International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 1st Dec 2022, 07:45:04am CET
MS 24: Recent advances in high-order mesh generation
The use of high-order methods within the scientific community continues to expand,
owing to their attractive numerical properties and potential to efficiently utilise modern
hardware when compared against lower-order methods. However, the starting point for
any high-order solver is a valid, high-quality mesh of the desired geometry, in order to
provide good quality solutions to the problem at hand. At present, the generation of such
meshes for complex configuations poses a significant challenge, imposing a barrier on the
more widespread uptake of these methods.
This minisymposium will bring together experts from academia and industry to
discuss progress on the latest techniques in high-order mesh generation. The proposed
talks include topics of significant interest in challenging problem areas, including boundary
layer mesh generation, parallel distributed mesh curving and techniques that can be used
to enable adaptive simulations in terms of mesh size (h-adaptation), element location
(r-adaptation) and polynomial order (p-adaptation).
1:50pm - 2:20pm
The generation of unit $P^2$ meshes: Error estimation and mesh adaptation
Université catholique de Louvain, Belgium
We propose a new framework for the generation and adaptation of unit $P^2$ meshes in dimension $2$. Meshes under consideration are curvilinear and mesh curvatures are not only used for capturing curved boundaries but also to capture complex features of solutions. Our study starts from a smooth function $f(x,y)$. We construct a metric field -- based on $f$ and its derivatives up to order $3$ -- that handles the curvilinear nature of the mesh, and generate a unit $P^2$ mesh with respect to this metric field, i.e. a mesh that has edges within an adimensional length range of $[0.7, 1.4]$. The mesh generation - that is done in a non-standard fashion - allows to generate meshes that are not achievable by simply curving a straight-sided mesh. In our approach, points are first placed in such a way that their mutual geodesic distance allows to draw edges of unit size, and connected in a very standard “isotropic” fashion; a curvilinear mesh quality criterion is proposed that allows to drive the mesh optimization process; the triangulation is subsequently modified using straight-sided edge swap, straight-sided edge curving, curvilinear edge swap and Curvilinear Small Polygon Reconnection(CSPR) to form the desired unit mesh; a unit curvilinear mesh containing only valid ``Geodesic Delaunay triangles'' is generated at the end.
A number of examples are finally presented that demonstrate the capabilities of the mesh adaptation procedure. The resulting adapted meshes allow to significantly reduce the approximation error compared with straight-sided $P^2$ meshes of thesame density.
2:20pm - 2:50pm
Nodal interpolation of subdivision features for curved geometry modeling
Barcelona Supercomputing Center, Spain
A nodal interpolation method to approximate a subdivision model is presented. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, the technique is devised to maintain the necessary sharp features and smooth the indicated ones. This sharp-to-smooth modeling capability handles unstructured configurations of the simulation points, curves, and surfaces. The surfaces are described by the initial linear triangles sharing the same surface identifier, and they determine the sharp point and curve features. The method detects a subset of sharp features to smooth, which the user modifies to obtain a limit model preserving the initial points. This model reconstructs the curvature by subdividing the initial triangular mesh, with no need for an underlying curved geometry model. Finally, given a polynomial degree and a nodal distribution, the method generates a piece-wise polynomial representation interpolating the limit model. Numerical evidence suggests that this approximation, naturally aligned to the subdivision features, converges to the model geometrically with the polynomial degree for fair nodal distributions. We also apply the method to prescribe the curved boundary of a high-order volume mesh. We conclude that our sharp-to-smooth modeling capability leads to curved geometry representations with enhanced preservation of the simulation intent.
2:50pm - 3:20pm
Stretching and aligning high-order meshes to match curved anisotropic features
Barcelona Supercomputing Center, Spain
To explore the possibilities of curved meshes to match curved anisotropic features, we present a mesh curving method to adapt piece-wise polynomial meshes to the stretching and alignment determined by a target metric. Our approach has two main ingredients. First, we propose a differentiable and regularized shape distortion measure for curved high-order elements. It measures the deviation of a given element, straight-sided or curved, from the stretching and alignment determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements for point-wise varying metrics. The second ingredient is a specific-purpose inexact Newton solver devised to minimize the proposed distortion efficiently. The solver allows curving (deforming) the elements of a given high-order (linear) mesh to match the curved anisotropic features determined by the metric. To enhance robustness, the solver improves the standard backtracking line-search globalization. To improve efficiency, we also propose a specific-purpose pre- conditioned conjugate-gradient solver. The examples show that our solver compared with the standard inexact Newton conjugate gradient solver, reduces up to one order of magnitude the total number of sparse-matrix vector products. We conclude that optimized curved high-order meshes, compared with optimized straight-edged meshes having the same resolution, match curved anisotropic features with higher fidelity.
3:20pm - 3:50pm
Recent advances in high-order mesh adaptivity using the target-matrix optimization paradigm
1Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550; 2Dihedral LLC, Bozeman, MT 59715
In previous work, we have developed geometry- and simulation-driven r-adaptivity of high-order meshes using the Target-Matrix Optimization Paradigm (TMOP). This method for optimizing mesh nodal positions is based on minimizing a functional that depends on each element’s current and target geometric parameters: aspect-ratio, size, skew, and rotation. Since fixed mesh topology limits the ability to achieve the target shape and size at each nodal position, we augment the r-adaptivity framework with nonconforming adaptive mesh refinement. The proposed formulation, referred to as hr-adaptivity, introduces TMOP-based quality estimators to satisfy the geometric targets via anisotropic and isotropic refinements in each element of the mesh. We have also improved the r-adaptivity component to enable 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 parameterizations, and can be beneficial in computations with dynamically changing geometry, for example shape optimization and moving mesh multimaterial simulations.
Performed under the auspices of the U.S. Department of Energy under Contract DE-AC52-07NA27344 (LLNL-ABS-821390)
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