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ABSTRACT
A new high-precision boundary meshfree method, namely virtual boundary meshfree Galerkin method (VBMGM), for calculating the multi-domain constant coefficient heat conduction with a heat source problem is given. In the paper, the radial basis function interpolation is used to solve the virtual source function of virtual boundary and the heat source within each subdomain. Simultaneously, the equation of VBMGM for multi-domain constant coefficient heat conduction with a heat source problem is obtained by the Galerkin method. Therefore, the proposed method has common advantages of the boundary element method, meshfree method, and Galerkin method. Coefficient matrix of this specific expression is symmetrical and the specific expression of VBMGM for the multi-domain constant coefficient heat conduction with a heat source problem is given. Two numerical examples are given. The numerical results are also compared with other numerical methods. The accuracy and feasibility of the method for the multi-domain constant coefficient heat conduction with a heat source problem are proved.
Nomenclature
| uh(x) | = | the approach function |
| u(x) | = | the real-valued function |
| Ri(x) | = | the radial basis function |
| uS | = | the vector for function values |
| N | = | the shape function |
| Ωl | = | the lth subdomain |
| Γl | = | the real boundary of Ωl |
| Sl | = | the virtual boundary of Ωl |
| Γls | = | the public boundary of Ωl and Ωs |
| φ(l)(ξ) | = | the virtual source function of Ωl |
| = | the fundamental solutions of temperature | |
| = | the fundamental solutions of heat flux | |
| ξ | = | virtual source point |
| Q(l)(x) | = | heat source of the lth subdomain |
| k(l) | = | heat conductivity of the lth subdomain |
| = | the known temperature real boundary of the lth subdomain | |
| = | the known heat flux real boundary of the lth subdomain | |
| = | the known temperature value of the lth subdomain | |
| = | the homogeneous solution of the temperature about lth subdomain | |
| = | the particular solution of the temperature about lth subdomain | |
| = | the known heat flux value of the lth subdomain | |
| = | the homogeneous solution of the heat flux about lth subdomain | |
| = | the particular solution of the heat flux about lth subdomain | |
| Ni(ξ) | = | the shape function of the virtual node |
| me′ | = | the number of elements on the virtual boundary |
| J′ | = | the element Jacobian on the virtual boundary |
| = | the weighting coefficient for Gauss numerical integral within virtual element | |
| ee′ | = | the number of Gauss points on virtual element |
| mT | = | the element number on the real and known temperature boundary |
| mq | = | the element number on the real and known heat flux boundary |
| eT | = | the Gauss point number on the real and known temperature boundary |
| eq | = | the Gauss point number on the real and known heat flux boundary |
| J | = | the element Jacobian on the real boundary |
| w1 | = | the weighted residual coefficient of the heat source |
| w2 | = | the weighted residual coefficient of the temperature |
| w3 | = | the weighted residual coefficient of the heat flux |
| w4 | = | the weighted residual coefficient of the temperature connection condition |
| w5 | = | the weighted residual coefficient of the heat flux connection condition |
| A | = | the known and symmetric coefficient matrix |
| B | = | the known matrix based on the boundary conditions |
| β | = | the entire unknown function value vector |
Nomenclature
| uh(x) | = | the approach function |
| u(x) | = | the real-valued function |
| Ri(x) | = | the radial basis function |
| uS | = | the vector for function values |
| N | = | the shape function |
| Ωl | = | the lth subdomain |
| Γl | = | the real boundary of Ωl |
| Sl | = | the virtual boundary of Ωl |
| Γls | = | the public boundary of Ωl and Ωs |
| φ(l)(ξ) | = | the virtual source function of Ωl |
| = | the fundamental solutions of temperature | |
| = | the fundamental solutions of heat flux | |
| ξ | = | virtual source point |
| Q(l)(x) | = | heat source of the lth subdomain |
| k(l) | = | heat conductivity of the lth subdomain |
| = | the known temperature real boundary of the lth subdomain | |
| = | the known heat flux real boundary of the lth subdomain | |
| = | the known temperature value of the lth subdomain | |
| = | the homogeneous solution of the temperature about lth subdomain | |
| = | the particular solution of the temperature about lth subdomain | |
| = | the known heat flux value of the lth subdomain | |
| = | the homogeneous solution of the heat flux about lth subdomain | |
| = | the particular solution of the heat flux about lth subdomain | |
| Ni(ξ) | = | the shape function of the virtual node |
| me′ | = | the number of elements on the virtual boundary |
| J′ | = | the element Jacobian on the virtual boundary |
| = | the weighting coefficient for Gauss numerical integral within virtual element | |
| ee′ | = | the number of Gauss points on virtual element |
| mT | = | the element number on the real and known temperature boundary |
| mq | = | the element number on the real and known heat flux boundary |
| eT | = | the Gauss point number on the real and known temperature boundary |
| eq | = | the Gauss point number on the real and known heat flux boundary |
| J | = | the element Jacobian on the real boundary |
| w1 | = | the weighted residual coefficient of the heat source |
| w2 | = | the weighted residual coefficient of the temperature |
| w3 | = | the weighted residual coefficient of the heat flux |
| w4 | = | the weighted residual coefficient of the temperature connection condition |
| w5 | = | the weighted residual coefficient of the heat flux connection condition |
| A | = | the known and symmetric coefficient matrix |
| B | = | the known matrix based on the boundary conditions |
| β | = | the entire unknown function value vector |
Acknowledgments
This work was supported by Guangxi Key Laboratory of information materials open research foundation (161005-K), Guizhou Provincial Science and Technology Department, Guizhou Normal University Joint Fund Project (Guizhou cooperation LH word [2016] 7216), Guizhou Province undergraduate teaching construction project (2016JG16), Civil Engineering Excellence Engineer Training Program (Guizhou higher education [2014] 378) and Civil Engineering Professional Comprehensive Reform Project (Guizhou higher education incidence [2015] 337), and Guizhou Normal University 2015 scientific research training program of college students (20151604).