SUMMARYThe 'variable-element-topology finite element method' (VETFEM) is a finite-element-like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement-based formulation is also given, which effectively treats the case of near-incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight-node hex elements in a 3D finite-deformation elastic-plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter.

An analysis system for solid mechanics applications is described in which a new finite element method that can accommodate general polyhedral elements is exploited. The essence of the method is direct polynomial approximation of the shape functions on the physical element, without transformation to a canonical element. The main motive is elimination of the requirement that all elements be similar to a canonical element via the usual isoparametric mapping. It is this topological restriction that largely drives the design of mesh-generation algorithms, and ultimately leads to the considerable human effort required to perform complex analyses. An integrated analysis system is described in which the flexibility of the polyhedral element method is leveraged via a robust computational geometry processor. The role of the latter is to perform rapid Boolean intersection operations between hex meshes and surface representations of the body to be analyzed. A typical procedure is to create a space-filling structured hex mesh that contains the body, and then extract a polyhedral mesh of the body by intersecting the hex mesh and the body’s surface. The result is a mesh that is directly usable in the polyhedral finite element method. Some example applications are: 1) simulation on very complex geometries; 2) rapid geometry modification and re-analysis; and 3) analysis of material-removal process steps following deformation processing. This last class of problems is particularly challenging for the conventional FE methodology, because the element boundaries are, in general, not aligned with the cutting geometry following the deformation (e.g. forging) step.

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