The Cell Method

While the author is not aware of the existence of what he calls the cell method, he thought he would try it out because of its easy to understand and simple approach. Namely, the governing partial diffential equations merely are evaluated at the points of a computational grid. Unlike in the finite element and boundary element methods, the cell method does not require that any integrations be performed. The author considers the cell method to be a hybrid of sorts between the finite element and finite difference methods. For all three of the equations listed directly below, the cell method programs are based on a bi-quadratic, 9-noded differentiation cell.

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 cell method solutions.

Theory Document

In the theory document, these two analytical solutions are presented. Also in this document, the theory of the cell method is explained, and the cell method solutions are compared to the exact solutions. For these two problems, for all practical purposes, the cell method basically reproduces the exact solutions.

Program

Here is the source code for the Java program (166,467 bytes) that the author wrote and used to perform the cell method analyses.

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

Theory Document

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

Program

Here is the source code for the Java program (236,646 bytes) that the author wrote and used to perform the cell method analyses.

The methodology of cell method for classical plate theory is substantially different than those of either the finite element and boundary element methods. Instead of using the governing fourth order partial differential equation directly, the governing equations used are six coupled second and first order partial differential equations. This allows for only bi-quadratic interpolation of the field variables to be used. The author considers that the cell method is at least (if not more) accurate than either the finite element or boundary element methods, and in fact, he considers the cell method as the best numerical method to use for classical plate theory.

Theory Document

In the theory document, two analytical solutions are presented. Also in this document, the theory of the cell method is explained, and the cell method solutions are compared to the exact solutions. For the first problem (which has straight boundaries), the cell method solution is highly accurate. For the second problem, though, which has curved boundaries, the numerically calculated values of the internal shear vector, in some instances, are inaccurate near the boundaries of the domain (although to a lesser extent than occurs in the finite element and boundary element methods).

Program

Here is the source code for the Java program (295,598 bytes) that the author wrote and used to perform the cell method analyses.