Finite elements based on the piece-wise linear weight functions in contact problems

Abstract: The Finite Element Method (FEM) has been applied widely in solving many engineering problems. The FEM is an approximate method, which makes use of a spatial discretization and a weighted residual formulation to arrive at a system of matrix equations. The most common weighted residual formulation employed is the Galerkin method, in which the applied shape functions and weight functions are identical. However, the Galerkin method performs poorly in certain areas, e.g. contact problems when quadratic elements are used. One possible explanation of the difficulties is that the resulting nodal reaction forces are unevenly distributed between mid-nodes and corner nodes. The piece-wise linear weight functions are especially developed to resolve this problem present in the standard quadratic element. The method of applying the piece-wise linear weight functions belongs to the Petrov-Galerkin method, which is the general name of the weighted residual methods that apply different shape functions and weight functions. The piece-wise linear weight functions at present are applied to quadratic elements only, both in 2D and in 3D. These quadratic elements are divided into some subelements, which are 3-node triangles in 2D case and 4-node tetrahedrons in 3D case. The piece-wise linear weight function for a certain node in a element is linear in the subelements connecting to the node and is zero in all other subelements, from which the name piece-wise comes. The piece-wise linear weight functions guarantee C0-continuity both for inter-element variables and for intra-element variables and C1-continuity for intra-subelement variables. In derivation of the finite element formulations an extra procedure to assemble contributions from each subelement is necessary. A new finite element family based on the piece-wise linear weight functions has been developed. It consists of 7 elements, four 2D elements: TRI6, QUAD8TC, QUAD8C and QUAD8D and three 3D elements: TETRA10, TETRA11 and BRICK27. Detailed procedures on deriving the finite element formulations of these new elements are given in the appended papers. The stiffness matrices and mass matrices of these elements are unsymmetric due to application of different shape and weight functions. An unsymmetric solver and extra calculations are required. Large number of examples presented in the appended papers demonstrate the applicability of these new finite elements.