Reliability of Computational Analysis

小行星9802367 该项目将解决以下主题：1）有限元方法中的后验误差估计。 它将侧重 工程利益值作为误差的误差估计 解，整个域中的梯度（应力）以及 子域，值的泛函等的上，下后验界将得到解决。 的有效性和稳健性， 这些估计将在理论上和计算上进行研究。 线性椭圆型方程及工程中的一些问题 实践将集中在，也非线性和抛物线的将 追究 研究结果也将应用于自适应过程。 2)广义有限元法。 有限元的一种灵活形式 允许使用特殊形状功能的设计， 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，并且可能稍后， 也是三维的。 该方法通常可用于 非均质材料问题的求解，有边界的求解 层和振荡特征。3)复合材料的问题。 这是一 多尺度问题，特别关注复合纤维材料。 主要的兴趣将是在纤维规模。 该研究将同时 确定性和随机性。随机公式是必要的 因为纤维尺度的位置具有统计特性。 这项工作将 将与瑞典航空研究所合作进行。四、 p版本的FEM。 这项工作的重点将是解决各种 在3个方面的p版本，虽然首先是方法， 将用于二维设置。 该理论将利用新的 解决出现的问题所需的功能空间 在工程学方面。 这项工作是成果的延续 过去获得的。 计划中的研究涉及高度重要的问题 在工程计算中。 它将专注于可靠和准确的 后验误差估计和自适应程序 元素方法，这是必不可少的计算数据的信心。 这样一来，各种意外就可以避免了。 虽然有限 单元法是工程计算的主要工具， 一些重要的问题实际上是无法用标准方法解决的。 因此，将设计一种新的广义柔性有限元方法 其目的是在求解时提高方法的有效性 异常困难的问题。复合材料的问题在于， 对纤维尺度的关注是这些类型的问题之一， 数以百万计的纤维只具有统计特征， 了解他们的立场。 这里的新方法非常严格， 实验研究是必要的，将得到处理。

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.