StinessMatricesofIsoparametric.PDFVIP

Stiffness Matrices of Isoparametric Four-node Finite Elements by Exact Analytical Integration Gautam Dasgupta, Member ASCE Columbia University, New York, NY Key words: C++ code, convex quadrilateral element, divergence theorem, exact integra- tion, FORTRAN code, isoparametric shape functions, Taig isoparametric map- ping, triangle with a side node, Wachspress irrational expression. Abstract Explicit algebraic expressions needed to compute element stiffness matrices using procedural (FORTRAN) and object oriented (C++) computer programs are presented. Numerical illustrations for a convex quadrilateral and a triangle with a side node are included. The wide controversy due to conventional element level approximate numerical quadrature within the computational square domain, in η and ξ coordinates, is completely resolved here by the closed form analytical integration within the physical element, in x and y coordinates. May 30, 2006 1 1 Introduction The finite element basis functions, Φ, are so chosen that their linear com- bination u will faithfully approximate solutions, within a domain ? with boundary Γ, for field equations (subjected to a forcing function f) of math- ematical physics (with appropriate boundary conditions g): Lu = f ; where L is an elliptic spatial operator in ? B u = g; where B is the boundary operator on Γ u = ∑ i ai Φi (1) In the light of the Ritz pioneering formulation, ref. [6], the constants ai are determined by minimizing an energy type scalar. These norms are well known for the Laplacian, and elastostatic and vibration problems, ref. [2]. In order to cover an arbitrary geometrical domain with tiles (i.e., their one, two and three-dimensional analogs) ?i, finite element shape functions φ are constructed as linear combinations of the bases: φi = ∑ j αij Φj with local support: φi = 0 outside ?i, αij are constants (2) The values at each element’s node constitute the discrete representation of the continuum solution u of equation (1). These nodal re