A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problems

References

• Bird, M. D., and C. R. Steele. 1992. “A Solution Procedure for Laplace’s Equation on Multiply Connected Circular Domains.” Journal of Applied Mechanics 59 (2): 398–404. doi:10.1115/1.2899533.
   Web of Science ®Google Scholar
• Chen, J. T., Y. L. Chang, S. Y. Leu, and J. W. Lee. 2018. “Static Analysis of the Free-Free Trusses by Using a Self-Regularization Approach.” Journal of Mechanics 34 (4): 505–518. doi:10.1017/jmech.2017.15.
   Web of Science ®Google Scholar
• Chen, J. T., H. Han, S. R. Kuo, and S. K. Kao. 2014. “Regularization Methods for Ill-conditioned System of the Integral Equation of the First Kind with the Logarithmic Kernel.” Inverse Problems in Science and Engineering 22 (7): 1176–1195. doi:10.1080/17415977.2013.856900.
   Web of Science ®Google Scholar
• Chen, J. T., W. S. Huang, Y. T. Lee, and S. K. Kao. 2016. “A Necessary and Sufficient BEM/BIEM for Two-dimensional Elasticity Problems.” Engineering Analysis with Boundary Elements 67: 108–114. doi:10.1016/j.enganabound.2016.03.007.
   Web of Science ®Google Scholar
• Chen, J. T., W. S. Huang, Y. T. Lee, S. R. Kuo, and S. K. Kao. 2017a. “Revisit of Degenerate Scales in the BIEM/BEM for 2D Elasticity Problems.” Mechanics of Advanced Materials and Structures 24 (1): 1–15. doi:10.1080/15376494.2015.1091526.
   Web of Science ®Google Scholar
• Chen, J. T., S. K. Kao, and J. W. Lee. 2017. “Analytical Derivation and Numerical Experiment of Degenerate Scale by Using the Degenerate Kernel of the Bipolar Coordinates.” Engineering Analysis with Boundary Elements 85: 70–86. doi:10.1016/j.enganabound.2017.08.006.
   Web of Science ®Google Scholar
• Chen, J. T., C. F. Lee, I. L. Chen, and J. H. Lin. 2002. “An Alternative Method for Degenerate Scale Problems in Boundary Element Methods for the Two-dimensional Laplace Equation.” Engineering Analysis with Boundary Elements 26 (7): 559–569. doi:10.1016/S0955-7997(02)00024-3.
   Web of Science ®Google Scholar
• Chen, J. T., Y. T. Lee, Y. L. Chang, and J. Jian. 2017b. “A Self-regularized Approach for Rank-deficient Systems in the BEM of 2D Laplace Problems.” Inverse Problems in Science and Engineering 25 (1): 89–113. doi:10.1080/17415977.2016.1138948.
   Web of Science ®Google Scholar
• Chen, J. T., S. R. Lin, and K. H. Chen. 2005. “Degenerate Scale Problem When Solving Laplace Equation by BEM and Its Treatment.” International Journal for Numerical Methods in Engineering 62 (2): 233–261. doi:10.1002/nme.1184.
   Web of Science ®Google Scholar
• Chen, J. T., L. W. Liu, and H.-K. Hong. 2003. “Spurious and True Eigensolutions of Helmholtz BIEs and BEMs for a Multiply-Connected Problem.” Proceedings of the Royal Society Series A 459 (2036): 1891–1925. doi:10.1098/rspa.2002.1084.
   Google Scholar
• Chen, Y. W., C. S. Liu, and J. R. Chang. 2009. “Applications of the Modified Trefftz Method for the Laplace Equation.” Engineering Analysis with Boundary Elements 33 (2): 137–146. doi:10.1016/j.enganabound.2008.05.008.
   Web of Science ®Google Scholar
• Christiansen, S. 1974. “On Green’s Third Identity as a Basis for Derivation of Integral Equations.” Journal of Applied Mathematics and Mechanics/Zeitschrift Für Angewandte Mathematik Und Mechanik 54 (12): 185–186. doi:10.1002/zamm.197405412100.
   Google Scholar
• Christiansen, S. 1982. “On Two Methods for Elimination of Non-unique Solutions of an Integral Equation with Logarithmic Kernel.” Applicable Analysis 13 (1): 1–18. doi:10.1080/00036818208839372.
   Web of Science ®Google Scholar
• Constanda, C. 1993. “On the Solution of the Dirichlet Problem for the Two Dimensional Laplace Equation.” Proceedings of the American Mathematical Society 119 (3): 877–884. doi:10.2307/2160526.
   Web of Science ®Google Scholar
• Feng, K. 1980. “Differential versus Integral Equations and Finite versus Infinite Elements.” Mathematica Numerica Sinica 2 (1): 100–105.
   Google Scholar
• Fichera, G. 1959. Boundary Problems in Differential Equations. Madison: University of Wisconsin Press.
   Google Scholar
• He, W. J., H. J. Ding, and H. C. Hu. 1996. “A Necessary and Sufficient Boundary Integral Formulation for Plane Elasticity Problems.” Communications in Numerical Methods in Engineering 12 (7): 413–424. doi:10.1002/(SICI)1099-0887(199607)12:7<413::AID-CNM988>3.0.CO;2-2.
   Google Scholar
• Hsiao, G., and R. C. MacCamy. 1973. “Solution of Boundary Value Problems by Integral Equations of the First Kind.” SIAM Review 15 (4): 687–705. doi:10.1137/1015093.
   Web of Science ®Google Scholar
• Hu, H. C. 1992a. “Necessary and Sufficient Boundary Integral Equations for Plane Harmonic Functions.” Science in China Series A-Mathematics 35 (7): 861–869. doi:10.1360/ya1992-35-7-861.
   Google Scholar
• Hu, H. C. 1992b. “A Necessary and Sufficient Set of Boundary Integral Equations with Indirect Unknowns for Plate Bending Problem.” Acta Mechanica Sinica 8 (2): 127–135. doi:10.1007/BF02487159.
   Google Scholar
• Kuo, S. R., J. T. Chen, and S. K. Kao. 2013. “Linkage between the Unit Logarithmic Capacity in the Theory of Complex Variables and the Degenerate Scale in the BEM/BIEMs.” Applied Mathematics Letters 26 (9): 929–938. doi:10.1016/j.aml.2013.04.011.
   Web of Science ®Google Scholar
• Langtangen, H. P., and G. K. Pedersen. 2016. Scaling of Differential Equations. Berlin: Springer. ISBN 978-3-319-32725-9.
   Google Scholar
• Lin, C. K., and K. C. Wu. 2007. “Fundamental Solution of the Laplace Equation (Dimensional Analysis Viewpoint).” Journal of the Mathematical Society of the Philippines 30 (2–3): 94–102.
   Google Scholar
• Liu, C. S. 2007a. “An Effectively Modified Direct Trefftz Method for 2D Potential Problems considering the Domain’s Characteristic Length.” Engineering Analysis with Boundary Elements 31 (12): 983–993. doi:10.1016/j.enganabound.2007.04.006.
   Web of Science ®Google Scholar
• Liu, C. S. 2007b. “A Modified Trefftz Method for Two-dimensional Laplace Equation considering the Domain’s Characteristic Length.” Computer Modeling in Engineering and Sciences 21 (1): 53–65. doi:10.3970/cmes.2007.021.053.
   Web of Science ®Google Scholar
• Liu, C. S. 2007c. “A Highly Accurate Solver for the Mixed-Boundary Potential Problem and Singular Problem in Arbitrary Plane Domain.” Computer Modeling in Engineering and Sciences 20 (2): 111–122. doi:10.3970/cmes.2007.020.111.
   Web of Science ®Google Scholar
• Liu, C. S. 2009. “Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation.” Computer Modeling in Engineering and Sciences 851 (1): 1–17. doi:10.3970/cmes.2008.028.077.
   Google Scholar
• Romano, G., R. Barretta, and M. Diaco. 2018. “A Geometric Rationale for Invariance, Covariance and Constitutive Relations.” Continuum Mechanics and Thermodynamics 30 (1): 175–194. doi:10.1007/s00161-017-0595-5.
   Web of Science ®Google Scholar