Mesh Smoothing for the Spectral Element Method


Laplacian- and optimization-based mesh-improvement methods are developed for high-order finite- and spectral-element based on 2D quadrilateral and 3D hexahedral meshes in general domains. A robust high-order interpolation library is used during the mesh smoothing process to improve the quality of the surface mesh while retaining the integrity of the original surface approximation. Boundary layer resolution in the original mesh is preserved through various controls in the smoothing process, including weighted interpolation between the optimized and original mesh. All mesh motion and gradient evaluations are performed on an element-by-element basis to ensure that all elements in a large mesh can be smoothed in parallel with minimum communication between different processors. Mesh quality improvements are shown to reduce the condition number of the preconditioned linear systems governing the numerical solution of the discretized partial differential equations, with corresponding reductions in iteration counts. The mesh smoother is tested on various meshes and is found to significantly improve the computational efficiency of calculations.

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    The actual vertex count can vary from \((N-1)^3\) to \((N+1)^3\) per element, depending on boundary conditions and domain connectivity. Throughout the text, we will use the simpler expression of \(N^3\) unless otherwise noted.


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The authors would like to thank Patrick Knupp for helpful discussions. This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under Contract DE-AC02-06CH11357, and in part by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering, and early testbed platforms, in support of the nation’s exascale computing imperative. The research also used resources of the Argonne Leadership Computing Facility, which is supported by the U.S. Department of Energy, Office of Science, under Contract DE-AC02-06CH11357, and from the Blue Waters sustained-petascale computing Project, which is supported by the National Science Foundation (Awards OCI-0725070 and ACI-1238993) and the state of Illinois.

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Mittal, K., Fischer, P. Mesh Smoothing for the Spectral Element Method. J Sci Comput 78, 1152–1173 (2019).

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  • High-order
  • Mesh smoothing
  • Optimization
  • Boundary layer
  • Condition number
  • Parallel