Verification of Contact Stress with Surrogate Duality

Abstract

We present a method for verifying the contact stress of elastic bodies. An explicit formulation of the total contact force, a fraction function with the numerator as a linear function and the denominator as a quadratic convex function, is derived by using the surrogate model of quadratic optimization for the contact problem. Then a bound formulation is obtained for the sum of the nodal contact forces, which is an explicit formulation of matrices of the finite element model, derived by maximizing the fraction function under the constraint that the sum of the normalized nodal contact forces is one. The bound is solved with the problem dimensions being only the number of contact nodes or node pairs, which are much smaller than the dimension for the original problem. Next, a scheme for constructing an upper bound on the contact stress is proposed that uses the bound on the sum of the nodal contact forces obtained on a fine finite element mesh and the nodal contact forces obtained on a coarse finite element mesh. Finally, the proposed method is verified through an example to demonstrate its feasibility and robustness. Keywords-contact stress; bound; surrogate; verification; finite elements I. INTRODUCTION Based on the principle of minimum potential energy, an elastic contact problem is essentially an optimization problem with the contact condition as a constraint. In the Karush-Kuhn- Tucker (KKT) conditions for these optimization problems, the Lagrange multipliers that represent the nodal forces are in an unbounded positive space. It is fortunate that when aggregating the constraints with a so-called surrogate constraint (1,2), we obtain an explicit formulation of the sum of the nodal contact forces, and the variable — the normalized nodal contact force — is only constrained in a bounded simplex. Therefore, it is possible to find an upper bound on the sum of nodal contact forces, and when this is done, we can obtain a bound on the contact stress with the information provided by the normalized nodal contact forces solved with a coarse mesh. For construction of the bounded feasible region, we have to solve an optimization problem with an objective being a fraction function. We first prove that the objective is pseudo concave in a neighborhood of the optimum, and then, with a suitable initial solution, optimization methods can be used to solve the fractional programming problem to obtain the optimum value. In this paper, we construct a bound that is explicitly formulated in terms of matrices regarding the finite element model, by maximizing the fraction function with a larger constraint field.