Accurate Computations On Unstructure3d Meshes

Abstract

The need to perform computations on irregularly distributed nets of nodes arises in
many applications of solid and fluid computational mechanics. This is specially problematic
in three dimensions. Typically, the finite element method with tetrahedral elements is used for
such purpose. However, this poses a number of problems. On one hand some elements are
considerably distorted – with eventually some null-volume elements – leading to poor
solutions. Also, in this method only h-refinement is feasible so that solution improvement
demands to refine the mesh. In this work we describe a meshless method which we designate
as Functional Integral Method (FIM) based on the use of blurred derivatives, that allows to
overcome the above mentioned difficulties. The method only requires the connectivity of each
node given by first neighbors (Voronoi cells) for discretization yielding the same structure of
non-zeros as FEM with tetrahedral elements. The matrix is nevertheless non-symmetric so
that storage and solution of the linear system increases by a factor close to two. However,
results of several numerical simulations indicate that the error is systematically much smaller
than with FEM and it is rather insensitive to node irregularity so that relation cost-benefit is
finally enhanced substantially. Also, it allows to perform p-refinement in a trivial manner by
just adding more neighbors to the local cloud of each node thus increasing the order of
interpolation. In this way the error can be further reduced without re-meshing