Interval arithmetic in modeling and solving Laplace's equation problems with uncertainly defined boundary shape

Eugeniusz Zieniuk; Marta Kapturczak

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Interval arithmetic in modeling and solving Laplace's equation problems with uncertainly defined boundary shape

机译：不确定定义边界形状的建模与求解拉普拉斯方程问题的区间算术

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摘要

The paper presents the problem of modeling the uncertainty of the boundary shape in boundary problems (described by Laplace's equation) and proposes a method for solving so-defined problems. The uncertainty of the boundary shape is modeled using interval numbers. The interval coordinates of control points, defined using classical and directed interval arithmetic, are considered in this paper. However, because of some disadvantages of well-known interval arithmetic, the authors propose a modification of directed interval arithmetic. This arithmetic is also used in the proposed modification of the traditional parametric integral equation system (PIES) method (previously used for exactly defined problems) to solve so-defined problems. Control points of the appropriate curves, necessary to define the boundary shape, are directly included in the mathematical formalism of the mentioned method. The developed interval method is tested on problems of various shapes, modeled with linear and curvilinear segments. The correctness of the obtained solutions is verified using proposed alternative methods. The obtained solutions indicate a very high potential of the proposed method in solving problems with an uncertainly defined boundary shape.

机译：本文提出了建模边界问题中边界形状的不确定性的问题（由拉普拉斯方程描述），并提出了一种解决如下问题的方法。边界形状的不确定性使用间隔数进行建模。本文考虑了使用经典和定向间隔算法定义的控制点的间隔坐标。然而，由于众所周知的间隔算法的一些缺点，作者提出了一种定向间隔算法的修改。该算法也用于传统参数积分方程系统（PIE）方法的建议修改（以前用于精确定义的问题）以解决所定义的问题。确定边界形状所需的适当曲线的控制点直接包括在所提到的方法的数学形式中。开发的间隔方法对各种形状的问题进行了测试，用线性和曲线段建模。使用所提出的替代方法验证所得溶液的正确性。所获得的解决方案表示求解不确定限定的边界形状的问题的提出方法的非常高的潜力。

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来源
《Engineering analysis with boundary elements》 |2021年第4期|110-123|共14页
作者
Eugeniusz Zieniuk; Marta Kapturczak;
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作者单位

Institute of Computer Science University of Bialystok 15-245 Konstantego Ciolkowskiego 1M Bialystok Poland;

Institute of Computer Science University of Bialystok 15-245 Konstantego Ciolkowskiego 1M Bialystok Poland;

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收录信息美国《科学引文索引》(SCI);美国《工程索引》(EI);
原文格式 PDF
正文语种 eng
中图分类
关键词
Uncertainty modeling; Boundary value problems; Interval arithmetic; Parametric integral equations system;

机译：不确定性建模;边值问题;间隔算法;参数积分方程系统;

Interval arithmetic in modeling and solving Laplace's equation problems with uncertainly defined boundary shape

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