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

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.

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.

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.

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

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

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

Time Analysis Time Number of Nodes 1.5 1 0.5 100 200 300 400 500 600 100 200 300 400 500 600 700 800 900 1000 Number of Nodes

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.

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

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

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.

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.

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.

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

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.

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.

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

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

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.

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.

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

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.

Definition sketch of domain for phase change of soil water

Boundary Element method nodal definitions

Boundary Element method node numbering scheme

Finite Element model discretization

Comparison between FEM and BEM results

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