Multivalued Law Research Articles

Numerical methods for determination the boundaries of unrecovered visco-plastic oil

In the present work, numerical methods are proposed for solving the problem of determining the boundaries of the unrecovered visco-plastic oil. This problem reduces to a stationary filtration problem for an incompressible fluid with an effective multi-valued law consisting in finding pressure fields and filtration velocity that satisfy the continuity equation and boundary conditions. A generalized formulation is formulated for pressure in the form of a mixed variational inequality with non-differentiable functional. To solve the variational inequality, an iterative splitting method is proposed that does not require inversion of the original operator. For the numerical implementation of this iterative method, finite element approximations of the variational inequality and the iteration method were constructed. A software package was developed in the Matlab environment. For model problems, numerical experiments were carried out for various initial data. The experimental results indicate the effectiveness of the iterative method.

Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law

We study the stationary incompressible flow of a generalized Newtonian fluid described by a nonlinear multivalued maximal monotone constitutive law and a multivalued nonmonotone frictional boundary condition. We provide results on the existence and uniqueness of a solution to the variational form of the problem. When the multivalued laws are of a subdifferential form, we prove the existence of a solution to a variational-hemivariational inequality for the flow’s velocity field.

Modelling contact with isotropic and anisotropic friction by the bipotential approach

Based on an extension of Fenchel's inequality, the bipotential approach is a non smooth mechanics tool used to model various non associative multivalued constitutive laws of dissipative materials (friction contact, soils, cyclic plasticity of metals, damage). Generally, such constitutive laws are given by a graph $M$. We propose a simple necessary and sufficient condition for the existence of a bipotential $b$ for which $M$ is the set of couples $(x,y)$ of dual variables such that $b(x,y) = \langle x,y \rangle$, and a method to construct such a bipotential by covering $M$ with cyclically monotone graphs which are not necessarily maximal (bipotential convex cover). As application, we show how to obtain the bipotential of the law of unilateral contact with Coulomb's friction by a bipotential convex cover. Introduced to extend the classical calculus of variation, the bipotential concept is also useful to construct numerical schemes for friction contact laws. In recents works, we extended the bipotential approach to a certain class of orthotropic frictional contact with a non-associated sliding rule proposed by Michalowski and Mroz. The bipotential suggests a predictor-corrector numerical scheme.

Solvability of a multivalued filtering problem in a heterogeneous environment with a distributed source

In this paper we formulate a generalized filtering problem in a heterogeneous environment in the presence of a source distributed along a line. Incompressible fluids obey a multivalued law with a linear growth at infinity. In this study we use the additive singularity extraction in the right-hand side of the problem constraint. We represent the pressure field as the sum of a known solution to a certain linear problem and an unknown “additive term”. We reduce the problem under consideration to a variational inequality of the second kind in a Hilbert space (with respect to the mentioned “additive term”) and prove its solvability.

Embedded Multivalued Control for Ceramic Manufacturing

This paper presents the realization of a prototypal embedded system which implements a new control law characterized by a fixed and quantized control set. This effort is motivated by the observation that many industrial plants with high parametric uncertainties, both for constructive simplicity and/or in order to minimize the operation cost, are usually commanded with signals that can only assume a finite number of values. This paper proposes a new multivalued control algorithm which is robust with respect to disturbances and plant's uncertain parameters and consents the plant's output to practically track a given reference trajectory, with preassigned maximum values of the tracking error and its first derivative. Moreover, this paper deals with the digital realization of this new multivalued control law and the key issues associated with its microprocessor implementation. The efficiencies of the methodology and of the design procedure utilized for the realization of the embedded system are shown through a very interesting application: a temperature control system for a ceramic kiln.

Decay of energy for second-order boundary hemivariational inequalities with coercive damping

In this article we consider the asymptotic behavior of solutions to second-order evolution inclusions with the boundary multivalued term u ″ ( t ) + A ( t , u ′ ( t ) ) + B u ( t ) + γ ̄ ∗ ∂ J ( t , γ ̄ u ′ ( t ) ) ∋ 0 and u ″ ( t ) + A ( t , u ′ ( t ) ) + B u ( t ) + γ ̄ ∗ ∂ J ( t , γ ̄ u ( t ) ) ∋ 0 , where A is a (possibly) nonlinear coercive and pseudomonotone operator, B is linear, continuous, symmetric and coercive, γ ̄ is the trace operator and J is a locally Lipschitz integral functional with ∂ denoting the Clarke generalized gradient taken with respect to the second variable. For both cases we provide conditions under which the appropriately defined energy decays exponentially to zero as time tends to infinity. We discuss assumptions and provide examples of multivalued laws that satisfy them.

Approximation of thermoelasticity contact problem with nonmonotone friction

The paper presents the formulation and approximation of a static thermoelasticity problem that describes bilateral frictional contact between a deformable body and a rigid foundation. The friction is in the form of a nonmonotone and multivalued law. The coupling effect of the problem is neglected. Therefore, the thermic part of the problem is considered independently on the elasticity problem. For the displacement vector, we formulate one substationary problem for a non-convex, locally Lipschitz continuous functional representing the total potential energy of the body. All problems formulated in the paper are approximated with the finite element method.

Solution of plane seepage problems for a multivalued seepage law when there is a point source

The steady seepage of an incompressible fluid in a uniform porous medium, occupying an arbitrary bounded two-dimensional region, when there is a point source present is considered. Part of the boundary of the region is free, while the remaining part is impermeable for the fluid. It is assumed that the function defining the seepage law is multivalued and has a linear increase at infinity. A generalized formulation of the problem is proposed in the form of a variational inequality of the second kind. An approximate solution of the problem is obtained by an iterative splitting method, which enables approximate values of both the solution itself (the pressure) and its gradient to be found. Analytic expressions describing the boundaries of the region where the modulus of the pressure gradient takes a constant value are obtained for model problems of a line of bore holes. Numerical experiments are carried out for model problems, which confirm the effectiveness of the proposed method. Good agreement is observed between the results of calculations obtained analytically and by approximate methods.

Thin-film modelling of poroviscous free surface flows

Thin-film models for the flow of a low reduced-Reynolds-number poroviscous droplet over a planar substrate are developed. One of the formulations is used to develop a minimal model for active animal cell motion in which the microscopic mechanisms of polymerisation and depolymerisation near the outer cell periphery are modelled by specifying the rate of mass transfer between the phases at the contact-line in terms of the velocity of the latter. An asymptotic analysis in the limit corresponding to strong cell-substrate adhesion is shown to lead to a novel class of multi-valued contact-line laws, a qualitative analysis of which leads in two dimensions to some intriguing behaviour, including (i) periodic contraction and expansion (pulsation), (ii) steady propagation at a contant speed, (iii) an unsteady combination of pulsation and propagation, and (iv) a bistable regime in which both non-motile and motile solutions are admissible, each of them being stable to sufficiently small perturbations, but with sufficiently large perturbations being able to ‘prod’ a stationary cell into motion or halt a moving one; these qualitative predictions are where possible compared with experiment. The contact-line behaviour is likely to be highly sensitive to environmental signals; the formulation may, therefore, provide a useful ‘minimal’ modelling framework for investigation of chemotactic effects at the cell scale. The corresponding extensional flow formulations are also noted.

Analysis of the Stationary Filtration Problem with a Multivalued Law in the Presence of a Point Source

Analysis of the Stationary Filtration Problem with a Multivalued Law in the Presence of a Point Source

Analysis of steel bolted connections by means of a nonsmooth optimization procedure

A nonsmooth optimization procedure is herein proposed for the investigation of the bearing resistance of steel bolted connections. Taking into account the presence of nonlinear effects such as e.g. plasticity and interface interaction, and describing the resistance in bearing strength by means of a nonmonotone multi-valued reaction-displacement law obtained by experimental testing, the problem is formulated as a hemivariational inequality one. The latter is equivalent to a substationarity problem of the potential energy of the connection under investigation. This problem can be effectively treated numerically by applying an appropriately chosen nonconvex, nonsmooth optimization algorithm and in particular, the NSOLIB optimization algorithm based on the proximal bundle method has been applied. Two numerical examples whose results are compared to experimental testing results demonstrate the effectiveness of the proposed method.

A variational inequality-based formulation of the frictional contact law with a non-associated sliding rule

We present a variational formulation of a complex frictional contact law with anisotropic friction condition and a non-associated sliding rule. The distinguishing characteristic of the proposed formulation is that the interface law, as well as its inverse, derive from a single scalar-valued function called bi-potential. This function, which depends on both the velocities and the associated forces, is split into the sum of two dual pseudo-potentials for standard multivalued laws. The main advantages of the formalism are the compact form taken by the present complex law and convexity property of the bi-potential that can be exploited for numerical purposes.

A multivalued boundary integral equation for adhesive contact problems

The present paper proposes a boundary integral equation (BIE) formulation for adhesive contact interface problems, i.e. problems involving interfaces glued by adhesives where unilateral contact conditions also hold. A non-monotonic, multi-valued law is assumed to describe the behaviour of the adhesive tangential to the interface direction, which leads to a hemivariational inequality problem. For the numerical treatment of the non-convex-non-smooth optimization problem, a new method is proposed which reduces the initial problem to a sequence of simple quadratic programming problems.

On Debonding and Delamination Effects in Adhesively Bonded Cracks on Fractal Type

The present paper studies the influence of adhesives on the behaviour of cracks in deformable bodies. Especially the delamination and debonding effects are studied. The adhesive material is assumed to introduce nonmonotone possibly multivalued laws which can be described via nonconvex superpotentials. Cracks of fractal geometry are considered. Approximating the fractal crack by a sequence of smooth surfaces or curves and combining this procedure with an algorithm based on the optimization of the potential and of the complementary energy for each approximation of the fractal crack, we get the solution of the problem. Numerical examples, using singular elements for the consideration of the crack singularity, illustrate the theory.

Numerical treatment of problems involving nonmonotone boundary or stress-strain laws

In order to describe the softening behavior of the materials, nonmonotone possible multivalued laws have been recently introduced. These laws are derived by nonconvex, generally nonsmooth energy functions called superpotentials that give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying superpotentials these problems have generally nonunique solutions. On the other hand, problems involving monotone laws lead to variational inequalities that can be easily treated using modern convex minimization algorithms. The present paper proposes a new method for the solution of the nonmonotone problem by approximating it using monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. This approach leads to effective reliable and versatile numerical alogrithms for large-scale hemivariational inequalities. The numerical method proposed is illustrated using appropriate examples.

Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms

Nonmonotone, possibly multivalued stress-strain or reaction-displacement laws give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying (super)potentials the problems generally have nonunique solutions (stable or unstable). In this paper we propose two methods for the solution of the hemivariational inequality problem. The first method is based on the decomposition of the nonconvex superpotential into convex constituents. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. Both methods are based on the solution of convex subproblems and constitute an effective, reliable and versatile family of numerical algorithms for large scale hemivariational inequalities. Finally, the two methods are applied to solve the same problem and the obtained results are compared.

Numerical treatment of hemivariational inequalities in mechanics: two methods based on the solution of convex subproblems

Hemivariational inequality problems describe equilibrium points (solutions) for structural systems in mechanics where nonmonotone, possibly multivalued laws or boundary conditions are involved. In the case of problems which admit a potential function this is a nonconvex, nondifferentiable one. In order to avoid the difficulties that arise during the calculation of equilibria for such mechanical systems, methods based on sequential convex approximations have recently been proposed and tested by the authors. The first method is based on ideas developed in the fields of quasidifferential and difference convex (d.c.) optimization and transforms the hemivariational inequality problem into a system of convex variational inequalities, which in turn leads to a multilevel (two-field) approximation technique for the numerical solution. The second method transforms the problem into a sequence of variational inequalities which approximates the nonmonotone problem by an iteratively defined sequence of monotone ones. Both methods lead to convex analysis subproblems and allow for treatment of large-scale nonconvex structural analysis applications. The two methods are compared in this paper with respect to both their theoretical assumptions and implications and their numerical implementation. The comparison is extended to a number of numerical examples which have been solved by both methods.

A numerical method for a class of hemivariational inequalities

Nonmonotone possibly multivalued stress-strain or reaction-displacement laws give rise to hemivariational inequalities. due to the lack of convexity and the nonsmoothness of the underlying (super) potentials the problems have generally nonunique solutions (stable or unstable). Here we propose a method for the decomposition of the hemivaritional inequality into a finite number of variational inequalities (monotone problems). The method is based on the decomposition of the contingent cone of the superpotential into convex constituents and gives rise to a very effective and reliable numerical algorithm for large scale hemivariational inequalities. Numerical results from the delamination of composites and discussion of the numerical aspects illustrate the theory.

A new class of multilevel decomposition algorithms for non monotone problems based on the quasidifferentiability concept

A convex, multilevel decomposition approach is proposed for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for model problems of structures having non-monotone interface or boundary conditions. First we decompose appropriately the non-monotone laws writing them as a difference of monotone constituents. In the general case, this is related to the quasidifferentiability concept. This permits us to obtain a system of convex variational inequalities, the solution s) of which describe the position s) of static equilibrium of the considered structure. Then the problems are formulated as min-min problems for appropriately defined Lagrangian functions. Convex optimization algorithms of various complexity are used in a multilevel scheme for the numerical solution of the considered structural analysis problem. Numerical results concerning the calculation of an elastic contact problem and an elastic stamp problem illustrate the theory.

On the approximation of nonmonotone multivalued problems by monotone subproblems

Structures involving zig-zag type reaction, displacement or stress-strain laws, i.e. nonmonotone possibly multivalued laws, cannot be effectively treated by the classical numerical methods for nonlinear laws. In order to calculate the arising free boundaries accurately, we propose a new approximation method of the nonmonotone problem by monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. Also the case of combining this procedure with fixed point algorithms to treat more complicated problems is considered. The numerical methods proposed are illustrated mainly by numerical examples which are described in detail.