1. The CVBEM: A Complex Variable Boundary Element Method for Mixed Boundaries
Investigators
Tony Johnson, T. V. Hromadka II and Steve Horton
Department of Mathematical Sciences, United States Military Academy, West
Point, NY 10996, United States

2. Background
The Complex Variable Boundary Element Method or CVBEM is a numerical method, which solves partial differential equations of the Laplace and Poisson type.
It is a generalization of the Cauchy integral formula into a boundary integral equation method.
This generalization allows an immediate and extremely valuable transfer of the modeling techniques used in real variable boundary integral equation methods (or boundary element methods) to the CVBEM.
Modeling techniques for dissimilar materials, anisotropic materials, and time advancement, can be directly applied without modification to the CVBEM.

3. Advantages of CVBEM over FDM/FEM
CVBEM does not require modeling nodal points to be defined on the problem boundary or in the interior of the problem domain.
Both the FDM and FEM techniques require nodal points to be defined on the problem boundary and within the problem domain.
The CVBEM also solves the governing PDE, while FEM and FDM are only able to develop approximations.

4. Advantages of CVBEM over FDM/FEM
The CVBEM provides an approximation function that is continuous throughout the problem domain where the other methods do not.
The CVBEM approximation function exists outside the problem domain, whereas the FEM and FDM approximations do not.
The CVBEM is more useful than analytical methods such as a Fourier series expansion because it is not limited to a specific domain.

6. Consider the twisting behavior of a elliptically oriented, homogeneous, isotropic shaft.

7. By the Cauchy integral formula, the value of w(z) at any point inside the closed contour C is determined by the values of the function along the boundary contour C.
w(z*)
w(z) C

8. Mixed Boundary condition
Evaluation points
Potentials
Mixed Boundary condition
Problem Domain x
Problem Boundary
Streamlines
Simple closed contour, C
Nodes

9. Time Analysis Time Number of Nodes 1.5 1 0.5 100 200 300 400 500 600
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500
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Number of Nodes

10. Two dimensional linear diffusion PDE is solved using CVBEM.
The approach is applicable modeling diffusion problems with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions.

11. Solution Methodology
The global initial-boundary value problem is decomposed into a steady-state component and a transient component.
The steady state part is modeled using CVBEM.
The transient part is modeled using by a linear combination of basis functions that are the products of a two-dimensional Fourier sine series and an exponential function

12. Solution Methodology
The global initial-boundary value problem is decomposed into a steady-state component and a transient component.
The steady state part is modeled using CVBEM.
The transient part is modeled using by a linear combination of basis functions that are the products of a two-dimensional Fourier sine series and an exponential function

13. Steady State Results
The Figure shows the transient approximation function as it converges to the transient initial condition using various numbers of basis functions (specified by n). The error distribution is shown below.

14. Steady State Results
Contour plot of CVBEM steady-state approximation. Collocation points are shown in red, and potential and streamlines are depicted.
Absolute error of CVBEM steady-state approximation on problem boundary.

15. Transient State Results
The Figure depicts the evolution of the global approximation function at various model times.
The approximations in this figure were created using 64 terms in the transient approximation function and 32 terms in the CVBEM approximation function.

16. Application of CVBEM for solving Diffusion and Wave equations was presented at the first colloquia at US Military West Point Academy, 2016.

17. Finite Volume Method (FVM) is popular in groundwater modeling community.
Complex Variable Boundary Element Method (CVBEM) solution is compared with that of bench mark Finite Volume method.
This is the first published comparison in literature.

18. Test Problem
The test problem is of two-dimensional ideal fluid flow in a 90-degree horizontal bend.
This is chosen due availability of the analytic solution, and the challenge of developing the flow field vector trajectories for a highly spatially variable flow field problem.
The focus is on comparing flow field trajectory vectors of the fluid flow, with respect to vector magnitude and direction.

19. Comparison of Vector Trajectories
Overlay of CVBEM velocity vectors
on FVM model velocity vectors
Overlay of CVBEM velocity vectors on
FVM model streamlines

20. Comparison of Vector Trajectories
Error measurement of velocity
magnitude of FVM & CVBEM solution
Error measurement of velocity vector
angle of FVM & CVBEM solution

21. Conclusions
Comparison of the CVBEM and FVM flow field trajectory vectors for the target problem of flow in a 90-degree bend shows good agreement between the considered methodologies.
The average relative error in velocity magnitude is 1.1% and in velocity direction is 0.15%.
This is the first such work in which velocity vectors developed by the CVBEM are compared to the results from an FVM model and the results indicate that the flow trajectory vectors developed from the CVBEM are correctly determined and properly represent the ideal fluid flow velocity and direction.

23. Objectives Five new advances in the CVBEM are presented. These are
The use of Mathematica and Matlab in tandem to calculate and plot the flow net of a boundary value problem.
The magnitude of the size of the problem domain is extended.
The modeling results include direct computation and development of a flow net.
The graphical displays of the total flownet are developed simultaneously.
The nodal point location as an additional degree of freedom in the CVBEM modeling approach is extended to mixed boundaries.

24. Wolfram Mathematica MATLAB 8 4 6 3 4 2 2 1 -2 -4 -1 -4 -2 2 4 6 8 10 2-2 -4 -1 -4 -2 2 4 6 8 10 2 4 6
Wolfram Mathematica
MATLAB

25. The Complex Variable Boundary Element Method or CVBEM is used to develop a computer model for estimating the location of the freezing front in soil-water phase change problems.
The model was applied over two-dimensional domain to predict the thermal regime of the soil system.

26. Definition sketch of domain for phase change of soil water

27. Boundary Element method nodal definitions

28. Boundary Element method node numbering scheme

29. Finite Element model discretization

30. Comparison between FEM and BEM results

31. Thank You