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TECHNICAL PAPERS: Thermal Systems

A Finite Element Formulation for the Determination of Unknown Boundary Conditions for Three-Dimensional Steady Thermoelastic Problems

[+] Author and Article Information
Brian H. Dennis

Institute of Environmental Studies, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 113-8656e-mail: dennis@garlic.q.t.u-tokyo.ac.jp

George S. Dulikravich

Department of Mechanical and Materials Engineering, Florida International University, 10555 West Flagler Street, Miami, FL 33174, USAe-mail: dulikrav@fiu.edu

Shinobu Yoshimura

Institute of Environmental Studies, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 113-8656e-mail: yoshi@q.t.u-tokyo.ac.jp

J. Heat Transfer 126(1), 110-118 (Mar 10, 2004) (9 pages) doi:10.1115/1.1640360 History: Received July 31, 2002; Revised September 08, 2003; Online March 10, 2004
Copyright © 2004 by ASME
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References

Larsen, M. E., 1985, “An Inverse Problem: Heat Flux and Temperature Prediction for a High Heat Flux Experiment,” Technical Report, SAND-85-2671, Sandia National Laboratories, Albuquerque, NM.
Hensel,  E. H., and Hills,  R., 1989, “Steady-State Two-Dimensional Inverse Heat Conduction,” Numer. Heat Transfer, 15, pp. 227–240.
Martin,  T. J., and Dulikravich,  G. S., 1996, “Inverse Determination of Boundary Conditions in Steady Heat Conduction,” ASME J. Heat Transfer, 3, pp. 546–554.
Dennis,  B. H., and Dulikravich,  G. S., 1999, “Simultaneous Determination of Temperatures, Heat Fluxes, Deformations, and Tractions on Inaccessible Boundaries,” ASME J. Heat Transfer, 121, pp. 537–545.
Olson, L. G., and Throne, R. D., 2000, “The Steady Inverse Heat Conduction Problem: A Comparison for Methods of Inverse Parameter Selection,” in 34th National Heat Transfer Conference-NHTC’00, No. NHTC2000-12022, Pittsburg, PA.
Martin,  T. J., Halderman,  J., and Dulikravich,  G. S., 1995, “An Inverse Method for Finding Unknown Surface Tractions and Deformations in Elastostatics,” Comput. Struct., 56, pp. 825–836.
Martin, T. J., and Dulikravich, G. S., 1995, “Finding Temperatures and Heat Fluxes on Inaccessible Surfaces in Three-Dimensional Coated Rocket Nozzles,” in 1995 JANNAF Non-Destructive Evaluation Propulsion Subcommittee Meeting, Tampa, FL, pp. 119–129.
Dennis, B. H., and Dulikravich, G. S., 2001, “A Three-Dimensional Finite Element Formulation for the Determination of Unknown Boundary Conditions in Heat Conduction,” in Proc. of Internat. Symposium on Inverse Problems in Eng. Mechanics, M. Tanaka, ed., Nagano City, Japan.
Hughes, T. J. R., 2000, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, Inc., New York.
Huebner, K. H., Thorton, E. A., and Byrom, T. G., 1995, The Finite Element Method for Engineers, third edition, John Wiley and Sons, New York, NY.
Neumaier,  A., 1998, “Solving Ill-Conditioned and Singular Linear Systems: A Tutorial on Regularization,” SIAM Rev., 40, pp. 636–666.
Boschi, L., and Fischer, R. P., 1996, “Iterative Solutions for Tomographic Inverse Problems: LSQR and SIRT,” technical report, Seismology Dept., Harvard University, Cambridge, MA.
Paige,  C. C., and Saunders,  M. A., 1982, “LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares,” ACM Trans. Math. Softw., 8, pp. 43–71.
Saad, Y., 1996, Iterative Methods for Sparse Linear Systems, PWS Publishing Co., Boston, MA.
Matstoms, P., 1991, The Multifrontal Solution of Sparse Least Squares Problems, Ph.D. thesis, Linköping University, Sweden.
Golub, G. H., and Van Loan, C. F., 1996, Matrix Computations, Johns Hopkins University Press, Baltimore, MD.

Figures

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Surface mesh for cylinder test case
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Direct problem: computed isotherms when both inner and outer boundary temperatures were specified
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Inverse problem: computed isotherms when only outer boundary temperatures and fluxes were specified
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Direct problem: computed normal stress magnitude when both inner and outer boundary conditions were specified
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Inverse problem: computed normal stress magnitude when only outer boundary conditions were specified
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Average percent error of predicted temperatures on unknown boundaries for regularization method 1 for cylinder region
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Average percent error of predicted temperatures on unknown boundaries for regularization method 2 for cylinder region
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Average percent error of predicted temperatures on unknown boundaries for regularization method 3 for cylinder region
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Surface mesh for multiply connected domain test case
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Direct problem: computed isotherms when both inner and outer boundary temperatures were specified
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Inverse problem: computed isotherms when only outer boundary temperatures and fluxes were specified and using regularization method 1
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Inverse problem: computed isotherms when only outer boundary temperatures and fluxes were specified and using regularization method 2
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Inverse problem: computed isotherms when only outer boundary temperatures and fluxes were specified and using regularization method 3 (Inverse and Direct contours plotted together)
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Inverse problem: computed isotherms on x−y plane at z=0.5 m when only outer boundary temperatures and fluxes were specified
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Inverse problem: computed displacement magnitude on x−y plane at z=0.5 m when only outer boundary displacements and tractions were specified
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Inverse problem: computed isotherms on x−y plane at z=2.5 m when only outer boundary temperatures and fluxes were specified
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Inverse problem: computed displacement magnitude on x−y plane at z=2.5 m when only outer boundary displacements and tractions were specified
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Inverse problem: computed isotherms on x−y plane at z=4.5 m when only outer boundary temperatures and fluxes were specified
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Inverse problem: computed displacement magnitude on x−y plane at z=4.5 m when only outer boundary displacements and tractions were specified

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