SIAM IMR22: Registration & Schedule

Local Decomposition of Hexahedral Singular Nodes into Singular Curves

Paul Zhang, MIT
Judy Chiang,
Xinyi Fan,
Klara Mundilova, MIT

Keywords: Singular graph, Mesh modification, Singularity untangling

Abstract

Hexahedral (hex) meshing is a long studied topic in geometry processing with many fascinating and challenging associated problems. Hex meshes vary in complexity from structured to unstructured depending on application or domain of interest. Fully structured meshes require that all interior mesh edges are adjacent to exactly four hexes. Edges not satisfying this criteria are considered singular and indicate an unstructured hex mesh. Singular edges join together into singular curves that either form closed cycles, end on the mesh boundary, or end at a singular node, a complex junction of more than two singular curves. While all hex meshes with singularities are unstructured, those with more complex singular nodes tend to have more distorted elements and smaller scaled Jacobian values. In this work, we study the topology of singular nodes. We show that all eight of the most common singular nodes are decomposable into just singular curves. We further show that all singular nodes, regardless of edge valence, are locally decomposable. Finally we demonstrate these decompositions on hex meshes, thereby decreasing their distortion and converting all singular nodes into singular curves. With this decomposition, the enigmatic complexity of 3D singular nodes becomes effectively 2D.

INCREMENTAL DECOMPOSITION FOR HEX-MESHING IN CAD USING VIRTUAL TOPOLOGY

Benoit Lecallard, Queen's University Belfast
Trevor Robinson, Queen's University Belfast
Cecil Armstrong, Queen's University Belfast
Declan Nolan, Queen's University Belfast
Harsha Ramesh, Rolls-Royce plc

Keywords: mesh generation, analysis topology, virtual topology, CAD

Abstract

This paper presents methods for preparing a geometry model for finite element mesh generation in a Mechanical Computer-Aided Design (MCAD) environment. It works by creating a new representation of the model through the application of virtual topology operators. The resulting “analysis topology” description is used to abstract the analysis model, enabling automated tools and experts to apply an incremental strategy to decompose the model for meshing, without perturbing the original CAD model. This work also demonstrated how virtual topology enables the integration of multiple model decomposition tools to expand the capabilities of the hosting CAD environment, providing support for more meshing strategies and more freedom in how they are applied, while bridging the gap between the CAD and analysis models. Herein, the virtual topology operators used to decompose the model are checked and propagated based on the required mesh constraints to ensure the resulting mesh is conformal at the interfaces. Finally, the methods required to decompose the original CAD model using the analysis topology description and “virtual geometry curves” are presented, enabling downstream automation of the mesh.

Towards Tangled Finite Element Analysis over Partially Inverted Hexahedral Elements

Bhagyashree Prabhune, University of Wisconsin-Madison
Krishnan Suresh, University of Wisconsin-Madison

Keywords: concave hexahedral elements, tangled mesh, finite element method, TFEM

Abstract

If a finite element mesh contains concave elements, it is said to tangled. Tangled meshes can occur during mesh generation, mesh optimization, and large deformation simulations, and will lead to erroneous results during finite element analysis. Recently, the authors introduced the \emph {tangled finite element method} (TFEM) to accurately handle tangled 2D concave quadrilateral elements. In this paper, TFEM is extended to 3D hexahedral elements. We demonstrate that TFEM leads to accurate results over tangled hexahedral elements, requiring minimal changes to existing FEM framework.

Machine Learning Classification for Rapid CAD-to-Simulation

Steven J. Owen, Sandia National Laboratories
Armida J. Carbajal, Sandia National Laboratories
Corey D. Ernst, Sandia National Laboratories
Matthew G. Peterson, Sandia National Laboratories
Timothy M. Shead, Sandia National Laboratories

Keywords: machine learning, mesh generation, CAD part classification, ensembles of decision trees, neural networks

Abstract

We identify machine learning methods that can rapidly distinguish specific categories of mechanisms within a complex CAD assembly with the objective of appreciably reducing user time in preparing models for analysis. We propose a new random forest model, and compare with a similar deep learning approaches. Custom geometric features are computed using an embedded 3rd party CAD kernel based on common queries often used in meshing applications. We also demonstrate comparable or improved accuracy over current published deep learning classification procedures.

Constructing a Volume Geometry Map for Hexahedra with Curved Boundary Geometries

Sandro Elsweijer, German Aerospace Center (DLR)
Johannes Holke, German Aerospace Center (DLR)
Jan Kleinert, German Aerospace Center (DLR)
Dirk Reith, Bonn-Rhein-Sieg University of Applied Sciences

Keywords: adaptive mesh refinement, computational geometry, hexahedron, high-order meshes, mesh generation

Abstract

In (dynamic) adaptive mesh refinement (AMR), a given input mesh is refined and coarsened during the computation to optimally adapt the resolution of the computational mesh to specific requirements. The input mesh is often the output of a mesh generator and provides information on the geometry of the domain. It is desired to keep its resolution as coarse as possible in order to benefit from the AMR mesh hierarchy and efficient mesh indexing algorithms. We present a novel approach to equip the coarse mesh with high-order geometry data and evaluate this geometry on the fine mesh elements in order to ensure geometric accuracy of the refined mesh elements, even for coarse input meshes. To this end, we construct a volume geometry map for hexahedral cells from given curved boundary geometry data and discuss our implementation in a state-of-the-art AMR library.

CORNER FEATURE PRESERVING QUAD MESH GENERATION BASED ON SURFACE FOLIATION

Xiaopeng Zheng, Dalian University of Technology
Na Lei, Dalian University of Technology
Peng Zheng, CAEP Software Center for High Performance Numerical Simulation
Hao Wang, Dalian University of Technology
Zhihui Qi, Dalian University of Technology

Keywords: Quad mesh generation, surface foliation, Rectangle corner preserving, modified double cover technique

Abstract

Quadrilateral meshes with high level structure and feature preserving property benefit industrial applications the most. Generation of such quad mesh remains a challenge. Quad meshes generated using surface foliation have the highest level structure, however they lack of the feature preserving ability. In this paper, we analyze the boundary curvature with Gauss-Bonnet theorem to determine whether a boundary rectangle corner preserving foliation based method exists. When it exists, we adopt a modified double cover technique together with surface foliation method to generate the corner feature preserving quad mesh. The experiments demonstrate the efficacy of our algorithm.

Existence of structured meshes based on Abel-Jacobi theory

Na Lei, Dalian Technology University
Xiaopeng Zheng, Dalian Technology University
Wei Chen, Dalian Technology University
Yiming Zhu, Dalian Technology University
Zhongxuan Luo, Dalian Technology University

Keywords: structured mesh, singular, Abel-Jacobi, holonomy

Abstract

Structured meshes play crucial roles in engineering fields. This work studies the configurations of singular vertices on structured meshes based on Abel-Jacobi theory. It discovers the fundamental relations between the structured meshes and the meromorphic differentials, and give a unified, simpler proof for the non-existence of some structured meshes with special singularities and coloring schemes.

Interactive Visualization of Large and Arbitrary Polygonal and Polyhedral Meshes with OpenGL 4

Matthieu Maunoury, Inria
Rémi Feuillet, Inria
Adrien Loseille, Inria

Keywords: Visualization, Polygonal Meshes, Polyhedral Meshes, OpenGL 4, GLSL, Shaders

Abstract

This paper describes an efficient strategy to visualize polygons and polyhedra using OpenGL 4 flexibility. Such meshes offer flexibility as the number of vertices and faces are arbitrary. Dual meshes are examples of polygonal and polyhedral meshes. We give explanations on how polygons and polyhedra can efficiently be stored in mesh files. Algorithms to tessellate polygons into triangles are described. Many examples and comparisons with another visualization software show that our methodology is efficient (about 40 times faster than ParaView). Interactivity is also ensured with post-processing tools such as picking and cut planes.

P3 Bézier CAD Surrogates for Anisotropic Mesh Adaptation

Adrien Loseille, Inria
Lucien Rochery, Inria

Keywords: BREP, CAD surrogate, $P^3$ Bézier triangles, anisotropic mesh adaptation, surface mesh generation

Abstract

Mesh generation and adaptation rely heavily on piecewise parametric descriptions of geometry known as CAD systems. A number of problems arises: continuity is not enforced (high tolerances), lines may degenerate, costly inverse projection, uninteded small features (strong local variations)... Furthermore, CAD systems are often proprietary and the complexity of the resulting geometrical descriptions poses a high barrier of entry for alternatives. High-order surface meshes offer an advantageous convergence speed over degree of freedom ratio over $P^1$ meshes. $P^3$ meshes, in particular, are the first degree for which $\mathcal{G}^1$ continuity at the vertices may be forced. In this paper, we compare two methods to construct $P^3$ meshes from a CAD model. The resulting $P^3$ meshes are then used instead of the CAD model in a full converging adaptation loop on a complex geometry, the HL-CRM wing with flaps with a highly anisotropic metric field.

Bisecting with optimal similarity bound on 3D unstructured conformal meshes

Guillem Belda-Ferrin, Barcelona Supercomputing Center
Eloi Ruiz-Girones, Barcelona Supercomputing Center
Xevi Roca, Barcelona Supercomputing Center

Keywords: bisection, marked bisection, newest vertex bisection

Abstract

We propose a new method to mark for bisection the edges of an arbitrary 3D unstructured conformal mesh. For these meshes, the approach conformingly marks all the tetrahedra with coplanar edge marks. To this end, the method needs three key ingredients. First, we propose a specific edge ordering. Second, marking with this ordering, we guarantee that the mesh becomes conformingly marked. Third, we also ensure that all the marks are coplanar in each tetrahedron. To demonstrate the marking method, we implement an existent marked bisection approach. Using this implementation, we mark and then locally refine 3D unstructured conformal meshes. We conclude that the resulting marked bisection features an optimal bound of $36$ similarity classes per tetrahedron.

Parallel four-dimensional anisotropic mesh adaptation

Philip Caplan, Middlebury College

Keywords: mesh adaptation, anisotropic, parallel, four-dimensional, pentatope, spacetime

Abstract

Anisotropic mesh adaptation is important for accurately predicting engineering quantities of interest with high-order finite element discretizations. Spacetime approaches, whereby time is treated as an additional spatial dimension and the mesh is fully coupled in space and time have also shown to be effective in reducing the number of degrees of freedom needed to achieve a certain level of accuracy. Previous efforts have successfully demonstrated four-dimensional mesh adaptation capabilities so as to support adaptive simulations in 3d+t, however, these algorithms have been restricted to sequential implementations. Parallel mesh adaptation strategies have become increasingly important in 3d numerical simulations, therefore, the goal of this work is to extend those strategies to perform parallel four-dimensional anisotropic mesh adaptation. We employ a functionality-first approach to parallelize an existing sequential mesh adaptation tool. This approach iteratively adapts the mesh while keeping partition interfaces fixed, and then migrates partition interfaces into the interior so they can be adapted during a subsequent pass within an adaptation iteration. We demonstrate that the algorithm scales well as the number of processors increases while maintaining good metric conformity for three- and four-dimensional problems.

Shrink wrap mesh generation using morphological operators with selected applications

Vijai Kumar Suriyababu, Delft Institute of Applied Mathematics, Technical University of Delft
Cornelis Vuik, Delft Institute of Applied Mathematics, Technical University of Delft
Matthias Möller, Delft Institute of Applied Mathematics, Technical University of Delft

Keywords: mesh simplification, shrink wrapping, boolean operations, fluid volume extraction

Abstract

Triangulated meshes discretized from commercial CAD applications often possess a considerable level of complexity. However, when conducting external aerodynamics simulations at an earlier design stage, these meshes are way too complex and contain complex features and topological holes. We propose a practical and fast algorithm to shrink wrap triangulated surfaces with the sole intent of topology and surface simplification. Building upon the concepts of mathematical morphology and newer advancements in geometry processing, such as generalized winding numbers, we show that it is possible to build a straightforward and robust algorithm that can guarantee genus-zero surfaces. Our approach uses a Cartesian background mesh (fixed and adaptive) to approximate an input triangulated surface's interior and exterior volume. We use an octree data structure for adaptive mesh refinement. Although we demonstrate our algorithm exclusively on triangulated meshes, they are equally applicable to general polyhedral meshes. They are also well suited for handling point clouds (oriented and unoriented), and we show some examples of the same with some unoriented point clouds. We built our algorithms with a wide variety of applications in mind. However, we showcase the applicability of our algorithms for aerodynamic simulations, fluid volume extraction, and surface simplification. We also emphasize the practicality and ease of implementation of the proposed algorithms. We also compare our algorithms with existing literature.

Smoothing of shell meshes on faceted B-rep geometry

Harold J Fogg, Siemens Digital Industries Software
Jonathan E Makem, Siemens Digital Industries Software

Keywords: mesh smoothing, mesh optimisation

Abstract

Most smoothing methods are designed to move nodes in the interior of a domain whilst holding the nodes on boundaries fixed. By incorporating a stage to move nodes along edges a further improvement in mesh quality may be achieved. This is particularly true of meshes on complex B-rep models with many thin faces and curved edges. The work presented here describes a two-stage process for smoothing shell meshes on the edges and faces of faceted a B-rep geometry. The first stage involves the global smoothing of the mesh to generally improve its quality by Laplacian smoothing along edges with tangent line constraints and variational smoothing on faces. The second local optimisation smoothing stage is designed to make additional adjustments to the globally smoothed mesh by targeting a specific element quality metric subject to constraints. Again, nodes on edges as well as faces are smoothed. The approach is demonstrated on a selection of geometries of varying complexity.

Robust Volumetric Optimal Transportation Algorithm

Na Lei, Dalian Technology University
Wei Chen, Dalian Technology University
David Xianfeng Gu, Stony Brook University

Keywords: optimal transport, variational, robust, computational geometry

Abstract

Optimal transportation map finds the most economical way to transfer one probability measure to another, it plays an important role in many engineering and medical fields. Semi-discrete optimal transportation can be obtained using computational geometric methods. One of the challenges of this method is the robustness. In this work, we show that the admissible solution space is convex and propose an admissibility testing algorithm to ensure the searching path is inside the admissible space, this greatly improve the computational stability. Experimental result demonstrates the robustness of the proposed algorithm.

Combinatorial Local Mesh Generation

Henrik Stromberg, Institute of Mechanical Engineering TU Clausthal
Valentin Mayer-Eichberger, TU Berlin
Armin Lohrengel, Institute of Mechanical Engineering TU Clausthal

Keywords: Mesh generation, Cube Decomposition, Answer-Set-Programming, Combinatorial Optimization

Abstract

We are currently working on generating meshes by solving a combinatorial problem with an Answer Set Programming (ASP) solver. This approach offers the advantage of proven optimality for the resulting mesh in given metrics. We show strengths and limitations of this method and illustrate how we are implementing it.

Mesh quantities in 3D and dimensions of finite element spaces

L. Ridgway Scott, The University of Chicago
Tabea Tscherpel, Bielefeld University

Keywords: mesh refinement, mixed finite elements, splits

Abstract

We present a way of counting the number of tetrahedra, faces, edges and vertices for simplicial meshes in 3D. Those mesh quantities can be represented in terms of the number of vertices, the average number of edges emanating from a vertex and the average number of tetrahedra sharing a vertex in the mesh. For uniform mesh refinement such as the 3D red refinement and the bisection based refinement the average number of edges converges to the corresponding value of certain regular meshes. This gives a practical tool to determine the dimensions of finite element spaces. For example, we apply it to compare the dimensions of exactly divergence-free finite element spaces. Such spaces are suited for pressure robust approximation of fluid equations. They can either be constructed by using low polynomial degree on certain split meshes or by higher order polynomials on the original mesh.

GPU PARALLEL ALGORITHM FOR THE GENERATION OF POLYGONAL MESHES BASED ON TERMINAL-EDGE REGIONS

Sergio Salinas, University of Chile
Nancy Hitschfeld, University of Chile
José Ojeda, University of Chile

Keywords: Polygonal meshing, GPU programming, Delaunay triangulations

Abstract

This paper presents a GPU parallel algorithm to generate a new kind of polygonal meshes obtained from Delaunay triangulations. To generate the polygonal mesh, the algorithm first uses a classification system to label each edge of the input triangulation and then builds polygons (simple or not) from terminal-edge regions implicitly created using the label system. The proposed kernels can also be used to build gpu-parallel Lepp-based algorithms.