# C03Shearlet approximation of brittle fracture evolutions

A brittle material, subjected to a force, first deforms itself elastically, then it breaks without any intermediate phase. A model of brittle fractures was proposed by Francfort and Marigo, where the displacement is typically a smooth function except on a relatively smooth jump set determining the fracture. This approach has the advantage not to require a pre-defined crack path, but has the drawback that any mesh discretization is a geometrical bias. Despite encouraging numerical results, FEM are expected to retain a certain geometrical bias and so far a proof of convergence of the proposed algorithm remains out of reach. Differently from finite elements, shearlets are frames with rotation invariance and optimal nonlinear approximation properties for the class of functions, which are smooth except on smooth lower dimensional sets. In this project we intend to compare anisotropic and adaptive mesh refinements with adaptive frame methods based on shearlets. In particular, by taking advantage of the shearlet property of optimally approximating piecewise smooth functions, we aim at reaching not only a proof of the convergence of the frame adaptive algorithms but also their optimal complexity.

#### Mission-

Our main goal is the robust numerical simulation of brittle fractures, implementing a provably convergent and geometrically unbiased adaptive frame scheme based on shearlet discretizations.

#### Scientific Details+

A brittle material, subjected to an external force, first deforms itself elastically, then it breaks without any intermediate phase. A mathematical model of brittle fractures has been proposed by Francfort and Marigo. The quasi-static evolution of the fracture is based on the successive minimization of an energy on the material displacement, designed according to the Griffith?s principle of energy balance between elastic energy and a fictitious crack energy. For a fixed time the energy functional can be approximated by an Ambrosio-Tortorelli functional, which can be solved by an alternating minimization algorithm involving the solution of elliptic PDEs.

Despite the realistic and physically sound modeling, this approach has the drawback that any actual discretization is a bias towards a proper fracture propagation. Using traditional isotropic finite element discretization this problem can only be minimized by applying very fine meshes, which drastically increases the computation time as well as the memory demand of the algorithms. Another remedy was recently proposed using adaptive anisotropic remeshing, with the following advantages:

[1] The number of degrees of freedom and the computational times are dramatically reduced, despite the remeshing.

[2] The remeshing does not alter the energy profile evolution.

[3] On the crack tip the automatically generated mesh is nearly isotropic and does not constitute an artificial bias for the crack evolution.

As a consequence we always obtain physically acceptable crack evolutions. Besides, it remains an open and very challenging problem to provide rigorous proofs for the reduced complexity as well as for the more fundamental properties (2) and (3). These very serious theoretical difficulties motivate us to consider the by now standard framework of frame discretizations for elliptic operator equations. Let us recall that a frame is a spanning system of elements in a Hilbert space with certain redundancy properties, i.e., the frame elements are not in general linearly independent. Adaptive numerical methods based on frames have been first addressed in the context of wavelet frame discretizations. However, a theory for general frames is available. The solutions of the brittle fracture evolution at different times  are expected to be smooth functions away from a smooth discontinuity set. This class of functions can be nearly optimally represented by so-called shearlet frames. In particular the approximation is far better than that using isotropic systems. By developing an adaptive frame method based on shearlet frames for the simulation of brittle fracture evolutions we expect the following fundamental advantages:

Adaptive frame methods solve the original problem defined on function spaces (Sobolev spaces) by adaptively solving an infinite dimensional problem defined over sequences; in a some sense, they perform a digitization of the analog problem.

Once this equivalence is established, the methods can be rather easily analyzed in terms of their convergence and complexity; While FEM are based on (adaptive) grids (which are refined or remeshed) and they introduce a natural bias towards the evolution of the fracture, shearlets are an infinite dimensional system which is simultaneously offering to the adaptive scheme all the possible directions and degrees of freedom which are needed to a certain accuracy. In other words shearlets are instantaneously providing us with all the possible futures with no present bias!

#### Publications+

##### Papers
###### Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

Authors: Petersen, Philipp and Raslan, Mones
Journal: Advances in Computational Mathematics, 45(3):1581--1606
Date: Jun 2019
DOI: 10.1007/s10444-019-09679-9
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###### Consistent Finite-Dimensional Approximation of Phase-Field Models of Fracture

Author: Stefano Almi, Sandro Belz
Journal: Ann. Mat. Pura Appl. (4), 198(4):1191--1225
Date: 2019
DOI: 10.1007/s10231-018-0815-z
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###### Optimal Approximation with Sparsely Connected Deep Neural Networks

Authors: Bölcskei, Helmut and Grohs, Philipp and Kutyniok, Gitta and Petersen, Philipp
Journal: SIAM J. Math. Data Sci., 1(1):8--45
Date: 2019
DOI: 10.1137/18M118709X
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###### Convergence of discrete and continuous unilateral flows for Ambrosio-Tortorelli energies and application to mechanics

Authors: Negri, Matteo and Almi, Stefano and Belz, Sandro
Journal: ESAIM: Mathematical Modelling and Numerical Analysis
Date: Sep 2018
DOI: 10.1051/m2an/2018057
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###### Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks

Authors: Almi, Stefano and Lucardesi, Ilaria
Journal: Nonlinear Differ. Equ. Appl. (2018) 25: 43.
Date: Aug 2018
DOI: 10.1007/s00030-018-0536-4
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###### Optimal approximation of piecewise smooth functions using deep ReLU neural networks

Authors: Petersen, Philipp and Voigtlaender, Felix
Journal: Neural Networks, 108:296--330
Date: 2018
DOI: 10.1016/j.neunet.2018.08.019
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###### Bendlets: A second-order shearlet transform with bent elements

Authors: Lessig, Christian and Petersen, Philipp and Schäfer, Martin
Journal: Applied and Computational Harmonic Analysis
Date: 2017
DOI: 10.1016/j.acha.2017.06.002
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###### Memory-optimal neural network approximation

Authors: Bölcskei, Helmut and Grohs, Philipp and Kutyniok, Gitta and Petersen, Philipp
In Proceedings: Wavelets and Sparsity XVII
Date: 2017
DOI: 10.1117/12.2272490
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###### Shearlet approximation of functions with discontinuous derivatives

Author: Petersen, Philipp
Note: preprint
Date: Aug 2015
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###### Anisotropic multiscale systems on bounded domains

Authors: Grohs, P. and Kutyniok, G. and Ma, J. and Petersen, P.
Note: preprint
Date: 2015
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###### Classification of edges using compactly supported shearlets

Authors: Kutyniok, Gitta and Petersen, Philipp
Journal: Applied and Computational Harmonic Analysis
Date: 2015
DOI: 10.1016/j.acha.2015.08.006
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###### Linear independence of compactly supported separable shearlet systems

Authors: Ma, Jackie and Petersen, Philipp
Journal: Journal of Mathematical Analysis and Applications, 428:238-257
Date: 2015
DOI: 10.1016/j.jmaa.2015.03.001
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###### $\alpha$-Molecules

Authors: Grohs, Philipp and Keiper, Sandra and Kutyniok, Gitta and Schäfer, Martin
Note: preprint
Date: Jul 2014
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###### Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames

Authors: Kutyniok, Gitta and Mehrmann, Volker and Petersen, Philipp
Note: preprint
Date: Jul 2014
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###### Dualizable Shearlet Frames and Sparse Approximation

Authors: Kutyniok, Gitta and Lim, Wang-Q
Note: preprint
Date: 2014
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###### ShearLab 3D: Faithful Digital Shearlet Transforms based on Compactly Supported Shearlets

Authors: Kutyniok, Gitta and Lim, Wang-Q and Reisenhofer, Rafael
Note: Preprint
Date: Jan 2014
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###### Gabor Shearlets

Authors: Bodmann, Bernhard G. and Kutyniok, Gitta and Zhuang, Xiaosheng
Journal: Appl. Comput. Harmon. Anal.
Note: submitted
Date: Mar 2013
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#### Prof. Dr. Massimo Fornasier   +

Projects: C03
University: TU München
E-Mail: massimo.fornasier[at]ma.tum.de

#### Prof. Dr. Gitta Kutyniok   +

Projects: C03, C02
University: TU Berlin
E-Mail: kutyniok[at]math.tu-berlin.de
Website: http://www.math.tu-berlin.de/?108957

#### Mones Raslan   +

Projects: C03
University: TU Berlin
E-Mail: raslan[at]math.tu-berlin.de