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IKP - Mechanics - Undergraduate Education

Numerical Methods for Nonsmooth Mechanics


Researcher: Peter W. Christensen

Funding: Swedish Research Council (Vetenskapsrådet)


Many problems in mechanics are modeled using constitutive laws where the force potential is nondifferentiable, i.e. nonsmooth. Examples are frictional contact problems, plasticity problems and damage problems. The engineering importance of these problems is great; applications include e.g. ball bearings, shrinkfits, bolted joints, adhesive joints, sheet metal forming and impact problems.

The aim of this project is to develop theoretically sound and numerically efficient algorithms for the solution of nonsmooth problems in mechanics. The algorithms are based on recent developments of Mathematical Programming (MP) algorithms for complementarity problems. Nonsmooth mechanics problems are most often solved using methods where the nondifferentiability is ignored. This implies that convergence results cannot be worked out. In our mind, it is essential to take the nondifferentiability into consideration in the design of algorithms, in order to develop more reliable methods with proven convergence properties. The benefit of using such methods is not only theoretical; they can compete against traditional algorithms in speed as well. By studying these theoretically sound algorithms, one may also give certain trial and error methods a theoretical justification. This is the case, for instance, for the simplest possible active-set strategy for frictionless contact problems, and the radial return method for J2-plasticity. In addition, one may give explanations as to why certain trial and error methods may fail for some problems. A typical example of this is three-dimensional frictional contact problems, where it is found that the ``Jacobian matrix'' of nonsmooth Newton methods contain complex elements impossible to conceive by trial and error methods.


Publications

P.W. Christensen, ``A Nonsmooth Newton Method for Elastoplastic Problems'', Computer Methods in Applied Mechanics and Engineering, 191 (2002) 1189-1219.

P.W. Christensen, ``Computational Nonsmooth Mechanics: Contact, Friction and Plasticity'', Linköping Studies in Science and Technology. Dissertations. No. 657, 2000.

P.W. Christensen, A. Klarbring, J.S. Pang & N. Strömberg, ``Formulation and Comparison of Algorithms for Frictional Contact Problems'', International Journal for Numerical Methods in Engineering, 42 (1998) 145-173.

P.W. Christensen & J.S. Pang, ``Frictional Contact Algorithms Based on Semismooth Newton Methods'', in M. Fukushima & L. Qi (editors), Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, Dordrecht (1998).

P.W. Christensen, ``Algorithms for Frictional Contact Problems Based on Mathematical Programming'', Licentiate thesis, LiU-TEK-LIC-1997:29, 1997.



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