The Boundary Element Method

The boundary element method is a method for solving second and fourth order partial differential equations in solid mechanics. The boundary element method is more complicated than the finite element method, and in the author's opinion, is not really a general procedure, since the method requires somewhat different approaches depending on the equations which are being solved. Notwithstanding, on this page, example problems are solved for the equations listed directly below. For each equation, both a theory document and a Java program are included. You are free to use the programs for personal and/or educational purposes.

• Poisson's Equation
• Two-Dimensional Linear Elasticity
• Classical Plate Theory

Poisson's equation is the simplest second order partial differential equation, and it governs the deflection of a (e.g., soap) film subjected to a transverse pressure loading. Two problems were solved analytically and compared to the boundary element solutions.

Theory Document

In the theory document, these two analytical solutions are presented. Also in this document, the theory of the boundary element method is explained, and the boundary element solutions are compared to the exact solutions. For these two problems, the boundary element method gives accurate results.

Program

Here is the source code for the Java program (211,952 bytes) that the author wrote and used to perform the numerical analyses. The program uses a 9-noded bi-quadratic integration cell to integrate the pressure loading over the domain. The boundary element is a 5 d.o.f. curved element, with the displacement being interpolated as a quadratic, and the normal derivative being interpolated as being linear. All integrations are carried out numerically with Gauss-Legendre quadrature.

The equations for two-dimensional linear elasticity are somewhat more complicated than those for Poisson's equation, but they are still relatively easy to solve with the boundary element method. Again, two problems were solved analytically and compared to the boundary element solutions.

Theory Document

In the theory document, these two analytical solutions are presented. Also in this document, the theory of the boundary element method is explained, and the boundary element solutions are compared to the exact solutions. For these two problems, the boundary element method gives sufficiently accurate results.

Program

Here is the source code for the Java program (242,400 bytes) that the author wrote and used to perform the numerical analyses. The program uses a 2-noded linear boundary element, i.e., both the displacements and boundary tractions are interpolated as being linear. While this interpolation scheme is counter-intuitive (the tractions should be interpolated one order lower than the displacements), the author found that this approach gives the most reliable numerical solutions. Finally, this program, perhaps, is unusual in that all integrations are performed analytically.

Classical plate theory is governed by a fourth order partial differential equation, and problems in this theory are more difficult to solve numerically via the boundary element method than those concerning either Poisson's equation or two-dimensional linear elasticity. In any case, as above, two problems of this theory were solved analytically, and their solutions compared to the numerical boundary element solutions.

Theory Document

In the theory document, these two analytical solutions are presented. Also in this document, the theory of the boundary element method is explained, and the boundary element solutions are compared to the exact solutions. For these two problems, for the most part, the boundary element method gives sufficiently accurate results. An exception to this is (which also occurs in the finite element method, and to some degree in the cell method), that, in some instances, at or near the boundaries of the domain, the numerically calculated shear vectors can be inaccurate.

Program

Here is the source code for the Java program (375,862 bytes) that the author wrote and used to perform the numerical analyses. The program uses a bi-linear integration cell to integrate the distributed load over the domain of the plate. The boundary element used is a curved 14 d.o.f. element, with: the displacements being interpolated with a quartic; the boundary rotations as a cubic; the edge moments as a quadratic; and finally, the Kirchhoff edge shears as a linear function. All integrations are carried out numerically with Gauss-Legendre quadrature.