Abstract : This minisymposium highlights recent developments in the mathematical analysis, numerical solution, and real-world applications of variational and hemivariational inequalities. The focus is on the modeling of problems leading to such inequalities, the well-posedness and properties of their solutions, numerical analysis, optimal control and optimization, and their applications in mechanics and physics. The related topics include, but are not limited to, dynamical systems, fixed points, nonconvex and nonsmooth analysis, nonlinear inclusions, and numerical methods. The overall goal of the minisymposium is to foster collaboration and knowledge-sharing among researchers working in the area of variational and hemivariational inequalities and their applications.
[04082] Recent Advances on Partial Differential Variational Inequalities
Format : Talk at Waseda University
Author(s) :
zhenhai Liu (guangxi minzu university)
Abstract : In this talk, we consider a partial differential complex system obtained by mixing an evolution partial differential equation and a variational inequality. This kind of problems may be regarded as a special feedback control problem. Firstly, we give our research motivation and examples. Then, based on the theory of semigroups, Filippov implicit function lemma and fixed point theory for set-valued mappings, we show several existence results of solutions to the mentioned problem. Finally we point out some problems for the further research.
[03283] Optimal contol for variational inequalities of obstacle type
Format : Talk at Waseda University
Author(s) :
Zijia Peng (Guangxi Minzu University)
Abstract : This talk is concerned with optimal control of obstacle problems whose weak formulations are nonlinear variational inequalities. Under appropriate assumptions, existence of optimal solutions is proved. Moreover, the necessary optimality conditions of first order are derived by regularization techniques.
[03248] Approximation techniques for solving hemivariational inequalities arising from contact mechanics
Format : Talk at Waseda University
Author(s) :
Li Zhibao (Central South University)
Abstract : Frictional contact phenomena are common in various industrial processes and engineering applications, and they can be described by variational or hemivariational inequalities. Most of these inequality problems lack analytical solutions. Hence, developing effective numerical methods to solve these inequalities is important. Hemivariational inequality is a useful tool to study nonlinear boundary value problems with nonsmooth and nonconvex functionals. With finite element discretization, HVIs become nonconvex optimization problems. In this talk, I will present some numerical methods based on approximation techniques to solve the nonconvex optimization problem for the hemivariational inequality arising from the frictional contact mechanics problem. These methods include the smooth quadratic regularization method, as well as the first and second order approximation methods for the nonconvex functions. I will also evaluate and compare these methods using numerical experiments at the end.
[03265] A virtual element method for a quasistatic frictional contact problem
Format : Talk at Waseda University
Author(s) :
fang feng (East China Normal University )
Weimin Han (University of Iowa)
Jianguo Huang (Shanghai Jiao Tong University)
Abstract : We consider the numerical solution of an abstract quasistatic variational inequality arising in the study of quasistatic physical processes. The temporal discretization is carried out by the backward difference method, while the spatial discretization is based on the virtual element method. A general framework is provided and a quasistatic contact problem is studied and the optimal order error estimates are derived for the lowest-order virtual element method.
[04032] The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality
Format : Talk at Waseda University
Author(s) :
Jiali Qiu (Xi'an Jiaotong University)
Fei Wang (Xi'an Jiaotong University)
Min Ling (School of Mathematical Sciences, Peking University)
Jikun Zhao (Zhengzhou University)
Abstract : We develop the interior penalty virtual element method (IPVEM) for solving a Kirchhoff plate contact problem, which can be described by a fourth-order elliptic hemivariational inequality (HVI). With certain assumptions, the well-posedness of the discrete problem is proved. Furthermore, a priori error estimation is established for the IPVEM for the fourth-order elliptic HVI, and we show that the lowest-order VEM achieves optimal convergence order. Finally, some numerical examples are presented to support the theoretical results.
[03325] Virtual element method for a frictional contact problem with normal compliance
Format : Talk at Waseda University
Author(s) :
Bangmin Wu (Xinjiang University)
Abstract : We study the virtual element method for solving the frictional contact problem with the normal compliance condition, which can be modeled by a quasi-variational inequality. Existence and uniqueness results are obtained for the discretized scheme. Furthermore, a priori error analysis is established, and an optimal order error bound is derived for the lowest order virtual element method. One numerical example is given to show the efficiency of the method and to illustrate the theoretical error estimate.
[03328] Well-posedness and Numerical Analysis of a Stokes Hemivariational Inequality
Format : Online Talk on Zoom
Author(s) :
Min Ling (School of Mathematical Sciences, Peking University)
Abstract : This talk is devoted to the development and analysis of a pressure projection stabilized mixed finite element method, with continuous piecewise linear approximations of velocities and pressures, for solving a hemivariational inequality of the stationary Stokes equations with a nonlinear non-monotone slip boundary condition. We present an existence
result for an abstract mixed hemivariational inequality and apply it for a unique solvability analysis of the numerical
method for the Stokes hemivariational inequality. An optimal order error estimate is derived for the numerical solution under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical
prediction of the convergence order.
[03157] Well-posedness of parabolic variational-hemivariational inequalites with unilateral constraints
Format : Online Talk on Zoom
Author(s) :
Stanislaw Migorski (Jagiellonian University in Krakow)
Dong-ling Cai (School of Mathematical Sciences, University of Electronic Science and Technology of China)
Abstract : In this talk we discuss a novel class of variational-hemivariational inequalites with a unilateral constraint of parabolic type. Results on existence, uniqueness and the continuous dependence of the weak solution with respect to perturbations in the data are proved. As an application we examine a mathematical model of nonsmooth quasistatic viscoelastic frictional contact problem with the Signorini unilateral contact condition and a generalization of the static Coulomb law of dry friction.
[03234] Duality Arguments in Analysis of Viscoelastic Contact Problem
Format : Online Talk on Zoom
Author(s) :
Anna Ochal (Jagiellonian University in Krakow)
Abstract : We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We provide three different variational formulation of the model in which the unknowns are the displacement, the stress and the strain, respectively. We prove that they are pairwise dual of each other and they are well-posedness. The proofs are based on recent results on history-dependent variational inequalities and inclusions.
[03636] Frictional contact problem for electrorheological fluid flows
Format : Online Talk on Zoom
Author(s) :
Dariusz Pączka ( Warsaw University of Technology)
Abstract : We study the stationary flow of an isothermal, homogeneous and incompressible electrorheological fluid with slip boundary conditions of frictional type. The variational formulation of the flow problem has the form of a hemivariational inequality. Existence and uniqueness of a weak solution to the hemivariational inequality is proved for a material function $p(\cdot)$ of an electric field without any condition of monotonicity type on the extra stress tensor. This result is established by abstract theorems on existence and uniqueness of a solution to a subdifferential operator inclusion and a hemivariational inequality in the variable exponent Sobolev space.
[03327] Numerical analysis of history-dependent variational-hemivariational inequality by virtual element method
Format : Online Talk on Zoom
Author(s) :
Wenqiang Xiao (School of Mathematical Sciences, Peking University)
Abstract : In this talk, we mainly introduce the virtual element method for solving two types of history-dependent variational-hemivariational inequalities arising in contact problems. In the first model, we consider an elastic material. The contact is modelled with a total slip-dependent version of Coulomb’s law of dry friction. In the second model, we consider viscoelastic material. We introduce fully discrete schemes for the above two problems, where the temporal integration is approximated by trapezoidal rule and the spatial variable is approximated by the virtual element methods. An optimal order error estimates is derived under appropriate solution regularity assumptions. Finally, numerical examples are reported, providing numerical evidence of the optimal convergence order theoretically predicted.
[03250] Stability analysis for nonstationary Stokes hemivariational inequality
Format : Online Talk on Zoom
Author(s) :
Changjie Fang (Chongqing University of Posts and Telecommunications)
Abstract : In this talk, we consider hemivariational inequality problem for the evolutionary Stokes equations with a nonlinear slip boundary condition. We assume the slip boundary condition together with a Clarke subdifferential
relation between the stress tensor and velocity. The existence of weak solutions is obtained by Galerkin
method. In addition, stability is analyzed for a perturbed hemivariational inequality. We also present a result on the existence of a solution to an optimal control problem for the nonstationary Stokes hemivariational inequality
