High-order mesh optimization for accurate and efficient simulations
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ETHOS: Enabling Technologies for High-Order Simulations

The Enabling Technologies for High-Order Simulations (ETHOS) project performs research of fundamental mathematical technologies for next-generation high-order simulations algorithms. The current focus is on advancing the theoretical understanding and practical utility of unstructured meshes with arbitrarily high-order curvilinear elements by researching key challenges associated with high-order mesh quality optimization and simulation-driven adaptivity.

With support from the Applied Mathematics research program in the Department of Energy (DOE) Office of Science, the ETHOS team performs basic research that is broadly applicable to many high-order simulation approaches, including high-order finite elements and tightly coupled arbitrary Lagrangian-Eulerian (ALE) simulations of interest to the DOE. Our research builds on the Target-Matrix Optimization Paradigm (TMOP) and nonlinear variational minimization to develop new node-movement strategies for high-order mesh quality optimization and adaptation in both purely geometric settings, as well as in the context of a given physical simulation.

Project summary

High-order meshes are difficult to control

  • High-order simulations rely on high-order meshes, and bad mesh quality leads to small time steps and applications failures.
  • In addition to good geometric quality, applications can require high-order mesh adaptivity to dynamic simulation features.

Our solution: Develop theory that rigorously defines high-order mesh quality

  • We are extending the TMOP of Knup to high-order meshes, defining pointwise quality based on sub-zonal information, and optimizing the mesh node positions with respect to an aggregated quality measure.
  • We are exploring (nonlinear) solvers for the global optimization problem, incorporating research in constrained optimization, linear and nonlinear solvers, and preconditioners.

Impact: Mesh optimization is relevant to a wide range of applications

  • Our research targets moving mesh applications (e.g., ALE methods) and applications where symmetry preservation or adaptation to physics is important (e.g., inertial confinement fusion [ICF] and tokamak magnetohydrodynamics).
  • We are also interested in mesh generation, surface optimization, h- and hp-refinement, and more.
  • The ETHOS algorithms are freely available on the MFEM website.



  • Many of the high-order mesh optimization algorithms developed in the ETHOS project are freely available in a user-friendly form in the MFEM finite element library.
  • See in particular the Mesh Optimizer miniapp in the miniapps/meshing directory and the TMOP sources in the fem directory.


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