Finite element stabilization matrices-a unification approach

Abstract In this paper, we show that the stabilization vector γ can be obtained naturally by taking the partial derivatives with respect to the natural coordinates. Hence, the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates. With these definitions of stresses and strains, a general form of the finite element stabilization matrices is then constructed for underintegrated, irregular shape and anisotropic elements. The explicit expressions of the stabilization matrices for the 4-, 9- and 8-node Laplace and continuum elements and the 4-node plate element are then derived from the general form. These are sufficiently general for the developments herein. The generalization of the stabilization-matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear, finite-deformation constitutive equations is also presented. The computer implementation aspects and numerical evaluation of these stabilized elements are also considered. Numerical tests confirm the stability and accuracy characteristics of the resulting elements. In particular, the numerical experiments reported here also show that the rates of convergence in the L2-norm and in the energy norm agree well with the expected convergence rates.

[1]  J. M. Kennedy,et al.  Hourglass control in linear and nonlinear problems , 1983 .

[2]  Wing Kam Liu,et al.  Improvement of mixed time implicit-explicit algorithms for thermal analysis of structures , 1983 .

[3]  Ted Belytschko,et al.  A stabilization matrix for the bilinear mindlin plate element , 1981 .

[4]  B. Irons,et al.  Techniques of Finite Elements , 1979 .

[5]  R. F. Kulak.  Finite element formulation for fluid-structure interaction in three-dimensional space , 1979 .

[6]  T. Hughes.  Generalization of selective integration procedures to anisotropic and nonlinear media , 1980 .

[7]  Ted Belytschko,et al.  Implementation and application of a 9-node Lagrange shell element with spurious mode control , 1985 .

[8]  T. Belytschko,et al.  A uniform strain hexahedron and quadrilateral with orthogonal hourglass control , 1981 .

[9]  G. A. Frazier,et al.  Treatment of hourglass patterns in low order finite element codes , 1978 .

[10]  Samuel W. Key,et al.  A finite element procedure for the large deformation dynamic response of axisymmetric solids , 1974 .

[11]  G. Strang.  VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD , 1972 .

[12]  Thomas J. R. Hughes,et al.  Nonlinear finite element analysis of shells: Part I. three-dimensional shells , 1981 .

[13]  Ted Belytschko,et al.  Implementation of mixed-time partition algorithms for nonlinear thermal analysis of structures , 1985 .

[14]  J. Oden,et al.  Analysis of hourglass instabilities and control in underintegrated finite element methods , 1984 .

[15]  Ted Belytschko,et al.  A consistent control of spurious singular modes in the 9-node Lagrange element for the laplace and mindlin plate equations , 1984 .

[16]  T. Belytschko,et al.  A stabilization procedure for the quadrilateral plate element with one-point quadrature , 1983 .

[17]  Wing Kam Liu,et al.  Implementation and accuracy of mixed-time implicit-explicit methods for structural dynamics , 1984 .

[18]  Ted Belytschko,et al.  Efficient linear and nonlinear heat conduction with a quadrilateral element , 1983 .

[19]  Ted Belytschko,et al.  COMPUTER MODELS FOR SUBASSEMBLY SIMULATION , 1978 .