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Uniform convergence of multigrid v-cycle on adaptively refined finite element meshes for second order elliptic problems

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Science China Mathematics 2006年第10期49卷 1405-1429页

作者： WU Haijun CHEN Zhiming Department of Mathematics Nanjing University Nanjing 210093 China Institute of Computational Mathematics Chinese Academy of Sciences Beijing 100080 China.

In this paper we prove the uniform convergence of the standard multigrid v-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.

关键词： Multigrid v-cycle algorithm, adaptive finite element meshes, local relaxation, Scott-Zhang interpolation.

The v-cycle Multigrid Method for a Hermite-type Rectangular Element

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Chinese Quarterly Journal of Mathematics 2010年第1期25卷 140-145页

作者： ZHAO Yan-min SHI Dong-yang School o.f Mathematical Sciences Xuchang University Xuehang 461000 China Department of Mathematics Zhengzhou University Zhengzhou 450052 China

In this paper, a v-cycle multigrid method is presented for a Hermite rectangular element. By defining proper mesh-dependent inner product and transfer operator, we obtain its convergence property and the uniform conve... 详细信息

关键词： v-cycle Hermite-type multigrid method transfer operator

AN OPTIMAL v-cycle MULTIGRID METHOD FOR CONFORMING AND NONCONFORMING PLATE ELEMENTS

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1997年第1期6卷 115-119页

作者：许学军李立康

In this paper, an optimal v-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.

关键词： v-cycle multigrid method conforming plate elements nonconforming plate elements.

Leveraging Robust Artificial Intelligence for Mechatronic Product Development—A Literature Review

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International Journal of Intelligence Science 2024年第1期14卷 1-21页

作者： Alexander Nüßgen René Degen Marcus Irmer Fabian Richter Cecilia Boström Margot Ruschitzka CAD CAM Center Cologne Institute of Automotive Engineering Cologne (IFK) Faculty of Automotive Systems and Production Cologne University of Applied Science Cologne Germany Division of Electricity Department of Electrical Engineering Uppsala University Uppsala Sweden

Mechatronic product development is a complex and multidisciplinary field that encompasses various domains, including, among others, mechanical engineering, electrical engineering, control theory and software engineering. The integration of artificial intelligence technologies is revolutionizing this domain, offering opportunities to enhance design processes, optimize performance, and leverage vast amounts of knowledge. However, human expertise remains essential in contextualizing information, considering trade-offs, and ensuring ethical and societal implications are taken into account. This paper therefore explores the existing literature regarding the application of artificial intelligence as a comprehensive database, decision support system, and modeling tool in mechatronic product development. It analyzes the benefits of artificial intelligence in enabling domain linking, replacing human expert knowledge, improving prediction quality, and enhancing intelligent control systems. For this purpose, a consideration of the v-cycle takes place, a standard in mechatronic product development. Along this, an initial assessment of the AI potential is shown and important categories of AI support are formed. This is followed by an examination of the literature with regard to these aspects. As a result, the integration of artificial intelligence in mechatronic product development opens new possibilities and transforms the way innovative mechatronic systems are conceived, designed, and deployed. However, the approaches are only taking place selectively, and a holistic view of the development processes and the potential for robust and context-sensitive artificial intelligence along them is still needed.

关键词： Artificial Intelligence Mechatronic Product Development Knowledge Management Data Analysis Optimization Human Experts Decision-Making Processes v-cycle

3-D direct current resistivity forward modeling by adaptive multigrid finite element method

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维普期刊数据库

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Journal of Central South University 2010年第3期17卷 587-592页

作者：汤井田王飞燕任政勇郭荣文 School of Info-physics and Geomatics Engineering Central South University Institute of Geophysics ETH Zurich-Swiss Federal Institute of Technology School of Earth and Ocean Sciences University of Victoria

Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, v-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.

关键词： adaptive multigrid a-posteriori error estimator unstructured mesh v-cycle finite element method

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先进ECU开发技术及其应用

先进ECU开发技术及其应用