Scalable Algorithms for Contact Problems

This book presents a comprehensive treatment of the authors' newly developed scalable algorithms for the solutions of multibody contact problems of linear elasticity. The brand-new feature of these algorithms is theoretically supported numerical scalability and parallel scalability demonstrated on problems discretized by billions of degrees of freedom. The theory supports solving multibody frictionless contact problems, contact problems with possibly orthotropic Tresca's friction, and transient contact problems. It covers BEM discretization, jumping coefficients, floating bodies, mortar non-penetration conditions, etc. This second edition includes new and updated content, including a chapter on hybrid domain decomposition methods for contact problems (Chap. 14.12). New sections showcase the latest improvements in algorithms, particularly those relating to the fast reconstruction of displacements (Sect. 11.9) and the adaptive reorthogonalization of dual constraints, which speeds up augmented Lagrangians iterations (Sects. 17.4 and 17.5.3). This edition updates the chapter (Chap. 19.5) on parallel implementation and extends several chapters. Additionally, new sections cover stable orthogonalization (Sect. 2.8), angles of subspaces (Sect. 2.11), bounds on the Schur complements' spectrum of stiffness matrices (Sects. 10.7 and 11.7), a benchmark for Coulomb orthotropic friction (Sect. 12.9.2), and bounds on the spectrum of the mass matrix (Sect. 13.6.1). Other sections are extended by a new staff on the Kronecker product (Sect. 2.1), CS-decomposition (Sect. 2.10), and details of discretization (Sects. 10.3 and 11.4).The exposition is divided into four parts, the first of which reviews facets of linear algebra, optimization, and analysis. The most important algorithms and optimality results are presented in the third. The presentation is complete, including continuous formulation, discretization, decomposition, optimality results, and numerical experiments. The final part includes extensions to contact shape optimization, plasticity, and HPC implementation. Graduate students and researchers in mechanical engineering, computational engineering, and applied mathematics, will find this book of great value and interest.