Proximal Bundle Method for Contact Shape Optimization Problem

Nikola Plivova,Petr Beremlijski

doi:10.15598/aeee.v15i2.2320

Abstract

From the mathematical point of view, the contact shape optimization is a problem of nonlinear optimization with a specific structure, which can be exploited in its solution. In this paper, we show how to overcome the difficulties related to the nonsmooth cost function by using the proximal bundle methods. We describe all steps of the solution, including linearization, construction of a descent direction, line search, stopping criterion, etc. To illustrate the performance of the presented algorithm, we solve a shape optimization problem associated with the discretized two-dimensional contact problem with Coulomb's friction.

Highlights

Shape optimization problems arise naturally in mechanical engineering whenever the design requirements include an optimal performance of a machine or a gadget comprising several bodies in mutual contact
Let us deal with the shape optimization of a discretized two-dimensional contact problem with Coulomb friction only briefly
The experiment was carried out in Mathworks Matlab. In this contribution we have briefly introduced the proximal bundle method for nonsmooth optimization problems with linear constraints Cx ≤ b and with some simple bounds xmin, xmax

Summary

Introduction

Shape optimization problems arise naturally in mechanical engineering whenever the design requirements include an optimal performance of a machine or a gadget comprising several bodies in mutual contact. The hardest difficulty is the direction searching in Step 2 since the cost function f mentioned in the above section is not differentiable but only locally Lipschitz continuous. This implies that to minimize the function f , we can choose an algorithm from the following two classes: derivative-free methods (like genetic algorithms) and methods that use the subgradient information (like subgradient or bundle methods). The proximal bundle method (see [4]) is presented This method needs the function value f (x) and one (arbitrary) Clarke subgradient of f at x in every step of the iteration process. To show the functionality of the presented algorithm, we solve a shape optimization problem with the discretized two-dimensional contact problem with Coulomb’s friction in the last part

Nonsmooth Analysis - Clarke Calculus

Optimality Conditions

Derivation of the Method

Conclusion