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

• Published in: Computational mathematics and mathematical physics Vol. 64; no. 11; pp. 2680 - 2695
• Main Author: Bobylev, A. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.11.2024; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Conjugate gradient method; Constraints; Contact stresses; Elastic half spaces; Mathematical Physics; Mathematics; Mathematics and Statistics; Normal stress; Quadratic programming; unilateral discrete contact; quadratic programming problem; conjugate gradient method; Poincaré–Steklov operator; boundary variational inequality
• ISSN: 0965-5425; 1555-6662
• DOI: 10.1134/S0965542524701525

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.