Functionality in Solids Obtained from Partial Differential Equations

Abstract

A method is presented in this paper which allows solids, defined in terms of parametric bounding surfaces, to be mapped using a partial differential equation on to some simple object in parametric space, typically a cuboid. The isoparametric surfaces within the solid define a mesh system and it is shown how various features of the chosen PDE may be used to modify the mapping, and in particular the generated mesh, and thus facilitate analysis of mass properties or physical processes required of the object. The physics can be conveniently solved in parameter space, after a suitable transformation of the governing equations, and an example of this involving heat transfer is presented. Alternatively, physical properties of the object could be calculated via a finite element analysis, the mesh generated by the mapping forming the basis for a suitably defined finite element mesh. Thus the geometry used in the design of the object is used directly in the calculation of the physical properties or functional performance of the object.