A constrained optimization approach to finite element mesh smoothing

Finite Elements in Analysis and Design, Elsevier, Vol 9, pp.309-320, 1991

MESHING
RESEARCH
CORNER

V.N. Parthasarathy
General Electric Consulting Services, Albany, NY 1220S, USA

Srinivas Kodiyalaxn
Solid Mechanics Laboratory, General Electric Corporate R&D Center, Schenectady, NY 12301, USA

Abstract
The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the "smoothness" or "shape" of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a "smooth" final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using QUADTREE/OCTREE automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.