Nonsmooth Lagrangian mechanics and variational collision integrators

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Mobility of bodies in contact. I. A 2nd-order mobility index formultiple-finger grasps

Using a configuration-space approach, the paper develops a 2nd-order mobility theory for rigid bodies in contact. A major component of this theory is a coordinate invariant 2nd-order mobility index for a body, B, in frictionless contact with finger bodies A1,...A k. The index is an integer that captures the inherent mobility of B in an equilibrium grasp due to second order, or surface curvature, effects. It differentiates between grasps which are deemed equivalent by classical 1st-order theories, but are physically different. We further show that 2nd-order effects can be used to lower the effective mobility of a grasped object, and discuss implications of this result for achieving new lower bounds on the number of contacting finger bodies needed to immobilize an object. Physical interpretation and stability analysis of 2nd-order effects are taken up in the companion pape

Numerical analysis of a non-clamped dynamic thermoviscoelastic contact problem

In this work we analyze a non-clamped dynamic viscoelastic contact problem involving thermal effect. The friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. This leads to a system of hyperbolic inclusion for displacement and parabolic equation for temperature. We provide a fully discrete approximation of studied problem and find optimal error estimates without any smallness assumption on the data. The theoretical result is illustrated numerically.Comment: 19 pages, unfinishe

A globally convergent filter-trust-region method for large deformation contact problems

We present a globally convergent method for the solution of frictionless large deformation contact problems for hyperelastic materials. The discretization uses the mortar method which is known to be more stable than node-to-segment approaches. The resulting nonconvex constrained minimization problems are solved using a filter--trust-region scheme, and we prove global convergence towards first-order optimal points. The constrained Newton problems are solved robustly and efficiently using a truncated nonsmooth Newton multigrid method with a monotone multigrid linear correction step. For this we introduce a cheap basis transformation that decouples the contact constraints. Numerical experiments confirm the stability and efficiency of our approach