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Pollution studies for high order isogeometric analysis and finite element for acoustic problems

机译：高等距几何分析的污染研究和声学问题的有限元分析

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摘要

It is well known that Galerkin finite element methods suffer from pollution error when solving wave problems. To reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave-like functions so that the exact wavenumber is incorporated into the finite element approximation space. Solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied. Recently, a plane wave enriched isogeometric analysis was introduced for acoustic problems. However, it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other. In this work, we show that isogeometric analysis outperforms finite element method in dealing with pollution. We observe similar behaviour when both the methods are enriched with plane waves. Using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions. However, the latter still leads to smaller errors using similar number of degrees of freedom. In conclusion, we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re-meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour. (C) 2019 Elsevier B.V. All rights reserved.
机译：众所周知，Galerkin有限元方法在求解波动问题时会遇到污染误差。为了减少污染对解决方案的影响，提出了各种方法来丰富具有波动函数的有限元方法，从而将精确的波数合并到有限元逼近空间中。在文献中还研究了用等几何分析解决波动问题的问题，其中研究了由于等高基础的连续性而导致的等几何分析的优越行为。最近，针对声学问题引入了富含平面波的等几何分析。但是，仍然没有量化这些不同方法对污染的影响或它们彼此之间如何表现。在这项工作中，我们表明在处理污染方面，等几何分析的性能优于有限元方法。当两种方法都富含平面波时，我们观察到类似的行为。与具有更多函数的低阶多项式相比，使用具有较少富集函数的高阶多项式似乎可以改善污染。但是，使用相似数量的自由度，后者仍会导致较小的误差。总而言之，我们提出统一等几何分析的划分可以成为解决波浪问题的有效工具，因为富集消除了在更高频率下对网格进行重新网格划分的需要，而且由于其即使在粗糙网格以及其改善的污染行为。 （C）2019 Elsevier B.V.保留所有权利。

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