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A new algorithm for direct and backward problems of heat conduction equation. (English) Zbl 1201.80012
Summary: Direct heat conduction problem (DHCP) and backward heat conduction problem (BHCP) are numerically solved by employing a new idea of fictitious time integration method (FTIM). The DHCP needs to consider the stability of numerical integration in the sense that the solution may be divergent for a specific time stepsize and specific spatial stepsize. The BHCP is renowned as strongly ill-posed because the solution does not continuously depend on the given data. In this paper, we transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time variable, and adding a fictitious viscous damping coefficient to enhance the stability of numerical integration of the discretized equations by employing a group preserving scheme. When 10 numerical examples are amenable, we find that the FTIM is applicable to both the DHCP and BHCP. Even under seriously noisy initial or final data, the FTIM is also robust against disturbance. More interestingly, when we use the FTIM, we do not need to use different techniques to treat DHCP and BHCP as that usually employed in the conventional numerical methods. It means that the FTIM can unifiedly approach both the DHCP and BHCP, and the gap between direct problems and inverse problems can be smeared out.

MSC:
80A20 Heat and mass transfer, heat flow (MSC2010)
80A23 Inverse problems in thermodynamics and heat transfer
80M99 Basic methods in thermodynamics and heat transfer
Software:
FEAPpv
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[1] Ă–zisik, M. Necati: Finite difference methods in heat transfer, (1994) · Zbl 0855.65097
[2] Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z.: The finite element method: its basis and fundamentals, (2005) · Zbl 1307.74005
[3] Zhu, S. P.: Solving transient diffusion problems: time-dependent fundamental solution approaches versus LTDRM approaches, Eng. anal. Bound. elem. 21, 87-90 (1998) · Zbl 0979.76525 · doi:10.1016/S0955-7997(97)00102-1
[4] Zhu, S. P.; Liu, H. W.; Lu, X. P.: A combination of LTDRM and ATPS in solving diffusion problems, Eng. anal. Bound. elem. 21, 285-289 (1998) · Zbl 0941.80008 · doi:10.1016/S0955-7997(98)00009-5
[5] Bulgakov, V.; Sarler, B.; Kuhn, G.: Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation, Int. J. Numer. meth. Eng. 43, 713-732 (1998) · Zbl 0948.76050 · doi:10.1002/(SICI)1097-0207(19981030)43:4<713::AID-NME445>3.0.CO;2-8
[6] Zerroukat, M.: A boundary element scheme for diffusion problems using compactly supported radial basis functions, Eng. anal. Bound. elem. 23, 201-209 (1999) · Zbl 0968.76568 · doi:10.1016/S0955-7997(98)00089-7
[7] Sutradhar, A.; Paulino, G. H.; Gray, L. J.: Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method, Eng. anal. Bound. elem. 26, 119-132 (2002) · Zbl 0995.80010 · doi:10.1016/S0955-7997(01)00090-X
[8] Bialecki, R. A.; Jurgas, P.; Kuhn, G.: Dual reciprocity BEM without matrix inversion for transient heat conduction, Eng. anal. Bound. elem. 26, 227-236 (2002) · Zbl 1002.80019 · doi:10.1016/S0955-7997(01)00097-2
[9] Walker, S. P.: Diffusion problems using transient discrete source superposition, Int. J. Numer. meth. Eng. 35, 165-178 (1992) · Zbl 0764.76044 · doi:10.1002/nme.1620350111
[10] Chen, C. S.; Golberg, M. A.; Hon, Y. C.: The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, Int. J. Numer. meth. Eng. 43, 1421-1435 (1998) · Zbl 0929.76098 · doi:10.1002/(SICI)1097-0207(19981230)43:8<1421::AID-NME476>3.0.CO;2-V
[11] Young, D. L.; Tsai, C. C.; Murugesan, K.; Fan, C. M.; Chen, C. W.: Time-dependent fundamental solutions for homogeneous diffusion problems, Eng. anal. Bound. elem. 28, 1463-1473 (2004) · Zbl 1098.76622 · doi:10.1016/j.enganabound.2004.07.003
[12] Chantasiriwan, S.: Methods of fundamental solutions for time-dependent heat conduction problems, Int. J. Numer. meth. Eng. 66, 147-165 (2006) · Zbl 1124.80313 · doi:10.1002/nme.1549
[13] Johansson, B. T.; Lesnic, D.: A method of fundamental solutions for transient heat conduction, Eng. anal. Bound. elem. 32, 697-703 (2008) · Zbl 1244.80021
[14] Han, H.; Ingham, D. B.; Yuan, Y.: The boundary element method for the solution of the backward heat conduction equation, J. comput. Phys. 116, 292-299 (1995) · Zbl 0821.65064 · doi:10.1006/jcph.1995.1028
[15] Mera, N. S.; Elliott, L.; Ingham, D. B.; Lesnic, D.: An iterative boundary element method for solving the one-dimensional backward heat conduction problem, Int. J. Heat mass transfer 44, 1937-1946 (2001) · Zbl 0979.80008 · doi:10.1016/S0017-9310(00)00235-0
[16] Mera, N. S.; Elliott, L.; Ingham, D. B.: An inversion method with decreasing regularization for the backward heat conduction problem, Numer. heat transfer B 42, 215-230 (2002)
[17] Jourhmane, M.; Mera, N. S.: An iterative algorithm for the backward heat conduction problem based on variable relaxation factors, Inv. prob. Eng. 10, 293-308 (2002)
[18] Muniz, W. B.; De Campos Velho, H. F.; Ramos, F. M.: A comparison of some inverse methods for estimating the initial condition of the heat equation, J. comput. Appl. math. 103, 145-163 (1999) · Zbl 0952.65068 · doi:10.1016/S0377-0427(98)00249-0
[19] Muniz, W. B.; Ramos, F. M.; De Campos Velho, H. F.: Entropy- and Tikhonov-based regularization techniques applied to the backward heat equation, Int. J. Comput. math. 40, 1071-1084 (2000) · Zbl 1016.80016 · doi:10.1016/S0898-1221(00)85017-8
[20] Kirkup, S. M.; Wadsworth, M.: Solution of inverse diffusion problems by operator-splitting methods, Appl. math. Model. 26, 1003-1018 (2002) · Zbl 1014.65095 · doi:10.1016/S0307-904X(02)00053-7
[21] Liu, J.: Numerical solution of forward and backward problem for 2-D heat conduction equation, J. comput. Appl. math. 145, 459-482 (2002) · Zbl 1005.65107 · doi:10.1016/S0377-0427(01)00595-7
[22] Mera, N. S.: The method of fundamental solutions for the backward heat conduction problem, Inv. prob. Sci. eng. 13, 65-78 (2005) · Zbl 1194.80107 · doi:10.1080/10682760410001710141
[23] Iijima, K.: Numerical solution of backward heat conduction problems by a high order lattice-free finite difference method, J. chin. Inst. eng. 27, 611-620 (2004)
[24] Liu, C. -S.; Chang, C. -W.; Chang, J. -R.: Past cone dynamics and backward group preserving schemes for backward heat conduction problems, CMES: comput. Model. eng. Sci. 12, 67-81 (2006) · Zbl 1232.65129 · doi:10.3970/cmes.2006.012.067
[25] Liu, C. -S.: Group preserving scheme for backward heat conduction problems, Int. J. Heat mass transfer 47, 2567-2576 (2004) · Zbl 1100.80005 · doi:10.1016/j.ijheatmasstransfer.2003.12.019
[26] Liu, C. -S.: Cone of non-linear dynamical system and group preserving schemes, Int. J. Non-linear mech. 36, 1047-1068 (2001) · Zbl 1243.65084
[27] Chang, J. -R.; Liu, C. -S.; Chang, C. -W.: A new shooting method for quasi-boundary regularization of backward heat conduction problems, Int. J. Heat mass transfer 50, 2325-2332 (2007) · Zbl 1123.80005 · doi:10.1016/j.ijheatmasstransfer.2006.10.050
[28] Chang, C. -W.; Liu, C. -S.; Chang, J. -R.: A new shooting method for quasi-boundary regularization of multi-dimensional backward heat conduction problems, J. chin. Inst. eng. 32, 307-318 (2009)
[29] Chang, C. -W.; Liu, C. -S.; Chang, J. -R.: A quasi-boundary semi-analytical method for backward heat conduction problems, J. chin. Inst. eng. 33, 163-175 (2010)
[30] Chang, C. -W.; Liu, C. -S.; Chang, J. -R.: A quasi-boundary semi-analytical approach for two-dimensional backward heat conduction problems, CMC: comput. Mater. contin. 15, 45-66 (2010)
[31] Liu, C. -S.: A highly accurate LGSM for severely ill-posed BHCP under a large noise on the final time data, Int. J. Heat mass transfer 53, 4132-4140 (2010) · Zbl 1194.80106 · doi:10.1016/j.ijheatmasstransfer.2010.05.036
[32] Chang, C. -W.; Liu, C. -S.; Chang, J. -R.: A group preserving scheme for inverse heat conduction problems, CMES: comput. Model. eng. Sci. 10, 13-38 (2005) · Zbl 1232.80005 · doi:10.3970/cmes.2005.010.013
[33] Liu, C. -S.: Solving an inverse Sturm – Liouville problem by a Lie-group method, Bound. value prob. 2008 (2008) · Zbl 1154.34005 · doi:10.1155/2008/749865
[34] Liu, C. -S.: Identifying time-dependent damping and stiffness functions by a simple and yet accurate method, J. sound vib. 318, 148-165 (2008)
[35] Liu, C. -S.: A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients, CMES: comput. Model. eng. Sci. 27, 137-149 (2008) · Zbl 1232.74027 · doi:10.3970/cmes.2008.027.137
[36] Liu, C. -S.; Chang, J. -R.; Chang, K. -H.; Chen, Y. -W.: Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data, CMC: comput. Mater. contin. 7, 97-107 (2008)
[37] Liu, C. -S.; Atluri, S. N.: A novel time integration method for solving a large system of non-linear algebraic equations, CMES: comput. Model. eng. Sci. 31, 71-83 (2008) · Zbl 1152.65428
[38] Liu, C. -S.: A time-marching algorithm for solving non-linear obstacle problems with the aid of an NCP-function, CMC: comput. Mater. contin. 8, 53-65 (2008)
[39] Liu, C. -S.: A fictitious time integration method for two-dimensional quasilinear elliptic boundary value problems, CMES: comput. Model. eng. Sci. 33, 179-198 (2008) · Zbl 1157.35384
[40] Liu, C. -S.; Atluri, S. N.: A novel fictitious time integration method for solving the discretized inverse Sturm – Liouville problems, for specified eigenvalues, CMES: comput. Model. eng. Sci. 36, 261-285 (2008) · Zbl 1232.74007 · doi:10.3970/cmes.2008.036.261
[41] Liu, C. -S.; Atluri, S. N.: A novel time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization, CMES: comput. Model. eng. Sci. 34, 155-178 (2008) · Zbl 1232.90334 · doi:10.3970/cmes.2008.034.155
[42] Liu, C. -S.: A fictitious time integration method for solving m-point boundary value problems, CMES: comput. Model. eng. Sci. 39, 125-154 (2009) · Zbl 1257.65046
[43] Liu, C. -S.: A fictitious time integration method for solving delay ordinary differential equations, CMC: comput. Mater. contin. 10, 97-116 (2009) · Zbl 1175.65087 · www.techscience.com
[44] Liu, C. -S.: A fictitious time integration method for a quasilinear elliptic boundary value problems, defined in an arbitrary plane domain, CMC: comput. Mater. contin. 11, 15-32 (2009) · Zbl 1231.65244 · doi:10.3970/cmc.2009.011.015
[45] Liu, C. -S.; Atluri, S. N.: A fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation to the filter theory, CMES: comput. Model. eng. Sci. 41, 243-261 (2009) · Zbl 1357.65070
[46] Ku, C. -Y.; Yeih, W.; Liu, C. -S.; Chi, C. C.: Applications of the fictitious time integration method using a new time-like function, CMES: comput. Model. eng. Sci. 43, 173-190 (2009) · Zbl 1232.65118 · doi:10.3970/cmes.2009.043.173
[47] Chi, C. -C.; Yeih, W.; Liu, C. -S.: A novel method for solving the Cauchy problem of Laplace equation using the fictitious time integration method, CMES: comput. Model. eng. Sci. 47, 167-190 (2009) · Zbl 1231.65235 · doi:10.3970/cmes.2009.047.167
[48] Chang, C. -W.; Liu, C. -S.: A fictitious time integration method for backward advection – dispersion equation, CMES: comput. Model. eng. Sci. 51, 261-276 (2009) · Zbl 1231.65156 · doi:10.3970/cmes.2009.051.261
[49] Ames, W. F.: Numerical methods for partial differential equations, (1992) · Zbl 0759.65059
[50] Crank, J.; Nicolson, P.: A practical method for numerical evaluation of solution of partial differential equations of the heat conduction type, Proc. camb. Phil. soc. 43, 50-67 (1947) · Zbl 0029.05901
[51] Lesnic, D.; Elliott, L.; Ingham, D. B.: An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation, Inv. prob. Eng. 6, 255-279 (1998)
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