NODE POSITIONING ALGORITHM
will be enforced. In this case, the coefficients of Equation (1) are determined so that the CVBEM approximation function matches the specified boundary conditions at the given colloca- tion points with the exception of possible truncation, round-off, and other numerically-introduced errors.
Importantly, since the basis functions are selected such
function is a linear combination of analytic functions and is, therefore, analytic itself. Hence, the real and imaginary parts of the CVBEM approximation function are related by the Cauchy-Riemann equations:
(2)
From the Cauchy-Riemann equations, it follows that the real and imaginary parts of the CVBEM approximation function are harmonic functions. That is,
(3)
Consequently, since the real and imaginary parts of the CVBEM approximation function satisfy Laplace’s equa- tion, the error of the approximation function is assessed by examining the closeness of fit of the approximation function to the specified boundary conditions. The maximum error of the CVBEM approximation function can be assessed using the maximum modulus principle for harmonic functions [17]. This is done by determining the maximum departure on the problem boundary of the CVBEM approximation function from the specified boundary conditions. That is, the maximum error, E, can be expressed as:
(4) where denotes the target potential function.
Additionally, since and are conjugate functions, their level curves are orthogonal, resulting in the well-known flow net graphical display that have been utilized in numerous applications in science, mathematics, engineering, and related fields [18]. This dual harmonic function outcome as a result of the CVBEM modeling process is of particular value when analyzing boundary value problems that incorporate both Dirichlet and Neumann boundary conditions. Such properties do not similarly exist for real valued functions and associated computational methods such as the finite difference method or finite element method (FEM). This is a principal motivation for using CVBEM approximation functions to solve boundary value problems of the Laplace and related types.
In this work, the coupled CVBEM/NPA approach is used to computationally solve difficult problems of groundwater flow where high-precision modeling outcomes are needed. The focus is to provide further evidence to supplement the usual geochemical evidence in order to determine the source of a contaminant in the groundwater from among a set of candidate source points given the location of the downgradient contamination site [19].
Summay of NPA Procedures and Apply- ing an NPA to the CVBEM
An important step in the CVBEM modeling procedure is the selection of computational nodes, which in this case, are the branch points of the basis functions described in [2]. Originally, the CVBEM methodology called for placing nodes on the problem boundary [2]. More recent implementations of the CVBEM have relaxed this constraint and used computa- to the approach taken in the MFS [20]. The next evolution of the procedure has involved the development and utilization of NPAs to systematically locate computational nodes with the goal of reducing computational model error [8, 9].
In general, the following steps were followed for the NPAs:
1. Generate an initial distribution of candidate compu- tational nodes and candidate collocation points. The candidate collocation points are located on the prob- lem boundary and are where the boundary conditions are enforced. The candidate computational nodes are
2. Initialize the NPA by selecting two collocation points to be used in the CVBEM model. Two collocation points are needed for the reason described above, which is that each complex coefficient of Equation (1) has two real parts. These unknown real values are determined by collocation. Therefore, each node corresponds to two collocation points.
3. For each of the candidate computational nodes, create a new CVBEM model by adding the candidate node to the current CVBEM model. Evaluate each of these models to determine the maximum error on the problem boundary using Equation (4) that corresponds to the use of the particular candidate node. Recall, the maximum error of the CVBEM approximation function is known to occur on the problem boundary as a consequence of the maximum modulus principle. After evaluating each of these models, select the model that resulted in the least maximum error with regard to satisfying the given boundary conditions, and add the appropriate node to the CVBEM model.
4. Evaluate the error of the CVBEM approximation function on the problem boundary using Equation (4). Select two new collocation points to be located on the problem boundary at the two greatest local maxima of the error function.
5. Repeat Steps 3 and 4 until 2n collocation points have been selected and n computational nodes have been selected. The final CVBEM model will consist of these 2n collocation points and n computational nodes. The resulting model can then be used to generate flownet graphical displays of the CVBEM approximation.
Variations of the procedure discussed in these steps have been considered. For example, the NPA discussed in [9] incor- porates a refinement procedure, which is used to re-evaluate the utility of previously-selected computational nodes and potentially replace those nodes with different nodes if it is found that the replacement further reduces the error of the CVBEM model.
However, the common feature of the recent NPAs is that the algorithm proceeds by alternately selecting two collocation points followed by one computational node. This is important because the challenge of selecting each additional node is
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