IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method
The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressi...
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MDPI AG
2023-02-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/4/898 |
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| author | Yini Jiang Hongwei Lin |
| author_facet | Yini Jiang Hongwei Lin |
| author_sort | Yini Jiang |
| collection | DOAJ |
| description | The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T08:28:51Z |
| publishDate | 2023-02-01 |
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| spelling | doaj.art-ea7b931fbd0d4fce8423a5111e51151b2023-11-16T21:55:33ZengMDPI AGMathematics2227-73902023-02-0111489810.3390/math11040898IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation MethodYini Jiang0Hongwei Lin1School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, ChinaSchool of Mathematical Sciences, Zhejiang University, Hangzhou 310058, ChinaThe isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method.https://www.mdpi.com/2227-7390/11/4/898progressive-iterative approximationisogeometric analysisisogeometric collocation methodnon-uniform rational B-splines |
| spellingShingle | Yini Jiang Hongwei Lin IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method Mathematics progressive-iterative approximation isogeometric analysis isogeometric collocation method non-uniform rational B-splines |
| title | IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method |
| title_full | IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method |
| title_fullStr | IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method |
| title_full_unstemmed | IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method |
| title_short | IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method |
| title_sort | ig lspia least squares progressive iterative approximation for isogeometric collocation method |
| topic | progressive-iterative approximation isogeometric analysis isogeometric collocation method non-uniform rational B-splines |
| url | https://www.mdpi.com/2227-7390/11/4/898 |
| work_keys_str_mv | AT yinijiang iglspialeastsquaresprogressiveiterativeapproximationforisogeometriccollocationmethod AT hongweilin iglspialeastsquaresprogressiveiterativeapproximationforisogeometriccollocationmethod |