Trefftz method for polynomial-based boundary identification in two-dimensional Laplacian problems

• Autorzy: Hożejowski L.

Identyfikatory

DOI

10.15632/jtam-pl.54.3.935

• Języki publikacji: EN

Abstrakty

EN

The paper addresses a two-dimensional boundary identification (reconstruction) problem in steady-state heat conduction. Having found the solution to the Laplace equation by superpositioning T-complete functions, the unknown boundary of a plane region is approximated by polynomials of an increasing degree. The provided examples indicate that sufficient accuracy can be obtained with a use of polynomials of a relatively low degree, which allows avoidance of large systems of nonlinear equations. Numerical simulations for assessing the performance of the proposed algorithm show better than 1% accuracy after a few iterations and very low sensitivity to small data errors.

Słowa kluczowe

EN

inverse geometry problem; boundary identification; Trefftz method; Laplace equation

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2016
• Tom: Vol. 54 nr 3
• Strony: 935--944
• Opis fizyczny: Bibliogr. 24 poz., rys., tab.

Twórcy

autor

Hożejowski L.

• Kielce University of Technology, Faculty of Management and Computer Modelling, Kielce, Poland

Bibliografia

• 1. Chan H.-F., Fan C.-M., 2013, The modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm for the inverse boundary determination problem for the biharmonic equation, Journal of Mechanics, 29, 2, 363-372
• 2. Chan H.-F., Fan C.-M., and Yeih W., 2011, Solution of inverse boundary optimization problem by Trefftz method and exponentially convergent scalar homotopy algorithm, CMC: Computers, Materials and Continua, 24, 2, 125-142
• 3. Chen J.-T., Wu Y.-T., Lee Y.-T., Chen K.-H., 2007, On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations, Computers and Mathematics with Applications, 53, 6, 851-879
• 4. Cheng C.-H., Chang M.-H., 2003a, A simplified conjugate-gradient method for shape identifi- cation based on thermal data, Numerical Heat Transfer, Part B: Fundamentals, 43, 5, 489-507
• 5. Cheng C.-H., Chang M.-H., 2003b, Shape identification by inverse heat transfer method, Journal of Heat Transfer, 125, 2, 224-231
• 6. Ciałkowski M., Grysa K., 2010, Trefftz method in solving the inverse problems, Journal of Inverse and Ill-Posed Problems, 18, 6, 595-616
• 7. Fan C.-M., Chan H.-F., 2011, Modified collocation Trefftz method for the geometry boundary identification problem of heat conduction, Numerical Heat Transfer, Part B: Fundamentals, 59, 1, 58-75
• 8. Fan C.-M., Chan H.-F., Kuo C.-L., Yeih W., 2012, Numerical solutions of boundary detection problems using modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm, Engineering Analysis with Boundary Elements, 36, 1, 2-8
• 9. Fan C.-M., Liu Y.-C., Chan H.-F., Hsiao S.-S., 2014, Solutions of boundary detection problems for modified Helmholz equation by Trefftz method, Inverse Problems in Science and Engineering, 22, 2, 267-281
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• 12. Huang C.-H., Liu C.-Y., 2010, A three-dimensional inverse geometry problem in estimating simultaneously two interfacial configurations in a composite domain, International Journal of Heat and Mass Transfer, 53, 1/3, 48-57
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• 14. Karageorghis A., Lesnic D., Marin L., 2011b, The MFS for inverse geometric problems, [In:] Inverse Problems and Computational Mechanics, L. Munteanu, L. Marin and V. Chiroiu (Eds.), Vol. 1, Editura Academiei, Bucharest, 191-216
• 15. Karageorghis A., Lesnic D., Marin L., 2014, Regularized collocation Trefftz method for void detection in two-dimensional steady-state heat conduction problems, Inverse Problems in Science and Engineering, 22, 3, 395-418
• 16. Kazemzadeh-Parsi M.J., Daneshmand F., 2009, Solution of geometric inverse conduction problems by smoothed fixed grid finite element method, Finite Elements in Analysis and Design, 45, 10, 599-611
• 17. Lesnic D., Berger J.R., Martin P.A., 2002, A boundary element regularization method for the boundary determination in potential corrosion damage, Inverse Problems in Engineering, 10, 2, 163-182
• 18. Li Z.-C., Lu T.-T., Hu H.-Y., Cheng A. H.-D., 2008, Trefftz and Collocation Methods, WIT Press, Southampton, Boston
• 19. Liu C.-S., 2007, A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length, CMES: Computer Modeling in Engineering and Sciences, 21, 1, 53-66
• 20. Liu C.-S., 2008, A highly accurate MCTM for inverse Cauchy problems of Laplace equation in arbitrary plane domains, CMES: Computer Modeling in Engineering and Sciences, 35, 2, 91-111
• 21. Liu C.-S., Chang C.-W., Chiang C.Y., 2008, A regularized integral equation method for the inverse heat conduction problem, International Journal of Heat and Mass Transfer, 51, 21/22, 5380-5388
• 22. Mera N.S., Lesnic D., 2005, A three-dimensional boundary determination problem in potential corrosion damage, Computational Mechanics, 36, 2, 129-138
• 23. Mera N.S., Eliott L., Ingham D.B., 2004, Numerical solution of a boundary detection problem using genetic algorithms, Engineering Analysis with Boundary Elements, 28, 4, 405-411
• 24. Trefftz E., 1926, Ein Gegenstck zum Ritzschen Verfahren, Verhandlungen des II. Kongress f¨ur technische Mechanik, Z¨urich, 131-137

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniajacą naukę.