SIAM IMR24: Short Courses

This year’s SIAM International Meshing Roundtable will feature three short courses, to be held on Tuesday March 5, 2024.

Courses are taught by internationally known experts. Instructors typically include an overview of the state of the art of their topic, and highlight their own research, but also include the current work of others. It is intended to be a “course” in the traditional sense of enabling attendees to go forth and produce new results of their own, rather than simply use existing knowledge.

This year we have three courses, which will be held on March 5, 2024. Each of the course details is outlined below.

Intelligent Mesh Generation

Slides

Prof. Na Lei

Abstract: Intelligent Mesh Generation (IMG) represents a novel and promising field of research, utilizing machine learning techniques to generate meshes. Despite its relative infancy, IMG has significantly broadened the adaptability and practicality of mesh generation techniques, delivering numerous breakthroughs and unveiling potential future pathways. In this course we will provide a systematic and thorough survey of the current IMG landscape. With a focus on more than one hundred preliminary IMG literatures, we undertake a meticulous analysis from various angles, encompassing core algorithm techniques and their application scope, agent learning objectives, data types, targeted challenges, as well as advantages and limitations. We have curated and categorized the literature, proposing three unique taxonomies based on key techniques, output mesh unit elements, and relevant input data types. We will also underscores several promising future research directions and challenges in IMG.

Biography: Dr. Na Lei is currently a professor at the School of Software, Dalian University of Technology. In 2002, She obtained a doctorate in science from the School of Mathematics, Jilin University, and then worked at the school as an assistant professor, promoted to full professor in 2009. In 2017, She was transferred to the Software School of Dalian University of Technology. She was an academic visiting professor at the University of Texas at Austin from 2007 to 2008 and the State University of New York at Stony Brook from 2014 to 2015. Her research directions mainly focus on computational conformal geometry, computational topology, computer mathematical algorithms and their applications in artificial intelligence, computer graphics, geometric modeling and medical images. She presides over the National Science Foundation for Distinguished Young Scholars, National Key Research and Development Programs, National Natural Science Foundation key projects, general projects, and some other Science and Technology Innovation Projects. Academic achievements have been introduced at international conferences many times by Fields Medalists and academicians of the National Academy of Sciences; the intellectual property rights obtained have been successfully transformed and applied in industry. She served as a committee member of IMR from 2019 to 2022.

Differential Geometry for Mesh Generation

Slides (I, II, III)

Prof. Xianfeng David Gu

Abstract: Mesh generation is crucial for CAD/CAE industries, which is intrinsically challenging due to its topological and geometrical nature. This short course focuses on the recent development in both unstructured and structured mesh generation based on model algebraic topology and differential geometry.

Unstructured surface mesh generation: According to geometric approximation theory, if the normal cycles of the discrete meshes converge to the normal cycle of a smooth surface under the Hausdorff distance, then the meshes converge to the smooth surface in terms of geodesics, curvature measures and Laplace-Beltrami spectrum and so on. According to the surface uniformization theorem, all metric surfaces can be conformally mapped onto one of three canonical shapes: the sphere, the Euclidean plane or the hyperbolic plane, the uniformization mapping can be achieved using surface Ricci flow. Then triangle meshes can be constructed on the canonical shapes using Delaunay refinement method and pulled back to the original surface. This procedure produces surface meshes with high geometric qualities.

Frame field and cross field generation: Global smooth frame field/cross field generation is important for structured mesh generation. This course will introduce the sufficient and necessary conditions for the singularities of surface cross fields and volume frame fields based on the topological obstruction theory of the characteristic classes of fiber bundles. Once the singularities are determined, by using surface Ricci flow one can obtain a flat metric with cone singularities, by parallel transportation and Hodge theory, the global smooth frame/cross fields on surfaces can be achieved.

Structured surface quadrilateral mesh generation and volume hexahedral mesh generation are crucial for geometric modeling and processing. But the structured meshing problem is intrinsically challenging. This talk focuses on the recent development to tackle these challenges based on modern topological and geometric theories.

Structured surface quadrilateral mesh generation: It is well known that a closed torus does not admit a quad-mesh with only two singularities, whose valences are 3 and 5 respectively. But the theoretic proof is highly non-trivial. Although it seems to be a combinatorial problem, it has deep roots in the characteristic class theory of holomorphic line bundles, especially the Abel-Jacobi theorem. This discovery leads to the governing equations of the configurations of quad-mesh singularities. By solving these equations using Hodge theory and surface Ricci flow, the high quality, automatic quad-mesh generation algorithms can be developed.

Volumetric hexahedral mesh generation: The singularity configuration of volumetric hex-meshes becomes more complicated, due to the fact that the fundamental group of the manifold of crosses is non-Abelian, hence the classical topological obstruction theory for fiber bundles can not be applied. Instead, the recent breakthrough on low dimensional topology leads to practical algorithms.

Biography: Dr. David Xianfeng Gu is a New York Empire Innovation Professor at the Department of Computer Science, Stony Brook University. Dr. Gu is a visiting professor of the Center of Mathematical Sciences and Applications at Harvard university. David received his Ph.D degree from the Department of Computer Science, Harvard University in 2003, supervised by the Fields medalist Prof. Shing-Tung Yau, and B.S. degree from Tsinghua University, Beijing, China in 1994.

David’s research focuses on applying modern topology and geometry theories in engineering and medical fields. With his collaborators, David systematically develops discrete theories and computational algorithms in the interdisciplinary field: Computational Conformal Geometry, and applies them for real problems, such as global surface parameterization based on Hodge decomposition theorem in graphics, deformable shape registration based on Teichmuller map in vision, structured mesh generation based on Abel-Jacobi theorem in geometric modelling, curvature convergence analysis in geometric processing, explainable generative models based on geometric optimal transportation in deep learning, brain mapping and virtual colonoscopy in medical imaging and so on.

Recently, David and his collaborators have proved the discrete surface uniformization theorem using surface Ricci flow, the Alexandrov theorem and Minkowski theorem using geometric optimal transportation, the singularity configuration of structured quad-mesh using Abel-Jacobi theorem. David is a recipient of Morningside Gold Medal of Applied Mathematics in 2013; National Science Foundation Faculty Early Career Award, 2005. David has broad collaborations with companies in the CAD/CAE industry, including Siemens, Cadence and Ansys.

High-Order Polytopal Methods

Slides (Disclaimer: not all figures are reference adequately)

Prof. Joaquim Peiró

Abstract: Polytopal meshes are spatial discretizations consisting of irregular polygons in 2D and polyhedra in 3D. These are general elemental shapes that include commonly used elemental types such as triangles, quads, tetrahedra, hexahedra, prisms, and pyramids. Therefore, they offer a greater geometric flexibility that will help in the generation of meshes for complex geometries. Polytopal mesh generators are widely available as a mesh generation option in commercial CFD software, and low-order methods implemented in these meshes can be more efficient than conventional methods.

The use of polytopal meshes facilitates mesh refinement, for instance in octree-based adaptive refinement, since cells are generic and do not require to ensure compatibility when employing a specific elemental shape or set of shapes, e.g. tetrahedra or hexahedra. Implementation of coarsening algorithms by agglomeration is also straightforward. The need to support discretization using general polytopal meshes has also led to the development of novel techniques for the numerical solution of PDEs.

The course will give a brief overview of current low-order polyhedral mesh generation and PDE discretization methods. This will be followed by a description of high-order polytopal discretization methods with a focus on the hybrid high-order and the virtual element methods. Finally, I will discuss the geometric requirements for the generation of suitable high-order polytopal meshes, and show how current a posteriori methods for the generation of curvilinear meshes can be adapted to the generation of such meshes.

Biography: Dr. Joaquim Peiró is a Professor of Engineering Computation in the Department of Aeronautics at Imperial College London, U.K. He received his MEng Civil Engineering from UPC, Barcelona, Spain, and his PhD from the University of Wales, Swansea. He is a Fellow of the Royal Aeronautical Society, a member of the Spanish Institution of Civil Engineers, and a member of INCOSE.

His research focus is the numerical simulation of physical phenomena governed by conservation laws in both complex geometries and reduced models. He specifically works on the development of pre-processors (mesh generation and CAD interaction), flow solvers (high-order discontinuous Galerkin methods) and post-processors (error estimation, adaptive methods and high-order visualization). Since 1985 he has developed simulation software systems based on unstructured meshes, e.g. FELISA (NASA) and FLITE3D (Rolls-Royce, Airbus, Arup and British Aerospace). His latest project is the open-source high-order mesh generator Nekmesh.

Contact details

For any question concerning the short courses, please contact the chair: