Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes. Existing techniques for high-order mesh generation typically output meshes with same polynomial order for all elements. However, high order elements away from curvilinear boundaries or interfaces increase the computational cost of the simulation without increasing geometric accuracy. In prior work, we have presented one such approach for generating body-fitted uniform-order meshes that takes a given mesh and morphs it to align with the surface of interest prescribed as the zero isocontour of a level-set function. We extend this method to generate mixed-order meshes such that curved surfaces of the domain are discretized with high-order elements, while low-order elements are used elsewhere. Numerical experiments demonstrate the robustness of the approach and show that it can be used to generate mixed-order meshes that are much more efficient than high uniform-order meshes. The proposed approach is purely algebraic, and extends to different types of elements (quadrilaterals/triangles/tetrahedron/hexahedra) in two- and three-dimensions.

Mesh optimization metrics of type Volume+Shape (VS) are critical in controlling local volume, skew, and aspect ratio in computational meshes. Such metrics can be viewed as being intermediate between pure shape metrics and pure volume metrics. In most applications using VS metrics one wants the optimal mesh to display a good balance between shape and volume characteristics. Many existing VS metrics are shape-dominated or volume-dominated, meaning that they do not give good balance between shape and volume. To achieve balance, some of the existing VS metrics contain a user parameter to adjust the balance in one direction or the other. Unfortunately this parameter is currently determined by trial and error because the parameter is not intuitively linked to a definition of balance. In this work we study the asymptotic properties of different compound metrics and find the ones that provide good balance between shape and volume. This asymptotic approach is motivated, presented, and tested by numerical experiments. The experiments confirm the expected behavior of the metrics and yields volume-shape balance in the optimized mesh.

When considering mesh quality improvement and optimization via node movement, a particularly important goal is to attempt to improve the worst quality in the mesh when that quality is unacceptable. Standard mesh optimization methods often do not address worst case quality and can make it worse. Three fundamental methods for addressing worst case quality in mesh optimization by node movement have appeared in the literature: (1) a shifted barrier approach by Barrera et. al. (Math Comput Simul 46(2):87–102, 1998), (2) a pseudo-barrier approach by Escobar et. al. (Comput Methods Appl Mech Eng 192:2775–2787, 2000), and (3) another barrier approach by Garanzha et. al. (In: IMR 2021-29th international meshing roundtable, 2021). The first two result in ‘simultaneous untangle-optimizers’ while the last addresses worst case quality in terms of the maximum value of the optimization metric. In terms of mesh optimization within the Target Matrix Optimization Paradigm (TMOP), worst case quality is defined by two quantities: the maximum value of the optimization metric ( maxμ ) and the minimum value of the local volume ( minτ ), both computed over the mesh sample points. In the present paper we show that the methods by the three authors cited above can be applied to the TMOP metrics and used on both linear and high-order element meshes. Unfortunately, the first two methods increase maxτ but do not address maxμ . The third method addresses maxμ , but fails to address minτ . Using a composition of functions approach, the present work creates new compound metrics that simultaneously increase minτ and decrease maxμ . This goal can also be accomplished by a two-stage optimization procedure in which the first stage untangles the initial mesh and the second stage decreases maxμ . Although none of these methods provide a guarantee that the worst case quality will be improved to the point that the quality becomes acceptable, it is shown by numerical examples that they can be very effective.

We propose a new approach for controlling the characteristics of certain mesh faces during optimization of high-order curved meshes. The practical goals are tangential relaxation along initially aligned curved boundaries and internal surfaces, and mesh fitting to initially non-aligned surfaces. The distinct feature of the method is that it utilizes discrete finite element functions (for example level set functions) to define implicit surfaces, which are used to adapt the positions of certain mesh nodes. The algorithm does not require CAD descriptions or analytic parametrizations, and can be beneficial in computations with dynamically changing geometry, for example shape optimization and moving mesh multimaterial simulations. The main advantage of this approach is that it completely avoids geometric operations (e.g., surface projections), and all calculations can be performed through finite element operations.

to as \(hr\)-adaptivity, introduces TMOP-based quality estimators to satisfy the aspect-ratio target via anisotropic refinements and size target via isotropic refinements in each element of the mesh. The methodology presented is purely algebraic, extends to both simplices and hexahedra/quadrilaterals of any order, and supports nonconforming isotropic and anisotropic refinements in 2D and 3D. Using a problem with a known exact solution, we demonstrate the effectiveness of \(hr\)-adaptivity over both r- and \(h\)-adaptivity in obtaining similar accuracy in the solution with significantly fewer mesh nodes. We also present several examples that show that \(hr\)-adaptivity can help satisfy geometric targets even when \(r\)-adaptivity fails to do so, due to the topology of the initial mesh.

We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of [P. Knupp, Eng. Comput., 28 (2012), pp. 419-429]. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Lagrangian-Eulerian (ALE) application [BLAST: High-order curvilinear finite elements for shock hydrodynamics, http://www.llnl.gov/CASC/blast] . Our implementation is freely available in an open-source library form [MFEM: Modular parallel finite element methods library, http://mfem.org].

We present a method for simulation-driven optimization of high-order curved meshes. This work builds on the results of Dobrev et al. (The target-matrix optimization paradigm for high-order meshes. ArXiv e-prints, 2018, https://arxiv.org/abs/1807.09807 ), where we described a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of Knupp (Eng Comput 28(4):419–429, 2012). In contrast to Dobrev et al. (2018), where all targets were based strictly on geometric information, in this work we blend physical information into the high-order mesh optimization process. The construction of target-matrices is enhanced by using discrete fields of interest, e.g., proximity to a particular region. As these discrete fields are defined only with respect to the initial mesh, their values on the intermediate meshes (produced during the optimization process) must be computed. We present two approaches for obtaining values on the intermediate meshes, namely, interpolation in physical space, and advection remap on the intermediate meshes. Our algorithm allows high-order applications to have precise control over local mesh quality, while still improving the mesh globally. The benefits of the new high-order TMOP methods are illustrated on examples from a high-order arbitrary Lagrangian-Eulerian application (BLAST, High-order curvilinear finite elements for shock hydrodynamics. LLNL code, 2018, http://www.llnl.gov/CASC/blast ).

SUMMARY A method for updating meshes in response to domain deformation is formulated within the target-matrix mesh optimization paradigm. By casting the problem within this paradigm, various formulations of the problem can be considered. Two local metrics are suggested for use in a multi-variable objective function to update meshes via numerical optimization. Both guarantee that if the deformation is null, the original mesh on the un-deformed domain will be replicated. Further, target matrices are constructed in a manner that ensures the updated mesh is as similar to the un-deformed mesh as possible. Numerical experiments confirm these properties on 2D examples. The update method has limitations, of course, the most important being that there is no guarantee that the deformed mesh will be invertible. Published in 2007 by John Wiley & Sons, Ltd.

Mesh smoothing is demonstrated to be an effective means of copying, morphing, and sweeping unstructured quadrilateral surface meshes from a source surface to a target surface. Construction of the smoother in a particular way guarantees that the target mesh will be a 'copy' of the source mesh, provided the boundary data of the target surface is a rigid body rotation, translation, and/or uniform scaling of the original source boundary data and provided the proper boundary node correspondence between source and target has been selected. Copying is not restricted to any particular smoother, but can be based on any locally elliptic second-order operator. When the bounding loops are more general than rigid body transformations the method generates high-quality, 'morphed' meshes. Mesh sweeping, if viewed as a morphing of the source surface to a set of target surfaces, can be effectively performed via this smoothing algorithm. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S, government work and is in the public domain in the United States.