A Simple Discretization of the Vector Dirichlet Energy
A Simple Discretization of the Vector Dirichlet Energy
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DOI：
10.1111/cgf.14070
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发表时间：
2020-08
影响因子：
2.5
通讯作者：
Oded Stein;M. Wardetzky;Alec Jacobson;E. Grinspun
中科院分区：
文献类型：
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作者：
Oded Stein;M. Wardetzky;Alec Jacobson;E. Grinspun
我们使用Crouzeix-Raviart有限元对3D三角形网格的协变导数向量Dirichlet能量进行了简单而简洁的离散。离散化是基于线性不连续的伽辽金元素，并且易于实现，而不会影响质量：每个网格边缘有两个自由度，并且稀疏狄利克雷能量矩阵可以使用一个简短的公式在所有三角形上构建，该公式仅取决于边缘长度，让人想起标量余切拉普拉斯算子。我们的矢量Dirichlet能量离散化可以用于各种应用，如Killing场的计算，矢量的并行传输和光滑矢量场设计。实验表明收敛性和适用性的应用类似于其他离散化的向量Dirichlet能量。
We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix‐Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.