116 research outputs found
Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics
In this paper an abstract evolutionary hemivariational inequality with a
history-dependent operator is studied. First, a result on its unique
solvability and solution regularity is proved by applying the Rothe method.
Next, we introduce a numerical scheme to solve the inequality and derive error
estimates. We apply the results to a quasistatic frictional contact problem in
which the material is modeled with a viscoelastic constitutive law, the contact
is given in the form of multivalued normal compliance, and friction is
described with a subgradient of a locally Lipschitz potential. Finally, for the
contact problem we provide the optimal error estimate
Tykhonov triples and convergence results for hemivariational inequalities
Consider an abstract Problem P in a metric space (X; d) assumed to have a unique solution u. The aim of this paper is to compare two convergence results u'n → u and u''n → u, both in X, and to construct a relevant example of convergence result un → u such that the two convergences above represent particular cases of this third convergence. To this end, we use the concept of Tykhonov triple. We illustrate the use of this new and nonstandard mathematical tool in the particular case of hemivariational inequalities in reflexive Banach space. This allows us to obtain and to compare various convergence results for such inequalities. We also specify these convergences in the study of a mathematical model, which describes the contact of an elastic body with a foundation and provide the corresponding mechanical interpretations
A class of differential hemivariational inequalities in Banach spaces
In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u) and compactness of C0-semigroup eA(t)
On optimal control in a nonlinear interface problem described by hemivariational inequalities
The purpose of this paper is three-fold. Firstly we attack a nonlinear
interface problem on an unbounded domain with nonmonotone set-valued
transmission conditions. The investigated problem involves a nonlinear monotone
partial differential equation in the interior domain and the Laplacian in the
exterior domain. Such a scalar interface problem models nonmonotone frictional
contact of elastic infinite media. The variational formulation of the interface
problem leads to a hemivariational inequality (HVI), which however lives on the
unbounded domain, and thus cannot analyzed in a reflexive Banach space setting.
By boundary integral methods we obtain another HVI that is amenable to
functional analytic methods using standard Sobolev spaces on the interior
domain and Sobolev spaces of fractional order on the coupling boundary.
Secondly broadening the scope of the paper, we consider extended real-valued
HVIs augmented by convex extended real-valued functions. Under a smallness
hypothesis, we provide existence and uniqueness results, also establish a
stability result with respect to the extended real-valued function as
parameter. Thirdly based on the latter stability result, we prove the existence
of optimal controls for four kinds of optimal control problems: distributed
control on the bounded domain, boundary control, simultaneous
distributed-boundary control governed by the interface problem, as well as
control of the obstacle driven by a related bilateral obstacle interface
problem.Comment: 26 pages, no figures. arXiv admin note: text overlap with
arXiv:2112.1217
Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics
In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate
On convergence of solutions to variational-hemivariational inequalities
In this paper we investigate the convergence behavior of the solutions to the time-dependent variational–hemivariational inequalities with respect to the data. First, we give an existence and uniqueness result for the problem, and then, deliver a continuous dependence result when all the data are subjected to perturbations. A semipermeability problem is given to illustrate our main results
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