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。在行业中，商业代码被广泛使用。它们是灵活的，可以调整以解决特定的感兴趣问题。解决数十万或数十万未知数的大问题变得常规。计算分析的目的是在成本和时间的限制内预测工程和物理现实的行为。如今，人类和时间成本比计算机成本更重要。这种趋势将持续到未来。计算结果与现实之间的介绍与两个因素有关，即问题的数学表述和数值解决方案的准确性。必须在分析目的的背景下理解准确性。不适当的规范中的一个小错误并不一定意味着计算结果可用于实际目的。通过不同的方法分析相同的工程问题有时可能导致差异显着的结果。这可能是由各种因素引起的，例如，使用了数学公式中的不同模型，或者数值解决方案无法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，可能会发生这种情况。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，但工程学的决策仍由人类而不是计算机做出。有必要意识到计算机始终提供数据，颜色图或电影，正确或不正确。因此，了解建模的基本数学背景至关重要。我的研究的主要目的是为计算机提供的数字是否可靠且值得信赖作为重要的工程决策的基础，为创造性评估做出重大贡献。