APPLICATION OF THE CONJUGATE GRADIENT METHOD TO SOLVING DISCRETE CONTACT PROBLEMS FOR AN ELASTIC HALF-PLANE

Problems of discrete contact of an elastic half-plane and a rigid punch with previously unknown areas of actual contact are considered. A variational formulation of the problems is obtained in the form of a boundary variational inequality using the Poincaré–Steklov operator for an elastic half-plane. An integral representation of this operator is constructed and conditions for its positive definiteness are indicated. A minimization problem equivalent to the variational inequality is presented, for the discretization of which the boundary element approach is used. As a result, a quadratic programming problem with constraints in the form of equalities and inequalities is obtained. A linear transformation of variables is proposed, which makes it possible to simplify the form of equality constraints that approximate the equations of the equilibrium of the punch. For the numerical solution of the problem based on the conjugate gradient method, a new computational algorithm has been developed that takes into account the specifics of the set of constraints. The computational efficiency of using the algorithm for solving the class of problems under consideration is investigated. One-parameter families of punches with surface relief are constructed, the parameter of which is the number of microprotrusions. As a result of computational experiments, it was established that a single contact pressure envelope exists for a one-parameter family of punches, that is, a curve passing through the local maxima of the contact pressure on individual contact spots. The position of the envelope for the punch family depends on the external load parameters.