SIAM IMR25: Program

High-Order Mesh Adaptivity Using Goal-Oriented Error Estimation

Tzanio Kolev, Lawrence Livermore National Laboratory
Ketan Mittal, Lawrence Livermore National Laboratory
Boyan Lazarov, Lawrence Livermore National Laboratory
Mathias Schmidt, Lawrence Livermore National Laboratory
Vladimir Tomov, Lawrence Livermore National Laboratory

Keywords: mesh adaptivity, goal-oriented, TMOP, adjoint sensitivity

Abstract

We explore a new approach to goal-oriented mesh optimization in PDE-driven computational simulations. Targeting high-order mesh adaptivity, the approach combines geometric quality optimization with control of the PDE solution error. By leveraging the Target-Matrix Optimization Paradigm (TMOP) in conjunction with adjoint sensitivity analysis, the method employs a goal-oriented error estimation framework to optimize the mesh adaptively. We demonstrate that the optimized mesh simultaneously achieves good geometric quality and reduces the PDE solution error in regions critical to a predefined computational objective.

High-Order Mesh Validity Detection

Tarik Dzanic, Lawrence Livermore National Laboratory
Tzanio Kolev, Lawrence Livermore National Laboratory
Ketan Mittal, Lawrence Livermore National Laboratory

Abstract

High-order meshes are essential for achieving optimal convergence rates on domains with curved surfaces, mesh alignment with physical flow/fields, moving mesh simulations, and symmetry preservation. A well known challenge with high-order curvilinear meshes is ensuring the positivity of the Jacobian determinant everywhere in the mesh. We present a novel technique to bound polynomial extrema in high-order finite element approximations. This technique is used to compute bounds on the Jacobian determinant in each element to determine mesh validity. We also incorporate these bounds in $r$-adaptivity to guarantee the positivity of Jacobians in the optimized mesh if the initial mesh is valid.

Simplified Dense and Coarse higher-order Mesh Generation Using Moving Least Squares

Cornelis Vuik, Delft University of Technology
Matthias Möller, Delft University of Technology
Vijai Kumar Suriyababu, Delft University of Technology

Keywords: surface reconstruction, moving least squares, higher order mesh, distance fields

Abstract

Higher-order surface and volume meshes are essential for better approximating PDEs. While a large body of literature focuses on PDE solvers for higher-order methods, there is still a need for robust grid generators that can produce higher-order surface/volume meshes. We propose practical workflows that provide fully automatic processes for higher-order mesh generation. We utilize the MLS algorithm, demonstrating its robustness in higher-order surface reconstruction. We also show its effectiveness in generating higher-order meshes in geometries with non-uniform triangulation, aided by intermediate support structures. Building on the recent advancements in surface multigrid methods, we propose the simplest coarse higher-order mesh generation approach to date. We illustrate the proposed workflows with a handful of examples to showcase their utility and effectiveness.

Hexahedral Mesh Generation for Swept Splines Using Foliations

Kendrick M. Shepherd, Brigham Young University
Caleb B. Goates, Brigham Young University

Keywords: splines, sweeps, meshing, computational geometry

Abstract

We introduce a method to create trivariate parameterizations of swept geometries from which we can create hexahedral cell complexes for trivariate splines and hex meshes. Starting with a linear tetrahedral mesh, we solve a discrete Laplace equation and use the level sets of the resulting discrete harmonic function as a two-dimensional foliation of the swept geometry. We can then extend an unstructured bivariate mesh over the sweep source surface to the entire swept geometry using bijective maps between foliation leaves. Our approach overcomes several difficulties found in harmonic function gradient tracing approaches. We also discuss under what conditions this method creates sweep-appropriate foliations, and the applications of these limitations to other harmonic-function-based approaches.

Large-Scale Mesh Curving for the 5th High-Lift Prediction Workshop

Xevi Roca, Barcelona Supercomputing Center - BSC
Eloi Ruiz-Gironés, Barcelona Supercomputing Center - BSC

Keywords: Mesh curving; Large scale; Complex geometry

Abstract

To take advantage of unstructured high-order methods, practitioners require a high-order curved mesh that discretizes the computational domain. To address this need, we have developed a curving technique to generate geometrically accurate distributed meshes composed of optimal-quality elements. Remarkably, our curving method has allowed us to provide meshes for the challenging geometries of the High-Lift Prediction Workshop. Our meshes have been tested and shown to be suitable for simulation purposes, yielding accurate solutions consistent with wind tunnel data.

In this work, we will describe the development of our curving method and its unique features, such as using floating meshes to approximate a virtual model and fully converge the optimization process to obtain optimal-quality curved meshes. We will also detail the challenges of generating meshes for the 5th High-Lift Prediction Workshop and our conclusions on large-scale mesh curving for complex geometries.

A flexible and path-preserving algorithm for curvilinear $P^2$ mesh generation based on Riemann geometry

Shengqi Zhang, Ningbo Institute of Digital Twin, Eastern Institute of Technology
Ruili Zhang, Ningbo Institute of Digital Twin, Eastern Institute of Technology

Keywords: Curvilinear mesh generation, Mesh adaptation, Riemannian metric field, Geodesic, Constrained delaunay triangulation, High-order error

Abstract

We propose a new framework for curvilinear mesh generation and adaptation that consists of several key components: A new algorithm for generating boundary mesh points ensures a uniform distribution along each boundary line. A flexible algorithm for generating mesh points in the computational domain allows meshing in optional numbers of directions or combinations thereof. A constrained Delaunay triangulation method that preserves the vertex generation paths. An enhanced algorithm for curving straight-sided edges improves the accuracy and efficiency of curvilinear edge triangles. Improvements in both the straight-sided edge swapping and curvilinear edge swapping algorithms increase the overall efficiency.

Higher-order surface mesh generation and its applications

KaiXin Yu, The School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China.
Ying He, The College of Computing and Data Science, Nanyang Technological University, Singapore.
Jianjun Chen, The School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China.
Yifu Wang, The School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China.

Keywords: Higher-order mesh generation; curvilinear mesh; mesh optimization; surface mesh

Abstract

This paper presents a novel method for generating higher-order meshes for CAD surfaces by leveraging minimal surface theory to improve the shape of mesh elements. The mesh generation process begins by segmenting CAD surfaces into linear elements, followed by the insertion of higher-order nodes within these elements. These nodes are then projected onto the CAD surface to form the initial higher-order elements. By optimizing energy functions related to minimal surfaces and projection distances, we achieve high-quality, geometrically accurate higher-order surface meshes. Building on this, we further conducted numerical simulations of the Poisson equation on the higher-order surface meshes, demonstrating that higher-order meshes achieve higher computational accuracy compared to linear meshes. This method has been validated on complex geometries, showcasing its potential in creating efficient higher-order meshes and improving computational accuracy for industrial CAD models.

A geometric method for automatic generation of orthogonal grids on meander-like regions

Gustavo Adolfo Restrepo Arboleda, Universidad Nacional de Colombia
Marco Paluszny, Universidad Nacional de Colombia

Keywords: orthogonal mesh, meander-like region, automatic grid generation

Abstract

We present a method to generate approximately orthogonal meshes over meander-like regions. Essentially, it depends on two conformal maps f(z) = 0.5(z + 1/z) and f(z) = (z + 1)(z − 1) that allow to transfer orthogonally gridded regions in an annulus to tiles that populate the meander-like region. The tiles are built using confocal lemniscates, ellipses and hyperbolas which are stitched together to create an approximately orthogonal grid on the whole meander-like region which is adapted to both boundaries. The quality of the grid is assessed using standard criteria involving angles and areas. As opposed to previous work we do not use sequential non linear optimization, instead we compare normalized segments of boundary with elements in a database. The database consists of approximately 50 thousand discretized arcs of lemniscates, ellipses and hyperbolas. The computational efficiency of the comparison process is improved by clusterizing the database, this reduces the comparison to the fitting of normalized segments of the meander-like region with cluster representatives.

Quad Mesh Density Control using Optimal Transport

Yuyang Cao, Stony Brook University
Yue Wang, Stony Brook University
Yiming Zhu, Dalian University of Technology
Hang Si, Cadence Design Systems Inc
Zhongxuan Luo, Dalian University of Technology
Xianfeng Gu, Stony Brook University
Na Lei, Dalian University of Technology

Keywords: quad mesh, density control, optimal transport

Abstract

Surface quadrilateral mesh generation plays a fundamental role in CAD/CAE fields. It is crucial to control the density of the meshing structures in practice.

In this work, a novel method for controlling the density of quad meshes is proposed based on the Optimal Mass Transport theory (OT). This method has a rigorous mathematical foundation and efficient and robust computational algorithms. It provides the designers with the flexibility to control the meshing density by prescribing arbitrary probability density functions on the surface. This work demonstrates the efficiency and efficacy of the method using topological disk models, in principle, the method can be generalized to surfaces with arbitrary topologies.

Quadrilateral Mesh Generation for High Genus Surfaces with Multiple Boundaries Based on Symmetric Abel Differentials

Yiming Zhu, Dalian University of Technology
Xiaopeng Zheng, Dalian University of Technology
Zhongxuan Luo, Dalian University of Technology
Hang Si, Cadence Design Systems Inc
Xianfeng David Gu, Stony Brook University
Na Lei, Dalian University of Technology

Keywords: Quad Mesh, Symmetric Abel Differential, Hodge Decomposition

Abstract

This study introduces a quadrilateral mesh generation method tailored for high-genus surfaces with multiple boundaries, utilizing symmetric Abel differentials. Grounded in robust theoretical principles, this method generates high-quality quad meshes characterized by exceptional orthogonality and minimal singularity presence. Additionally, the grid lines are strategically aligned with the boundaries, enhancing the mesh’s structural integrity and applicability in complex modeling scenarios.

Polycubes via Dual Loops

Kevin Verbeek, TU Eindhoven
Bettina Speckmann, TU Eindhoven
Maxim Snoep, TU Eindhoven

Keywords: polycube, polycube map, polycube segmentation

Abstract

We present a complete characterization of polycubes of any genus based on their dual structure: a collection of oriented loops which run in each of the axis directions and capture polycubes via their intersection patterns. A polycube loop structure uniquely corresponds to a polycube. We also describe all combinatorially different ways to add a loop to a loop structure while maintaining its validity; similarly, we show how to identify loops that can be removed from a polycube loop structure without invalidating it. Our characterization gives rise to an iterative algorithm to construct provably valid polycube-maps for a given input surface; polycube-maps are frequently used to solve texture mapping, spline fitting, and hexahedral meshing. We showcase some results of a proof-of-concept implementation of this iterative algorithm.

Validity-first automatic polycube labeling for CAD models

Christophe Bourcier, CEA
Franck Ledoux, CEA
Sébastien Mestrallet, CEA

Keywords: hexahedra, hex-mesh, block-structured, polycube, labeling, CAD models

Abstract

For many simulation codes, block-structured hex meshes remain preferred while their automatic generation is unsolved. We investigate the usage of a polycube-based approach. More specifically, we focus on the labeling stage which consists in assigning each boundary facet to one of the 6 signed principal axis. Similar works are confronted with 2 challenges: over-constraining validity criteria, and the mixed-up processing of validity criteria with quality metrics. We tackled these obstacles with automatic routines based on semi-global labeling operators. Our approach is successfully tested on CAD models, which are of interest for many numerical simulation problems.

C1-Smooth Parametrization of Polynomial Shapes over Polygonal Domains

Marcel Campen, Osnabrück University
Payam Khanteimouri, Osnabrück University

Keywords: high-order, triangle meshing, smoothness, continuity

Abstract

The generation of meshes and mesh-based parametrizations for given geometric domains is a cornerstone of modeling and analysis. Of particular interest in this context are spaces of increased smoothness, in terms of the continuity order across elements. We describe a method to, by means of higher-order mesh generation, construct a parametrization of 2D shapes with curved boundaries, with a focus on C1-continuity. Different from previous work, this method guarantees the absence of inversions or degeneracies as well as exact conformance to the given polynomial boundary. It leverages recent results on higher-order triangle mesh generation with guaranteed quality bounds. These bounds enable us to formulate an optimization problem aiming for continuity while maintaining validity.

Fast practical untangling of curvilinear meshes

Guillaume Coiffier, Université catholique de Louvain
Amaury Johnen, Université catholique de Louvain
Jean-François Remacle, Université catholique de Louvain

Keywords: High order meshing

Abstract

What is proposed here is to systematically reduce the untangling of any high-order elements to the problem of untangling simplices (triangles in 2D and tetrahedra in 3D). First, we present a general way of expressing the validity of a high-order element by calculating linear combinations of areas of well-chosen simplices. We then show how to adapt [3] to these linear combinations of simplices. Examples of 2D boundary layer untangling are presented with P2 and P3 elements. The algorithm is then adapted to P2 tetrahedra.

Layered voxel-dominant meshing for additive manufacturing simulation

Jianhui Yang, Voxshell Limited
Liang Yang, Cranfield University

Keywords: Voxel-dominant mesh, additive manufacturing simulation

Abstract

Additive manufacturing (AM) simulations demand meshing techniques that balance computational efficiency with geometric accuracy to capture the complexity of layer-by-layer fabrication. Here, we present a novel layered voxel-dominant meshing approach, leveraging the structured simplicity of Cartesian voxel grids while integrating enhancements to better represent curved surfaces and intricate features. This approach addresses the limitations of traditional voxel meshes, such as poor fidelity in fine or closely packed features, by incorporating adaptive strategies that align with AM’s layered nature.

Single Surface Spline Reconstruction by Distortion-Minimizing Flattening

Rebecca Rasmussen, Brigham Young University
Kendrick Shepherd, Brigham Young University

Keywords: Spline reconstruction, isogeometric analysis

Abstract

Structural integrity of a variety of engineering structures is frequently best assessed using a two-dimensional shell midsurface embedded in three-dimensional space, rather than the full volumetric representation of the model. However, computer-aided geometric design models are seldom suitable for downstream operations, including structural analysis, because the models are comprised of multiple trimmed surfaces that prove problematic when coupled using known solver methods. In this research, we present automated, theory-based techniques to convert open B-rep and mesh surfaces into analysis-ready spline models represented as a single trimmed spline surface—a representation used in many state-of-the-art isogeometric analysis representations. Here, a parametric domain of an open geometric model is defined by flattening a surface into the Euclidean plane and then optimizing its geometry to approximate an isomorphism. An approximating spline is then defined based on the correspondence between the original surface and its parametric domain. Interpolation and extrapolation operations are defined to naturally extend the spatial domain of the original geometry to be valid for the entire reconstructed spline surface. We demonstrate the success of the method in reconstructing and analyzing CAD models of both aerospace and automotive interest.

Automatic Multi-Material Mesh Generation for the Simulation of Metal Casting Processes

Jan Thomsen, MAGMA Giessereitechnologie GmbH
Andreas Schulz, MAGMA Giessereitechnologie GmbH
Robert Schneiders, MAGMA Giessereitechnologie GmbH

Keywords: Mixed element mesh; multi material; automatic; robust

Abstract

We describe a robust algorithm for the automatic generation of mixed-element meshes for multimaterial domains. Geometries are given as a set of overlapping triangulations, with minimal requirements to geometric quality. The algorithm overlays a geometry with a simple base mesh which is then adapted to the internal and external boundaries. Invalid elements are dealt with by smoothing, splitting, topology changes and optimization so that only valid elements remain. No user intervention is required.

Tusqh: Topological Control of Volume-Fraction Meshes Near Small Features and Dirty Geometry

Kendrick Shepherd, Brigham Young University Department of Civil and Construction Engineering
Scott Mitchell, Sandia National Laboratories, Center for Computing Research
Brian Shawcroft, Brigham Young University Department of Civil and Construction Engineering and Sandia National Laboratories, Center for Computing Research

Abstract

This work develops a framework through which meshes with user-specified homology can be created from potentially dirty geometry by coupling background grids, persistent homology, and a generalization of volume fractions. For a mesh with fixed grid size, the topology of the output mesh changes predictably and monotonically as its volume-fraction threshold decreases. Topological anti-aliasing methods are introduced to resolve pinch points and disconnected regions that are artifacts of user choice of grid size and orientation, making the output meshes suitable for downstream processes including analysis. The methodology is demonstrated on geographical, mechanical, and graphics models in 2D and 3D using a custom-made software called Tusqh. The work demonstrates that the proposed framework is viable both for generating meshes on topologically invalid geometries and for automatic defeaturing of small geometric artifacts. Finally, the work shows that although subdividing the background grid frequently improves the topological and geometrical fidelity of the output mesh, there are simple 2D examples for which the topology does not converge under refinement for volume-fraction codes.

Parallelization of the Finite Element-based Mesh Warping Algorithm Using Hybrid Parallel Programming

Suzanne Shontz, University of Kansas
Abir Haque, University of Kansas

Keywords: Mesh Warping, Finite Element, Hybrid Parallel Programming, Message Passing Interface

Abstract

Warping large volume meshes has applications in biomechanics, aerodynamics, image processing, and cardiology. However, warping large, real-world meshes is computationally expensive. Existing parallel implementations of mesh warping algorithms do not take advantage of shared-memory and one-sided communication features available in the MPI-3 standard. We describe our parallelization of the finite element-based mesh warping algorithm for tetrahedral meshes. Our implementation is portable across shared and distributed memory architectures, as it takes advantage of shared memory and one-sided communication to precompute neighbor lists in parallel. We then deform a mesh by solving a Poisson boundary value problem and the solution of the resulting linear system with multiple right-hand sides in parallel. Our results demonstrate excellent efficiency and strong scalability on up to 32 cores on a single node. Furthermore, we show a 90% increase in speedup with 256 cores distributed uniformly across 64 nodes versus our largest single node speedup while observing sublinear speedups overall.

Linear-algebraic representation and transformation of unstructured meshes

Daniel Shapero, University of Washington

Keywords: mesh generation, computational geometry, algebraic topology

Abstract

This paper will show some new approaches for implementing common transformations to the connectivity or topology of an unstructured mesh. The key enabling technology for our approach is to borrow ideas from algebraic topology: we use the boundary operators of a chain complex to represent the mesh. Boundary operators are really just integer matrices. By representing the objects of study using the language of linear algebra, we can use linear algebraic reasoning and intuition to define transformations. We show how to implement a few common transformations on the boundary operator representation: merging cells, splitting cells on a new vertex, and splitting a cell on facets. The source code for these transformations becomes short and easy to test.