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[1] Hromadka II, T. & Guymon, G., Application of a boundary integral equation to predic- tion of freezing fronts in soil. Cold Regions Science and Technology, 6, pp. 115–121, 1982. [Crossref]
[2] Hromadka II, T. V. & Guymon, G. L., A complex variable boundary element method: development. International Journal for Numerical Methods in Engineering, 20, pp. 25–37, 1984. [Crossref]
[3] Hromadka II, T. V. & Guymon, G. L., The complex variable boundary element method. International Journal of Numerical Methods Engineering, 1984.
[4] Johnson, A. N., Hromadka II, T. V., Hughes, M. T. & Horton, S. B., Modeling mixed boundary problems with the complex variable boundary element method (CVBEM) using matlab and mathematica. International Journal of Computational Methods and Experimental Measurements, 3(3), pp. 269–278, 2015. v3-n3-269-278 [Crossref]
[5] Wilkins, B. D., Hromadka II, T. V., Johnson, A. N., Boucher, R., McInvale, H. D. & Hor- ton, S., Assessment of complex variable basis functions in the approximation of ideal fluid flow problems. International Journal of Computational Methods and Experimen- tal Measurements, 7(1), pp. 45–56, 2019. [Crossref]
[6] Wilkins, B. D. & Hromadka II, T. V., Using the digamma function for basis functions in mesh-free computational methods. Engineering Analysis with Boundary Elements, 2021 (in press).
[7] Demoes, N. J., Bann, G. T., Wilkins, B. D., Grubaugh, K. E. & Hromadka II, T. V., Optimization algorithm for locating computational nodal points in the method of fun- damental solutions to improve computational accuracy in geosciences modelling. The Professional Geologist, 2019.
[8] 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). International Journal of Computational Methods and Experimental Mea- surements, 8(4), 2020.
[9] Demoes, N. J., Bann, G. T., Wilkins, B. D., Hromadka II, T. V. & Boucher, R., 35 years of advancements with the complex variable boundary element method. International Journal of Computational Methods and Experimental Measurements, 7(1), pp. 1–13, 2018. [Crossref]
[10] Wilkins, B. D., Greenberg, J., Redmond, B., Baily, A., Flowerday, N., Kratch, A., Hro- madka II, T. V., Boucher, R., McInvale, H. D. & Horton, S., An unsteady two-dimen- sional complex variable boundary element method. SCIRP Applied Mathematics, 8(6), pp. 878–891, 2017. [Crossref]
[11] Johnson, A. N. & Hromadka II, T. V., Modeling mixed boundary conditions in a Hilbert space with the complex variable boundary element method (CVBEM). MethodsX, 2, pp. 292–305, 2015. [Crossref]
[12] Tikhonov, A., On the stability of inverse problems. In Proceedings of the USSR Acad- emy of Sciences, 1943.
[13] Tikhonov, A., Solution of incorrectly formulated problems and the regularization method. Soviet Mathematics Doklady, 4, pp. 1035–1038, 1963. https://doi.org/10.1134/ s1064562418030250
[14] Hromadka II, T. V., Linking the complex variable boundary-element method to the analytic function method. Numerical Heat Transfer, 7, pp. 235–240, 1984. https://doi. org/10.1080/01495728408961822
[15] Kirchhoff, R. H., Potential Flows Computer Graphic Solutions, CRC Press, 1985.
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Acadlore takes over the publication of IJCMEM from 2025 Vol. 13, No. 3. The preceding volumes were published under a CC BY 4.0 license by the previous owner, and displayed here as agreed between Acadlore and the previous owner. ✯ : This issue/volume is not published by Acadlore.

Open Access
Research article

A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function

bryce d. wilkins1,
theodore v. hromadka ii2
1
Carnegie Mellon University, USA
2
Hromadka & Associates, USA
International Journal of Computational Methods and Experimental Measurements
|
Volume 10, Issue 3, 2022
|
Pages 237-259
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
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Abstract:

Two approaches for formulating a computational Complex Variable Boundary Element Method (CVBEM) model are examined. In particular, this paper considers a collocation approach as well as a least squares approach. Both techniques are used to fit the CVBEM approximation function to given boundary conditions of benchmark boundary value problems (BVPs). Both modeling techniques provide satisfactory computational results, when applied to the demonstration problems, but differ in specific outcomes depending on the number of nodes used and the type of BVP being examined. Historically, the CVBEM has been implemented using the collocation approach. Therefore, the novelty of this work is in formulating the least squares approach and applying the least squares formulation to a Dirichlet BVP as well as a mixed BVP. This work does not claim that one technique should always be used over the other, but rather it seeks to demonstrate the viability of the least squares approach and assert that both techniques for determining the coefficients of the CVBEM approximation function should be considered during the modeling process.

Keywords: Applied complex variables, Complex Variable Boundary Element Method (CVBEM), Least squares, Mesh-reduction methods, Potential flow

1. Introduction

2. The Collocation Approach

3. The Least Squares Adpproach

4. Demonstration Problems and Numerical Results

5. Conclusions

References
[1] Hromadka II, T. & Guymon, G., Application of a boundary integral equation to predic- tion of freezing fronts in soil. Cold Regions Science and Technology, 6, pp. 115–121, 1982. [Crossref]
[2] Hromadka II, T. V. & Guymon, G. L., A complex variable boundary element method: development. International Journal for Numerical Methods in Engineering, 20, pp. 25–37, 1984. [Crossref]
[3] Hromadka II, T. V. & Guymon, G. L., The complex variable boundary element method. International Journal of Numerical Methods Engineering, 1984.
[4] Johnson, A. N., Hromadka II, T. V., Hughes, M. T. & Horton, S. B., Modeling mixed boundary problems with the complex variable boundary element method (CVBEM) using matlab and mathematica. International Journal of Computational Methods and Experimental Measurements, 3(3), pp. 269–278, 2015. v3-n3-269-278 [Crossref]
[5] Wilkins, B. D., Hromadka II, T. V., Johnson, A. N., Boucher, R., McInvale, H. D. & Hor- ton, S., Assessment of complex variable basis functions in the approximation of ideal fluid flow problems. International Journal of Computational Methods and Experimen- tal Measurements, 7(1), pp. 45–56, 2019. [Crossref]
[6] Wilkins, B. D. & Hromadka II, T. V., Using the digamma function for basis functions in mesh-free computational methods. Engineering Analysis with Boundary Elements, 2021 (in press).
[7] Demoes, N. J., Bann, G. T., Wilkins, B. D., Grubaugh, K. E. & Hromadka II, T. V., Optimization algorithm for locating computational nodal points in the method of fun- damental solutions to improve computational accuracy in geosciences modelling. The Professional Geologist, 2019.
[8] 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). International Journal of Computational Methods and Experimental Mea- surements, 8(4), 2020.
[9] Demoes, N. J., Bann, G. T., Wilkins, B. D., Hromadka II, T. V. & Boucher, R., 35 years of advancements with the complex variable boundary element method. International Journal of Computational Methods and Experimental Measurements, 7(1), pp. 1–13, 2018. [Crossref]
[10] Wilkins, B. D., Greenberg, J., Redmond, B., Baily, A., Flowerday, N., Kratch, A., Hro- madka II, T. V., Boucher, R., McInvale, H. D. & Horton, S., An unsteady two-dimen- sional complex variable boundary element method. SCIRP Applied Mathematics, 8(6), pp. 878–891, 2017. [Crossref]
[11] Johnson, A. N. & Hromadka II, T. V., Modeling mixed boundary conditions in a Hilbert space with the complex variable boundary element method (CVBEM). MethodsX, 2, pp. 292–305, 2015. [Crossref]
[12] Tikhonov, A., On the stability of inverse problems. In Proceedings of the USSR Acad- emy of Sciences, 1943.
[13] Tikhonov, A., Solution of incorrectly formulated problems and the regularization method. Soviet Mathematics Doklady, 4, pp. 1035–1038, 1963. https://doi.org/10.1134/ s1064562418030250
[14] Hromadka II, T. V., Linking the complex variable boundary-element method to the analytic function method. Numerical Heat Transfer, 7, pp. 235–240, 1984. https://doi. org/10.1080/01495728408961822
[15] Kirchhoff, R. H., Potential Flows Computer Graphic Solutions, CRC Press, 1985.
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Wilkins, B. D. & Hromadka II, T. V. (2022). A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function. Int. J. Comput. Methods Exp. Meas., 10(3), 237-259. https://doi.org/10.2495/CMEM-V10-N3-237-259
B. D. Wilkins and T. V. Hromadka II, "A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function," Int. J. Comput. Methods Exp. Meas., vol. 10, no. 3, pp. 237-259, 2022. https://doi.org/10.2495/CMEM-V10-N3-237-259
@research-article{Wilkins2022ALS,
title={A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function},
author={Bryce D. Wilkins and Theodore V. Hromadka Ii},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2022},
page={237-259},
doi={https://doi.org/10.2495/CMEM-V10-N3-237-259}
}
Bryce D. Wilkins, et al. "A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function." International Journal of Computational Methods and Experimental Measurements, v 10, pp 237-259. doi: https://doi.org/10.2495/CMEM-V10-N3-237-259
Bryce D. Wilkins and Theodore V. Hromadka Ii. "A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function." International Journal of Computational Methods and Experimental Measurements, 10, (2022): 237-259. doi: https://doi.org/10.2495/CMEM-V10-N3-237-259
WILKINS B D， HROMADKA II T V. A Least Squares Approach for Determining the Coefficients of the CVBEM Approximation Function[J]. International Journal of Computational Methods and Experimental Measurements， 2022， 10(3)： 237-259. https://doi.org/10.2495/CMEM-V10-N3-237-259