Tetrahedron VII Abstracts
Frédéric Alauzet, INRIA Saclay, France
Impact of anisotropic mesh adaptation for turbulent flows in aeronautics. Towards the certification of numerical solutions?
This presentation will discuss recent progress made in anisotropic mesh adaptation for turbulent flow (RANS) simulations. We will focus on drag and lift prediction, and vortex dominated flows in aeronautics. First, we will present each step of the mesh-adaptive solution platform: the flow solver Wolf, the RANS error estimates (feature-based and goal-oriented will be compared), the local remesher Feflo.a and the solution interpolation method. The key point is to propose robust numerical methods that are compatible with extreme anisotropy, ie element ratios of 1:100000. We will describe how, in practice, the mesh adaptation loop is implemented and all the advantages of doing it in this way (multigrid type effect,free convergence study, ...). In the presented examples, we will show the early capturing property of the mesh adaptation process on output functionals of interest, and, the fact that the mesh-convergence of the solution is achieved even in 3D. We will also emphasize the automaticity of the mesh adaptation process and its independence to the initial mesh. All these properties are a necessary first step towards the certification of numerical solutions.
This presentation will discuss recent progress made in anisotropic mesh adaptation for turbulent flow (RANS) simulations. We will focus on drag and lift prediction, and vortex dominated flows in aeronautics. First, we will present each step of the mesh-adaptive solution platform: the flow solver Wolf, the RANS error estimates (feature-based and goal-oriented will be compared), the local remesher Feflo.a and the solution interpolation method. The key point is to propose robust numerical methods that are compatible with extreme anisotropy, ie element ratios of 1:100000. We will describe how, in practice, the mesh adaptation loop is implemented and all the advantages of doing it in this way (multigrid type effect,free convergence study, ...). In the presented examples, we will show the early capturing property of the mesh adaptation process on output functionals of interest, and, the fact that the mesh-convergence of the solution is achieved even in 3D. We will also emphasize the automaticity of the mesh adaptation process and its independence to the initial mesh. All these properties are a necessary first step towards the certification of numerical solutions.
Pierre Alliez, INRIA Sophia Antipolis
Reconstruction of concise meshes from 3D point clouds
The popular quadric error metrics were initially designed for fine-to-coarse mesh decimation. I will present a concise mesh reconstruction approach for 3D point clouds that proceeds coarse-to-fine by clustering the input points enriched with quadric error metrics. Such a clustering approach favors the placement of generators on sharp features, and tends to equidistribute the error among clusters. This approach avoids dense reconstruction prior to simplification and produces immediately an optimized surface triangle mesh mesh.
The popular quadric error metrics were initially designed for fine-to-coarse mesh decimation. I will present a concise mesh reconstruction approach for 3D point clouds that proceeds coarse-to-fine by clustering the input points enriched with quadric error metrics. Such a clustering approach favors the placement of generators on sharp features, and tends to equidistribute the error among clusters. This approach avoids dense reconstruction prior to simplification and produces immediately an optimized surface triangle mesh mesh.
David Bommes, University of Bern, Switzerland
Quadrilateral and Hexahedral Mesh Generation with Integer-Grid Maps
Automatically generating quadrilateral and hexahedral meshes is a notoriously challenging task, specifically if alignment to freeform surfaces in combination with high mesh regularity and low distorted elements is required. Novel algorithms based on global optimization rely on the construction of integer-grid maps, which pull back a Cartesian grid of integer isolines from a 2D or 3D domain onto a structure-aligned quadrilateral or hexahedral mesh. Such global optimization algorithms do not suffer from limitations known from local advancing front methods, as for instance, a large number of irregularities, and enable meshes comparable to manually designed ones by finding a good compromise between regularity and element distortion. The key to efficiently finding high-quality solutions are frame fields that are employed to optimize the orientation and sizing of mesh elements globally. In my talk, I will give an overview of the state of the art and discuss the strengths and weaknesses of available algorithms, including recent advances on locally meshable frames fields for hexahedral meshing.
Automatically generating quadrilateral and hexahedral meshes is a notoriously challenging task, specifically if alignment to freeform surfaces in combination with high mesh regularity and low distorted elements is required. Novel algorithms based on global optimization rely on the construction of integer-grid maps, which pull back a Cartesian grid of integer isolines from a 2D or 3D domain onto a structure-aligned quadrilateral or hexahedral mesh. Such global optimization algorithms do not suffer from limitations known from local advancing front methods, as for instance, a large number of irregularities, and enable meshes comparable to manually designed ones by finding a good compromise between regularity and element distortion. The key to efficiently finding high-quality solutions are frame fields that are employed to optimize the orientation and sizing of mesh elements globally. In my talk, I will give an overview of the state of the art and discuss the strengths and weaknesses of available algorithms, including recent advances on locally meshable frames fields for hexahedral meshing.
Olivier Coulaud, CENAERO, Belgium
with Adrien Loseille and Pierre Schrooyen
High-order solution-based anisotropic mesh adaptation
When solving a PDE by the use of high-order numerical schemes, the minimization of the interpolation error induced by the projection of the solution onto a high-order finite elements space is a significant issue. Consequently, there is a need for mesh adaptation with respect to this high-order solution. A way to address this issue is the log-simplex method. From an initial mesh and a solution of arbitrary order, this later enables to produce a dedicated metric-field which is then used to lead an anisotropic mesh adaptation process. The method, which relies on a priori error estimates governed by the high-order differential of the solution, is actually a natural extension to the classical Hessian based mesh adaptation methods.
When solving a PDE by the use of high-order numerical schemes, the minimization of the interpolation error induced by the projection of the solution onto a high-order finite elements space is a significant issue. Consequently, there is a need for mesh adaptation with respect to this high-order solution. A way to address this issue is the log-simplex method. From an initial mesh and a solution of arbitrary order, this later enables to produce a dedicated metric-field which is then used to lead an anisotropic mesh adaptation process. The method, which relies on a priori error estimates governed by the high-order differential of the solution, is actually a natural extension to the classical Hessian based mesh adaptation methods.
Darren Engwirda, Los Alamos National Laboratory, USA
Optimising primal-dual discretisations and meshes for geophysical fluid dynamics
Leveraging unstructured solvers in global climate models is an emerging research area, with e.g. US-DOE's Energy Exascale Earth System Model capturing multiscale effects in ocean, atmosphere and land-surface processes using Voronoi-type meshes. Despite success in a range of applications, current generation models are based on low-order accurate discretisations, and suffer from a lack of convergence at high resolution and/or in complex unstructured configurations. In this work an alternative primal-dual mesh structure and numerical discretisation is described, taking Laguerre-Power meshes in place of Delaunay-Voronoi pairs and developing a structure-preserving finite-volume scheme that conserves energy and other high-order moments associated with the flow. Use of Laguerre- rather than Voronoi-based schemes is shown to improve both accuracy and robustness — increasing order of accuracy and reducing mesh imprinting in the solution, and enabling the construction of high quality non-obtuse primal-dual tessellations designed to minimise discretisation errors. A number of applications in geophysical modelling are presented.
Leveraging unstructured solvers in global climate models is an emerging research area, with e.g. US-DOE's Energy Exascale Earth System Model capturing multiscale effects in ocean, atmosphere and land-surface processes using Voronoi-type meshes. Despite success in a range of applications, current generation models are based on low-order accurate discretisations, and suffer from a lack of convergence at high resolution and/or in complex unstructured configurations. In this work an alternative primal-dual mesh structure and numerical discretisation is described, taking Laguerre-Power meshes in place of Delaunay-Voronoi pairs and developing a structure-preserving finite-volume scheme that conserves energy and other high-order moments associated with the flow. Use of Laguerre- rather than Voronoi-based schemes is shown to improve both accuracy and robustness — increasing order of accuracy and reducing mesh imprinting in the solution, and enabling the construction of high quality non-obtuse primal-dual tessellations designed to minimise discretisation errors. A number of applications in geophysical modelling are presented.
Tzanio Kolev, LLNL, USA
with J. Camier, V. Dobrev, P. Knupp, K. Mittal, and V. Tomov
Mesh Adaptivity for Scalable and Accurate High-Order Simulations
We discuss mesh adaptivity in the context of HPC simulations on general unstructured high-order meshes. Our approach relies on node movement with fixed topology, through the Target-Matrix Optimization Paradigm (TMOP) and uses a global nonlinear solve over the whole computational mesh. In this talk we will review recent progress in TMOP theory, adaptive surface fitting, GPU acceleration, and HPC benchmarks and applications. Our surface fitting approach utilizes discrete finite element functions (e.g., level set functions) to define implicit surfaces, which are used to adapt the positions of certain mesh nodes. The algorithm does not require CAD descriptions or analytic parameterizations and can be beneficial in computations with dynamically changing geometry, for example shape optimization and moving mesh multi-material simulations. The main advantage of this approach is that it completely avoids geometric operations (e.g., surface projections), and all calculations can be performed through finite element operations.
We discuss mesh adaptivity in the context of HPC simulations on general unstructured high-order meshes. Our approach relies on node movement with fixed topology, through the Target-Matrix Optimization Paradigm (TMOP) and uses a global nonlinear solve over the whole computational mesh. In this talk we will review recent progress in TMOP theory, adaptive surface fitting, GPU acceleration, and HPC benchmarks and applications. Our surface fitting approach utilizes discrete finite element functions (e.g., level set functions) to define implicit surfaces, which are used to adapt the positions of certain mesh nodes. The algorithm does not require CAD descriptions or analytic parameterizations and can be beneficial in computations with dynamically changing geometry, for example shape optimization and moving mesh multi-material simulations. The main advantage of this approach is that it completely avoids geometric operations (e.g., surface projections), and all calculations can be performed through finite element operations.
Franck Ledoux, CEA, France
with C. Bourcier, C. Dumery, N. Le Goff, S. Mestrallet, F. Protais, N. Ray, and D. Sokolov
Evolutional Polycubes for Hexahedral Mesh blocking
Several CEA simulation codes require hexahedral block-structured meshes to run efficiently. Such types of meshes are known as very challenging to generate, and in practice, CEA engineers and scientists use dedicated interactive 3D software to handcraft the appropriate mesh. This process can take many days or weeks, depending on the model complexity and the involved physics. At CEA DAM, we have developed for more than a decade now a meshing software named MGX, which is dedicated to hexahedral blocking. This software includes many blocking operations to handle and modify a coarse block structure that is constantly classified onto a geometric CAD model. In order to make the meshing process faster, we currently incorporate a blocking pipeline based on the polycube approach. In this presentation, I will describe the current state of this whole pipeline, its limitations, and the future direction we expect to follow.
To tackle the hex blocking problem, we consider polycubes as being aggregates of axis-aligned coarse cubes that provide a first non-optimal solution. Then, we topologically optimize this first solution to consider boundary-alignment constraints that fit engineers' requirements.
Unlike state-of-the-art algorithms that generate and optimize polycube solving non-convex mixed-integer problems using heuristical approaches, we made the choice to consider evolutionary algorithms. Our approach is also based on empirical choice, but it should give us the ability to explore a large landscape of solutions that we will expose to the end user for selecting the "best" one. More specifically, we designed and implemented a generic approach in the context of polycube labeling optimization, defining fitness, crossover, and mutations. We use Ant Colony Optimization to optimize polycube structure, considering the sheet insertion procedure as a kind of extension of the Travelling Salesman Problem (TSP) in 3D.
Several CEA simulation codes require hexahedral block-structured meshes to run efficiently. Such types of meshes are known as very challenging to generate, and in practice, CEA engineers and scientists use dedicated interactive 3D software to handcraft the appropriate mesh. This process can take many days or weeks, depending on the model complexity and the involved physics. At CEA DAM, we have developed for more than a decade now a meshing software named MGX, which is dedicated to hexahedral blocking. This software includes many blocking operations to handle and modify a coarse block structure that is constantly classified onto a geometric CAD model. In order to make the meshing process faster, we currently incorporate a blocking pipeline based on the polycube approach. In this presentation, I will describe the current state of this whole pipeline, its limitations, and the future direction we expect to follow.
To tackle the hex blocking problem, we consider polycubes as being aggregates of axis-aligned coarse cubes that provide a first non-optimal solution. Then, we topologically optimize this first solution to consider boundary-alignment constraints that fit engineers' requirements.
Unlike state-of-the-art algorithms that generate and optimize polycube solving non-convex mixed-integer problems using heuristical approaches, we made the choice to consider evolutionary algorithms. Our approach is also based on empirical choice, but it should give us the ability to explore a large landscape of solutions that we will expose to the end user for selecting the "best" one. More specifically, we designed and implemented a generic approach in the context of polycube labeling optimization, defining fitness, crossover, and mutations. We use Ant Colony Optimization to optimize polycube structure, considering the sheet insertion procedure as a kind of extension of the Travelling Salesman Problem (TSP) in 3D.
Bruno Levy, INRIA Nancy, France
Monge-Ampère Gravity
The standard model of cosmology, Lambda-CDM, based on Newtonian gravity and general relativity, fits amazingly well the observational data. However anomalies and tension are being discovered which call for a better theory. We present the first 3D cosmological simulation of Monge-Ampère gravity, where the Poisson equation is replaced by the nonlinear elliptic Monge-Ampère equation. We exploit its relations with Optimal Transport theory to design a novel large-scale solver, and we simulate Monge-Ampère gravity with 512^3 particles. We compare the results with a classical Newtonian N-body simulation. In the simulation results, we detect more abundant enhanced filamentary structures, haloes with larger angular momenta, less power on small scales and more on large scales, which may provide a novel solution to a few challenges of the standard model of cosmology.
The standard model of cosmology, Lambda-CDM, based on Newtonian gravity and general relativity, fits amazingly well the observational data. However anomalies and tension are being discovered which call for a better theory. We present the first 3D cosmological simulation of Monge-Ampère gravity, where the Poisson equation is replaced by the nonlinear elliptic Monge-Ampère equation. We exploit its relations with Optimal Transport theory to design a novel large-scale solver, and we simulate Monge-Ampère gravity with 512^3 particles. We compare the results with a classical Newtonian N-body simulation. In the simulation results, we detect more abundant enhanced filamentary structures, haloes with larger angular momenta, less power on small scales and more on large scales, which may provide a novel solution to a few challenges of the standard model of cosmology.
Joaquim Peiró, Imperial College London, UK
with Kaloyan Kirilov, Jingtian Zhou, Mashy Green, and David Moxey
On CAD non-compliance in high-order curvilinear meshing
The talk will discuss the effect of geometrical errors in high-order simulations. We will illustrate the relevant issues, and propose some remedies, for a high-order CFD pipeline where the initial straight-sided mesh is generated from CAD by a third-party software that does not retain the geometrical information required to curve the high-order mesh to be boundary conforming.
The talk will discuss the effect of geometrical errors in high-order simulations. We will illustrate the relevant issues, and propose some remedies, for a high-order CFD pipeline where the initial straight-sided mesh is generated from CAD by a third-party software that does not retain the geometrical information required to curve the high-order mesh to be boundary conforming.
Capucine Legentil and Jeanne Pellerin, TotalEnergies, France
Meshing for geological process simulations: where are we?
In the course of subsurface modeling projects, one often needs to revise an existing interpretation, integrate new spatial data, and perturb a geomodel to reflect subsurface uncertainty and to ultimately reduce this uncertainty using inversion methods. In this presentation we will present a mesh-based approach for local updating of meshed geomodels 2D and 3D. Local modifications are performed in a particular region of the model by changing the unstructured meshes of geomodels. The objective is to maintain a conformal, valid and good quality mesh for numerical simulations. The impact of local modifications on physical simulations is presented on a synthetic 3D reservoir for the injection and storage of CO2. In the second part of this presentation, we present challenges of unstructured mesh generation and numerical simulations for real case subsurface focusing on coupled flow-poromechanical simulations with GEOS an open-source simulation software.
In the course of subsurface modeling projects, one often needs to revise an existing interpretation, integrate new spatial data, and perturb a geomodel to reflect subsurface uncertainty and to ultimately reduce this uncertainty using inversion methods. In this presentation we will present a mesh-based approach for local updating of meshed geomodels 2D and 3D. Local modifications are performed in a particular region of the model by changing the unstructured meshes of geomodels. The objective is to maintain a conformal, valid and good quality mesh for numerical simulations. The impact of local modifications on physical simulations is presented on a synthetic 3D reservoir for the injection and storage of CO2. In the second part of this presentation, we present challenges of unstructured mesh generation and numerical simulations for real case subsurface focusing on coupled flow-poromechanical simulations with GEOS an open-source simulation software.
Per-Olof Persson, University of California, Berkeley, USA
with Arjun Narayanan and Lewis Pan
Learning topological operations on meshes with application to block decomposition of polygons
We present a learning based framework for mesh quality improvement on unstructured triangular and quadrilateral meshes. Our model learns to improve mesh quality according to a prescribed objective function purely via self-play reinforcement learning with no prior heuristics. The actions performed on the mesh are standard local and global element operations. The goal is to minimize the deviation of the node degrees from their ideal values, which in the case of interior vertices leads to a minimization of irregular nodes.
We present a learning based framework for mesh quality improvement on unstructured triangular and quadrilateral meshes. Our model learns to improve mesh quality according to a prescribed objective function purely via self-play reinforcement learning with no prior heuristics. The actions performed on the mesh are standard local and global element operations. The goal is to minimize the deviation of the node degrees from their ideal values, which in the case of interior vertices leads to a minimization of irregular nodes.
Trevor Robinson, Queen's University, Belfast, UK
Geometry for simulation
In this talk Trevor will talk about research he has led and been involved in at Queen's University Belfast over the last 20 years. He will describe the context of this research, and cover topics including Geometric Reasoning with a particular emphasis on identify regions in a geometry model to which specific mesh styles can be applied, and Geometry Handling to overcome the limitations imposed on the meshing process when working with the complex geometry formats produced by most CAD systems. He will also talk about work he has led on using geometry as an input to data processes, including ontologies, relational learning and machine learning. He will close with some perspective on the future of geometry for simulation.
In this talk Trevor will talk about research he has led and been involved in at Queen's University Belfast over the last 20 years. He will describe the context of this research, and cover topics including Geometric Reasoning with a particular emphasis on identify regions in a geometry model to which specific mesh styles can be applied, and Geometry Handling to overcome the limitations imposed on the meshing process when working with the complex geometry formats produced by most CAD systems. He will also talk about work he has led on using geometry as an input to data processes, including ontologies, relational learning and machine learning. He will close with some perspective on the future of geometry for simulation.
Eloi Ruiz-Gironés, Barcelona Supercomputing Center, Spain
with Xevi Roca
Generation of large-scale curved meshes for complex virtual geometries
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. During this presentation, we will describe the development of our curving method and how it has enabled us to significantly reduce computational time, memory footprint, and energy consumption when generating curved meshes. We will also highlight our unique approaches of using floating meshes to approximate a virtual model and fully converge the optimization process to obtain optimal-quality curved meshes.
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. During this presentation, we will describe the development of our curving method and how it has enabled us to significantly reduce computational time, memory footprint, and energy consumption when generating curved meshes. We will also highlight our unique approaches of using floating meshes to approximate a virtual model and fully converge the optimization process to obtain optimal-quality curved meshes.
Rubén Sevilla, Swansea University, UK
with Callum Lock, Oubay Hassan, and Jason Jones
Generating near-optimal meshes using green AI
Most techniques employed to solve partial differential equations need to generate a mesh that describes the geometry of the model. Unstructured mesh technology is nowadays widely adopted, but when optimising a design, many simulations for different operating conditions and geometric configurations are required. Generating the most appropriate mesh for each configuration becomes too time consuming, due to the requirement of excessive human intervention and expertise.
This talk will present our recent work on the use of artificial intelligence to predict near-optimal meshes suitable for simulation. The main idea is to make use of the data available in industry, to optimise the selection of a suitable spacing function. The proposed approach aims at exploiting the knowledge embedded in previous simulations to inform the mesh generation. The methodology proposed will be assessed in terms of accuracy of the predictions, efficiency, and environmental implications.
Most techniques employed to solve partial differential equations need to generate a mesh that describes the geometry of the model. Unstructured mesh technology is nowadays widely adopted, but when optimising a design, many simulations for different operating conditions and geometric configurations are required. Generating the most appropriate mesh for each configuration becomes too time consuming, due to the requirement of excessive human intervention and expertise.
This talk will present our recent work on the use of artificial intelligence to predict near-optimal meshes suitable for simulation. The main idea is to make use of the data available in industry, to optimise the selection of a suitable spacing function. The proposed approach aims at exploiting the knowledge embedded in previous simulations to inform the mesh generation. The methodology proposed will be assessed in terms of accuracy of the predictions, efficiency, and environmental implications.
Hang Si, CADENCE EDA
On Automatic robust and efficient mesh generation and adaptation for EDA
This talk focuses on the mesh generation problem in electric design automation (EDA). The PCB/IC/Package layout geometry in EM and thermal devices is highly complex. It consists of massively distributed components in multiple (possibly curved) thin layers. The geometry size varies from nanometers (1e-9) to meters. Automatic mesh generation and adaptation is challenging and a significant bottleneck in the simulation flow. We describe a mesh generation framework to generate high-quality unstructured and hybrid meshes from watertight Brep CAD geometries. In this method, surface and volume meshing are closely coupled to respect the CAD geometry faithfully, and self-intersections (due to mesh size and curved thin layers) are resolved by local mesh refinement and adaptation. Theoretical and practical algorithms are employed to handle arbitrarily complicated geometry.
This talk focuses on the mesh generation problem in electric design automation (EDA). The PCB/IC/Package layout geometry in EM and thermal devices is highly complex. It consists of massively distributed components in multiple (possibly curved) thin layers. The geometry size varies from nanometers (1e-9) to meters. Automatic mesh generation and adaptation is challenging and a significant bottleneck in the simulation flow. We describe a mesh generation framework to generate high-quality unstructured and hybrid meshes from watertight Brep CAD geometries. In this method, surface and volume meshing are closely coupled to respect the CAD geometry faithfully, and self-intersections (due to mesh size and curved thin layers) are resolved by local mesh refinement and adaptation. Theoretical and practical algorithms are employed to handle arbitrarily complicated geometry.
Dmitry Sokolov, Université de Lorraine, France
with Vladimir Garanzha, Igor Kaporin, Liudmila Kudryavtseva, and François Protais
Lowest distortion mappings
Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative length error to assess map quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We propose to solve a proxy problem approximating scale-optimal maps. To this end, we present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion. Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum.
Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative length error to assess map quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We propose to solve a proxy problem approximating scale-optimal maps. To this end, we present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion. Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum.
Vladimir Tomov, Lawrence Livermore National Lab, USA
with V. Dobrev, P. Knupp, T. Kolev, K. Mittal, R. Rieben, and M. Stees
Balancing Shape and Size: Asymptotic Analysis of Compound Volume+Shape Mesh Optimization Metrics
Mesh quality metrics of type Volume+Shape (VS) are critical in controlling local volume, skew, and aspect ratio in the context of adaptive mesh optimization. Such metrics can be viewed as being intermediate between pure shape metrics and pure volume metrics. Applications that rely on VS metrics expect the optimal mesh to display a good balance between shape and volume characteristics, while it is undesirable to be shape-dominated or volume-dominated. To achieve balance, some of the existing VS metrics contain a user parameter to adjust the balance in one direction or the other. Unfortunately this parameter is currently determined by trial and error, because it is not intuitively linked to a definition of balance. In this work we study the asymptotic properties of different VS metrics and find the ones that provide good balance between shape and volume. This asymptotic approach is motivated, presented, and tested by numerical experiments.
Mesh quality metrics of type Volume+Shape (VS) are critical in controlling local volume, skew, and aspect ratio in the context of adaptive mesh optimization. Such metrics can be viewed as being intermediate between pure shape metrics and pure volume metrics. Applications that rely on VS metrics expect the optimal mesh to display a good balance between shape and volume characteristics, while it is undesirable to be shape-dominated or volume-dominated. To achieve balance, some of the existing VS metrics contain a user parameter to adjust the balance in one direction or the other. Unfortunately this parameter is currently determined by trial and error, because it is not intuitively linked to a definition of balance. In this work we study the asymptotic properties of different VS metrics and find the ones that provide good balance between shape and volume. This asymptotic approach is motivated, presented, and tested by numerical experiments.
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