Mesh Quality Meets The Virtual Element Method
DescriptionThis course emphasizes the growing importance of mesh quality assessment for numerical methods in computer graphics, through a combination of theoretical notions and practical exercises.
We explore various approaches to define and measure "mesh quality".
The course starts with basic concepts about meshes, polytopes, and operators, then covers both common and novel quality indicators and metrics for evaluating a discretization and guiding mesh generation, coarsening, and refinement.
Various local and global indicators for two- and three-dimensional meshes, including simplicial, quadrangular/hexahedral, and generic polytopal elements, are discussed.
A live exercise using the software PEMesh demonstrates how to compute and compare mesh quality using these indicators.
Subsequently, we introduce the Virtual Element Method (VEM) for discretizing Partial Differential Equations (PDEs), focusing on the second-order elliptic PDE with homogeneous boundary conditions, specifically Poisson's equation.
We discuss the advantages of VEM over the traditional Finite Element Method (FEM), detailing the enhanced VEM local and global discrete space for a general polynomial degree k, and presenting theoretical error estimations analogous to FEM.
Mesh requirements for VEM are explained, followed by a demonstration using the VEM solver in PEMesh to optimize mesh quality.
We review notable mesh optimization algorithms and present a specific algorithm for optimizing polygonal or polyhedral meshes, utilizing the previously defined quality indicators.
This process involves analyzing local element quality and agglomerating elements to optimize global mesh quality, with a user-set parameter controlling the percentage of elements to be removed.
Practical applications to constrained domains and time-dependent problems are demonstrated, culminating in a live PEMesh exercise comparing optimizations driven by different quality indicators.
Attendees will learn which aspects of a mesh play a role in numerical simulations, how to measure the quality of a discretization, how to solve a numerical problem with the VEM, and how to optimize the quality of a mesh.
We explore various approaches to define and measure "mesh quality".
The course starts with basic concepts about meshes, polytopes, and operators, then covers both common and novel quality indicators and metrics for evaluating a discretization and guiding mesh generation, coarsening, and refinement.
Various local and global indicators for two- and three-dimensional meshes, including simplicial, quadrangular/hexahedral, and generic polytopal elements, are discussed.
A live exercise using the software PEMesh demonstrates how to compute and compare mesh quality using these indicators.
Subsequently, we introduce the Virtual Element Method (VEM) for discretizing Partial Differential Equations (PDEs), focusing on the second-order elliptic PDE with homogeneous boundary conditions, specifically Poisson's equation.
We discuss the advantages of VEM over the traditional Finite Element Method (FEM), detailing the enhanced VEM local and global discrete space for a general polynomial degree k, and presenting theoretical error estimations analogous to FEM.
Mesh requirements for VEM are explained, followed by a demonstration using the VEM solver in PEMesh to optimize mesh quality.
We review notable mesh optimization algorithms and present a specific algorithm for optimizing polygonal or polyhedral meshes, utilizing the previously defined quality indicators.
This process involves analyzing local element quality and agglomerating elements to optimize global mesh quality, with a user-set parameter controlling the percentage of elements to be removed.
Practical applications to constrained domains and time-dependent problems are demonstrated, culminating in a live PEMesh exercise comparing optimizations driven by different quality indicators.
Attendees will learn which aspects of a mesh play a role in numerical simulations, how to measure the quality of a discretization, how to solve a numerical problem with the VEM, and how to optimize the quality of a mesh.
Event Type
Courses
TimeWednesday, 4 December 20242:00pm - 5:45pm JST
LocationG602, G Block, Level 6
