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.
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...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 convergence rate independent of mesh size and level are established.展开更多
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 ellipt...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.展开更多
This paper provides a proof for the uniform convergence rate （independently of the number of mesh levels） for the nonnested V-cycle multigrid method for nonsymmetric and indefinite second-order elliptic problems.
Mechatronic product development is a complex and multidisciplinary field that encompasses various domains, including, among others, mechanical engineering, electrical engineering, control theory and software engineeri...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.展开更多
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...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.展开更多
In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discre...In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained.The stability and convergence of these numerical schemes are discussed.Next,a V-cycle multigrid method is proposed to solve the resulting linear systems.The convergence of the multigrid method is investigated.Finally,some numerical examples are given for verification of our theoretical analysis.展开更多
基金The rescarch was supported by the Doctoral Point Foundation of chinese Universities and NSF
文摘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.
基金Supported by NSF of China(10971203)Supported by the NSF of the education Department of Henan Province (2009A110017)
文摘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 convergence rate independent of mesh size and level are established.
基金The first author was supported in part by the National Natural Science Foundation of China(Grant No.10401016)by the National Basic Research Program(Grant No.2005CB321701)+1 种基金The second author was supported in part by the National Natural Science Foundation of China(Grant No.10025102)by China MOS(Grant No.G1999032802).
文摘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.
文摘This paper provides a proof for the uniform convergence rate （independently of the number of mesh levels） for the nonnested V-cycle multigrid method for nonsymmetric and indefinite second-order elliptic problems.
文摘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.
基金Projects(2006AA06Z105, 2007AA06Z134) supported by the National High-Tech Research and Development Program of ChinaProjects(2007, 2008) supported by China Scholarship Council (CSC)
文摘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.
基金This research is supported by the National Center for Mathematics and Interdisciplinary Sciences,CAS,and by the National Natural Science Foundation of China(Grant No.11371357).
文摘In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained.The stability and convergence of these numerical schemes are discussed.Next,a V-cycle multigrid method is proposed to solve the resulting linear systems.The convergence of the multigrid method is investigated.Finally,some numerical examples are given for verification of our theoretical analysis.
