数值流形法自被提出以来,在结构分析、渗流分析、裂纹扩展等多个方面都取得了众多应用。但这些问题的计算区域大多是有限区域,即所谓的内问题。对于地下和地表结构、波传导等一系列问题,需要考虑场变量在远场的行为,该类问题被称为无界域问题或外问题。基于数值流形法,构造了适用于无界域问题的有限元覆盖及其权函数,根据所求场变量在无穷处的渐进性质来构造局部逼近,以此反映解在趋于无限时的行为。不同于有限单元法中无限单元的形函数,本方法中权函数仅需满足单位分解,局部逼近反映场变量在片上的局部性质,这使得对场变量的逼近更加合理。经算例验证,结果表明:该方法构造方式合理,能够使用较少的计算单元,获得准确的计算结果。
Abstract
Since its invention, the Numerical Manifold Method has been applied to the analysis of a wide variety of problems, including the analysis of structures, seepage flow, and crack propagation. These problems typically involve bounded domains, or interior problems. However, for problems such as underground and surface structures, wave propagation, and other unbounded domain problems or exterior problems, the behavior of field variables in the far field needs to be considered in the numerical solution process. In this study, we constructed a finite element cover and weight functions suited to unbounded domain problems using the Numerical Manifold Method. With consideration of the asymptotic behavior of field variables at infinity, a local approximation was constructed to approach the behavior of the infinite domain. Unlike the shape function of infinite elements in the finite element method, the weight function in our proposed method only needs to satisfy the partition of unity, while the local approximation needs to approximate the behavior of the field variables, which makes the approximation of field variables more reasonable. The results from numerical examples demonstrate that the proposed method is effective and yields accurate results with fewer elements.
关键词
数值流形法 /
无界域问题 /
无限单元 /
无限片 /
线性水波问题
Key words
numerical manifold method /
unbounded domain problems /
infinite elements /
infinite patches /
linear water wave problem
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