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A boundary-precise method of numerical integration over implicitly defined regions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 468卷

作者： Zhang, Xuejing Ma, Huanhuan Wu, Xinyu Deng, Jiansong Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China Hefei Univ Sch Artificial Intelligence & Big Data Hefei 230601 Anhui Peoples R China

In this paper, we introduce a boundary-precise method of numerical integration over implicitly defined regions using the idea of isogeometry. This method ensures precise boundary representations during the integration process by establishing proper local parameterizations near these boundaries based on the implicit representations. We establish an adaptive algorithm for the method, and provide two examples for integration over different types of regions. We also calculate the error while integrating by this method and provide detailed error estimates. When solving partial differential equations (PDEs) using weighted extended B-spline (web) method, the augmented matrix in the linear system of Galerkin framework is derived through integration, where this boundary-precise method can be applied. Three different types of examples are displayed and the results demonstrate the convergence of error and the stability of order.

关键词： Numerical integration Error estimates web method NURBS Isogeometric analysis Implicit representations

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Solving higher order PDEs with isogeometric analysis on implicit domains using weighted extended THB-splines

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COMPUTER AIDED GEOMETRIC DESIGN 2019年 71卷 202-219页

作者： Qarariyah, Ammar Yang, Tianhui Deng, Jiansong Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China

We study the numerical solution for higher order partial differential equations defined on implicit domains. We represent isogeometric analysis on implicit domains and construct weighted extended truncated hierarchical B-splines. Specifically, the spline basis functions are classified, according to the cells occupied by their support inside the domain, using a level by level classification mechanism. A hierarchical extension is then proposed to connect basis functions with small support in the computational domain to more stable basis inside the domain. The basis functions are formulated to acquire local adaptive refinement and preserve stability to release the full approximation power. A numerical illustration is performed to solve the Biharmonic equation on a number of implicit domains with holes and reentrant corners. The error results and condition numbers of the numerical examples reflect the potential of the approach. (C) 2019 Published by Elsevier B.V.

关键词： isogeometric analysis web method Truncated hierarchical B-splines Higher order PDEs Implicit domains

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Isogeometric Analysis on Implicit Domains:Approximation,Stability and Error Estimates

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Communications in Mathematics and Statistics 2024年第4期12卷 633-658页

作者： Fang Deng Tianhui Yang Jingjing Liu Jiansong Deng School of Mathematics and Statistics North China University of Water Resources and Electric PowerZhengzhou 450045People’s Republic of China Department of Mathematics Harbin Institute of Technology WeihaiWeihai 264209People’s Republic of China School of Mathematics and Statistics Hefei Normal UniversityHefei 230061People’s Republic of China School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026People’s Republic of China

The gap between finite element analysis and computer-aided design derives the development of isogeometric analysis(IGA),which uses the same representation in the geometry and the ***,the parameterization in IGA is *** extended B-splines(web)method replaces grid generation and parameterization with weight function construction(R-function or distance function).By using implicit spline representation,isogeometric analysis on implicit domains(IGAID)adopts the merits of the“isoparametric”in IGA and“weight function generation”in *** the theoretical properties have not been fully studied *** this paper,we study the theoretical aspects of IGAID using tensor-product *** the approximation and stability properties of IGAID are *** setting appropriate constraints on the weight function,we can derive the optimal approximation order and *** examples show the effectiveness of the approach and validate the theoretical results.

关键词： Isogeometric analysis web method Implicit domain Weight function Error estimates

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Isogeometric analysis on implicit domains using weighted extended PHT-splines

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 350卷 353-371页

作者： Qarariyah, Ammar Deng, Fang Yang, Tianhui Liu, Yuan Deng, Jiansong Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China North China Univ Water Resources & Elect Power Sch Math & Stat Zhengzhou 450045 Henan Peoples R China Hefei Univ Technol Sch Math Hefei 230009 Anhui Peoples R China

The urging quest for diminishing the gap between finite element methods and computer aided design is growing every day. Isogeometric analysis is driving that search a step forward in contrast with traditional finite element methods. One of the areas that need more investigation is using isogeometric analysis on domains that contain singularities. In this case, the parameterization process is occasionally non-trivial, and some of the inherited basis functions are not well defined producing rough solutions. Moreover, NURBS basis functions are not the best choice to perform local refinement since adaptivity is indispensable. In this study, a new approach that serves as a generalization for some of the ideas in both the isogeometric analysis and the web method is introduced. Isogeometric analysis on implicit domains replaces mesh generation with implicit spline weight function generation and construct weighted extended PHT-splines for analysis. The geometry and the analysis both are built starting from the same function space avoiding parameterization. The PHT-splines are adaptive and allow local refinement naturally leading to a higher accuracy while dealing with singularities. An illustration with numerical examples demonstrates the efficiency and performance of the proposed method. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Isogeometric analysis web method PHT-splines Weight function

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Spline R-Function and Applications in FEM

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Numerical Mathematics(Theory,methods and Applications) 2020年第1期13卷 150-175页

作者： Tianhui Yang Ammar Qarariyah Jiansong Deng School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026AnhuiChina

R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base ***,there is square root operation in the *** that the use of splines will facilitate the calculations within the CAD system,in this paper,we propose a system of R-functions represented in spline form called Spline R-function(SR).After trans-forming the function ranges of two base primitives to a new coordinate system,a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise *** of SR in both B´ezier form and B-spline form have been *** which the B´ezier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of *** construction processes for both Boolean intersection and union operations with different smoothness are discussed in *** experiments are conducted to show the potential of the proposed spline R-function.

关键词： R-functions CSG Boolean operations B-splines B-net method web method

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AJAX technology applications in the network test system

AJAX technology applications in the network test system

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2nd Annual Conference on Electrical and Control Engineering, ICECE 2011

作者： Wang, Xiaofeng Institute of Mathematical Information Langfang Teachers College Langfang China

ISBN: (纸本)9781424481637

This paper introduces the AJAX technology and how it works, and is mainly involved in the application of technology and the current situation. It discusses the limitations of using the traditional web method and advantages of using AJAX technology in the network test system, and it also gives a specific implementation of AJAX in the network test system. Using AJAX technology greatly reduces the pressure on the server, improves the system response speed and reduces the waiting time. © 2011 IEEE.

关键词： AJAX AJAX technology application Browsers Cascading style sheets Engines Google HTML Internet Java Servers web web method XML XML XMLHttpRequest hypermedia markup languages network servers network test system network test system server pressure system res

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