The Diamond Laplace for Polygonal and Polyhedral Meshes
The Diamond Laplace for Polygonal and Polyhedral Meshes
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用于多边形和多面体网格的钻石拉普拉斯
DOI：
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
M. Alexa
中科院分区：
文献类型：
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作者：
A. Bunge;M. Botsch;M. Alexa
本文介绍了一种离散梯度算子的构造方法，它可以直接应用于任意多边形曲面和多面体体网格。其主要思想是将定义在网格顶点处的函数的梯度与菱形相关联：由对偶边及其相应的原始元素（表面网格的边和体积网格的面）所跨越的区域。我们把取梯度的散度所得到的算子称为钻石拉普拉斯算子。用于构造的附加顶点被表示为原始顶点的仿射组合，使得拉普拉斯算子从顶点处的值映射到顶点处的值，如在几何处理应用中常见的。构造是局部的，对于所有类型的网格完全相同，并且导致具有线性精度的对称负定算子。我们表明，钻石拉普拉斯算子的精度是类似或更好的相比，其他离散化。更大的通用性和一般良好的行为是以增加非零系数的数量为代价的，非零系数取决于网格单元的阶数。
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.
DOI：
10.1145/3243651
发表时间：
2018-05
期刊：
ACM Transactions on Graphics (TOG)
影响因子：
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作者：
Nicholas Sharp;Yousuf Soliman;Keenan Crane
通讯作者：
Nicholas Sharp;Yousuf Soliman;Keenan Crane
DOI：
10.1145/3313797
发表时间：
2018-04
期刊：
ACM Transactions on Graphics (TOG)
影响因子：
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作者：
T. Schneider;Jérémie Dumas;Xifeng Gao;M. Botsch;Daniele Panozzo;D. Zorin
通讯作者：
T. Schneider;Jérémie Dumas;Xifeng Gao;M. Botsch;Daniele Panozzo;D. Zorin