Christoph Wilking

Abstract

In this paper, a new approach is proposed to improve efficiency of the integration procedure for mortar integrals within finite element mortar methods for contact. Appropriate approaches subdivide polygonal integration segments into triangular integration cells where well-established quadrature rules can be applied for numerical integration. Here, a subdivision of segments into quadrilateral integration cells is proposed and investigated in detail. By this procedure, the numerical effort is decreased because the number of integration cells is smaller and less quadrature points are needed. In all the aforementioned methods, necessary projections of integration points result in rational polynomials in the integrand. Thus, an exact numerical integration is impossible. Using quadrilateral integration cells additionally involves non-constant Jacobian determinants which further increases the polynomial degree of the integrand. Numerical experiments indicate, that the resulting increase in the error is small enough to be acceptable in consideration of the gained speed-up.

Abstract

This work deals with computer simulations of contact problems using the finite element method. Two modifications are proposed for the mortar method, which is the method applied to discretise the contact. The first modification concerns the numerical calculation of so-called contact integrals. For the corresponding integration in general cases polynomial integrands have to be integrated over polygonal areas. In order to use ordinary numerical quadratures the polygonal areas are usually subdivided into triangular integration cells. In this work an alternative subdivision into quadrilateral integration cells is suggested, which yields less integrations points. With the numerical experiments described in this work it is shown that due to this reduction the numerical effort is decreased considerably without deteriorating the accuracy of integration significantly. The second modification improves the contact stresses of the dual mortar method. This method uses dual functions to approximate the Lagrange multiplier field, yielding the advantage that the dual mortar method is more efficient than the standard mortar method. However, the contact stresses of the dual mortar method are less accurate than the contact stresses of the standard mortar method. In this work a modification of the contact stresses based on an L2 projection is presented for the dual mortar method. Numerical experiments show that with the introduced L2 projection the accuracy of the contact stress of the dual mortar method is improved and comparable to the accuracy of the standard mortar method.