Reliability in finite element method for higher dimensional space problems

高维空间问题的有限元方法的可靠性

• 批准号: 229809-2011
• 负责人: Liu, Liping
• 金额: $ 0.73万
• 依托单位: Lakehead University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635665
• 关键词: Reliability; finite; element; method; higher

Mathematical modeling plays an essential role in science and engineering. Costly and time-consuming experiments are replaced by computational analysis. In industry, commercial codes are widely used. They are flexible and can be adjusted for solving specific problems of interest. Solving large problems with tens or hundreds of thousands of unknowns becomes routine. The objective of computational analysis is to predict the behavior of the engineering and physical reality usually within the constraints of cost and time. Today, human and time costs are more important than computer costs. This trend will continue into the future.Agreement between computational results and reality is related to two factors, namely the mathematical formulation of the problems and the accuracy of the numerical solution. The accuracy has to be understood in the context of the aim of the analysis. A small error in an inappropriate norm does not necessarily mean that the computed results are usable for practical purposes. Analyzing the same engineering problem by different methods could sometimes lead to results that differ significantly. This could be caused by various factors, e.g., different models in mathematical formulations were used or the numerical solution does not approximate the data of interest well. This can happen especially when various modern adaptive codes with a posteriori error estimation are used. Obviously, to understand the reasons for such discrepancies is very important. Decisions in engineering are still made by humans and not by computers, although computers are the main tools. It is necessary to realize that the computer always provides data, color graphs or movies, correct or incorrect. Hence, to understand the basic mathematical background of the modeling is of major importance. The main objective of my research is to contribute significantly to the creative assessment of whether the numbers provided by computers are reliable and trustworthy as the basis for crucial engineering decisions.

数学建模在科学和工程中起着至关重要的作用。昂贵且耗时的实验被计算分析所取代。在工业中，商业代码被广泛使用。它们很灵活，可以进行调整以解决感兴趣的特定问题。解决具有数万或数十万未知数的大问题已成为惯例。计算分析的目标是通常在成本和时间的限制内预测工程和物理现实的行为。如今，人力和时间成本比计算机成本更重要。这种趋势未来仍将持续。计算结果与现实的一致性与两个因素有关，即问题的数学表述和数值解的准确性。必须在分析目标的背景下理解准确性。不适当规范中的小误差并不一定意味着计算结果可用于实际目的。用不同的方法分析同一工程问题有时可能会导致截然不同的结果。这可能是由多种因素引起的，例如，使用了不同的数学公式模型或数值解不能很好地近似感兴趣的数据。当使用具有后验误差估计的各种现代自适应代码时，尤其会发生这种情况。显然，了解这种差异的原因非常重要。尽管计算机是主要工具，但工程决策仍然由人类而不是计算机做出。有必要认识到计算机总是提供正确或不正确的数据、彩色图表或电影。因此，了解建模的基本数学背景至关重要。我研究的主要目标是为创造性评估做出重大贡献，评估计算机提供的数字是否可靠且值得信赖，作为关键工程决策的基础。