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[1] Demoes, N.J., Bann, G.T., Wilkins, B.D., Grubaugh, K.E. & Hromadka II, T.V., Optimizationalgorithm for locating computational nodal points in the method of fundamentalsolutions to improve computational accuracy in geosciences modeling. TheProfessional Geologist, 2019.
[2] Bathe, K.J., Finite Element Procedures. Prentice Hall, Pearson Education, Inc., 1996.
[3] Young, D., Chen, K., Chen, J. & Kao, J., A modified method of fundamental solutionswith source on the boundary for solving laplace equations with circular and arbitrarydomains. Computer Modeling in Engineering and Sciences, 19(3), pp. 197–221, 2007. [Crossref]
[4] Fornberg, B. & Flyer, N., Fast generation of 2-D node distributions for mesh-free PDEdiscretizations. Computers and Mathematics with Applications, 69, pp. 531–544, 2015. [Crossref]
[5] Liu, Y., Nie, Y., Zhang, W. & Wang, L., Node placement method by bubble simulationand its application. Computer Modeling in Engineering and Sciences, 55(1), pp.89–109, 2010.
[6] Shankar, V., Kirby, R.M. & Fogelson, A.L., Robust Node Generation for Mesh-free discretizationson irregular domains and surfaces. SIAM Journal on Scientific Computing,40(4), pp. 2584–2608, 2018. [Crossref]
[7] Slak, J. & Kosec, G., Fast generation of variable density node distributions for meshfreemethods. Boundary Elements and other Mesh Reductions Methods XLI. WIT Press:Southampton and Boston, pp. 163–173, 2019.
[8] Chen, C., Karageorghis, A. & Li, Y., On choosing the location of the sources in the MFS.Numerical Algorithms (Springer), 72(1), pp. 107–130, 2016. [Crossref]
[9] Hromadka II, T. & Guymon, G., Application of a boundary integral equation to predictionof freezing fronts in soil. Cold Regions Science and Technology, 6, pp. 115–121,1982. [Crossref]
[10] Wilkins, B.D., Greenberg, J., Redmond, B., Baily, A., Flowerday, N., Kratch, A., HromadkaII, T.V., Boucher, R., McInvale, H.D. & Horton, S., An unsteady two dimensionalcomplex variable boundary element method. SCIRP Applied Mathematics, 8(6),pp. 878–891, 2017. [Crossref]
[11] Wilkins, B.D., Hromadka II, T.V. & Boucher, R., A conceptual numerical model of thewave equation using the complex variable boundary element method. Applied Mathematics,8(5), p. 724, 2017. [Crossref]
[12] Hromadka II, T.V. & Guymon, G.L., A complex variable boundary element method:Development. International Journal for Numerical Methods Engineering, 20, pp.25–37, 1984. [Crossref]
[13] Johnson, A.N. & Hromadka II, T.V., Modeling mixed boundary conditions in a Hilbertspace with the complex variable boundary element method (CVBEM). MethodsX, 2,pp. 292–305, 2015. [Crossref]
[14] Hromadka II, T. & Pardoen, G.C., Application of the CVBEM to non-uniform St.Venant torsion. Computer Methods in Applied Mechanics and Engineering, 53(2), pp.149–161, 1985. [Crossref]
[15] Hromadka II, T.V. & Lai, C., The Complex Variable Boundary Element Method.Springer-Verlag, New York, 1987.
[16] Hromadka II, T. & Whitley, R., A new formulation for developing CVBEM approximationfunctions. Engineering Analysis with Boundary Elements, 18(1), pp. 39–41, 1996. [Crossref]
[17] Hromadka II, T.V. & Whitley, R.J., Advances in the Complex Variable Boundary ElementMethod. Springer, New York, 1998.
[18] Hromadka II, T. & Whitley, R., Foundations of the Complex Variable Boundary ElementMethod. Springer, 2014.
[19] Brebbia, C.A., The Boundary Element Method for Engineers. Wiley, 1978.
[20] Brebbia, C. & Wrobel, L., Boundary element method for fluid flow. Advances in WaterResources, 2, pp. 83–89, 1979. [Crossref]
[21] Hromadka II, T.V., A Multi-Dimensional Complex Variable Boundary Element Method,volume 40 of Topics in Engineering. WIT Press: Southampton and Boston, 2002.
[22] Johnson, A.N., Hromadka II, T.V., Hughes, M.T. & Horton, S.B., Modeling mixedboundary problems with the Complex Variable Boundary Element Method (CVBEM)using matlab and mathematica. International Journal of Computational Methods andExperimental Measurements, 3(3), pp. 269–278, 2015. [Crossref]
[23] Wilkins, B.D., Hromadka II, T.V., Johnson, A.N., Boucher, R., McInvale, H.D. & Horton,S., Assessment of complex variable basis functions in the approximation of idealfluid flow problems. International Journal of Computational Methods and ExperimentalMeasurements, 7(1), pp. 45–56, 2019. [Crossref]
[24] Kirchhoff, R.H., Potential Flows Computer Graphic Solutions. CRC Press, 1985.
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Open Access
Research article

Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM)

Bryce D. Wilkins1,
Theodore V. Hromadka II2,
Jackson McInvale3
1
Carnegie Mellon University, USA
2
United States Military Academy, USA
3
Duke University, USA
International Journal of Computational Methods and Experimental Measurements
|
Volume 8, Issue 4, 2020
|
Pages 289-315
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
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Abstract:

In this paper, we introduce a new node positioning algorithm (NPA) for determining suitable locations of the computational nodes that are a typical feature of mesh reduction numerical methods for partial differential equations – specifically, the Complex Variable Boundary Element Method (CVBEM). The novelty of the introduced NPA is a ‘position refinement’ procedure, which facilitates the relocation of nodes that are already being used in the current CVBEM model when such relocation reduces the maximum error of the associated CVBEM model. The results of the new NPA (referred to as NPA2) are compared to the results obtained using the recent NPA described in [1] (referred to as NPA1). We compare NPA1 and NPA2 by modeling two example Dirichlet boundary value problems that have been selected due to having regions of extreme curvature in the analytic flow regime that are difficult to model computationally. Consequently, these problems serve as good benchmark problems for testing the efficacy of the current and future NPAs. Our empirical findings suggest that the use of NPA2 can reduce the maximum error of the associated CVBEM model by several orders of magnitude compared to the corresponding result obtained using NPA1.

Keywords: applied complex variables, Complex Variable Boundary Element Method (CVBEM), mesh reduction methods, node positioning algorithms (NPAs), potential flow
References
[1] Demoes, N.J., Bann, G.T., Wilkins, B.D., Grubaugh, K.E. & Hromadka II, T.V., Optimizationalgorithm for locating computational nodal points in the method of fundamentalsolutions to improve computational accuracy in geosciences modeling. TheProfessional Geologist, 2019.
[2] Bathe, K.J., Finite Element Procedures. Prentice Hall, Pearson Education, Inc., 1996.
[3] Young, D., Chen, K., Chen, J. & Kao, J., A modified method of fundamental solutionswith source on the boundary for solving laplace equations with circular and arbitrarydomains. Computer Modeling in Engineering and Sciences, 19(3), pp. 197–221, 2007. [Crossref]
[4] Fornberg, B. & Flyer, N., Fast generation of 2-D node distributions for mesh-free PDEdiscretizations. Computers and Mathematics with Applications, 69, pp. 531–544, 2015. [Crossref]
[5] Liu, Y., Nie, Y., Zhang, W. & Wang, L., Node placement method by bubble simulationand its application. Computer Modeling in Engineering and Sciences, 55(1), pp.89–109, 2010.
[6] Shankar, V., Kirby, R.M. & Fogelson, A.L., Robust Node Generation for Mesh-free discretizationson irregular domains and surfaces. SIAM Journal on Scientific Computing,40(4), pp. 2584–2608, 2018. [Crossref]
[7] Slak, J. & Kosec, G., Fast generation of variable density node distributions for meshfreemethods. Boundary Elements and other Mesh Reductions Methods XLI. WIT Press:Southampton and Boston, pp. 163–173, 2019.
[8] Chen, C., Karageorghis, A. & Li, Y., On choosing the location of the sources in the MFS.Numerical Algorithms (Springer), 72(1), pp. 107–130, 2016. [Crossref]
[9] Hromadka II, T. & Guymon, G., Application of a boundary integral equation to predictionof freezing fronts in soil. Cold Regions Science and Technology, 6, pp. 115–121,1982. [Crossref]
[10] Wilkins, B.D., Greenberg, J., Redmond, B., Baily, A., Flowerday, N., Kratch, A., HromadkaII, T.V., Boucher, R., McInvale, H.D. & Horton, S., An unsteady two dimensionalcomplex variable boundary element method. SCIRP Applied Mathematics, 8(6),pp. 878–891, 2017. [Crossref]
[11] Wilkins, B.D., Hromadka II, T.V. & Boucher, R., A conceptual numerical model of thewave equation using the complex variable boundary element method. Applied Mathematics,8(5), p. 724, 2017. [Crossref]
[12] Hromadka II, T.V. & Guymon, G.L., A complex variable boundary element method:Development. International Journal for Numerical Methods Engineering, 20, pp.25–37, 1984. [Crossref]
[13] Johnson, A.N. & Hromadka II, T.V., Modeling mixed boundary conditions in a Hilbertspace with the complex variable boundary element method (CVBEM). MethodsX, 2,pp. 292–305, 2015. [Crossref]
[14] Hromadka II, T. & Pardoen, G.C., Application of the CVBEM to non-uniform St.Venant torsion. Computer Methods in Applied Mechanics and Engineering, 53(2), pp.149–161, 1985. [Crossref]
[15] Hromadka II, T.V. & Lai, C., The Complex Variable Boundary Element Method.Springer-Verlag, New York, 1987.
[16] Hromadka II, T. & Whitley, R., A new formulation for developing CVBEM approximationfunctions. Engineering Analysis with Boundary Elements, 18(1), pp. 39–41, 1996. [Crossref]
[17] Hromadka II, T.V. & Whitley, R.J., Advances in the Complex Variable Boundary ElementMethod. Springer, New York, 1998.
[18] Hromadka II, T. & Whitley, R., Foundations of the Complex Variable Boundary ElementMethod. Springer, 2014.
[19] Brebbia, C.A., The Boundary Element Method for Engineers. Wiley, 1978.
[20] Brebbia, C. & Wrobel, L., Boundary element method for fluid flow. Advances in WaterResources, 2, pp. 83–89, 1979. [Crossref]
[21] Hromadka II, T.V., A Multi-Dimensional Complex Variable Boundary Element Method,volume 40 of Topics in Engineering. WIT Press: Southampton and Boston, 2002.
[22] Johnson, A.N., Hromadka II, T.V., Hughes, M.T. & Horton, S.B., Modeling mixedboundary problems with the Complex Variable Boundary Element Method (CVBEM)using matlab and mathematica. International Journal of Computational Methods andExperimental Measurements, 3(3), pp. 269–278, 2015. [Crossref]
[23] Wilkins, B.D., Hromadka II, T.V., Johnson, A.N., Boucher, R., McInvale, H.D. & Horton,S., Assessment of complex variable basis functions in the approximation of idealfluid flow problems. International Journal of Computational Methods and ExperimentalMeasurements, 7(1), pp. 45–56, 2019. [Crossref]
[24] Kirchhoff, R.H., Potential Flows Computer Graphic Solutions. CRC Press, 1985.
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Wilkins, B. D., Ii, T. V. H., & Mcinvale, J. (2020). Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM). Int. J. Comput. Methods Exp. Meas., 8(4), 289-315. https://doi.org/10.2495/CMEM-V8-N4-289-315
B. D. Wilkins, T. V. H. Ii, and J. Mcinvale, "Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM)," Int. J. Comput. Methods Exp. Meas., vol. 8, no. 4, pp. 289-315, 2020. https://doi.org/10.2495/CMEM-V8-N4-289-315
@research-article{Wilkins2020ComparisonOT,
title={Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM)},
author={Bryce D. Wilkins and Theodore V. Hromadka Ii and Jackson Mcinvale},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2020},
page={289-315},
doi={https://doi.org/10.2495/CMEM-V8-N4-289-315}
}
Bryce D. Wilkins, et al. "Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM)." International Journal of Computational Methods and Experimental Measurements, v 8, pp 289-315. doi: https://doi.org/10.2495/CMEM-V8-N4-289-315
Bryce D. Wilkins, Theodore V. Hromadka Ii and Jackson Mcinvale. "Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM)." International Journal of Computational Methods and Experimental Measurements, 8, (2020): 289-315. doi: https://doi.org/10.2495/CMEM-V8-N4-289-315
WILKINS B D， HROMADKA II T V， MCINVALE J. Comparison of Two Algorithms for Locating Computational Nodes in the Complex Variable Boundary Element Method (CVBEM)[J]. International Journal of Computational Methods and Experimental Measurements， 2020， 8(4)： 289-315. https://doi.org/10.2495/CMEM-V8-N4-289-315