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Rudiger Westermann - One of the best experts on this subject based on the ideXlab platform.
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a real time multigrid finite Hexahedra method for elasticity simulation using cuda
Simulation Modelling Practice and Theory, 2011Co-Authors: Christian Dick, Joachim Georgii, Rudiger WestermannAbstract:Abstract We present a multigrid approach for simulating elastic deformable objects in real time on recent NVIDIA GPU architectures. To accurately simulate large deformations we consider the co-rotated strain formulation. Our method is based on a finite element discretization of the deformable object using Hexahedra. It draws upon recent work on multigrid schemes for the efficient numerical solution of partial differential equations on such discretizations. Due to the regular shape of the numerical stencil induced by the Hexahedral regime, and since we use matrix-free formulations of all multigrid steps, computations and data layout can be restructured to avoid execution divergence of parallel running threads and to enable coalescing of memory accesses into single memory transactions. This enables to effectively exploit the GPU’s parallel processing units and high memory bandwidth via the CUDA parallel programming API. We demonstrate performance gains of up to a factor of 27 and 4 compared to a highly optimized CPU implementation on a single CPU core and 8 CPU cores, respectively. For Hexahedral models consisting of as many as 269,000 elements our approach achieves physics-based simulation at 11 time steps per second.
Remacle Jean-francois - One of the best experts on this subject based on the ideXlab platform.
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A 44-element mesh of schneiders’ pyramid
'Springer Science and Business Media LLC', 2019Co-Authors: Verhetsel K., Pellerin J., Remacle Jean-francoisAbstract:This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders’ pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders’ pyramid has no Hexahedral meshes with fewer than 18 interior vertices and 17 Hexahedra, and introduce a valid mesh with 44 Hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 Hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the Hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders’pyramid is obtained by exhaustively listing the Hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 Hexahedra. The number of elements was reduced using an algorithm which locally simplifies groups of Hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid Hexahedral mesh. © Springer Nature Switzerland AG 2019
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Finding Hexahedrizations for Small Quadrangulations of the Sphere
'Association for Computing Machinery (ACM)', 2019Co-Authors: Verhetsel Kilian, Pellerin Jeanne, Remacle Jean-francoisAbstract:This paper tackles the challenging problem of constrained Hexahedral meshing. An algorithm is introduced to build combinatorial Hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small Hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of Hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general Hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial Hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of Hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a Hexahedral mesh is known. When a small Hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed Hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable Hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 Hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n Hexahedra. This very significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201
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Finding hexahedrizations for small quadrangulations of the sphere
'Association for Computing Machinery (ACM)', 2019Co-Authors: Verhetsel Kilian, Pellerin Jeanne, Remacle Jean-francoisAbstract:This paper tackles the challenging problem of constrained Hexahedral meshing. An algorithm is introduced to build combinatorial Hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small Hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of Hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general Hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial Hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of Hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a Hexahedral mesh is known. When a small Hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed Hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable Hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54, 943 such quadrangulations were meshed using no more than 72 Hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n Hexahedra. This very significantly lowers the previous upper bound of 5396 n
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There are 174 subdivisions of the hexahedron into tetrahedra
'Association for Computing Machinery (ACM)', 2018Co-Authors: Pellerin Jeanne, Verhetsel Kilian, Remacle Jean-francoisAbstract:This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all Hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent Hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into Hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling.We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of Hexahedra with 15 tetrahedra that have a strictly positive Jacobian. © 2018 Association for Computing Machinery
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A 44-element mesh of Schneiders' pyramid: Bounding the difficulty of hex-meshing problems
2018Co-Authors: Verhetsel Kilian, Pellerin Jeanne, Remacle Jean-francois, 27th International Meshing RoundtableAbstract:This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders' pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders' pyramid has no Hexahedral meshes with fewer than 18 interior vertices and 17 Hexahedra, and introduce a valid mesh with 44 Hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 Hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the Hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders'pyramid is obtained by exhaustively listing the Hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 Hexahedra. The number of elements was reduced using an algorithm which locally simplifies groups of Hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid Hexahedral mesh
Yongjie Zhang - One of the best experts on this subject based on the ideXlab platform.
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conformal adaptive Hexahedral dominant mesh generation for cfd simulation in architectural design applications
Winter Simulation Conference, 2011Co-Authors: Rui Zhang, Khee Poh Lam, Yongjie ZhangAbstract:Mesh generation is a critical and probably the most manually intensive step in CFD simulations in the architectural domain. One essential feature is the large span of dimensional scales that is encountered in design, particularly if the model aims to simulate indoor and outdoor conditions concurrently, e.g., site at the magnitude of kilometers while building elements at the magnitude of centimeters. In addressing the challenge this paper presents an approach to generate adaptive Hexahedral-dominate meshes for CFD simulations in sustainable architectural design applications. Uniform all-Hexahedral meshes and adaptive Hexahedral-dominant meshes are both generated for natural ventilation simulation of a proposed retrofit building in Philadelphia. Simulation results show that adaptive Hexahedral-dominate meshes generate very similar results of air change rate in the space due to natural ventilation, compared to all-Hexahedral meshes yet with up to 90% reduction in number of elements in the domain, hence improve computation efficiency.
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surface smoothing and quality improvement of quadrilateral Hexahedral meshes with geometric flow
Communications in Numerical Methods in Engineering, 2009Co-Authors: Yongjie Zhang, Chandrajit L BajajAbstract:This paper describes an approach to smooth the surface and improve the quality of quadrilateral/Hexahedral meshes with feature preserved using geometric flow. For quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direction, and the aspect ratio is improved with feature preserved by adjusting vertex positions in the tangent direction. For Hexahedral meshes, besides the surface vertex movement in the normal and tangent directions, interior vertices are relocated to improve the aspect ratio. Our method has the properties of noise removal, feature preservation and quality improvement of quadrilateral/Hexahedral meshes, and it is especially suitable for biomolecular meshes because the surface diffusion flow preserves sphere accurately if the initial surface is close to a sphere. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations.
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patient specific vascular nurbs modeling for isogeometric analysis of blood flow
Computer Methods in Applied Mechanics and Engineering, 2007Co-Authors: Yongjie Zhang, Chandrajit L Bajaj, Yuri Bazilevs, Samrat Goswami, Thomas J R HughesAbstract:We describe an approach to construct Hexahedral solid NURBS (Non-Uniform Rational B-Splines) meshes for patient-specific vascular geometric models from imaging data for use in isogeometric analysis. First, image processing techniques, such as contrast enhancement, filtering, classification, and segmentation, are used to improve the quality of the input imaging data. Then, luminal surfaces are extracted by isocontouring the preprocessed data, followed by the extraction of vascular skeleton via Voronoi and Delaunay diagrams. Next, the skeleton-based sweeping method is used to construct Hexahedral control meshes. Templates are designed for various branching configurations to decompose the geometry into mapped meshable patches. Each patch is then meshed using one-to-one sweeping techniques, and boundary vertices are projected to the luminal surface. Finally, Hexahedral solid NURBS are constructed and used in isogeometric analysis of blood flow. Piecewise linear Hexahedral meshes can also be obtained using this approach. Examples of patient-specific arterial models are presented.
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patient specific vascular nurbs modeling for isogeometric analysis of blood flow
Computer Methods in Applied Mechanics and Engineering, 2007Co-Authors: Yongjie Zhang, Chandrajit L Bajaj, Yuri Bazilevs, Samrat Goswami, Thomas J R HughesAbstract:We describe an approach to construct Hexahedral solid NURBS (Non-Uniform Rational B-Splines) meshes for patient-specific vascular geometric models from imaging data for use in isogeometric analysis. First, image processing techniques, such as contrast enhancement, filtering, classification, and segmentation, are used to improve the quality of the input imaging data. Then, lumenal surfaces are extracted by isocontouring the preprocessed data, followed by the extraction of vascular skeleton via Voronoi and Delaunay diagrams. Next, the skeleton-based sweeping method is used to construct Hexahedral control meshes. Templates are designed for various branching configurations to decompose the geometry into mapped meshable patches. Each patch is then meshed using one-to-one sweeping techniques, and boundary vertices are projected to the lumenal surface. Finally, Hexahedral solid NURBS are constructed and used in isogeometric analysis of blood flow. Piecewise linear Hexahedral meshes can also be obtained using this approach. Examples of patient-specific arterial models are presented.
Christian Dick - One of the best experts on this subject based on the ideXlab platform.
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A real-time multigrid finite Hexahedra method for elasticity simulation using CUDA. Simulation Modelling Practice and Theory 2011;19(2):801
2015Co-Authors: Christian Dick, Joachim GeorgiiAbstract:We present a multigrid approach for simulating elastic deformable objects in real time on recent NVIDIA GPU architectures. To accurately simulate large deformations we consider the co-rotated strain formulation. Our method is based on a finite element discretization of the deformable object using Hexahedra. It draws upon recent work on multigrid schemes for the efficient numerical solution of partial differential equations on such discretizations. Due to the regular shape of the numerical stencil induced by the Hexahedral regime, and since we use matrix-free formulations of all multigrid steps, computations and data layout can be restructured to avoid execution divergence of parallel running threads and to enable coalescing of memory accesses into single memory transactions. This enables to effectively exploit the GPU’s parallel processing units and high memory bandwidth via the CUDA parallel programming API. We demonstrate performance gains of up to a factor of 27 and 4 compared to a highly optimized CPU implementation on a single CPU core and 8 CPU cores, respectively. For Hexahedral models consisting of as many as 269,000 elements our approach achieves physics-based simulation at 11 time steps per second. Keywords: Elasticity simulation, deformable objects, finite element methods, multigrid, GPU, CUD
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a real time multigrid finite Hexahedra method for elasticity simulation using cuda
Simulation Modelling Practice and Theory, 2011Co-Authors: Christian Dick, Joachim Georgii, Rudiger WestermannAbstract:Abstract We present a multigrid approach for simulating elastic deformable objects in real time on recent NVIDIA GPU architectures. To accurately simulate large deformations we consider the co-rotated strain formulation. Our method is based on a finite element discretization of the deformable object using Hexahedra. It draws upon recent work on multigrid schemes for the efficient numerical solution of partial differential equations on such discretizations. Due to the regular shape of the numerical stencil induced by the Hexahedral regime, and since we use matrix-free formulations of all multigrid steps, computations and data layout can be restructured to avoid execution divergence of parallel running threads and to enable coalescing of memory accesses into single memory transactions. This enables to effectively exploit the GPU’s parallel processing units and high memory bandwidth via the CUDA parallel programming API. We demonstrate performance gains of up to a factor of 27 and 4 compared to a highly optimized CPU implementation on a single CPU core and 8 CPU cores, respectively. For Hexahedral models consisting of as many as 269,000 elements our approach achieves physics-based simulation at 11 time steps per second.
Joachim Georgii - One of the best experts on this subject based on the ideXlab platform.
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A real-time multigrid finite Hexahedra method for elasticity simulation using CUDA. Simulation Modelling Practice and Theory 2011;19(2):801
2015Co-Authors: Christian Dick, Joachim GeorgiiAbstract:We present a multigrid approach for simulating elastic deformable objects in real time on recent NVIDIA GPU architectures. To accurately simulate large deformations we consider the co-rotated strain formulation. Our method is based on a finite element discretization of the deformable object using Hexahedra. It draws upon recent work on multigrid schemes for the efficient numerical solution of partial differential equations on such discretizations. Due to the regular shape of the numerical stencil induced by the Hexahedral regime, and since we use matrix-free formulations of all multigrid steps, computations and data layout can be restructured to avoid execution divergence of parallel running threads and to enable coalescing of memory accesses into single memory transactions. This enables to effectively exploit the GPU’s parallel processing units and high memory bandwidth via the CUDA parallel programming API. We demonstrate performance gains of up to a factor of 27 and 4 compared to a highly optimized CPU implementation on a single CPU core and 8 CPU cores, respectively. For Hexahedral models consisting of as many as 269,000 elements our approach achieves physics-based simulation at 11 time steps per second. Keywords: Elasticity simulation, deformable objects, finite element methods, multigrid, GPU, CUD
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a real time multigrid finite Hexahedra method for elasticity simulation using cuda
Simulation Modelling Practice and Theory, 2011Co-Authors: Christian Dick, Joachim Georgii, Rudiger WestermannAbstract:Abstract We present a multigrid approach for simulating elastic deformable objects in real time on recent NVIDIA GPU architectures. To accurately simulate large deformations we consider the co-rotated strain formulation. Our method is based on a finite element discretization of the deformable object using Hexahedra. It draws upon recent work on multigrid schemes for the efficient numerical solution of partial differential equations on such discretizations. Due to the regular shape of the numerical stencil induced by the Hexahedral regime, and since we use matrix-free formulations of all multigrid steps, computations and data layout can be restructured to avoid execution divergence of parallel running threads and to enable coalescing of memory accesses into single memory transactions. This enables to effectively exploit the GPU’s parallel processing units and high memory bandwidth via the CUDA parallel programming API. We demonstrate performance gains of up to a factor of 27 and 4 compared to a highly optimized CPU implementation on a single CPU core and 8 CPU cores, respectively. For Hexahedral models consisting of as many as 269,000 elements our approach achieves physics-based simulation at 11 time steps per second.