Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis
Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis
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基于复阶导数近似的有限应变下切线模量的鲁棒数值计算及其在局部化分析中的应用
DOI：
10.1016/j.cma.2013.11.005
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发表时间：
2014
影响因子：
7.2
通讯作者：
J. Schröder
中科院分区：
文献类型：
--
作者：
Masato Tanaka;M. Fujikawa;D. Balzani;J. Schröder
提出了一种非常稳健和高效的有限应变下材料和空间切线模数的数值逼近方法，该方法可以很容易地在标准有限元软件中实现。该方法基于复数阶导数近似(CSDA)。CSDA被证明是二阶精度的，它不受浮点运算中的舍入误差的影响，这些误差限制了其他经典数值方法的精度，例如有限差分近似。因此，当摄动值选择接近机器精度时，CSDA可以提供与解析解极其相似的近似值。给出了用CSDA进行应力计算的切线模强稳健数值逼近的实现细节，并通过有限变形的典型算例说明了它们的性能。除此之外，我们还将重点放在物质不稳定性的测定上。因此，执行伴随的局部化分析，其中声学张量直接从模数的近似计算。结果表明，经典数值近似对于微扰值是敏感的，因此只要稍微改变微扰，就可以人为地探测到材料的不稳定性。另一方面，CSDA方法在很宽的摄动值范围内提供了高精度和稳健的近似，从而可以精确地检测材料的不稳定性。
An extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard FEM software. This method is based on the complex-step derivative approximation (CSDA) approach. The CSDA is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e.g. finite difference approximation. Therefore, the CSDA can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision. Implementation details of the robust numerical approximation of tangent moduli from stress calculations using the CSDA are given and their performance is illustrated through representative examples involving finite deformations. In addition to that, we focus on the determination of material instabilities. Therefore, an accompanying localization analysis is performed, where the acoustic tensor is directly computed from the approximation of the moduli. It is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation. On the other hand, the CSDA approach provides high-accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely.
DOI：
10.1115/1.2979872
发表时间：
2008-12
期刊：
Journal of biomechanical engineering
影响因子：
--
作者：
Sun W;Chaikof EL;Levenston ME
通讯作者：
Levenston ME