Variational methods for mass scaling

拟议的研究项目的主题是在显式动态有限元分析的背景下发展新的变分方法，以实现现代工业应用的实质性加速。有三个基本的创新，我们寻求发展和推进：首先，一种方法的直接和稀疏构造一致的逆质量矩阵将被开发。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反演。初步工作表明，在这些反质量矩阵上应用选择性质量标度可以使临界时间步长增加两倍，同时保持计算结果的精度接近使用对角化质量矩阵获得的结果。在此，需要进一步努力将该方法扩展到更广泛的实体有限元范围，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准确性、临界时间步长和稳定性。其次，对几何非线性问题、单边接触、多点约束和规定速度边界条件以及新的质量矩阵进行了有效的处理。第三，将变分选择质量标度方法推广到实体壳单元。到目前为止，只有选择性质量标度的代数方法被应用于这类元素。

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.