Tetrahedron Workshop VIII

Rao Garimella
Los Alamos National Laboratory

Many real life assessments require simulations involving multiple physics spanning multiple regimes. However, it is not possible to develop a single code that can perform such a comprehensive simulation accurately. Instead multiple simulations that are optimized for particular physics in suitable regimes must be loosely coupled by transferring the output of one to the next. Each of these simulations have their particular requirements on dimensionality, level of geometric detail and the resolution and characteristics of meshes. Moreover, accurate transfer of the solution involves accurate conservative remapping, material interface reconstruction and material model matching. This talk will discuss the geometry and meshing challenges encountered in such coupling and propose some solutions.

Bob Haimes
Geocentric Technologies LLC

As attention is focused upon the “time to solution”, it becomes obvious that the entire process must be taken into account – not just the cost and efficiency of the solver. Amdahl’s Law tells us that any serial portion of the application will be the limiting factor in scalability. Therefore it does not matter how efficient a solver is if both the pre- and post-processing have not been given the same focus towards scalability. The most obvious way to insure that a scalable process exists is to view the process as an integrated whole and remove any serial portions. The work discussed here makes geometry available in a parallel environment to support parallel mesh generation, grid adaptation, and the curving of linear meshes to support high(er) order spacial discretizations.

EGADSlite has been put together with the EGADS functions that are used in mesh generation. This has been part of the ESP distribution for the last 5 years. But it has 2 basic assumptions that limit its usage: (1) All partitions get all of the geometry and (2) it is difficult to be performant in GPUs at the current API level. Solving this involves viewing the problem from a process level higher than the geometry kernel.

Lena Plötzke
German Aerospace Center (DLR)

This talk presents t8code, a modern software library for tree-based adaptive mesh refinement (AMR), outlining its current state and development roadmap. While AMR is essential for high-fidelity simulations, effective implementation often introduces substantial overhead and demands specialized expertise to harness efficiently. To address this challenge, we introduce a novel, simplified mesh handle architecture that significantly reduces implementation complexity without compromising computational performance. The presentation explores t8code’s practical strengths in managing complex 3D meshes, emphasizing parallel efficiency and advanced adaptivity strategies. Specifically, we highlight a subelement-based strategy for hanging-node resolution lowering the barrier to entry for codes that require conforming meshes. We discuss ongoing efforts to seamlessly integrate the library into established solver ecosystems like OpenFOAM and deal.II, bridging sophisticated meshing algorithms with industrial simulation workflows. The talk concludes with our vision for broader ecosystem adoption and extends an open invitation to partners in science and industry seeking robust, high-performance 3D mesh adaptation solutions for concrete engineering applications.

David Kopriva
Florida State University

High order spectral element solutions are only as accurate as the approximate geometries on which they are computed, yet mesh errors are under appreciated. We derive error bounds for hyperbolic systems solved on approximated domains to show that errors in the metric terms and the Jacobian dominate. We use this knowledge to choose a boundary error measure and construct procedures to compute the optimal piecewise polynomial approximation of a boundary curve. The piecewise polynomial satisfies smoothness constraints and the error measure to within a user-specified tolerance. We incorporate these ideas into HOHQMesh to drive the mesh generation process and to compute the best boundary approximation of the resulting mesh.

Hang Si
Dalian University of Technology

In this talk, we review recent generative AI techniques for 3D mesh generation. Our empirical findings reveal that these methods commonly face a core challenge—balancing topological complexity with geometric precision—which is critical for meeting engineering requirements. This challenge raises fundamental questions: What does a deep learning system truly learn? How does it learn? How well does it learn? And must a system forget certain information in order to generalize? In response, we offer a preliminary geometric perspective on generative AI, aimed at providing insight into how far current approaches fall short of meeting the demands of real-world industrial applications for 3D mesh generation.

John Verdicchio
Siemens

Within STARCCM+ , the “Resurfacer” code that takes an existing triangular mesh and creates a mesh that meets the users sizing and quality requirements only ran in serial. I will be discussing how we changed the code so it would run under a distributed memory parallel system whilst not re-writing the entire codebase.

Henry Bucklow
ITI Ltd

Every simulation starts with a geometry. Commercial CAD systems are the dominant source of geometry for industrial simulations. These systems provide boundary representation models, accurately describing the outer surfaces of the geometry for layout and manufacturing, but offer little structure for characterising the internal volume.

Mesh generation requires decisions using this internal volume; these decisions can be highly consequential to the success and quality of the mesh. To optimise these decisions, we create geometry reasoning structures: objects which propagate information through the domain, allowing reasoned decisions based on the volumetric shape.

We explore several such geometry reasoning methods, including 2D and 3D medial objects, cross- and frame-fields, and medial fields. We show how applying and combining these methods can provide automated mesh generation tools for industrial users. We demonstrate these tools for both structured and unstructured mesh generation, across aerospace, automotive, and medical applications.

Marcel Campen
Paderborn University

This talk highlights research advances on the topic of higher-order mesh generation, for 2D triangle and 3D tetrahedral meshes. Concretely, the focus is on techniques that offer guarantees regarding element validity, element quality, and domain boundary conformance, for meshes with polynomial or rational elements of arbitrary order. A common scheme in these techniques is favoring constructive techniques over optimization-centric approaches, and incrementally reducing a hard meshing problem to simpler problem classes. Using a careful algorithmic design, even the common gap between reliability in theory and in practice (in particular due to numerical inaccuracies) can be addressed.

Jessica Zhang
Carnegie Mellon University

Polycube structures provide structured parametric domains for all-hexahedral (all-hex) mesh generation and analysis-suitable volumetric spline construction in isogeometric analysis (IGA). Recent learning-based polycube pipelines have improved automation, yet several challenges remain when handling more complex CAD geometries. These challenges include the limited diversity of geometric primitives, restricted grid configurations, and the increasing cost of genus guided context search during inference as both the primitive set and the grid size grow. In this paper, we present extended DDPM-Polycube, a diffusion-based polycube construction method that addresses these limitations. First, we expand the primitive set from two geometric primitives to three by introducing a blind-hole cube primitive, thereby improving the representation of local hole-like features that do not change the global genus. Second, we extend the grid configuration from the previous 2x1 setting to an enlarged three-dimensional grid configuration, which increases representational capacity and reduces mapping distortion for complex geometries. Third, we develop a genus-guided context generation strategy together with a hierarchical verification procedure, enabling robust context generation in both user-guided and automated modes. Once a valid polycube structure is generated, it is used for parametric mapping, all-hex mesh generation, and volumetric spline construction. Experimental results demonstrate that scalable DDPM-Polycube improves the generality, scalability, and automation of diffusion-based polycube generation, and supports robust hex mesh generation and volumetric spline construction for IGA applications on complex geometries.

Stefano Berrone
Politecnico di Torino

Mesh generation for geometrically complex domains remains one of the major challenges in numerical analysis. This talk focuses on a posteriori error estimation for Virtual Element Methods, adaptive mesh refinement, and mesh generation within the framework of polygonal and polyhedral numerical methods. The talk also introduces a novel refinement strategy for convex polygonal meshes that is designed to preserve or improve mesh quality throughout the refinement process. After repeated refinement steps, the procedure naturally produces high-quality triangular or quadrilateral meshes. Starting from a very coarse mesh obtained directly from the geometric description of the domain, the proposed approach offers an efficient technique for generating high-quality polygonal meshes. Finally, a refinement strategy for polyhedral elements will be presented and assessed for its application in adaptive mesh refinement methods.

Jean-Francois Remacle
Universite catholique de Louvain

Suzanne M. Shontz
The University of Kansas

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 resulting linear system, which has 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 33.9% increase in speedup with 256 cores distributed uniformly across 64 nodes versus our largest single node speedup while observing sublinear speedups overall. This talk represents joint work with Abir Haque, formerly of The University of Kansas.

Per-Olof Persson
UC Berkeley

Traditional mesh generation often relies on rigid heuristics and manual tuning. In this talk, I will present our work on learning-based mesh generation, reframing the problem as an optimal decision-making process solved via deep reinforcement learning (RL) and graph neural networks (GNNs). First, I will review our early work on mesh topology optimization, where an RL agent learns sequences of topological moves to maximize regular nodes in triangular and quadrilateral meshes. Next, I will introduce a more general, fully self-supervised formulation based on edge and vertex insertions that produces arbitrary polygonal elements. Moving beyond topology, I will discuss our work on geometry and node placement, demonstrating how a GNN learns a policy to dynamically insert, delete, and move vertices within a domain. Finally, I will share our most recent results extending these frameworks to 3D tetrahedral mesh generation and integrating RL with the DistMesh algorithm, pointing toward the future of fully automated meshing pipelines.

Xevi Roca
Barcelona Supercomputing Center

To enable high-fidelity aerodynamic and electromagnetic simulations of future aviation designs, engineers will need methods that preserve relevant physical properties while delivering high numerical and geometrical accuracy on complex geometries. To meet these needs, computational scientists and engineers are developing a new generation of structure-preserving unstructured high-order methods that require curved simplicial meshes. Focusing on these meshes, this talk provides an overview of our efforts to address different geometric, meshing, and numerical challenges. Specifically, this talk presents our research on geometry and field approximation, curved adaptivity, bisection-based refinement, and large-scale curved meshing. In perspective, by contributing to curved simplicial meshing, we are helping to enable high-order methods for future high-fidelity aeronautical analysis.

Eloi Ruiz-Gironés
Barcelona Supercompting Center

We enhance our mesh curving method to generate high-order meshes suitable for solving the inviscid part of wall-modeled large-eddy simulations (WMLES) with an explicit time-marching scheme. The simulation waiting time is determined by the time-step, and this time-step is limited by the most restrictive distance between mesh points. Without a proper mesh, the time-step is so small that the simulation waiting time is prohibitively long. Unfortunately, in aeronautics, it is challenging to meet the mesh point spacing needs, especially next to aircraft surfaces exposed to high velocity flow. Examples of exposed regions are the highly pronounced tangencies in the slat recovery surfaces, and small element sizes in the narrow surfaces at the trailing edges.

To fulfill the point spacing requirements for explicit methods, we can accept a geometric discrepancy between the target geometry and the initial straight-edged mesh. However, this geometric discrepancy might have a negative impact on the mesh curving process and the quality of the curved mesh. Thus, solving this robustness problems is of major importance to obtain a curved mesh that fulfills the point spacing requirements.

Our solution is to enhance our distortion-based curving method by fixing the position of the high-order triangles that approximate problematic regions. To this end, the user provides an initial list of triangles to fix, and our process automatically detects the additional triangles that need to be fixed. The main idea is to fix the surface triangles that lead to an invalid mesh during the backtracking line-search step. Accordingly, around the problematic regions, the mesh curving automatically accepts a geometric discrepancy between the curved mesh and the target geometry. The main benefit of our enhancement is that the resulting curved high order meshes are valid while also featuring an increased mesh point spacing. Hence, the resulting meshes are suitable for explicit time-marching in WMLES.

Trevor Robinson
Queen's University Belfast

Mechanical Computer-Aided Design systems are heavily used by industry to create geometry models to support computational simulation processes. The de facto standard for these Mechanical Computer-Aided Design systems is to represent the geometry as boundary representation models. These are complex constructs that combine geometric and topological entities to represent shape. This presentation will summarise ongoing research that uses graphs based on these constructs to support simulation pre-processing activities, including mesh generation.

Kaloyan Kirilov
King's College London

One of the major bottlenecks in front of the industrialization of high-order methods has been the development of high-order mesh generators. To tackle the problem, we have previously presented our mesh generation workflow, which combines the robustness of arbitrary third-party straight-sided boundary layer mesh generation with the quality of NekMesh’s a-posteriori mesh curving and optimization modules.

In this work, we address some of the commonly encountered weaknesses of a-posteriori high-order mesh generation. First, we address the automation problem, where an initially unsuitable for mesh curving linear mesh can be modified internally with high-order mesh modifications, potentially reducing the need for slow manual iterations in the third-party tool.

Second, we propose a reformulation of the long-standing isoparametric prism layer splitting approach to allow for generic boundary layer meshes. This is further combined with ongoing work on AMR flow simulation. For this we present initial results on dynamic boundary layer mesh adaptation, which first benefits from the integration of NekMesh with the Spectral/hp framework Nektar++, but also from a novel, more efficient solution transfer method that leverages the isoparametric boundary layer structure.

Last, we give a brief overview of our current eLorts towards a parallel implementation of NekMesh and more efficient I/O mesh formats.

Xi Zou
Swansea University

Mesh generation remains one of the most labour-intensive stages of finite element analysis. Conventional NURBS-based CAD models often contain multiscale geometric features that require extensive geometry cleaning and de-featuring to prevent excessive local mesh refinement. Such refinement is particularly undesirable for numerical simulations employing explicit time integration, where the smallest elements impose severe restrictions on the stable time step.

The NURBS-Enhanced Finite Element Method (NEFEM) addresses this challenge by decoupling geometric representation from solution approximation within an anisoparametric finite element framework. The exact NURBS description is obtained directly from the CAD model to represent the geometry, while standard polynomial basis functions are used to approximate the solution. This enables NEFEM to preserve the exact CAD geometry throughout the analysis, significantly reducing or eliminating the need for geometry de-featuring.

To fully exploit these advantages, a dedicated mesh generation strategy is required, as conventional mesh generators cannot produce NEFEM elements. Unlike standard finite elements, NEFEM elements are designed to span multiple CAD curves or surfaces, allowing them to capture complex geometric features while satisfying user-prescribed mesh size requirements. This talk presents a hierarchical three-dimensional mesh generation framework for constructing NEFEM surface and volume meshes. Several examples are presented to demonstrate the effectiveness of the proposed mesh generation approach and the advantages it offers for high-order finite element analysis of complex CAD geometries.