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Real time boundary element node location optimization

机译：实时边界元素节点位置优化

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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 L_2 and also L_r 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.
机译：边界元方法（BEM）计算机模型通常涉及使用节点，这些节点是奇异势函数的位置，例如欧几里得距离函数的对数或倒数。就识别而言，这些奇异功能通常与节点本身关联。复变量边界元方法（CVBEM）是类似类型的奇异势函数的另一种应用，并且包括非奇异但是控制偏微分方程（PDE）的基本解的其他函数。这些各种奇异的势函数构成了一个基础，该线性组合的跨度（适当时为实数空间或复数空间）为向量空间。作为近似方法的一部分，可以确定向量空间中最接近PDE和相关边界条件的精确解的元素（通常以最小二乘残差度量）。对BEM或CVBEM近似中使用的基函数类型的最新研究表明，相对于问题边界上可能的位置以及外部位置，通过优化奇异基函数的位置可以大大提高计算精度和效率问题边界的范围（通常是问题域的外部）。为了为建模节点（和相关的奇异基函数）开发最佳位置，本文提出的方法是开发一个实时边界元素节点定位模块，使程序用户可以单击和拖动节点（一次）。整个问题域的外部（也就是说，允许将节点放置在问题边界上或任意靠近问题边界，也可以将其放置在问题域联合边界的外部）。所提供的模块与在计算机程序Mathematica中内置的CVBEM程序相接口，以便随着节点的实时移动，各种类型的信息流到显示模块。显示的信息包括问题边界和域的图形，域联合边界的外部，用于表示问题边界条件的评估点，节点位置，L_2以及L_r范数中的建模误差，以及问题边界条件与对问题边界进行建模估计，以使模型与问题边界条件的拟合紧密度可视化。当目标节点在屏幕上移动时，这些各种信息形式会更改并显示给程序用户，从而使用户能够快速将目标节点导航到首选位置。一旦在某个优化位置上建立了一个节点，就可以单击另一个节点并将其拖动到新位置，同时减少过程中的建模错误。

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