Unique decomposition of element stiffness numeric matrices for three‐noded plate bending triangles

Abstract

This work is concerned with the numeric stiffness matrices of three-noded triangular plate bending finite elements; in particular with those numeric stiffness matrices, which are freedom-deficient and comply with the conditions of the patch test Subsequent to initial transformation of the rotation connectors for such matrices it is evident that there must exist an unique decomposition to the stiffness matrix K̆6∈ℝ6×6 of the corresponding Kirchhoff six-noded constant bending moment triangle. In K̆6 all six trial functions are themselves synonymous with those which describe the patch test. The transformation matrices of decomposition, and subsequent restoration upon modification or design, are derived explicitly and are succinct in application. Decomposition of the numeric stiffness matrix leads to exceptional versatility in objective modification, e.g. design of the matrix by adaptive process. Attention is confined here to the stiffness matrices K̆9∈ℝ9×9 of nine-degree-of-freedom three-noded Kirchhoff plate bending triangles with their single-degree-of-freedom deficiency. The decomposition of the element stiffness matrix immediately reveals those six coefficients which are available for design. They control the effect of transverse shear and are the constituents of a symmetric positive-definite matrix M3∈ℝ3×3 which is designated the ‘mechanism restraint’ matrix. It is necessary only that the designed coefficients are such that the matrix M3 remains symmetric and positive definite so as to ensure retention of patch test satisfaction on restoration to the newly designed K9. The illustrative examples provide a first perception of the leap in expectation which is enabled by design of the numeric K9 when uninhibited by formal method. Thus, the feasibility is illustrated of simple adaptive design of K9 with objective to recover cubically varying w displacements over an equilateral patch of equal triangles. This recovery is readily achieved by ad hoc inverse method but raises the issue of uniqueness in design. In highlighting the characteristics of M3 it is evident that there remains a wealth of opportunity for further research before adaptive design of the element stiffness matrix, within an arbitrary prevailing w displacement field, can become a practical reality. An appendix lists the Fortran computer codings which are used in the examples to calculate the stiffness matrix K̆6 of the six-noded constant bending moment Kirchhoff triangle as well as the explicit transformation matrices for decomposition and restoration of the numeric K9 stiffness matrix. © 1998 John Wiley & Sons, Ltd.