the early work of Eshelby [1] who introduced the concept of the energy–momentum tensor to capture the driving forces for these energetic changes, a quantity that is now commonly denoted as Eshelbian stress tensor. More recently, the mechanics of material forces has been applied in the field of fracture and defect mechanics to study the effect of energy release upon crack propagation and the movement of defects with respect to its ambient material, see e.g. the works of Maugin [2,3] or the contributions in [4–6]. Moreover, the mechanics in material space is closely ∗ Corresponding author. Tel.: +49 0 9131 85 28501; fax: +49 0 9131 85 28503.

E-mail addresses: steinmann@ltm.uni-erlangen.de, paul.steinmann@ltm.uni-erlangen.de, paul.steinmann@ltm.uni-erlangen.de (P. Steinmann). http://dx.doi.org/10.1016/j.cma.2014.06.010 0045-7825/ c⃝ 2014 Elsevier B.V. All rights reserved.

S. Riehl, P. Steinmann / Comput. Methods Appl. Mech. Engrg. 278 (2014) 640–663 641 related to the field of shape optimization since in either case one is concerned with changes in the state of a continuum body due to variations in the material (or design) space [7,8]. For the well-established compliance minimization problem in shape optimization the material forces acting on the boundary denote the shape gradient function, i.e. they indicate the optimal shape of the design boundary. Furthermore, the development of adaptive finite element methods based on the concept of over-all energy minimization1 is among the most prominent fields of application for the mechanics of material forces and lies within the scope of so-called arbitrary Lagrangian–Eulerian formulations [9,10].

Herein, one aims at minimizing the discrete total potential energy with respect to both, the material and the spatial placement of all nodal points within a finite element mesh. This is achieved by considering not only a spatial but also a material version of the localized force balance to render two different residual statements upon discretization.

However, due to the numerical inaccuracies induced by discretization, a non-vanishing material residual is generally observed, once the spatial deformation map is established. These material residual forces can be utilized in order to allow for different types of adaptive mesh refinement. Primarily, a variety of r -adaptive schemes can be derived in which the nodal point positions are altered in order to reduce the total potential energy.2 Therein, the negative material forces denote the steepest descent directions. An early approach of this kind dates back to the work of

Sussman [11], in which a closed-form expression for the change in the total potential energy upon perturbations of the nodal point coordinates is derived for the case of linear elasticity. However, it was not until the work of

Braun [12] that the term material (or configurational) forces has been employed in order to characterize the spurious residual forces induced by finite element discretization. In the works of Mueller [13,14] a basic steepest descent approach is derived in which all interior nodal points are moved into the negative direction of the corresponding nodal material forces. In order to achieve convergence and to avoid ill-shaped elements, the step length parameter is to be chosen sufficiently small. A more flexible approach is proposed by Thoutireddy [15] wherein not only the material node positions are optimized using a Polak–Ribiere iteration procedure but also additional node-swap operations are taken into account. Therefore, optimal meshes are attainable that could not be reached for meshes having a fixed topological structure. In the work of Mosler [16] additional flexibility is achieved due to the possible migration of nodal points in and out of the boundary. Moreover, a viscous regularization term is used to augment the actual energy minimization problem in order to stabilize a Newton-type solution procedure. The crucial issue of mesh distortion within r -adaptive mesh optimization is explicitly addressed in the works of Scherer [17,18]. Therein, the energy minimization problem is augmented through the introduction of elementwise distortion constraints whereas a penaltybarrier method transforms the constrained energy minimization problem into a sequence of unconstrained problems.