Displacement control in time‐adaptive non‐linear finite‐element analysis
Displacement control in time‐adaptive non‐linear finite‐element analysis
复制标题
时间自适应非线性有限元分析中的位移控制
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Ahmad
中科院分区：
文献类型：
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作者：
S. Hartmann;K. Quint;Ahmad
一方面，位移控制过程经常应用于有限元法的计算力学中，另一方面，对于这种特殊情况的研究和介绍存在一些不足。在这篇文章中，我们集中我们的理论考虑准静态问题的演变型本构方程。在这种情况下，目前已知的是，“Newton-Raphson方法”中的全局和局部迭代的求解过程与线方法有关，其中通过有限元方法进行空间离散化后得到微分代数方程组。这可以通过后向欧拉方法来解决，或者更合适地使用时间自适应，严格精确，对角隐式Runge-Kutta方法与多级牛顿方法相结合。在这方面，目前应用的隐式非线性有限元分析的理论基础被扩展到位移控制过程。在这篇文章中，考虑到新的DAE‐方法，使用拉格朗日法和罚函数法计算反作用力特别有趣。因此，一个重要的目标在于一致的符号，使已知和未知变量的依赖性以及从局部到全局的过渡变得明显。这些研究对于有限元方法中的微观-宏观均匀化技术以及新的全局时空自适应积分方法的发展至关重要。
On the one hand, displacement controlled processes are frequently applied in Computational Mechanics using the finite‐element method, on the other hand, there are some shortcomings regarding the study and the presentation of this particular case. In this article, we focus our theoretical considerations on quasi‐static problems with constitutive equations of evolutionary type. In this case, it is currently known that the solution procedure of global and local iterations within the “Newton‐Raphson method” is related to the method of lines, where one arrives at a system of differential‐algebraic equations after the spatial discretization by means of the finite element method. This could be solved by means of the Backward‐Euler method or more appropriately using time‐adaptive, stiffly accurate, diagonally implicit Runge‐Kutta methods in combination with the Multilevel‐Newton method. In this respect the theoretical basis of currently applied implicit non‐linear finite element analyses is extended to displacement controlled processes. In this article, the calculation of the reaction forces using the method of Lagrange multipliers and the penalty method are of special interest in view of the new DAE‐approach. Accordingly, an important objective lies in a consistent notation so that the dependence of known and unknown variables as well as the transition from local to global level becomes obvious. These investigations are of utmost importance for micro‐macro homogenization techniques in FE$^2$ approaches and the development of new global time and space‐adaptive integration methods.