Variational Problem Research Articles

Hard interfaces with microstructure: the cases of strain gradient elasticity and micropolar elasticity.

As the size of a layered structure scales down, the adhesive layer thickness correspondingly decreases from macro- to micro-scale. The influence of the material microstructure of the adhesive becomes more pronounced, and possible size effect phenomena can appear. This paper describes the mechanical behaviour of composites made of two solids, bonded together by a thin layer, in the framework of strain gradient and micropolar elasticity. The adhesive layer is assumed to have the same stiffness properties as the adherents. By means of the asymptotic methods, the contact laws are derived at order 0 and order 1. These conditions represent a formal generalization of the hard elastic interface conditions. A simple benchmark equilibrium problem (a three-layer composite micro-bar subjected to an axial load) is developed to numerically assess the asymptotic model. Size effects and non-local phenomena, owing to high strain concentrations at the edges, are highlighted. The example proves the efficiency of the proposed approach in designing micro-scale-layered devices.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

A variational approach to frame-indifferent quasistatic viscoelasticity of rate type.

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

The homogenized dynamical model of a thermoelastic composite stitched with reinforcing filaments.

The dynamical problem of linear thermoelasticity for a body with incorporated thin rectilinear inclusions is studied. It is assumed that the inclusions (i.e. filaments and threads) are parallel to each other and the problem contains a small parameter [Formula: see text], which characterizes the distance between two neighbouring inclusions. Using the two-scale convergence approach, we find the limiting problem as [Formula: see text]. As a result, we get a well-posed homogenized model of an anisotropic inhomogeneous body with effective characteristics inheriting thermomechanical properties of inclusions.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Long-time behaviour of a porous medium model with degenerate hysteresis.

Hysteresis in the pressure-saturation relation in unsaturated porous media, owing to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. As an extension of previous existence and uniqueness results, we prove that under physically admissible initial conditions and without mass exchange with the exterior, the unique global solution of the fluid diffusion problem exists and asymptotically converges as time tends to infinity to a possibly non-homogeneous mass distribution and an a priori unknown constant pressure.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Thin inclusion at the junction of two elastic bodies: non-coercive case.

This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Exploring the performance of the topological energy method for object and damage detection from noisy and poor databases.

In this work, we study the performance of numerical methods based on the computation of topological energies to process data from synthetic and real experiments where only noisy limited-view data corresponding to a few frequencies are available. We show numerical experiments in two problems of practical interest. The first one corresponds to experimental measurements of the electromagnetic scattering produced by different objects extracted from the Fresnel database. The second one is related to synthetic experiments in a simplified model where steel welding joints are acoustically tested to identify possible flaws (air bubbles and inclusions) produced during the welding process. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

On the numerical corroboration of an obstacle problem for linearly elastic flexural shells.

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Mathematical modelling of fracture waves in a blocky medium with thin compliant interlayers.

Physical mechanisms that contribute to the generation of fracture waves in condensed media under intensive dynamic impacts have not been fully studied. One of the hypotheses is that this process is associated with the blocky structure of a material. As the loading wave passes, the compliant interlayers between blocks are fractured, releasing the energy of self-balanced initial stresses in the blocks, which supports the motion of the fracture wave. We propose a new efficient numerical method for the analysis of the wave nature of the propagation of a system of cracks in thin interlayers of a blocky medium with complex rheological properties. The method is based on a variational formulation of the constitutive relations for the deformation of elastic-plastic materials, as well as the conditions for contact interaction of blocks through interlayers. We have developed a parallel computational algorithm that implements this method for supercomputers with cluster architecture. The results of the numerical simulation of the fracture wave propagation in tempered glass under the action of distributed pulse disturbances are presented. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Forward and inverse problems for creep models in viscoelasticity.

This study examines a class of time-dependent constitutive equations used to describe viscoelastic materials under creep in solid mechanics. In nonlinear elasticity, the strain response to the applied stress is expressed via an implicit graph allowing multi-valued functions. For coercive and maximal monotone graphs, the existence of a solution to the quasi-static viscoelastic problem is proven by applying the Browder-Minty fixed point theorem. Moreover, for quasi-linear viscoelastic problems, the solution is constructed as a semi-analytic formula. The inverse viscoelastic problem is represented by identification of a design variable from non-smooth measurements. A non-empty set of optimal variables is obtained based on the compactness argument by applying Tikhonov regularization in the space of bounded measures and deformations. Furthermore, an illustrative example is given for the inverse problem of isotropic kernel identification. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle.

A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition.

The paper investigates a problem concerning the equilibrium of a solid body containing a thin rigid inclusion and a crack. It is assumed that the body is hyperelastic, therefore, it is described within the framework of finite strain theory. One of the peculiarities of this problem is a global injectivity constraint, which prevents the body, the crack faces and the inclusion from both mutual and self penetration. First, the paper deals with the differential formulation of the problem. Next, we consider energy minimization, showing that the latter provides the weak formulation of the former. Finally, the existence of the weak solution is demonstrated through the use of the variational technique.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Non-smooth variational problems and applications in mechanics.

Non-smooth variational problems and applications in mechanics.

On second-order tensor representation of derivatives in shape optimization.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

A Method with Double Inertial Type and Golden Rule Line Search for Solving Variational Inequalities

In this work, we study a new line-search rule for solving the pseudomonotone variational inequality problem with non-Lipschitz mapping in real Hilbert spaces as well as provide a strong convergence analysis of the sequence generated by our suggested algorithm with double inertial extrapolation steps. In order to speed up the convergence of projection and contraction methods with inertial steps for solving variational inequalities, we propose a new approach that combines double inertial extrapolation steps, the modified Mann-type projection and contraction method, and the line-search rule, which is based on the golden ratio (5+1)/2. We demonstrate the efficiency, robustness, and stability of the suggested algorithm with numerical examples.

A New Algorithm for Variational Inequality Problems in CAT(0) Spaces

A New Algorithm for Variational Inequality Problems in CAT(0) Spaces

Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.

A Galerkin Finite Element Method for the Reconstruction of a Time-Dependent Convection Coefficient and Source in a 1D Model of Magnetohydrodynamics

The mathematical analysis of viscous magnetohydrodynamics (MHD) models is of great interest in recent years. In this paper, a finite element Galerkin method is employed for the estimation of an unknown time-dependent convection coefficient and source in a 1D magnetohydrodynamics flow system. In this inverse problem, two integral observations are posed and used to transform the inverse problem to a non-classical direct problem with a non-local parabolic operator. Then, the non-classical strongly coupled parabolic system is studied in various settings. The equivalence of the inverse problem (IP) and the direct one are proven. The Galerkin procedure is analyzed to proove the existence and uniqueness of the solution. The finite element method (FEM) has been developed for the solution of the variational problem. Test examples are discussed.

Applications of a variable anchoring iterative method to equation and inclusion problems on Hadamard manifolds

In this paper, we introduce a new iterative technique with a variable anchoring operator for reckoning the solution of a variational inequality problem over the set of the common fixed points of a nearly nonexpansive sequence of operators in the framework of Hadamard manifolds. We also establish a convergence result on the proposed algorithm for approximating a solution of the problem, under suitable assumptions. We apply our results for finding the solutions of a system of nonlinear equations, and of inclusion problems to support their utility. Our work improves results in the recent literature. Numerical simulations are given for a better understanding of the effectiveness of our outcomes.

Inertial iterative method for solving generalized equilibrium, variational inequality, and fixed point problems of multivalued mappings in Banach space

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain.