Nonlinear energy functionals appearing in the calculus of variations can be discretized by the finite element (FE) method and formulated as a sum of energy contributions from local elements. A fast evaluation of energy functionals containing the first order gradient terms is a central part of this contribution. We describe a vectorized implementation using the simplest linear nodal (P1) elements in which all energy contributions are evaluated all at once without the loop over triangular or tetrahedral elements. Furthermore, in connection to the first-order optimization methods, the discrete gradient of energy functional is assembled in a way that the gradient components are evaluated over all degrees of freedom all at once. The key ingredient is the vectorization of exact or approximate energy gradients over nodal patches. It leads to a time-efficient implementation at higher memory-cost. Provided codes in MATLAB related to 2D/3D hyperelasticity and 2D p-Laplacian problem are available for download and structured in a way it can be easily extended to other types of vector or scalar forms of energies.

翻译：变化微量中出现的非线性能源功能可以通过有限元素(FE)方法分解,并按当地元素的能量贡献总和来拟订。快速评估含有第一顺序梯度术语的能源功能是这一贡献的一个核心部分。我们用简单最线性交点(P1)元素描述矢量化实施,其中所有能量贡献都一次性评价,而没有三角或四面元素的循环。此外,在第一阶优化方法方面,能源功能的离散梯度的组装方式是,梯度组件一次在所有自由度上进行评估,关键成分是节点段上精确或近似能量梯度的矢量化,导致在更高的记忆成本下实现时间效率化。MATLAB中提供的与2D/3D超弹性和2D p-斜方问题有关的代码可供下载,其结构可以很容易扩展至其他类型的矢量或量形式能源。