NODE POSITIONING ALGORITHM
reduced to a single variable optimization problem since the only quantity that is varying is the location of the computa- tional node to be added to the CVBEM model. Ordinarily, such optimization problems are solved by taking the partial derivative of an objective function and setting the result equal to zero. In this case, the objective function is Equation (4), and we solve this problem computationally by testing each candidate computational node to determine which one optimizes the objective function by resulting in the CVBEM model of least error.
While it can be computationally expensive to use these NPAs, the benefit of using them is that it is possible to achieve highly efficient uses of computational nodes and collocation points. That is, using these NPAs can result in CVBEM models that use relatively few nodes and collocation points, but still achieve the same or better accuracy compared to much larger CVBEM models where the locations of the collocation points and nodes have not been determined by an NPA.
Problem Description
In this work, we consider three different example problems in which the goal is to determine the source of accumulated groundwater contamination detected downgradient from a set of candidate source points. In these example problems, we have hypothesized that one of the candidate source points is a leaking underground storage tank (LUST), which is the true source of the detected groundwater contaminant. The task is to use computational modeling of the groundwater flow scenario in order to accurately identify the LUST. Of course, in real-world analyses, computational modeling such as this is just one prong of a multi-pronged analysis that includes sampling of the groundwater and conducting geochemical tests that provide aging information and chemical degradation information, among other metrics.
The problem geometries that are considered in this work were selected for two reasons. First, they are fundamental and can be used as “building blocks” in the development of more sophisticated potential flow models. Second, each problem incorporates areas of extreme curvature in the flow regime, which are difficult to model computationally and require highly precise numerical models in order to obtain satisfactory results. Thus, the example problems are both challenging and relevant to problems of interest in the geosciences and other fields. For each problem, we develop two numerical approxi- mations: (i) a coupled CVBEM/NPA model and (ii) an FEM model. The analytic solutions for these problems are known and adapted from [18]. The availability of the analytic solu- tions is important because it allows us to precisely quantify the computational error of the CVBEM and FEM models.
Table 1 - Example Problem 1 - Problem Description
Problem Domain: Governing PDE: Boundary Conditions:
Number of Candidate Computational Nodes for CVBEM Model: 500 Number of Candidate Collocation Points for CVBEM Model: Number of Nodes for FEM Model: Number of Elements for FEM Model:
56 TPG •
Apr.May.Jun 2021 1,000
154,769 306,176
www.aipg.org
Each of the problems is formulated as a Dirichlet boundary value problem with boundary conditions specified from the target potential function. As a consequence of the Cauchy- Riemann equations, the CVBEM approximation of the target flow function can be developed directly based on the CVBEM approximation of the target potential function. That is, the CVBEM procedure allows us to generate flowlines using only boundary data from the potential function and vice-versa. However, for the FEM approach, the target flowlines must be developed using a post-processing vector gradient procedure.
Once the flowlines are developed, we identify the flowline going through the contamination site. Then, this flowline can be traced upgradient to the source of the contamination. We assess accuracy by determining how well the flowlines gener- ated by each numerical method trace the flowlines generated by the analytic solution. In particular, we are interested in whether each numerical method accurately identifies the LUST.
In the following problems, we assume the groundwater is represented by an ideal and incompressible fluid with no vor- ticity. Under these conditions, the velocity potential describing the groundwater flow situation is a scalar function satisfying Laplace’s equation.
Additionally, these example problems each take advantage of symmetry in order to reduce the size of modeling area of interest. In particular, each problem is symmetric about the vertical axis. Therefore, the problem domain is considered to be the right half-plane corresponding to Re(z) computational outcomes depicted in the left half-plane were obtained by reflecting the computed outcomes in the right half- plane. By taking advantage of the symmetry of each problem situation, we are able to avoid branch cuts associated with the analytic solutions intersecting the problem domain. This feature makes these problems well-suited for being modeled by the CVBEM.
Example Problem 1
Potential Flow Around a Constrained Circular Obstruction
The analytic representation of the velocity potential for this problem is given in [18] as
(5)
The problem domain is defined in Table 1 such that the exact solution is analytic everywhere within . Thus, the real and imaginary parts of Equation (5) are harmonic functions in .
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