Reliability of Computational Analysis

9802367巴布斯卡该项目将涉及以下主题：1)有限元方法中的后验误差估计。它将集中于工程兴趣值作为解中误差的误差估计、整个区域和子域中的梯度(应力)、泛函的值等，并讨论a-后验上下界。这些估计的有效性和稳健性将从理论和计算上进行研究。重点介绍工程实践中出现的线性椭圆型方程和问题，以及非线性和抛物型方程。结果也将应用于自适应过程。2)广义有限元方法。设计了一种灵活的有限元形式，允许使用特殊的形函数，并对其进行了理论分析。经典的h-和p-版本是一个特例。该方法将在两个维度上实现，并且可能稍后在三个维度上实现。该方法特别适用于求解非均质材料问题、边界层问题和振荡问题。3)复合材料问题。这是一个多尺度的问题，特别关注复合纤维材料。人们的主要兴趣将放在纤维规模上。这项研究将既是确定性的，也是随机的。由于纤维鳞片的位置统计特性，随机公式是必不可少的。这项工作将与瑞典航空研究所合作进行。4)有限元的p-版本。这项工作将侧重于从3个方面解决p版本的各个方面问题，尽管首先将在二维环境中使用这一方法。该理论将利用新的功能空间，这些空间是解决工程中出现的问题所需的。这项工作是对过去取得的成果的延续。计划中的研究解决了工程计算中具有高度重要性的问题。它将集中于可靠和准确的后验误差估计和有限元方法中的自适应过程，这是计算数据的可信度所必需的。这样，就可以避免各种事故的发生。虽然有限元方法是工程计算中的一种主要工具，但用标准方法实际上无法解决各种重要问题。因此，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。当数以百万计的纤维仅具有关于其位置的知识的统计特征时，关注于纤维鳞片的复合材料问题是此类问题之一。在这方面，需要并将解决与实验研究非常密切相关的新方法。

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.