Variational methods for mass scaling

提出的研究项目的主题是在显式动态有限元分析的背景下开发用于选择性质量定标的新的变分方法，目的是为了实现现代工业应用的大幅加速。我们寻求发展和推进的基本创新有三个：第一，将开发一种直接和稀疏地构造一致逆质量矩阵的方法。这样的矩阵允许从全局力向量计算节点加速度，即不需要成本密集的矩阵求逆。初步工作表明，在这些逆质量矩阵上应用选择性质量标度，可以将临界时间步长增加两倍，同时使计算结果的精度保持在与对角化质量矩阵接近的水平。在这里，还需要进一步努力将该方法扩展到更广泛的实体有限元，并通过网格色散和特征值分析来评估新获得的质量矩阵的精度、关键时间步长和稳定性。其次，发展了一种有效的处理几何非线性问题、单向接触、多点约束和规定的速度边界条件以及新的质量矩阵。第三，将变分选择质量标度法推广到实心壳单元。到目前为止，只有选择质量标度的代数方法被应用于这一族元素。

The subject of the proposed research project is the development of new variational methods for selective mass scaling in the context of explicit dynamic finite element analysis with the aim to achieve substantial speed-up for modern industrial applications. There are three fundamental innovations which we seek to develop and advance: First, a method for direct and sparse construction of consistent inverse mass matrices will be developed. Such matrices allow trivial computation of the nodal accelerations from the global force vector, i. e. no cost-intensive matrix inversion is required. Preliminary work has shown that application of selective mass scaling onto these inverse mass matrices allows to increase the critical time step by a factor of two while keeping accuracy of thecomputational results close to the ones obtained with a diagonalized mass matrix. Here, further effort is needed to extend the method for a broader range of solid finite elements and to assess accuracy, critical time step and stability of the newly obtained mass matrices via grid dispersion and eigenvalue analyses. Second, an efficient treatment of geometrically non-linear problems, unilateral contact, multi-point constraints and prescribed velocity boundary conditions together with the new mass matrices is developed. Third, variational selective mass scaling methods are extended to solid shell elements. So far only algebraic methods for selective mass scaling have been applied to this family of elements.