Modeling Mixed Boundary Problems with the Complex Variable Boundary Element Method (Cvbem) Using Matlab and Mathematica

Modeling Mixed Boundary Problems with the Complex Variable Boundary Element Method (Cvbem) Using Matlab and Mathematica

Anthony N. Johnson T. V. Hromadka Li M. T. Hughes S. B. Horton 

Department of Mathematical Sciences, United States Military Academy, USA

Page: 
269-278
|
DOI: 
https://doi.org/10.2495/CMEM-V3-N3-269-278
Received: 
N/A
| |
Accepted: 
N/A
| | Citation

OPEN ACCESS

Abstract: 

The complex variable boundary element method or CVBEM is a numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy integral equation in complex analysis. Given the potential values (i.e. a Dirichlet problem) along the boundary, the typical problem is to use the potential function to solve the governing Laplace equation. In this approach, it is not necessary to know the streamline values on the boundary. The modeling approach can be extended to problems where the streamline function is needed because there are known streamline values along the problem boundary (i.e. a mixed boundary value problem). Two common problems that have such conditions are insulation on a boundary and fluid flow around a solid obstacle. In this paper, five advances in the CVBEM are made with respect to the modeling of the mixed boundary value problem; namely (1) the use of Mathematica and Matlab  in tandem to calculate and plot the flow net of a boundary value problem. (2) The magnitude of the size of the problem domain is extended. (3) The modeling results include direct computation and development of a flow net. (4) The graphical displays of the total flownet are developed simultaneously. And (5) the nodal point location as an additional degree of freedom in the CVBEM modeling approach is extended to mixed boundaries. A demonstration problem of fluid flow is included to illustrate the flownet development capability.

Keywords: 

approximate boundary, collocation, complex variable boundary element method (CVBEM), complex variables, mixed boundary conditions, Mathematica, Matlab, MATLink

  References

[1] Hromadka, T.V.  & Guymon, G.L., The complex variable boundary element method. International Journal for Numerical Methods in Engineering, 1984.

[2] Whitley, R.J. & Hromadka, T.V., Theoretical developments in the complex variable boundary element method. Engineering Analysis with  Boundary  Elements, 30(12), pp. 1020-1024, 2006. doi: http://dx.doi.org/10.1016/j.enganabound.2006.08.002

[3] Hromadka, T.V. & Lai, C. The Complex Variable Boundary Element Method, Springer- Verlag: New York, NY, 1987.

[4] Hromadka, T.V. & Whitley, R.J., Advances in the Complex Variable Boundary Element Method, Springer: New York, NY, 1998.

[5] Hromadka, T.V. & Whitley, R., Foundations of the Complex Variable Boundary Element Method, Springer, 2014.

[6] Hromadka, T.V.  A Multi-dimensional Complex Variable  Boundary Element   Method. Topics in Engineering, Vol. 40. WIT Press: Billerica, MA, 2002.

[7] Johnson, A.N., Hromadka, T.V., Carroll, M., Hughes, M., Jones, L., Pappas, N., Thomasy, C., Horton, S., Whitley, R. & Johnson, M., A computational approach to determining {CVBEM} approximate boundaries. Engineering Analysis with Boundary Elements, 41(0), pp. 83-89, 2014. doi: http://dx.doi.org/10.1016/j.enganabound.2013.12.011