Nonsmooth Variational Problems Research Articles

Multiple normalized solutions for a quasi-linear Schrödinger equation via dual approach

In this paper, we construct multiple normalized solutions of the following from quasi-linear Schrödinger equation: -\Delta u-\Delta(|u|^{2})u-\mu u=|u|^{p-2}u, \quad\text{in } \mathbb{R}^N, subject to a mass-subcritical constraint. In order to overcome non-smoothness of the associated variational formulation we make use of the dual approach. The constructed solutions possess energies being clustered at $0$ level which makes it difficult to use existing methods for non-smooth variational problems such as the variational perturbation approach.

Junction problem for thin elastic and volume rigid inclusions in elastic body.

The article concerns a junction problem for two-dimensional elastic body with a thin elastic inclusion and a volume rigid inclusion. It is assumed that the inclusions have a common point. A delamination of the thin inclusion from the surrounding elastic body is assumed thus forming an interfacial crack. Constraint-type boundary conditions are imposed at the crack faces to prevent interpenetration between the faces. Moreover, a connection between the crack faces is characterized by a positive damage parameter. Limit transitions are justified as the damage parameter tends to infinity and to zero. In addition to this, a transition to limit is analysed as a rigidity parameter of the thin inclusion tends to infinity. Limit models are investigated. In particular, junction conditions at the common point are found for all cases considered. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Fully discrete approximation schemes for rate-independent crack evolution.

Over the past two decades, several distinct solution concepts for rate-independent evolutionary systems driven by non-convex energies have been suggested in an attempt to model properly jump discontinuities in time. Much attention has been paid in this context to the modelling of crack propagation. This paper studies two fully discrete (in time and space) approximation schemes for the rate-independent evolution of a single crack in a two-dimensional linear elastic material. The crack path is assumed to be known in advance, and the evolution of the crack tip along it relies on the Griffith theory. On the time-discrete level, the first scheme is based on local minimization, whereas the second scheme is a regularized version of the first one. The crucial feature of the schemes is their adaptive time-stepping nature, with finer time steps at those points where the evolution of the crack tip might develop a discontinuity. The set of discretization parameters includes the mesh size, crack increment, locality parameter and regularization parameter. In both cases, we explore the interplay between the discretization parameters and derive sufficient conditions on them ensuring the convergence of discrete interpolants to parametrized balanced viscosity solutions of the continuous model. To illustrate the performance of the approximation schemes, we support our theoretical analysis with numerical simulations. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Non-smooth variational problems and applications.

Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Equilibrium problem for a thermoelastic Kirchhoff-Love plate with a delaminated flat rigid inclusion.

A new mathematical model describing an equilibrium of a thermoelastic heterogeneous Kirchhoff-Love plate is considered. A corresponding nonlinear variational problem is formulated with respect toa two-dimensional domain with a cut. This cut corresponds to an interfacial crack located on a given part of the boundary of a flat rigid inclusion. The flat inclusion is described by a cylindrical surface. Due tothe presence of the flat rigid inclusion in the plate, restrictions of the functions describing displacements to the corresponding curves satisfy special constraints having a linear form. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual non-penetration of opposite crack faces. Solvability of the problem is proved. Under the assumption that the solution of the variational problem is smooth enough, an equivalent differential formulation is found. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Interaction of scales for a singularly perturbed degenerating nonlinear Robinproblem.

We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in , , with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: in the limit case, the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model.

The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori, whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton-Hoff and Ramberg-Osgood materials. This article is part of the theme issue 'Non-smooth variational problems and applications'.

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations.

Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue 'Non-smooth variational problems and applications'.

A generalized Stokes system with a non-smooth slip boundary condition.

A class of quasi variational-hemivariational inequalities in reflexive Banach spaces is studied. The inequalities contain a convex potential, a locally Lipschitz superpotential and an implicit obstacle set of constraints. Results on the well-posedness including existence, uniqueness, dependence of solution on the data and the compactness of the solution set in the strong topology are established. The applicability of the results is illustrated by the steady-state generalized Stokes model of a generalized Newtonian incompressible fluid with a non-monotone slip boundary condition. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Lagrange multipliers and nonlinear variational inequalities with gradient constraints.

The paper deals with nonlinear monotone variational inequalities with gradient constraints. In particular, using a new strong duality principle, the equivalence between the problem under consideration and a suitable double obstacle problem is proved. Moreover, the existence of [Formula: see text] Lagrange multipliers is achieved. This article is part of the theme issue 'Non-smooth variational problems and applications'.

On solutions for a generalized Navier-Stokes-Fourier system fulfilling the entropy equality.

We consider a flow of a non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the Dirichlet boundary condition for the temperature. In three dimensions, for a power-law index greater or equal to , we show the existence of a solution fulfilling the entropy equality. The entropy equality can be formally deduced from the energy equality by renormalization. However, such a procedure can be justified by the DiPerna–Lions theory only for . The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

On a frictional unilateral contact problem in nonlinear elasticity-existence and smoothing procedure.

This note is devoted to a novel frictional unilateral contact problem in finite-strain elasticity. Here, we adopt the Tresca friction model from linear elasticity. Our analysis relies on the polyconvexity approach to nonlinear elasticity due to J. Ball. We include the delicate case where the elastic body neither is fixed nor has a deformation prescribed along some part of its boundary, but rests on a rigid foundation with a free boundary and is only submitted to forces and loads acting in the interior (like gravity) and at the boundary, respectively. This leads to a loss of coercivity and necessitates an extra condition that prevents the body from escaping by the geometry of the obstacle. This new condition extends a similar condition of Ciarlet and Nečas from the frictionless case to the case of Tresca friction. In addition, as a first step towards a numerical treatment of such nonlinear problems, we present a smoothing procedure that tackles the non-smooth term from Tresca friction and provide a convergence result for the novel smoothing method. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Multiscale analysis of stationary thermoelastic vibrations of a composite material.

The problem of description of stationary vibrations is studied for a planar thermoelastic body incorporating thin inclusions. This problem contains two small positive parameters [Formula: see text] and [Formula: see text], which describe the thickness of an individual inclusion and the distance between two neighbouring inclusions, respectively. Relying on the variational formulation of the problem, by means of the modern methods of asymptotic analysis, we investigate the behaviour of solutions as [Formula: see text] and [Formula: see text] tend to zero. As the result, we construct two models corresponding to the limit cases. At first, as [Formula: see text], by the version of the method of formal asymptotic expansions we derive a limit model in which inclusions are thin (of zero width). Then, from this limit model, as [Formula: see text], we derive a homogenized model, which describes effective behaviour on the macroscopic scale, i.e. on the scale where there is no need to take into account each individual inclusion. The limiting passage as [Formula: see text] is based on the use of the two-scale convergence theory. This article is part of the theme issue 'Non-smooth variational problems and applications'.

An inverse coefficient problem of identifying the flexural rigidity in damped Euler-Bernoulli beam from measured boundary rotation.

We present a new mathematical model and method for identifying the unknown flexural rigidity [Formula: see text] in the damped Euler-Bernoulli beam equation [Formula: see text] [Formula: see text], subject to the simply supported boundary conditions [Formula: see text], [Formula: see text], from the available measured boundary rotation [Formula: see text]. We prove the existence of a quasi-solution and derive an explicit gradient formula for the Fréchet derivative of the Tikhonov functional [Formula: see text]. The results obtained here also form the basis of gradient-based computational methods for solving this class of inverse coefficient problems. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body.

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Correction to '10 years of research on toughness enhancement of structural ceramics by graphene' (2022) by Cristina Ramírez.

Open AccessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Ramírez Cristina 2022Correction to ‘10 years of research on toughness enhancement of structural ceramics by graphene’ (2022) by Cristina RamírezPhil. Trans. R. Soc. A.3802022029920220299http://doi.org/10.1098/rsta.2022.0299SectionOpen AccessCorrectionCorrection to ‘10 years of research on toughness enhancement of structural ceramics by graphene’ (2022) by Cristina Ramírez Cristina Ramírez Cristina Ramírez Google Scholar Find this author on PubMed Search for more papers by this author Cristina Ramírez Cristina Ramírez Google Scholar Find this author on PubMed Search for more papers by this author Published:26 September 2022https://doi.org/10.1098/rsta.2022.0299This article corrects the followingReview Article10 years of research on toughness enhancement of structural ceramics by graphenehttps://doi.org/10.1098/rsta.2022.0006 Cristina Ramírez volume 380issue 2232journalTitle Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences01 August 2022Phil. Trans. R. Soc. A380, 20220006. (Published online 1 August 2022). (doi:10.1098/rsta.2022.0006)In the original version of this article, there was an error in the legend of figure 2c, where the chemical formula of silicon nitride was incorrectly provided as Si2N4 instead of Si3N4. This has now been corrected. Previous Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsRelated articles10 years of research on toughness enhancement of structural ceramics by graphene01 August 2022Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences This Issue14 November 2022Volume 380Issue 2236Theme issue ‘Non-smooth variational problems and applications’ compiled and edited by Victor A. Kovtunenko, Hiromichi Itou, Alexander M. Khludnev and Evgeny M. Rudoy Article InformationDOI:https://doi.org/10.1098/rsta.2022.0299PubMed:36154482Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Manuscript received30/08/2022Manuscript accepted30/08/2022Published online26/09/2022Published in print14/11/2022 License:© 2022 The Authors.Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Citations and impact PDF Download Subjectsmaterials science

An Investigation on Semismooth Newton based Augmented Lagrangian Method for Image Restoration

The augmented Lagrangian method (also called as method of multipliers) is an important and powerful optimization method for lots of smooth or nonsmooth variational problems in modern signal processing, imaging and optimal control. However, one usually needs to solve a coupled and nonlinear system of equations, which is very challenging. In this paper, we propose several semismooth Newton methods to solve arising nonlinear subproblems for image restoration in finite dimensional spaces, which leads to several highly efficient and competitive algorithms for imaging processing. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods with semismooth Newton solvers.

The subdifferential descent method in a nonsmooth variational problem

The paper is devoted to the classical variational problem with a nonsmooth integrand of the functional to be minimized. The integrand is supposed to be subdifferentiable. Under some natural conditions the subdifferentiability of the functional considered is proved. The problem of finding the subdifferential descent direction is solved and the subdifferential descent method is applied to solve the original problem. The algorithm developed is demonstrated by examples.

Formulation of branched transport as geometry optimization

The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different models used in the literature, another advantage of the urban planning formulation for branched transport is that it provides a more transparent interpretation of the overall cost by separation into a transport (Wasserstein-1-distance) and a network maintenance term, and it splits the problem into the actual transportation task and a geometry optimization.

A Lagrangian DG-Method for Wave Propagation in a Cracked Solid with Frictional Contact Interfaces

We developed a discontinuous Galerkin (DG) numerical scheme for wave propagation in elastic solids with frictional contact interfaces. This type of numerical scheme is useful in investigations of wave propagation in elastic solids with micro-cracks (cracked solid) that involve modeling the damage in brittle materials or architected meta-materials. Only processes with mild loading that do not trigger crack fracture extension or the nucleation of new fractures are considered. The main focus lies on the contact conditions at crack surfaces, including provisions for crack opening and closure and stick-and-slip with Coulomb friction. The proposed numerical algorithm uses the leapfrog scheme for the time discretization and an augmented Lagrangian algorithm to solve the associated non-linear problems. For efficient parallelization, a Galerkin discontinuous method was chosen for the space discretization. The frictional interfaces (micro-cracks), where the numerical flux is obtained by solving non-linear and non-smooth variational problems, concerns only a limited number the degrees of freedom, implying a small additional computational cost compared to classical bulk DG schemes. The numerical method was tested through two model problems with analytical solutions. The proposed Lagrangian approach of the nonlinear interfaces had excellent results (stability and high accuracy) and only required a reasonable additional amount of computational effort. To illustrate the method, we conclude with some numerical simulations on the blast propagation in a cracked material and in a meta-material designed for shock dissipation.