Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

Abstract

SUMMARYA singular function boundary integral method (SFBIM) is proposed for solving biharmonic problems with boundary singularities. The method is applied to the Newtonian stick-slip ow problem. The streamfunction is approximated by the leading terms of the local asymptotic solution expansion which are also used to weight the governing biharmonic equation in the Galerkin sense. By means of the divergence theorem the discretized equations are reduced to boundary integrals. The Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are calculated together with the singular coe cients. The method converges very fast with the number of singular functions and the number of Lagrange multipliers, and accurate estimates of the leading singular coe cients are obtained. Comparisons with the analytical solution and results obtained with other numerical methods are also made.

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

“…Table 1 presents the convergence in the values of the four leading coefficients, with respect to the number of Lagrange multipliers N λ and for N a =13. As in previous implementations of the method [9][10][11][12][13][14], one may observe that the values of singular coefficients converge rapidly with N λ . In fact, in [28] a theoretical analysis of the method proved algebraic convergence in N λ .…”

“…The implementation of the SFBIM to both a 2-D and a 3-D Laplacian model problems, yielded highly accurate results for the singular coefficients and the EFIFs and exhibited fast convergence, as in other two-dimensional applications [9][10][11][12] of the method. For the planar problem, the numerical results are favorably compared with the analytic solution.…”

“…Now, in order to obtain the optimum combination of N a and N λ , several series of runs have been made. Previous applications of the method in 2-D problems [9][10][11][12][13][14], indicated that N λ should be large enough. This means that we must have an adequate number of elements on the boundary which will also help in having a better accuracy in numerical integration.…”

“…Following the notation used in previous applications of the SFBIM (e.g. [9][10][11][12][13][14]), the above expansion is written as follows:…”

On solving the cracked‐beam problem by block method

“…Furthermore, the BAMs give high accurate results for the leading coefficients j , in particular, as shown in [11], for the Motz problem the number of converged significant digits for 1 is 13. The BAMs can be applied in solving the problems of biharmonic equation also (see [13]). However, as it happens in all versions of BAMs (see [3,11,14]), there is a loss of stability as N increases (i.e.…”