The Enabling Technologies for HighOrder Simulations (ETHOS) project performs research of fundamental mathematical technologies for nextgeneration highorder simulations algorithms. The current focus is on advancing the theoretical understanding and practical utility of unstructured meshes with arbitrarily highorder curvilinear elements by researching key challenges associated with highorder mesh quality optimization and simulationdriven adaptivity.
With support from the Applied Mathematics research program in the Department of Energy (DOE) Office of Science, the ETHOS team performs fundamental research that is broadly applicable to many highorder simulation approaches, including highorder finite elements and tightly coupled arbitrary LagrangianEulerian (ALE) simulations of interest to the DOE. Our research builds on the TargetMatrix Optimization Paradigm (TMOP) and nonlinear variational minimization to develop new nodemovement strategies for highorder mesh quality optimization and adaptation in both purely geometric settings, as well as in the context of a given physical simulation.
Project summary
Highorder meshes are difficult to control
 Highorder simulations rely on highorder meshes, and bad mesh quality leads to small time steps and applications failures.
 In addition to good geometric quality, applications can require highorder mesh adaptivity to dynamic simulation features.
Our solution: Develop theory that rigorously defines highorder mesh quality
 We have extended the Target Matrix Optimization Paradigm (TMOP) by Knupp to highorder meshes, defining pointwise quality based on subzonal information, and optimizing the mesh node positions with respect to an aggregated quality measure. The TMOP variational minimization technique is based on optimizing an objective function that depends on a quality metric measuring the difference between the current and target geometric parameters of the mesh.
 We are currently exploring (nonlinear) solvers for the global optimization problem, incorporating research in constrained optimization, linear and nonlinear solvers, and preconditioners.
 We are developing the TMOPbased variational formulation for the upcoming exascale architectures.
 We are also interested in mesh generation, surface optimization, hrprefinement, interface fitting, and more.
Impact: Mesh optimization is relevant to a wide range of applications. Some innovations under the ETHOS project include:
 Target construction: Methods for systematic generation of geometric target parameter values (size, aspectratio, skew, and orientation) that vary over mesh elements or sample points. Permits flexible and precise control over essential geometric quality features of mesh, tailored to any specific application.
 Simulationdriven radaptivity: Simulationbased adaptivity to optimize meshes during runtime using the solution of the PDE. This approach has already demonstrated significant impact in improving solution accuracy and computational efficiency for ALE hydrodynamics.
 hradaptivity: TMOPbased energy estimator to augment radaptivity with anisotropic mesh refinement. This allows us to satisfy size and aspectratio targets in cases where the topology of the original mesh does not allow the geometric targets to be met.
 Interface fitting: Penaltybased formulation to align elements in a mesh to internal interfaces. This approach allows us to take Cartesian meshes and align them to domains with complicated curvilinear interfaces.
 Adaptive node limiting: Penaltybased formulation to optimize the geometric quality of the mesh while minimizing the change associated with a discrete function defined on the original mesh. This functionality was inspired by multimaterial applications where it is crucial to preserve material mass for each zone while optimizing the mesh quality.
 TMOPbased adaptivity has been used for a wide variety of applications ranging from inertial confinement fusion, where symmetry preservation is important, to tokamak magnetohydrodynamics, where mesh alignment with the magnetic fields helps accelerate the convergence of the solution.
 The ETHOS algorithms are freely available on the MFEM website.
Team

Vladimir Tomov (LLNL)

Patrick Knupp (Dihedral, LLC)

Tzanio Kolev (LLNL) – Project Leader

Veselin Dobrev (LLNL)

Ketan Mittal (LLNL)
Software
 Many of the highorder mesh optimization algorithms developed in the ETHOS project are freely available in a userfriendly 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.
Publications
 P. Knupp, Tz. Kolev, K. Mittal, V. Tomov, Adaptive Surface Fitting and Tangential Relaxation for HighOrder Mesh Optimization. International Meshing Roundtable, (2021).
 V. Dobrev, P. Knupp, Tz. Kolev, K. Mittal, V. Tomov, hradaptivity for nonconforming highorder meshes with the targetmatrix optimization paradigm. Engineering with Computers, (2021).
 P. Knupp, Metric Type in the Targetmatrix Mesh Optimization Paradigm (No. LLNLTR817490). Lawrence Livermore National Laboratory, Livermore, CA (2020).
 V. Dobrev, P. Knupp, Tz. Kolev, K. Mittal, R. Rieben, and V. Tomov, SimulationDriven Optimization of HighOrder Meshes in ALE Hydrodynamics. Computers & Fluids, 104602 (2020).
 P. Knupp, Target formulation and construction in mesh quality improvement (No. LLNLTR795097). Lawrence Livermore National Laboratory, Livermore, CA (2019).
 V. Dobrev, P. Knupp, Tz. Kolev, and V. Tomov, Towards SimulationDriven Optimization of HighOrder Meshes by the TargetMatrix Optimization Paradigm, In International Meshing Roundtable (pp. 285302). Springer, Cham (2019).
 V. Dobrev, P. Knupp, Tz. Kolev, K. Mittal, and V. Tomov, The TargetMatrix Optimization Paradigm for HighOrder Meshes, SIAM J. Sci. Comput., 41(1), pp. B50–B68, (2019).
 R. Anderson, V. Dobrev, Tz. Kolev, R. Rieben, and V. Tomov, HighOrder MultiMaterial ALE Hydrodynamics, SIAM J. Sci. Comp., Vol. 40(1), pp. B32–B58, (2018).
 P. Knupp, Introducing the targetmatrix paradigm for mesh optimization by node movement, Engineering with Computers 28(4), pp. 419–429, (2012).
Presentations
 IMR 2022 talk
 IMR 2021 talk
 ICOSAHOM 2021 talk
 WCCM 2021 talk
 MultiMat 2019 talk
 SIAM CSE 2019 talk
 SIAM CSE 2019 talk
 ASCR 2019 PI poster
 ASCR 2018 PI poster
Gallery
 Surface fitting on internal surfaces.
 Size and aspectratio based adaptivity using a material indicator function defined on a cartesian mesh.
 Simulationbased radaptivity for the gas impact problem. Click here for a movie.
 Comparison of a cartesian mesh and an optimized mesh for the triplepoint problem. radaptivity using the solution at runtime enables mesh alignment with material interfaces which reduces numerical discretization error.
 Comparison of a mesh obtained from multimaterial ALE hydrodynamics calculation in Laghos with radaptivity and hradaptivity meshes.
 Some fun mesh optimization using images. The meshes shown below are colored by element sizes.