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首页> 外文期刊>Engineering analysis with boundary elements >Performance of iterative solvers for acoustic problems. Part Ⅰ. Solvers and effect of diagonal preconditioning
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Performance of iterative solvers for acoustic problems. Part Ⅰ. Solvers and effect of diagonal preconditioning

机译：声学问题迭代求解器的性能。第一部分。对角线预处理的求解器和效果

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Boundary element discretization of the Kirchhoff-Helmholtz integral equation gives rise to a linear system of equations. This system may be solved directly or iteratively. Application of direct solvers is quite common but turns out to be inefficient for large scale problem with 10,000 unknowns and more. These systems can be solved on behalf of iterative methods. This paper is dedicated to testing performance of four iterative solvers being the Restarted Bi-Conjugate Gradient Stabilized algorithm, the Conjugate Gradient method applied to the normal equations (CGNR), the Generalized Minimal Residual (GMRes) and the Transpose Free Quasi Minimal Residual. For that, we distinguish between internal and external problems. Performance of iterative solvers with respect to problem size, polynomial degree of interpolation, wave-number, wave-number over problem size, absorption at surface, and smoothness of the surface is investigated. Furthermore, the effect of diagonal preconditioning is illuminated. All examples consist of different meshes of up to more than 100,000 elements. In general, the methods perform well for the internal problems, a duct problem, a sedan cabin compartment and a fictitious small concert hall. GMRes proves to solve the problems most efficiently. External problems appear more challenging due to the hypersingular operator of the Burton and Miller formulation. Scattering of a plane wave at a sphere and at a cat's eye are investigated as well as a tire noise problem. The first two are remarkably efficiently solved in the medium and high frequency range by CGNR whereas the tire noise example is only solved by GMRes. In all examples, at least one or two solution methods turn out to require less operations than a direct solver. The effect of diagonal preconditioning is marginal especially for higher frequencies.
机译：Kirchhoff-Helmholtz积分方程的边界元离散化产生了一个线性方程组。该系统可以直接解决或迭代解决。直接求解器的应用非常普遍，但对于具有10,000或更多未知数的大规模问题却证明效率不高。这些系统可以代表迭代方法来求解。本文致力于测试四个迭代解算器的性能，这些算法是：重新启动的双共轭梯度稳定算法，适用于正态方程的共轭梯度方法（CGNR），广义最小残差（GMRes）和转置自由拟最小残差。为此，我们区分内部问题和外部问题。研究了迭代求解器在问题大小，多项式插值度，波数，超过问题大小的波数，表面吸收和表面光滑度方面的性能。此外，照亮了对角线预处理的效果。所有示例均包含多达100,000个元素的不同网格。通常，这些方法对于内部问题，管道问题，轿车厢室和虚拟小型音乐厅效果很好。 GMRes被证明可以最有效地解决问题。由于伯顿和米勒公式的超奇异算子，外部问题似乎更具挑战性。研究了平面波在球体和猫眼处的散射以及轮胎噪声问题。 CGNR在中高频范围内可以有效地解决前两个问题，而轮胎噪声示例仅可以通过GMRes解决。在所有示例中，与直接求解器相比，至少一种或两种求解方法需要的操作更少。对角预处理的影响微乎其微，特别是对于较高的频率。

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