The Diamond Laplace for Polygonal and Polyhedral Meshes
We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge togeth...
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| Vydáno v: | Computer graphics forum Ročník 40; číslo 5; s. 217 - 230 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford
Blackwell Publishing Ltd
01.08.2021
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element — an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non‐zero coefficients that depends on the degree of the mesh elements. |
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| AbstractList | We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element — an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient
Diamond
Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non‐zero coefficients that depends on the degree of the mesh elements. We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element — an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non‐zero coefficients that depends on the degree of the mesh elements. |
| Author | Bunge, A. Botsch, M. Alexa, M. |
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| Cites_doi | 10.1051/m2an:2005047 10.1016/j.finel.2007.05.009 10.1111/j.1467-8659.2008.01292.x 10.1007/BF02187870 10.1142/S0218202514400065 10.1111/cgf.12709 10.1016/j.jcp.2009.05.002 10.1145/3414685.3417776 10.1111/j.1467-8659.2007.01058.x 10.21236/ADA081688 10.1007/978-3-662-05105-4_2 10.1145/3072959.3073666 10.1145/3313797 10.1145/2010324.1964997 10.1142/S0218202505000832 10.1201/9781315153452 10.1145/2516971.2516977 10.1093/imanum/drr046 10.1145/1391989.1391995 10.1137/120874746 10.1080/10586458.1993.10504266 10.1145/311535.311576 10.1007/s10483-009-1003-3 10.1111/cgf.13931 10.1006/jcph.2000.6466 10.1515/cmam-2013-0011 10.1137/100786046 10.1111/j.1467-8659.2008.01293.x 10.1145/3243651 10.1142/S0218202514400041 10.1111/cgf.14068 |
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| SubjectTerms | Apexes CCS Concepts Computer graphics Computing methodologies → Mesh models DDFV Diamonds Discrete Differential Geometry Discrete Laplace Operator Divergence Mathematics of computing → Discretization Operators (mathematics) Polygons |
| Title | The Diamond Laplace for Polygonal and Polyhedral Meshes |
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