Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems

It has been shown that the significance of the positivity conditions in the collocation methods (CM), and the violation of the positivity conditions can significantly result in a large error in the numerical solution. For boundary points, however, the positivity conditions cannot be satisfied, obvio...

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Autores principales:	Yong-Ming GUO, Kouji SHIOYA, Kei OOBUCHI, Genki YAGAWA, Shunpei KAMITANI
Formato:	article
Lenguaje:	EN
Publicado:	The Japan Society of Mechanical Engineers 2014
Materias:	meshless method collocation method positivity conditions over-range points kronecker-delta Mechanical engineering and machinery TJ1-1570
Acceso en línea:	https://doaj.org/article/e1291205ebbd4b6cbb35c8d0074fb5ae
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spelling	oai:doaj.org-article:e1291205ebbd4b6cbb35c8d0074fb5ae2021-11-26T06:04:02ZAccuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems2187-974510.1299/mej.2014cm0008https://doaj.org/article/e1291205ebbd4b6cbb35c8d0074fb5ae2014-04-01T00:00:00Zhttps://www.jstage.jst.go.jp/article/mej/1/2/1_2014cm0008/_pdf/-char/enhttps://doaj.org/toc/2187-9745It has been shown that the significance of the positivity conditions in the collocation methods (CM), and the violation of the positivity conditions can significantly result in a large error in the numerical solution. For boundary points, however, the positivity conditions cannot be satisfied, obviously. To overcome the demerit of the CM, the over-range collocation method (ORCM) has been proposed. In the ORCM, some over-range collocation points are introduced which are located at the outside of the domain of an analyzed body, and at the over-range collocation points no satisfaction of any governing partial differential equation or boundary condition is needed. In this paper, it is shown that the positivity conditions of boundary points in the ORCM are satisfied by calculated results on the positivity conditions, while the positivity conditions of boundary points in the CM are not satisfied. The boundary value problems on the 2-D and 3-D Poisson’s equations and the 3-D Helmholtz’s equation are analyzed by using the ORCM and the CM. The numerical solutions by using both the ORCM and the CM are compared with the exact solutions. The relative errors by using the ORCM are smaller than those by using the CM, for both the unknown variables and their derivatives of 2-D problems and for the unknown variables of 3-D problems, and the relative errors of the unknown's derivatives of 3-D problems by using the ORCM are about same as those by using the CM. Convergence studies in the numerical examples show that the ORCM possesses good convergence for both the unknown variables and their derivatives. Because the ORCM does not demand any specific type of partial differential equations, it is concluded that the ORCM promises the wide engineering applications.Yong-Ming GUOKouji SHIOYAKei OOBUCHIGenki YAGAWAShunpei KAMITANIThe Japan Society of Mechanical Engineersarticlemeshless methodcollocation methodpositivity conditionsover-range pointskronecker-deltaMechanical engineering and machineryTJ1-1570ENMechanical Engineering Journal, Vol 1, Iss 2, Pp CM0008-CM0008 (2014)
institution	DOAJ
collection	DOAJ
language	EN
topic	meshless method collocation method positivity conditions over-range points kronecker-delta Mechanical engineering and machinery TJ1-1570
spellingShingle	meshless method collocation method positivity conditions over-range points kronecker-delta Mechanical engineering and machinery TJ1-1570 Yong-Ming GUO Kouji SHIOYA Kei OOBUCHI Genki YAGAWA Shunpei KAMITANI Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems
description	It has been shown that the significance of the positivity conditions in the collocation methods (CM), and the violation of the positivity conditions can significantly result in a large error in the numerical solution. For boundary points, however, the positivity conditions cannot be satisfied, obviously. To overcome the demerit of the CM, the over-range collocation method (ORCM) has been proposed. In the ORCM, some over-range collocation points are introduced which are located at the outside of the domain of an analyzed body, and at the over-range collocation points no satisfaction of any governing partial differential equation or boundary condition is needed. In this paper, it is shown that the positivity conditions of boundary points in the ORCM are satisfied by calculated results on the positivity conditions, while the positivity conditions of boundary points in the CM are not satisfied. The boundary value problems on the 2-D and 3-D Poisson’s equations and the 3-D Helmholtz’s equation are analyzed by using the ORCM and the CM. The numerical solutions by using both the ORCM and the CM are compared with the exact solutions. The relative errors by using the ORCM are smaller than those by using the CM, for both the unknown variables and their derivatives of 2-D problems and for the unknown variables of 3-D problems, and the relative errors of the unknown's derivatives of 3-D problems by using the ORCM are about same as those by using the CM. Convergence studies in the numerical examples show that the ORCM possesses good convergence for both the unknown variables and their derivatives. Because the ORCM does not demand any specific type of partial differential equations, it is concluded that the ORCM promises the wide engineering applications.
format	article
author	Yong-Ming GUO Kouji SHIOYA Kei OOBUCHI Genki YAGAWA Shunpei KAMITANI
author_facet	Yong-Ming GUO Kouji SHIOYA Kei OOBUCHI Genki YAGAWA Shunpei KAMITANI
author_sort	Yong-Ming GUO
title	Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems
title_short	Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems
title_full	Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems
title_fullStr	Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems
title_full_unstemmed	Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems
title_sort	accuracy improvement of collocation method by using the over-range collocation points for 2-d and 3-d problems
publisher	The Japan Society of Mechanical Engineers
publishDate	2014
url	https://doaj.org/article/e1291205ebbd4b6cbb35c8d0074fb5ae
work_keys_str_mv	AT yongmingguo accuracyimprovementofcollocationmethodbyusingtheoverrangecollocationpointsfor2dand3dproblems AT koujishioya accuracyimprovementofcollocationmethodbyusingtheoverrangecollocationpointsfor2dand3dproblems AT keioobuchi accuracyimprovementofcollocationmethodbyusingtheoverrangecollocationpointsfor2dand3dproblems AT genkiyagawa accuracyimprovementofcollocationmethodbyusingtheoverrangecollocationpointsfor2dand3dproblems AT shunpeikamitani accuracyimprovementofcollocationmethodbyusingtheoverrangecollocationpointsfor2dand3dproblems
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Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems

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