C03
Shearlet 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
  • Philipp Petersen and Mones Raslan.
    Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces.
    Advances in Computational Mathematics, 45(3):1581–1606, June 2019.
    arXiv:1712.01047, doi:10.1007/s10444-019-09679-9.
  • Sandro Belz Stefano Almi.
    Consistent Finite-Dimensional Approximation of Phase-Field Models of Fracture.
    Ann. Mat. Pura Appl. (4), 198(4):1191–1225, 2019.
    arXiv:1707.00578, doi:10.1007/s10231-018-0815-z.
  • Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, and Philipp Petersen.
    Optimal Approximation with Sparsely Connected Deep Neural Networks.
    SIAM J. Math. Data Sci., 1(1):8–45, 2019.
    arXiv:1705.01714, doi:10.1137/18M118709X.
  • Matteo Negri, Stefano Almi, and Sandro Belz.
    Convergence of discrete and continuous unilateral flows for Ambrosio-Tortorelli energies and application to mechanics.
    ESAIM: Mathematical Modelling and Numerical Analysis, September 2018.
    doi:10.1051/m2an/2018057.
  • Stefano Almi and Ilaria Lucardesi.
    Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks.
    Nonlinear Differ. Equ. Appl. (2018) 25: 43., August 2018.
    doi:10.1007/s00030-018-0536-4.
  • Philipp Petersen and Felix Voigtlaender.
    Optimal approximation of piecewise smooth functions using deep ReLU neural networks.
    Neural Networks, 108:296–330, 2018.
    arXiv:1709.05289, doi:10.1016/j.neunet.2018.08.019.
  • Christian Lessig, Philipp Petersen, and Martin Schäfer.
    Bendlets: A second-order shearlet transform with bent elements.
    Applied and Computational Harmonic Analysis, 2017.
    arXiv:1607.05520, doi:10.1016/j.acha.2017.06.002.
  • Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, and Philipp Petersen.
    Memory-optimal neural network approximation.
    In Wavelets and Sparsity XVII, volume 10394, 103940Q. International Society for Optics and Photonics, 2017.
    URL: https://www.math.tu-berlin.de/fileadmin/i26_fg-kutyniok/Kutyniok/Papers/EfficientMemoryDNNs.pdf, doi:10.1117/12.2272490.
  • Philipp Petersen.
    Shearlet approximation of functions with discontinuous derivatives.
    preprint, August 2015.
    arXiv:1508.00409.
  • P. Grohs, G. Kutyniok, J. Ma, and P. Petersen.
    Anisotropic multiscale systems on bounded domains.
    preprint, 2015.
    arXiv:1510.04538.
  • Gitta Kutyniok and Philipp Petersen.
    Classification of edges using compactly supported shearlets.
    Applied and Computational Harmonic Analysis, 2015.
    arXiv:1411.5657, doi:10.1016/j.acha.2015.08.006.
  • Jackie Ma and Philipp Petersen.
    Linear independence of compactly supported separable shearlet systems.
    Journal of Mathematical Analysis and Applications, 428:238–257, 2015.
    arXiv:1404.1690, doi:10.1016/j.jmaa.2015.03.001.
  • Philipp Grohs, Sandra Keiper, Gitta Kutyniok, and Martin Schäfer.
    α-Molecules.
    preprint, July 2014.
    arXiv:1407.4424.
  • Gitta Kutyniok, Volker Mehrmann, and Philipp Petersen.
    Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames.
    preprint, July 2014.
    arXiv:1407.7349.
  • Gitta Kutyniok and Wang-Q Lim.
    Dualizable Shearlet Frames and Sparse Approximation.
    preprint, 2014.
    dgd:130.
  • Gitta Kutyniok, Wang-Q Lim, and Rafael Reisenhofer.
    ShearLab 3D: Faithful Digital Shearlet Transforms based on Compactly Supported Shearlets.
    Preprint, January 2014.
    arXiv:1402.5670, dgd:49.
  • Bernhard G. Bodmann, Gitta Kutyniok, and Xiaosheng Zhuang.
    Gabor Shearlets.
    Appl. Comput. Harmon. Anal., March 2013. submitted.
    URL: http://www.math.tu-berlin.de/fileadmin/i26_fg-kutyniok/Kutyniok/Papers/GaborShearlets.pdf, arXiv:1303.6556.

Team+

Prof. Dr. Gitta Kutyniok   +

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