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.
(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
(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
(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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