Assessment of methods for computing the closest point projection, penetration, and gap functions in contact searching problems

Abstract

Summary In computational contact mechanics problems, local searching requires calculation of the closest point projection of a contactor point onto a given target segment. It is generally supposed that the contact boundary is locally described by a convex region. However, because this assumption is not valid for a general curved segment of a three‐dimensional quadratic serendipity element, an iterative numerical procedure may not converge to the nearest local minimum. To this end, several unconstrained optimization methods are tested: the Newton–Raphson method, the least square projection, the sphere and torus approximation method, the steepest descent method, the Broyden method, the Broyden–Fletcher–Goldfarb–Shanno method, and the simplex method. The effectiveness and robustness of these methods are tested by means of a proposed benchmark problem. It is concluded that the Newton–Raphson method in conjunction with the simplex method significantly increases the robustness of the local contact search procedure of pure penalty contact methods, whereas the torus approximation method can be recommended for contact searching algorithms, which employ the Lagrange method or the augmented Lagrangian method. Copyright © 2015 John Wiley & Sons, Ltd.

Explicit frictional stick–slip dynamics of elastic contact problem incorporating the LuGre model

“…For the determination of the contact interface γ c in the current configuration, the distance function d(x e , x g ) is introduced as d(x e , x g ) := x e − x g (ξ, η) , (8) which describes the distance between a fixed point x e of the elastic body on γ c and an arbitrary point x g (ξ, η) on the ground surface. Among all arbitrary points, denote xg = x g ( ξ, η) that minimises d as the projection of x e , the determination of which usually entails the Closest Point Projection (CPP) method [34,35]. In this method, the solution can be found by solving the following equations…”

Asynchronously in Time Integrated Interface Dynamics Problem While Maintaining Zero Interface Energy

“…The numerical results obtained by the 3D solver with the bi-penalty contact method have been used for validation. Details about the bi-penalty method and the solver itself can be found in [15,19,20,21].…”

A coupled 3D discrete elements/isogeometric method for particle/structure interaction problems

“…Because of its local convergence nature, the initial values of the parameter coordinates, ξ 1 c and ξ 2 c , need to be estimated by a robust method. Because only the squared distance function s d = ∥d − x (ξ 1 c , ξ 2 c )∥ 2 is needed, rather than its derivatives [40,41], the simplex method is considered to be a robust optimization method, and thus is employed for the initial value estimation of the Brent iteration. Refer to [24] for more detail on how to solve Eqs.…”