A new shape optimization approach for fracture propagation

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A new shape optimization approach for fracture

propagation

Tim Suchan1, Kathrin Welker2, and Winnifried Wollner3

1Helmut-Schmidt-Universität / Universität der Bundeswehr

Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany,

suchan@hsu-hh.de

2Technische Universität Bergakademie Freiberg, Akademiestraße 6,

09599 Freiberg, Germany, Kathrin.Welker@math.tu-freiberg.de

3Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany,

winnifried.wollner@uni-hamburg.de

Abstract

Within this work, we present a novel approach to fracture simula-

tions based on shape optimization techniques. Contrary to widely-used

phase-ﬁeld approaches in literature the proposed method does not require

a speciﬁed ’length-scale’ parameter deﬁning the diﬀused interface region

of the phase-ﬁeld. We provide the formulation and discuss the used so-

lution approach. We conclude with some numerical comparisons with

well-established single-edge notch tension and shear tests.

1 Introduction

We consider an alternative approach for the solution of quasi-static brittle frac-

ture propagation due to Griﬃth’s [12] model. Based on the variational formu-

lation proposed initially by Francfort & Marigo [8]. In its simplest form this

model consists of a minimization problem in displacement ¯

wand fracture ¯

which need to solve

min

w,UEbulk(w,U) + GcH1(U)

over all admissible displacements wand fractures Uwhich will be speciﬁed

in Section 2. Here Gc∈Rdenotes the fracture toughness and H1is the 1-

dimensional Hausdorﬀ-measure. Due to the diﬃculty of discretizing the lower-

dimensional fracture a common approach is based on Ambrosio & Tortorelli [3].

Here the lower-dimensional fracture is replaced by a, smooth, phase-ﬁeld func-

tion whose values indicate fractured or non fractured regions, see, e.g., [8, 5,

18, 2, 19] for application to fracture problems. The price to be paid in such

phase-ﬁeld problems is the introduction of (at least) two regularization param-

eters, one for the approximation of the Hausdorﬀ-measure and one to assert the

arXiv:2210.05502v2 [math.OC] 9 Nov 2022

coercivity of the bulk-energy Ebulk in regions of vanishing phase-ﬁeld. To ob-

tain meaningful numerical results the precise choice of the parameters and their

balance with the discretization error must be carefully addressed, see, e.g., [28],

and a necessary sharp resolution of the transition zone requires adaptive dis-

cretizations with appropriate a posteriori error indicators [6, 4, 15, 25].

Within this article, we propose a new approach avoiding the replacement of

the lower dimensional fracture by a phase-ﬁeld method. The fracture evolution

is then realized by means of techniques from shape optimization. In order

to obtain an eﬃcient shape optimization algorithm, we consider the so-called

Steklov-Poincaré metric [21] in this work. The Steklov-Poincaré metric has

some numerical advantages over other types of metrics as shown in [23, 26].

In addition, the Steklov-Poincaré metric allows to work with so-called weak

formulations of shape derivatives, i.e., volume expression of shape derivatives.

In the past, e.g., [7, 24], major eﬀort in shape calculus has been devoted towards

expressions for shape derivatives in the Hadamard form, i.e., in the boundary

integral form. An equivalent and intermediate result in the process of deriving

Hadamard expressions is a volume expression of the shape derivative, called the

weak formulation. Thus, working with weak formulations saves analytical eﬀort.

The rest of the paper is structured as follows, in Section 2, we will describe

the considered problem in more detail. In Section 3, we will discuss how this

problem can be restated as a shape optimization problem and provide the for-

mulas needed in the computation of descent directions. Finally, in Section 4

we will provide numerical results for the proposed approach for the well known

single-edge notch tension and shear test from [17] and the setup given in [15].

2 Problem description

We focus on a two-dimensional setup. Here a hold-all domain D⊂R2is con-

sidered which is decomposed into a fracture Uand a remaining domain Ω⊂R2

such that D=ΩtU. A sketch can be found in Fig. 1.

Given an initial fracture U0the quasi-static evolution of the fracture Uis

governed by the energy minimization

min

w,U

2(C:ε(w),ε(w))L2(Ω)−(f,w)L2(Ω)+GcH1(U).(1)

Here w∈H1

D(Ω;R2) + wD={w∈H1(Ω;R2)|w=wDon ΓD}are the ad-

missible displacements, where ΓD⊂∂Ω \Uis a given boundary part, and wD

denotes prescribed Dirichlet data. The fracture Uis required to be monoton-

ically increasing in time, i.e., U(t+δt)⊃U(t)for any δt > 0. The ﬁrst two

terms in (1) denote the elastic bulk energy, i.e., ε(w) = 1

2∇w+ (∇w)>is

the symmetric gradient, and the stress-strain relation is given by σ(w) = C:

ε(w)=2µε(w) + λtr(ε(w))Iwith the Lamé parameters µ, λ. The last term

in (1) is the 1-dimensional Hausdorﬀ measure which describes the length of the

fracture.

In order to solve (1) with shape optimization techniques, we need to ﬁnd a

shape deﬁnition of the fracture. One possibility is to deﬁne the fracture by a

curve u: [0,1] →R2and assume that uis an element of

e=B0

e[0,1],R2:=Emb0([0,1],R2)/Diﬀ0([0,1]),

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摘要：

AnewshapeoptimizationapproachforfracturepropagationTimSuchan1,KathrinWelker2,andWinnifriedWollner31Helmut-Schmidt-Universität/UniversitätderBundeswehrHamburg,Holstenhofweg85,22043Hamburg,Germany,suchan@hsu-hh.de2TechnischeUniversitätBergakademieFreiberg,Akademiestraÿe6,09599Freiberg,Germany,Kathrin....

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A new shape optimization approach for fracture propagation

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