Nodal variables for complete conforming finite elements of arbitrary polynomial order

AbstractNodal variables are given for a new family of complete conforming triangular finite elements of arbitrary polynomial order p for use in linear stress analysis. This family has two important properties: (1) hierarchic property, i.e. the elemental stiffness matrix corresponding to an approximation of order p is a submatrix of the elemental stiffness matrix corresponding to an approximation of order p + 1; (2) the family enforces exactly the degree of smoothness across interelement boundaries which is required by the problem (C0 continuity for plane elasticity, C1 continuity for plate bending) even at vertices. It is shown how to use precomputed arrays in an efficient manner in calculating elemental stiffness matrices. Results from a numerical example in plane stress analysis are presented. These results demonstrate the efficiency of a p-convergence procedure which uses the new family of finite elements

Block diagonal dominance for systems of nonlinear equations with application to load flow calculations in power systems

AbstractThe concept of a pointwise strict (or Ω) diagonally dominant nonlinear function, first introduced by Moré, is generalized to the blockwise case. A sufficient condition is obtained for the convergence of underrelaxed block Jacobi and block Gauss– Seidel iterations for a nonlinear system of equations in terms of the strict (or Ω) diagonal dominance of an associated matrix. A new formulation for the determination of the steady-state load flow in lossless electric power systems is described and it is shown that this formulation leads to the solution of a system of quadratic equations in the unknown (complex-valued) voltages. Under suitable assumptions on the power system the sufficient condition is satisfied. Numerical examples, consisting of an illustrative three bus system and a realistic thirty bus system, are presented. Results of our block Gauss–Seidel iteration are compared with those of Newton–Raphson iteration