Reliability of Computational Analysis

该项目将解决以下问题：1)有限元法中的后验误差估计。它将着重于工程兴趣值的误差估计，如解决方案中的误差，整个域以及子域的梯度（应力），泛函值等。上后验界和下后验界将被处理。这些估计的有效性和稳健性将在理论上和计算上进行研究。本课程将着重于工程实践中出现的线性椭圆型方程和问题，并对非线性和抛物型方程和问题进行研究。结果也将应用于自适应过程。2)广义有限元法。设计了一种允许使用特殊形状函数的柔性有限元形式，并进行了理论分析。经典的h-和p-版本是一个特例。该方法将在二维中实现，以后可能还会在三维中实现。该方法可用于求解非均质材料、边界层和振荡性质的问题。3)复合材料问题。这是一个多尺度问题，特别关注复合纤维材料。主要的兴趣将集中在纤维的规模上。研究将是确定性和随机的。随机公式是必不可少的，因为纤维尺度的位置具有统计特性。这项工作将与瑞典航空研究所合作进行。这项工作将侧重于解决三维p-版本的各个方面，尽管首先该方法将用于二维设置。该理论将利用解决工程中出现的问题所需的新功能空间。这项工作是对过去取得的成果的延续。计划的研究解决了工程计算中具有高度重要性的问题。它将侧重于可靠和准确的后验误差估计和自适应程序在有限元方法中，这是对计算数据的信心至关重要。这样可以避免各种事故。虽然有限元法是工程计算的主要工具，但许多重要问题实际上是标准方法无法解决的。因此，本文将设计一种新的广义柔性有限元方法，以提高该方法在求解异常困难问题时的有效性。以纤维尺度为重点的复合材料的问题是其中一种类型的问题，当数以百万计的纤维存在时，它们的位置知识仅具有统计特征。在这里，需要并将解决与实验研究非常紧密的新方法。

9802367 Babuska The project will address the following topics: 1) A posteriori error estimation in the finite element method. It will focus on the error estimation of the values of engineering interest as error in the solution, the gradients (stresses) in the entire domain as well as subdomains, the values of the functionals, etc. The upper and lower a- posteriori bounds will be addressed. The effectivity and robustness of these estimates will be theoretically and computationally investigated. The linear elliptic equations and problems occurring in engineering practice will be focussed on and also the nonlinear and parabolic ones will be investigated. The results will also be applied in the adaptive procedure. 2) The generalized finite element method. A flexible form of FEM allowing one to use special shape functions will be designed and theoretically analyzed. The classical h- and p- version is a special case. The method will be implemented in two dimensions and possibly later, three dimensions as well. This method can typically be applied for solving problems of heterogeneous material, solutions with boundary layers and oscillatory character. 3) The problem of composites. This is a multi-scale problem with particular focus on composite fibrous materials. The main interest will be on a fiber scale. The study will be both deterministic and stochastic. The stochastic formulation is essential because the position of the fiber scale statistical character. The work will be performed in collaboration with the Aeronautics Institute of Sweden. 4) The p-version of FEM. This work will be focused on resolving various aspects of the p-version in 3 dimensions although first the methodology will be used on the two-dimensional setting. The theory will utilize new functional spaces which are needed for addressing the problems occurring in engineering. The work is a continuation of the results of results obtained in the past. The planned research addresses problems with a high level of importance in the engineering computations. It will focus on the reliable and accurate a-posteriori error estimations and adaptive procedures in the Finite Element Method, which are essential for confidence in the computed data. In this way, various accidents could be avoided. Although the finite element method is a major tool in engineering computations, various important problems are practically unsolvable by the standard approach. Hence a new generalized flexible finite element method will be designed with the goal to increase the effectivity of the method when solving unusually difficult problems. The problem of the composite material with the focus on the fiber scale is one of these types of problems when millions of fibers are present with only statistical character in regard to knowledge of their position. Here new approaches which are very tight to the experimental studies are needed and will be addressed.