Maple in discretizing equations by the Finite Volume Element method

The Finite Volume Element FVE method combines the exact conservation of nite volumes with the continuous representation of nite elements One drawback of this approach is that to evaluate the coe cients in the equations the stencils many simple integrals involving unknowns must be evaluated This problem is compounded when working with systems of di erential equations involving several di erent operators and or variable coe cients This talk will examine one such project for which Maple was used to perform the integrals and much of the algebra in the calculation of the stencils INTRODUCTION In ongoing research we are investigating ow in a weld pool where the temperature variations along the free surface of the liquid metal pool cause di erences of surface tension that drive the ow These thermocap illary e ects would be dominant in laser welding for example For arc welding the electromagnetic forces predominate but even there the thermocapillary forces are dominant near the edge of the pool because of the positive feedback between the temperature gradient driving the ow and the ow compressing the thermal gradient Previous work analyzed this feedback mechanism by isolating the cold corner region where the liquid free surface meets the solid as a model problem The current research considers a more realistic model for the whole pool Here the global heat transfer problem is considered which is coupled to the ow in the liquid to investigate how the thermocapillary feedback mechanism a ects the shape of the solid liquid interface bounding the pool This process is modeled computationally and the steady solution sought for a wide range of the two governing parameters The choice of numerical methods was motivated by the requirements of this problem The heat balance is very important both in driving the ow and in determining the shape of the liquid region so we wanted a method that exactly satis es conservation laws Also because small local length scales are important along the pool boundary which is to be determined in the solution an adaptive grid was required The Finite Volume Element FVE discretization was chosen which combined the exact conservation of Finite Volumes with the continuous representation of Finite Elements The FVE approach is also well suited to our chosen solver the Fast Adaptive Composite Grid FAC method for multilevel solution with local re nement Implementing this approach on our system of coupled partial di erential equations in axisymmetric geometry necessitated the evaluation of a large number of integrals to nd the coe cients stencil for each discrete operator Maple was used to compute these stencils which not only saved a great deal of tedious e ort but also assured accurate results Without the automation of stencil calculation through Maple the FVE discretization would have been far less feasible for this complicated problem PROBLEM STATEMENT A half space of a pure material is subjected to concentrated heating on the at horizontal surface giving a pool of molten material surrounded by solid The total heat ux Q is constant and far away the solid