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Analytical- and Boundary Elements based integral representation for numerical solution of 3-D potential problems in heterogeneous media containing singularities

机译：基于奇异点和边界元的积分表示法，用于包含奇异性的非均质介质中3-D势问题的数值解

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摘要

Mathematical formulation of processes in different areas of physics, e. g. fluid flows and transport phenomena through porous media, electrostatics and electrodynamics, heat conduction etc. lead to boundary value problems for partially differential equation elliptical type in term of a potential function which is characteristic for the considered process. In many engineering applications, the processes mentioned above, have behaviours that mathematically represent singularities (point sources/sinks or line sources/sinks etc. in 2-D or 3-D domain). Generally the numerical solutions with classical numerical methods like FVM or FVM lead to difficulties. In the paper a solution that eliminates these difficulties and shortcomings are present a method based on integral representation of the solution, combining the Analytical Elements and Boundary Elements . The presentation will be focused especially on the flow problems in porous media where the singularities are wells or drains shaped as thin objects of finite length etc. The 3-D flow domain is constituted of several arbitrary distributed sub-domains with different conductivity's (e.g. hydraulic conductivity) called Non-Singularity-Objects (NSO) and several point- source/sink line-source/sink as sub-domains called Singularity-Objects (SO) which can be located arbitrarily regarding the NSO. On the boundary of the NSO transition conditions between the internal and external flow should be satisfied while on the boundary of the SO only boundary conditions for the external process are required. The singular integral representation of the solution of boundary value problems for each object will be used as theoretical base for a BEM 3D software implementation.
机译：不同物理领域中过程的数学公式，例如G。流体通过多孔介质的流动和传输现象，静电和电动力学，热传导等，导致了偏微分方程椭圆型边值问题，这是所考虑过程的特征。在许多工程应用中，上述过程的行为在数学上表示奇点（2-D或3-D域中的点源/汇点或线源/汇点等）。通常，使用经典数值方法（例如FVM或FVM）进行数值求解会带来困难。在本文中，解决了这些困难和缺点的解决方案提出了一种基于解决方案的整体表示方法，结合了分析元素和边界元素的方法。本演讲将特别关注多孔介质中的流动问题，其中奇异点是井或排水管，形状像是有限长度的薄物体等。3-D流域由几个具有不同电导率的任意分布子域组成（例如，水力电导率）称为非奇异性对象（NSO）和几个点源/汇点线源/汇聚作为子域，称为奇异性对象（SO），可以相对于NSO任意放置。在NSO的边界上，应满足内部和外部流之间的过渡条件，而在SO的边界上，仅需要外部过程的边界条件。每个对象的边值问题解决方案的奇异积分表示将用作BEM 3D软件实现的理论基础。

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