The Automatic Generation of Difference Approximations
of the Biharmonic Equations

George Trapp and Bohe Wang

Computer Science and Electrical Engineering Department,
West Virginia University, Morgantown WV 26506

Abstract

The biharmonic equation is used to model the deflections arising in two dimensional rectangular orthotropic symmetric laminate plates. The plate can be subjected to external perpendicular force(q) and one is interested in the resulting deflections. Various boundary conditions can be applied to the problem. The edges can be simply supported, clamped, or free. In this work, an automatic procedure is defined which produces the difference equations for each edge boundary case. The procedure is illustrated, using MATLAB,with numerical results for problems with and without exact analytical solutions. We define an automatic procedure to generate the difference equations for approximating the solution to the biharmonic equation under various boundary conditions.

After reviewing the biharmonic equation and its relevance to two dimensional rectangular orthotropic symmetric laminate plates, we work the various boundary conditions that can be applied to the edges of the plate. Subsequently, we derive the difference equation approximations for the basic equation and boundary conditions. We then work the procedure to automatically generate the finite difference approximations. Finally, using MATLAB, we present numerical results which models the bending of an orthotropic symmetric laminated plates with small deflection. Here are some cases.

 

Case 1. Four edges are simply supproted

(1)

For this case, we have the true solution. First we give the true solution:

(2)

Using 10x10 mesh, the difference solution is

(3)

Using 30x30 mesh, the difference solution is

(4)

Using 40x40 mesh and "surf" technic, the difference solution is

 

Case 2. One edge is clamped, one edge is simply supported and other two edges are free

(1)

For this case, we have not true solution. Using 5x5 mesh, the difference solution is

(2)

Using 15x15 mesh, the difference solution is

(3)

Using 25x25 mesh, the difference solution is

(4)

Using 40x40 mesh and "surf" technic, the difference solution is

 

Case 3. Saddle

(1)

Using 10x10 mesh, the difference solution is

(2)

Using 20x20 mesh, the difference solution is

(3)

Using 30x30 mesh, the difference solution is

(4)

Using 40x40 mesh, the difference solution is

(5)

Using 40x40 mesh and "surf" technic, the difference solution is

 

Concluding Remarks

The solution of the partial differential equation of the orthotropic symmetric laminated plates is of great importance in engineering applications. Based on the difference method, the patterns of the finite difference with the boundary conditions of simply supported, clamped, free and their combinations has been developed. With this pattern and MATLAB, we can get very precise numerical solution(case 1) automatically. It is especially suited to calculate the various boundary conditions problem without analysis solution(for examples: case 2 and 3). Further, with the calculated deflection, it is possible to calculate interior forces and vibration problems.

 

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