Application of the Conjugate Gradient Method for Solving Unilateral Discrete Contact Problems for an Elastic Half-Space

Problems of unilateral discrete contact between an elastic half-space and a rigid punch of finite size with a surface microrelief are considered. A variational formulation of the problems is obtained in the form of a boundary variational inequality using the Poincaré–Steklov operator, which maps normal stresses into normal displacements on a part of the boundary of the elastic half-space. A minimization problem equivalent to the variational inequality is presented, as a result of the approximation of which a quadratic programming problem subject to equality and inequality constraints is obtained. To solve this problem, a new computational algorithm based on the conjugate gradient method is proposed, which includes three equations of punch equilibrium in the calculation. The algorithm belongs to the class of active set methods and takes into account the specifics of the set of constraints. Some patterns of contact interaction of surfaces with a regular microrelief are established.