A novel space–time meshless method for solving the backward heat conduction problem
•A novel spacetime meshless method for solving the backward heat conduction problem.•Numerical approximation using the Trefftz basis function of the heat equation.•Collocating the boundary points in the spacetime coordinate system in the Trefftz method.•Highly accurate numerical solutions can be obt...
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| Published in: | International journal of heat and mass transfer Vol. 130; pp. 109 - 122 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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Oxford
Elsevier Ltd
01.03.2019
Elsevier BV |
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| ISSN: | 0017-9310, 1879-2189 |
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| Abstract | •A novel spacetime meshless method for solving the backward heat conduction problem.•Numerical approximation using the Trefftz basis function of the heat equation.•Collocating the boundary points in the spacetime coordinate system in the Trefftz method.•Highly accurate numerical solutions can be obtained comparing to that from conventional time-marching scheme.•Boundary data on the inaccessible boundary can be recovered even the partial data on the final time boundary are absent.
This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent. |
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| AbstractList | •A novel spacetime meshless method for solving the backward heat conduction problem.•Numerical approximation using the Trefftz basis function of the heat equation.•Collocating the boundary points in the spacetime coordinate system in the Trefftz method.•Highly accurate numerical solutions can be obtained comparing to that from conventional time-marching scheme.•Boundary data on the inaccessible boundary can be recovered even the partial data on the final time boundary are absent.
This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent. This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent. |
| Author | Yeih, Weichung Ku, Cheng-Yu Liu, Chein-Shan Fan, Chia-Ming Liu, Chih-Yu |
| Author_xml | – sequence: 1 givenname: Cheng-Yu surname: Ku fullname: Ku, Cheng-Yu email: chkst26@mail.ntou.edu.tw organization: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan – sequence: 2 givenname: Chih-Yu orcidid: 0000-0002-2018-3401 surname: Liu fullname: Liu, Chih-Yu organization: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan – sequence: 3 givenname: Weichung surname: Yeih fullname: Yeih, Weichung organization: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan – sequence: 4 givenname: Chein-Shan orcidid: 0000-0002-5077-865X surname: Liu fullname: Liu, Chein-Shan organization: Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu 210098, China – sequence: 5 givenname: Chia-Ming surname: Fan fullname: Fan, Chia-Ming organization: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan |
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| Cites_doi | 10.1080/10407790.2015.1125215 10.1016/j.ijheatmasstransfer.2016.01.008 10.1002/fld.1650200824 10.3390/w9120954 10.1080/02533839.2004.9670908 10.3846/16486897.2017.1326925 10.1002/nme.1620370205 10.1007/s12665-017-6902-4 10.1016/j.ijheatmasstransfer.2016.03.003 10.1142/S0219876209001759 10.1016/S0017-9310(00)00235-0 10.1080/10407790.2017.1420317 10.1016/0041-5553(64)90006-0 10.1016/j.jmaa.2006.08.040 10.1016/S0898-1221(00)85017-8 10.1016/j.enganabound.2014.08.007 10.1515/ijnsns-2015-0060 10.4134/JKMS.2007.44.6.1281 10.1006/jcph.1995.1028 10.1016/j.ijheatmasstransfer.2003.12.019 10.1080/10682760410001710141 10.1080/02533839.2009.9671510 10.1016/j.enganabound.2010.03.010 10.1016/S0377-0427(98)00249-0 10.1080/02533839.2010.9671608 10.1016/S0307-904X(02)00053-7 |
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| Keywords | Trefftz method Backward heat conduction problem Meshless method Inverse problem |
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| Snippet | •A novel spacetime meshless method for solving the backward heat conduction problem.•Numerical approximation using the Trefftz basis function of the heat... This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the... |
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| SubjectTerms | Backward heat conduction problem Basis functions Boundary conditions Boundary value problems Collocation methods Conduction heating Conductive heat transfer Coordinates Finite difference method Finite element method Heat conductivity Inverse problem Inverse problems Meshless method Meshless methods Numerical analysis Thermodynamics Time domain analysis Trefftz method |
| Title | A novel space–time meshless method for solving the backward heat conduction problem |
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