Real time boundary element node location optimization

Boundary Element Method (BEM) computer models typically involve use of nodal points that are the locations of singular potential functions such as the logarithm or reciprocal of the Euclidean distance function. These singular functions are typically associated with the nodes themselves as far as identification. The Complex Variable Boundary Element Method (CVBEM) is another application of similar types of singular potential functions and includes other functions that are not singular but are fundamental solutions of the governing partial differential equation (PDE). These various singular potential functions form a basis whose span of linear combinations (either real or complex space, as appropriate) is a vector space. As part of the approximation approach, one determines that element in the vector space that is closest (usually in a least squares residual measure) to the exact solution of the PDE and related boundary conditions. Recent research on the types of basis functions used in a BEM or CVBEM approximation has shown that considerable improvement in computational accuracy and efficiency can be achieved by optimizing the location of the singular basis functions with respect to possible locations on the problem boundary and also locations exterior of the problem boundary (in general, exterior of the problem domain). To develop such optimum locations for the modeling nodes (and associated singular basis functions), the approach presented in this paper is to develop a Real Time Boundary Element Node Location module that enables the program user to click and drag nodes (one at a time) throughout the exterior of the problem domain (that is, nodes are allowed to be positioned on or arbitrarily close to the problem boundary, and also to be positioned exterior of the problem domain union boundary). The provided module interfaces with the CVBEM program, built within computer program Mathematica, so that various types of information flows to the display module as the node is moved, in real time. The information displayed includes a graphic of the problem boundary and domain, the exterior of the domain union boundary, evaluation points used to represent problem boundary conditions, nodal locations, modeling error in L2 and also L∞ norms, and a plot of problem boundary conditions versus modeling estimates on the problem boundary to enable a visualization of closeness of fit of the model to the problem boundary conditions. As the target node is moved on the screen, these various information forms change and are displayed to the program user, enabling the user to quickly navigate the target node towards a preferred location. Once a node is established at some optimized location, another node can then be clicked upon and dragged to new locations, while reducing modeling error in the process.