Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity
Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity
复制标题
误差控制自适应目标导向建模和弹性有限元近似
DOI：
10.1016/j.cma.2006.10.032
复制
发表时间：
2007
影响因子：
7.2
通讯作者：
S. Ohnimus
中科院分区：
文献类型：
--
作者：
E. Stein;M. Rüter;S. Ohnimus
在我们的早期工作[E。Stein，S. Ohnimus，有限元法中的耦合模型和解自适应性，Comput。方法应用机械工程150（1997）327-350; E. Stein，S. Ohnimus，各向异性离散化和模型误差估计在固体力学的局部诺依曼问题，计算机。方法应用机械工程176（1999）363-385; E. Stein，M. Rüter，S. Ohnimus，固体和结构的自适应有限元分析和建模。发现、问题和趋势，国际J. Numer。方法Engrg. 60（2004）103-138.]，基于残差的组合模型和有限元离散误差估计作为自适应性的基础，依赖于诺依曼问题的解决方案的元素水平。特别是，这些估计控制的误差，同时扩大模型的层次结构从1D到2D-和从2D到3D-弹性理论（全球）能量规范。在本文中，我们扩展这两种类型的估计目标为导向的后验误差估计的框架内的非线性结构力学。结构模型自适应性的特点是尺寸，一般的本构方程，特别是相关的子域，从而测试和解决方案的空间考虑的过程相关的模型是不同的。这具有重要的后果，即必须将较粗模型“m”的有限元的运动和静态节点量与精细模型“m+1”的有限元的运动和静态节点量进行一致的换算。因此，这类延拓算子进行了详细的讨论。此外，对偶技术用于模型和离散化误差估计，使用系统和负载相关的感兴趣的量。在本文中，我们还考虑了原始问题和对偶问题的不同有限元网格。最后，一些数值例子展示了预期的效果，实现了结构力学的验证甚至确认，即接近计算力学中实现可靠性和效率的新范式。
In our earlier work [E. Stein, S. Ohnimus, Coupled model- and solution-adaptivity in the finite-element method, Comput. Methods Appl. Mech. Engrg. 150 (1997) 327–350; E. Stein, S. Ohnimus, Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Engrg. 176 (1999) 363–385; E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends, Int. J. Numer. Methods Engrg. 60 (2004) 103–138.], residual-based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of Neumann problems on the element level. In particular, these estimators control the errors obtained while expanding the model hierarchically from 1D- to 2D- and from 2D- to 3D-elastic theories in the (global) energy norm. In this paper, we extend both types of estimates to goal-oriented a posteriori error estimation within the framework of nonlinear structural mechanics. Structural model adaptivity is characterized by the fact that the dimensions, in general the constitutive equations and especially the related subdomains—and thus the test and solution spaces of the process-dependent models considered—are different. This has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model “m” to those of the refined model “m+1” are necessary. Therefore, such prolongation operators are discussed in detail. Furthermore, duality techniques are used for the model and discretization error estimates, using system and load-dependent quantities of interest. In this paper, we also consider different finite element meshes for the primal and the dual problem. Finally, some numerical examples demonstrate the expected effects, realizing verification and even validation in structural mechanics, i.e. approaching a new paradigm in Computational Mechanics for achieving reliability and efficiency.