Hemivariational inequalities are nonsmooth and nonconvex problems.  They arise in a variety of applications in sciences and engineering. For applications in mechanics, through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. In the recent years, substantial progress has been made on numerical analysis of hemivariational inequalities.  In this talk, a summarizing account will be given on recent and new results on the numerical solution of hemivariational inequalities with applications in contact mechanics.

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