Scalable algorithms for contact problems (Q345423)

The main purpose of book is to present effective scalable algorithms for the solution of multibody contact problems of linear elasticity, including the problems with friction and dynamic contacts. The presentation of the algorithms starts from the formulation of contact problems of elasticity (Chapter 1), briefly describes their discretization and the properties of the discretized problems, concludes with the analysis, numerical experiments. The book is arranged into four parts, the first of which (Chapters 2--4) reviews some well-known facts of linear algebra, optimization and functional analysis. The second part (Chapters 5--9) is concerned with the algorithms for minimizing a quadratic function subject to linear equality constraints and/or convex separable constraints. The description of the algorithms starting with the conjugate gradient method (Chapter 5) for unconstrained optimization and the results on gradient projection (Chapter 6). Chapters 7 and 8 describe Modified Proportioning with Gradient Projections for minimizing strictly convex quadratic functions subject to separable constraints and its adaptation Modified Proportioning with Reduced Gradient Projections for bound constrained problems. Chapter 9 combines the algorithms for solving problems with separable constraints and a variant of the augmented Lagrangian method in order to minimize a convex quadratic function subject to separable and equality constraints. The third part (Chapters 10--16) includes the scalable algorithms for solving multibody frictionless contact problems, contact problems with friction, and transient contact problems. Chapter 10 presents the basic ideas of the scalable algorithms in a simplified setting of multidomain scalar variational inequalities. Chapters 11--13 develop the ideas presented in Chapter 10 to the solution of multibody frictionless contact problems, contact problems with friction, and transient contact problems. Chapter 14 extends the results of Chapters 10 and 11 to solving the problems discretized by the boundary element methods in the framework of Total Boundary Element Tearing and Interconnecting method. Chapters 15 and 16 extend the results of Chapters 10--14 to solving the problems with varying coefficients and/or with non-penetration conditions implemented by mortars. The last part begins with Chapters 17 and 18 dealing with the extension of the optimality results to some applications, in particular to contact shape optimization and contact problems with plasticity. The book is completed by Chapter 19 on massively parallel implementation and parallel scalability. The methods presented in the book can be used for solving many problems, as demonstrated by the numerical results. The book can serve as an introductory text for anybody interested in contact problems. Graduate students and researchers in mechanical engineering, computational engineering, and applied mathematics, also will find this book of big value and interest.