Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Dynamic frictionless contact with adhesion

A model for the dynamic, adhesive, frictionless contact between a viscoelastic body and a deformable foundation is described. The adhesion process is modeled by a bonding field on the contact surface. The contact is described by a modified normal compliance condition. The tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field. The existence of a unique weak solution for the problem is established, together with a partial regularity result. The existence proof proceeds by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space.     

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.     

The dynamics of elastic closed curves under uniform high pressure

We consider the dynamics of an inextensible elastic closed wire in the plane under uniform high pressure. In 1967, Tadjbakhsh and Odeh (J. Math. Anal. Appl. 18:59–74, 1967) posed a variational problem to determine the shape of a buckled elastic ring under uniform pressure. In order to comprehend a dynamics of the wire, we consider the following two mathematical questions: (i) can we construct a gradient flow for the Tadjbakhsh–Odeh functional under the inextensibility condition?; (ii) what is a behavior of the wire governed by the gradient flow near every critical point of the Tadjbakhsh–Odeh variational problem? For (i), first we derive a system of equations which governs the gradient flow, and then, give an affirmative answer to (i) by solving the system involving fourth order parabolic equations. For (ii), we first prove a stability and instability of each critical point by considering the second variation formula of the Tadjbakhsh–Odeh functional. Moreover, we give a lower bound of its Morse index. Finally we prove a dynamical aspects of the wire near each equilibrium state.     

Optimization problems for weighted Sobolev constants

In this paper, we study a variational problem under a constraint on the mass. Using a penalty method we prove the existence of an optimal shape. It will be shown that the minimizers are Hölder continuous and that for a large class they are even Lipschitz continuous. Necessary conditions in form of a variational inequality in the interior of the optimal domain and a condition on the free boundary are derived.     

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.      

Stress formulation for frictionless contact of an elastic-perfectly-plastic body

A mathematical model for frictionless contact of a deformable body with a rigid moving obstacle is analyzed. The Prandtl–Reuss elastic-perfectly-plastic constitutive law is used to describe the material's behavior, and contact is modeled with a unilateral condition imposed on the surface velocity. The problem is motivated by the process of the plowing of the ground. A variational formulation of the problem is derived in terms of the stresses and the existence of the unique weak solution is proven. The proof is based on arguments for differential inclusions obtained in A. Amassad, M. Shillor and M. Sofonea (2001). A quasistatic contact problem for an elastic perfectly plastic body with Tresca's friction. Nonlin. Anal., 35, 95–109. Finally, a study of the continuous dependence of the solution on the data is presented.     

Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory

We consider a mathematical model which describes the bilateral contact between a deformable body and an obstacle. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the friction is modeled with Tresca’s law. The problem has a unique weak solution. Here we study spatially semi-discrete and fully discrete schemes using finite differences and finite elements. We show the convergence of the schemes under the basic solution regularity and we derive order error estimates. Finally, we present an algorithm for the numerical realization and simulations for a two-dimensional test problem.     

Front propagation in infinite cylinders. II. The sharp reaction zone limit

This paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction–diffusion–advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples.     

A frictionless contact problem for elastic–viscoplastic materials with normal compliance: Numerical analysis and computational experiments

Summary. In this paper we consider a frictionless contact problem between an elastic–viscoplastic body and an obstacle. The process is assumed to be quasistatic and the contact is modeled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using strongly monotone operators arguments and Banach's fixed point theorem. We also study the numerical approach to the problem using spatially semi-discrete and fully discrete finite elements schemes with implicit and explicit discretization in time. We show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions. Finally, we present some numerical results involving examples in one, two and three dimensions. Received May 20, 2000 / Revised version received January 8, 2001 / Published online June 7, 2001     

Recovering complex elastic scatterers by a single far-field pattern

We consider the inverse scattering problem of reconstructing multiple impenetrable bodies embedded in an unbounded, homogeneous and isotropic elastic medium. The inverse problem is nonlinear and ill-posed. Our study is conducted in an extremely general and practical setting: the number of scatterers is unknown in advance; and each scatterer could be either a rigid body or a cavity which is not required to be known in advance; and moreover there might be components of multiscale sizes presented simultaneously. We develop several locating schemes by making use of only a single far-field pattern, which is widely known to be challenging in the literature. The inverse scattering schemes are of a totally “direct” nature without any inversion involved. For the recovery of multiple small scatterers, the nonlinear inverse problem is linearized and to that end, we derive sharp asymptotic expansion of the elastic far-field pattern in terms of the relative size of the cavities. The asymptotic expansion is based on the boundary-layer-potential technique and the result obtained is of significant mathematical interest for its own sake. The recovery of regular-size/extended scatterers is based on projecting the measured far-field pattern into an admissible solution space. With a local tuning technique, we can further recover multiple multiscale elastic scatterers.     

A discretization theory for a class of semi-coercive unilateral problems

Summary. In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a Signorini-Fichera unilateral contact model and some obstacle problem with frictions are studied. The theoretical conditions proved are in good agreement with the numerical ones. Received January 14, 1999 / Revised version received June 24, 1999 / Published online July 12, 2000     

Quasistatic Frictional Problems for Elastic and Viscoelastic Materials

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational inequalities and fixed-point arguments. We also prove that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero. Finally, we describe a number of concrete contact and friction conditions to which our results apply.     

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results. Received: February 16, 2005     

Ginzburg-Landau equation with magnetic effect in a thin domain

We study the Ginzburg-Landau equation with magnetic effect in a thin domain in , where the thickness of the domain is controlled by a parameter . This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in , we consider a reduced problem as and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in near the solution to the reduced equation if is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros. Received: 11 May 2001 / Accepted: 11 July 2001 /Published online: 19 October 2001     

A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy

We study a quasistatic evolution problem for a nonconvex elastic energy functional. Due to lack of convexity, the natural energetic formulation can be obtained only in the framework of Young measures. Since the energy functional may present multiple wells, an evolution driven by global minimizers may exhibit unnatural jumps from one well to another one, which overcome large potential barriers. To avoid this phenomenon, we study a notion of solution based on a viscous regularization. Finally we compare this solution with the one obtained with global minimization.     

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.     

Quasistatic crack growth in finite elasticity with non-interpenetration

We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking into account the non-interpenetration condition.     

Partitions with prescribed mean curvatures

We consider a certain variational problem on Caccioppoli partitions with countably many components, which models immiscible fluids as well as variational image segmentation, and generalizes the well-known problem with prescribed mean curvature. We prove existence and regularity results, and finally show some explicit examples of minimizers. Received: 7 June 2001 / Revisied version: 8 October 2001