In the field of structural dynamics, spectral finite elements are well known for their appealing approximation properties. Based on a special combination of shape functions and quadrature points, a diagonal mass matrix is obtained. More recently, the so-called optimally blended spectral element method was introduced, which further improves the accuracy but comes at the cost of a non-diagonal mass matrix. In this work, we study and compare the approximation properties of the different spectral and finite element methods. For each method, an h-version (fine meshes and low-order shape functions) as well as a p-version (coarse meshes and high-order shape functions) are considered. Special attention is paid to the influence of the quadrature rule used to compute the stiffness matrix and the element distortion on the convergence behavior. The investigations reveal the importance of a correct (full) integration of the stiffness matrix in order to achieve the theoretically predicted convergence rates. However, looking at the full spectrum, novel variants of the method that apply only a single (reduced) quadrature rule for mass and stiffness matrix show a higher accuracy.

中文翻译：

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结构动力学中的最佳混合光谱元素：选择性积分和网格失真

在结构动力学领域，谱有限元以其吸引人的近似特性而闻名。基于形状函数和正交点的特殊组合，得到对角质量矩阵。最近，引入了所谓的最佳混合光谱元素方法，进一步提高了准确性，但代价是非对角质量矩阵。在这项工作中，我们研究和比较了不同光谱和有限元方法的近似特性。对于每种方法，一个H-版本（精细网格和低阶形状函数）以及p-版本（粗网格和高阶形状函数）被考虑。特别注意用于计算刚度矩阵的求积法则和单元畸变对收敛行为的影响。研究揭示了刚度矩阵的正确（完全）积分的重要性，以实现理论上预测的收敛速度。然而，从全谱来看，仅对质量和刚度矩阵应用单个（简化）正交规则的方法的新变体显示出更高的精度。