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The practical interest in contact problems stems from the fundamental role of contact in mechanics of solids and structures. Indeed, the contact of one body with another is a typical way how loads are delivered to a structure and the typical mechanism which supports structures to sustain the loads. Thus we do not exaggerate much if we say that the contact problems are in the heart of mechanical
engineering.
The contact problems are also interesting from the computational point of view. The conditions of equilibrium of a system of bodies in mutual contact enhance a priori unknown boundary conditions, which make the contact problems strongly nonlinear, and when some of the bodies are “floating,” then the boundary conditions admit rigid body motions and a solution need not be unique. After the discretization, the contact problem can be reduced to a finite dimensional problem, such as the minimization of a possibly non-differentiable convex function in many variables (currently from tens of thousands to billions) subject to linear or nonlinear inequality constraints for surface variables, with a specific sparse structure. Due to the floating bodies, the cost function can have a positive semidefinite quadratic part. Thus the solution of large discretized multibody contact problems still remains a challenging task, which can hardly be solved by general algorithms.
Contact Problems and Their Solution
Basic Concepts
Linear Algebra
Optimization
Analysis
Optimal QP and QCQP Algorithms
Conjugate Gradients
Gradient Projection for Separable Convex Sets
MPGP for Separable QCQP
MPRGP for Bound-Constrained QP
Solvers for Separable and Equality QPQCQP Problems
Scalable Algorithms for Contact Problems
TFETI for Scalar Problems
Frictionless Contact Problems
Contact Problems with Friction
Transient Contact Problems
TBETI
Mortars
Preconditioning and Scaling
Other Applications and Parallel Implementation
Contact with Plasticity
Contact Shape Optimization
Massively Parallel Implementation