Free Discontinuity Problems Research Articles

Manifold-constrained free discontinuity problems and Sobolev approximation

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain Ω⊂R2) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano (Conti et al., 2017; Conti et al., 2019) for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.

Learning the effective adhesive properties of heterogeneous substrates

Adhesion is a fundamental phenomenon that plays a role in many engineering and biological applications. This paper concerns the use of machine learning to characterize the effective adhesive properties when a thin film is peeled from a heterogeneous substrate. There has been recent interest in the use of machine learning in multiscale modeling where macroscale constitutive relations are learnt from data gathered from repeated solution of the microscale problem. We extend this approach to peeling; this is challenging because peeling from heterogeneous substrates is characterized by pinning where the peel front gets stuck at a heterogeneity followed by an abrupt depinning. This results in a heterogeneity dependent critical force and a singular peel force vs. overall peel rate relationship. We propose a neural architecture that is able to accurately predict both the critical peel force and the singular nature of the peel force vs. overall peel rate relationship from the heterogeneous adhesive pattern. Similar issues arise in other free boundary and free discontinuity problems, and the methods we develop are applicable in those contexts.

Integral representation and Gamma -convergence for free-discontinuity problems with p(cdot )-growth

An integral representation result for free-discontinuity energies defined on the space GSBV^{p(cdot )} of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-Hölder continuity for the variable exponent p(x). Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitté et al. (Arch Ration Mech Anal 165:187–242, 2002) for a constant exponent. We prove Gamma -convergence of sequences of energies of the same type, we identify the limit integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions.

Accounting for weak interfaces in computing the effective crack energy of heterogeneous materials using the composite voxel technique

AbstractWe establish a computational methodology to incorporate interfaces with lower crack energy than the surrounding phases when computing the effective crack energy of brittle composite materials. Recent homogenization results for free discontinuity problems are directly applicable to the time-discretized Francfort-Marigo model of brittle fracture in the anti-plane shear case, and computational tools were introduced to evaluate the effective crack energy on complex microstructures using FFT-based solvers and a discretization scheme based on a combinatorially consistent grid. However, this approach only accounts for the crack resistance per volume and is insensitive to the crack resistance of the interface which is expected to play a significant role by considerations from materials science. In this work we introduce a remedy exploiting laminate composite voxels. The latter were originally introduced to enhance the accuracy of solutions for elasticity problems on regular voxel grids. We propose an accurate approximation of the effective crack energy of a laminate with weak interface where an explicit solution is available. We incorporate this insight into an efficient algorithmic framework. Finally, we demonstrate the capabilities of our approach on complex microstructures with weak interfaces between different constituents.

Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation

We analyse the$\Gamma$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.

Regularity of minimizers for free-discontinuity problems withp(·)-growth

A regularity result for free-discontinuity energies defined on the spaceSBVp(·)of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Hölder continuity for the variable exponentp(x). Our analysis expands on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper N. Fusco et al.,J. Convex Anal.8 (2001) 349-367, dealing with a constant exponent.

\Gamma-convergence for free-discontinuity problems in linear elasticity: homogenization and relaxation

\Gamma-convergence for free-discontinuity problems in linear elasticity: homogenization and relaxation

Degenerate Free Discontinuity Problems and Spectral Inequalities in Quantitative Form

We introduce a new geometric–analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with a negative sign. This is motivated by searching quantitative inequalities for the best constants of Sobolev–Poincaré inequalities with trace terms in \({\mathbb {R}}^n\) which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.

Multiphase free discontinuity problems: Monotonicity formula and regularity results

The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.

Quantitative analysis of finite-difference approximations of free-discontinuity problems

Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio–Tortorelli functional. Denoted by \varepsilon the elliptic-approximation parameter and by \delta the discretisation step-size, we fully describe the relative impact of \varepsilon and \delta in terms of \Gamma -limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when \varepsilon and \delta are of the same order, the underlying lattice structure affects the \Gamma -limit which turns out to be an anisotropic free-discontinuity functional.

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

We analyze integral representation and $\Gamma$-convergence properties of functionals defined on \emph{piecewise rigid functions}, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for $\Gamma$-convergence and a careful adaption of the global method for relaxation (Bouchitt\'e et al. 1998, 2001) to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.

Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity problems, we prove existence of minimizers for boundary value problems. We then pass to a small strain limit in terms of suitably rescaled displacement fields and show that the nonlinear energies can be identified with a linear Griffith model in the sense of Γ-convergence. This complements the study in [ 39 ] by providing a linearization result in arbitrary space dimensions.

Minimization of the k-th eigenvalue of the Robin-Laplacian

The paper is concerned with the minimization of the k-th eigenvalue of the Laplace operator with Robin boundary conditions, among all open sets of RN satisfying a volume constraint. We prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets. The main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation. One of the key difficulties (for k≥3) comes from the fact that the eigenvalues are critical points.

A compactness result in $$GSBV^p$$ and applications to $$\varGamma $$-convergence for free discontinuity problems

We present a compactness result in the space $$GSBV^p$$ which extends the classical statement due to Ambrosio (Arch Ration Mech 111:291–322, 1990) to problems without a priori bounds on the functions. As an application, we revisit the $$\varGamma $$ -convergence results for free discontinuity functionals established recently by Cagnetti et al. [Ann Inst H Poincare Anal Non Lineaire (to appear)]. We investigate sequences of boundary value problems and show convergence of minimum values and minimizers.

Γ-convergence of free-discontinuity problems

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

Second-order approximation of free-discontinuity problems with linear growth

Motivated by applications to image denoising, we propose an approximation of functionals of the form F ( u ) = ∫ Ω | ∇ u | d x + ∫ S u g ( | u + − u − | ) d H n − 1 + | D c u | ( Ω ) , u ∈ BV ( Ω ) , with g : [ 0 , + ∞ ) → [ 0 , + ∞ ) increasing and

6th International Workshop on New Computational Methods for Inverse Problems

ForewordThis volume of Journal of Physics: Conference Series is dedicated to the scientific contributions presented during the 6th International Workshop on New Computational Methods for Inverse Problems, NCMIP 2016 (http://complement.farman.ens-cachan.fr/NCMIP 2016.html). This workshop took place at Ecole Normale Supérieure de Cachan, on May 20, 2016. The prior editions of NCMIP also took place in Cachan, France, firstly within the scope of ValueTools Conference, in May 2011, and secondly at the initiative of Institut Farman, in May 2012, May 2013, May 2014 and May 2015.The New Computational Methods for Inverse Problems (NCMIP) workshop focused on recent advances in the resolution of inverse problems. Indeed, inverse problems appear in numerous scientific areas such as geophysics, biological and medical imaging, material and structure characterization, electrical, mechanical and civil engineering, and finances. The resolution of inverse problems consists in estimating the parameters of the observed system or structure from data collected by an instrumental sensing or imaging device. Its success firstly requires the collection of relevant observation data. It also requires accurate models describing the physical interactions between the instrumental device and the observed system, as well as the intrinsic properties of the solution itself. Finally, it requires the design of robust, accurate and efficient inversion algorithms. Advanced sensor arrays and imaging devices provide high rate and high volume data; in this context, the efficient resolution of the inverse problem requires the joint development of new models and inversion methods, taking computational and implementation aspects into account. During this one- day workshop, researchers had the opportunity to bring to light and share new techniques and results in the field of inverse problems.The topics of the workshop were: algorithms and computational aspects of inversion, Bayesian estimation, Kernel methods, learning methods, convex optimization, free discontinuity problems, metamodels, proper orthogonal decomposition, reduced models for the inversion, non-linear inverse scattering, image reconstruction and restoration, and applications (bio-medical imaging, nondestructive evaluation...).NCMIP 2016 was a one-day workshop held in May 2016 which attracted around seventy attendees. Each of the submitted papers has been reviewed by two reviewers. There have been eleven accepted papers. In addition, three international speakers were invited to present a longer talk.The workshop was supported by Institut Farman (ENS Cachan, CNRS) and endorsed by the following French research networks: GDR ISIS, GDR MIA, GDR MOA, GDR Ondes. The program committee acknowledges the following research laboratories: CMLA, LMT, LURPA and SATIE.Eric Vourc'h and Thomas RodetWorkshop co-chairs:Eric Vourc'h, SATIE laboratory, Ecole Normale Supérieure de Cachan, CNRS, France Thomas Rodet, SATIE laboratory, Ecole Normale Supérieure de Cachan, CNRS, FranceTechnical program committee:Alexandre Baussard, ENSTA Bretagne, Lab-STICC, France Marc Bonnet, ENSTA, ParisTech, FranceLaure Blanc-Féraud, I3S laboratory and INRIA Nice Sophia-Antipolis, FranceAntonin Chambolle, CMAP, Ecole Polytechnique, CNRS, FranceOliver Dorn, School of Mathematics, University of Manchester, UKCécile Durieu, SATIE, ENS Cachan, CNRS, FranceLaurent Fribourg, LSV, ENS Cachan, CNRS, FranceJerôme Idier, IRCCyN Laboratory, Ecole Centrale de Nantes, FrancePierre-Yves Joubert, IEF, Paris-Sud University, CNRS, FranceMarc Lambert, Geeps Laboratory, CNRS, CentraleSupElec, Paris-Sud University, France Giacomo Oliveri, eledia research center/eledia@L2S group, University of Trento, Italy Dominique Lesselier, L2S Laboratory, CNRS, CentraleSupElec, Paris-Sud University, France Matteo Pastorino, DIBE, University of Genoa, ItalyGabriel Peyré, Ceremade laboratory, University of Paris Dauphine, France Anthony Quinn, Trinity College Dublin, Ireland Christian Rey, Safran Tech, FranceMarco Salucci, eledia research center/eledia@L2S group, University of Trento, Italy Jean-Yves Tourneret, IRIT laboratory, ENSEEIHT (Toulouse), FranceLocal chairs:Virginie Pauchont, CMLA Laboratory, ENS Cachan, Paris-Sud University, CNRS, France Sophie Abriet, SATIE Laboratory, ENS Cachan, Paris-Sud University, CNRS, France Béatrice Bacquet, SATIE Laboratory, ENS Cachan, Paris-Sud University, CNRS, France Aurore Gracia, SATIE Laboratory, ENS Cachan, Paris-Sud University, CNRS, FranceInvited speakersLaure Blanc-Féraud, I3S Laboratory, CNRS UMR 6070, Sophia Antipolis, France Michel Defrise, Department of Nuclear Medicine, Vrije Universiteit Brussel, Belgium Houssem Haddar, INRIA, CMAP, Ecole Polytechnique, Palaiseau, France.

Fine numerical analysis of the crack-tip position for a Mumford–Shah minimizer

A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford–Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping \tilde{z}=\sqrt{z} in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.

Faber–Krahn Inequalities for the Robin-Laplacian: A Free Discontinuity Approach

Isoperimetric inequalities for the principal eigenvalues of the Robin-Laplacian are interpreted as free discontinuity problems (of unusual type). We prove a full range of Faber–Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincare inequalities with trace terms in \({\mathbb{R}^N}\).

Monotonicity Formula and Regularity for General Free Discontinuity Problems

We give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d − 1. This result allows us to prove partial regularity results (that is closure and density estimates for the jump set) for a large class of free discontinuity problems involving general energies associated to the jump set, as for example free boundary problems with Robin conditions. In particular, we give a short proof to the De Giorgi–Carriero–Leaci result for the Mumford–Shah functional.