Peter W. Christensen

Formulation and Comparison of Algorithms for Frictional Contact Problems

This paper presents two algorithms for solving the discrete, quasi-static, small-displacement, linear elastic, contact problem with Coulomb friction. The algorithms are adoptions of a Newton method for solving B-differentiable equations and an interior point method for solving smooth, constrained equations. For the application of the former method, the contact problem is formulated as a system of B-differentiable equations involving the projection operator onto sets with simple structure; for the application of the latter method, the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non-negativity of variables treated as constraints. The two algorithms are numerically tested for two-dimensional problems containing up to 100 contact nodes and up to 100 time increments. Results show that at the present stage of development, the Newton method is superior both in robustness and speed. Additional comparison is made with a commercial finite element code.

An introduction to structural optimization

Mechanical and structural engineers have always strived to make as efficient use of material as possible, e.g. by making structures as light as possible yet able to carry the loads subjected to the ...

A semi-smooth newton method for elasto-plastic contact problems

In this paper we reformulate the frictional contact problem for elasto-plastic bodies as a set of unconstrained, non-smooth equations. The equations are semi-smooth so that Pangs Newton method for B-differentiable equations can be applied. An algorithm based on this method is described in detail. An example demonstrating the efficiency of the algorithm is presented.

A nonsmooth Newton method for elastoplastic problems

In this work we reformulate the incremental, small strain, J2-plasticity problem with linear kinematic and nonlinear isotropic hardening as a set of unconstrained, nonsmooth equations. The reformul ...

Frictional Contact Algorithms Based on Semismooth Newton Methods

In this paper, we establish that the discrete, three-dimensional, quasistatic, small-displacement, elastic-body frictional contact problem can be formulated as a system of semismooth equations. We give two such formulations, one unconstrained (i.e., no additional restriction on the variables of the equation) and the other constrained (that is, with additional nonnegativity constraints on some variables). A potential reduction Newton method for solving a constrained semismooth equation is developed and its convergence is established. This method is applied to the two formulations of the frictional contact problem and experimentally tested on several realistic contact problems with over 400 contact nodes. The numerical results demonstrate that the unconstrained formulation yields performance far superior to the constrained formulation.

Examples of Optimization of Discrete Parameter Systems

1. Minimization of the weight of a two-bar truss subject to stress constraints. 2. Minimization of the weight of a two-bar truss subject to stress and instability constraints. 3. Minimization of the weight of a two-bar truss subject to stress and displacement constraints. 4. Minimization of the weight of a two-beam cantilever subject to a displacement constraint. 5. Minimization of the weight of a three-bar truss subject to stress constraints. 6. Minimization of the weight of a three-bar truss subject to a stiffness constraint.