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The generated 3D surface mesh is optimized using the Laplacian smoothing technique. The smoothing is accomplished directly in the real space using the following iterative formula


where is the node being smoothed, are nodes connected to node by an edge, and is a relaxation parameter given by


The relaxation parameter can be either kept constant, by appropriately modifying weight for each node with respect to the particular number of surrounding nodes, or varying with , while keeping the constant. The first alternative is usually used for the smoothing of planar surfaces by setting to . The second alternative is applied for the smoothing of curved surfaces by setting to . The optimized grid is usually obtained after only a few cycles of the smoothing (typically up to six). Note that only nodes which are classified to the discretized surface are subjected to the smoothing. Since the smoothing of a node is likely to shift the node out of the surface, the projection must be employed to satisfy the surface constraint. This makes the Laplacian smoothing computationally more demanding when compared to its planar version. However, since the original position of the node being smoothed may be used as the initial approximation, better convergence rates of the projection are achieved. The stability of the projection can be further improved (especially when very coarse meshes are required) by increasing the weight resulting in the reduction of the relaxation parameter .

The Laplacian smoothing can be optionally extended by weighting based on the required element size, nodal connectivity, or both. In this case, the first part of Eq. (81) can be rewritten as


with equal to for the element size weighting, or to for the connectivity weighting, or to for the combined weighting. The element size based weight and connectivity based weight are given by



where and denote the actual and ``optimal'' valency (number of connected nodes) of node and and are conveniently selected coefficients (currently set to and ). The optimal valency can be expressed as


where represents the angle (in degrees) in the tangent plane at node which is filled in by the surface and is the number of gaps in that filling. Thus is equal to for internal nodes and to for boundary nodes on a  continuous boundary curve. In the presented work, a simplified approach has been adopted, in which is equal to for nodes classified to the surface or to otherwise.

Next: Computational Complexity Up: Real Space Meshing Previous: Surface Triangulation

Daniel Rypl