Higher order finite and infinite elements for the solution of Helmholtz problems

Biermann, Jan; Estorff, Otto von; Petersen, Steffen; Wenterodt, Christina

doi:10.1016/j.cma.2008.11.009

Cited by 26 publications

(12 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The first approach is the stabilized finite element method [2,3] which was developed by the modifications of the classical Galerkin method, such as the Galerkin/least-squares finite element method (GLS) [14][15][16], the residual-free bubbles method (RFB) [17] and the quasi-stabilized finite element method (QSFEM) [18]. Another effective strategy is the high-order finite element method, including the p-version FEM [19][20][21][22] and the partition of unity method (PUM) [23]. Although these methodologies are very effective in controlling the pollution effect, the dispersion still exists when resorting to these methods for acoustic analysis.…”

Section: Introductionmentioning

confidence: 99%

Dispersion Error Analysis of Stable Node-Based Finite Element Method for the Helmholtz Equation

Hu¹,

Cui²,

Zhang³

et al. 2018

CICP

View full text Add to dashboard Cite

Numerical dispersion error is inevitable when the finite element method is employed to simulate acoustic problems. Studies have shown that the dispersion error is essentially rooted at the "overly-stiff" property of the standard FEM model. To reduce the dispersion error effectively, a discrete model that provides a proper softening effects is needed. Thus, the stable node-based smoothed finite element method (SNS-FEM) which contains a stable item is presented. In this paper, the SNS-FEM is investigated in details with respect to the pollution effect. Different kinds of meshes are employed to analyze the relationship between the dispersion error and the parameter involved in the stable item. To ensure the SNS-FEM can be applied in the practical engineering problems effectively, the relationship is finally constructed based on the hexagonal patch for the commonly used unstructured mesh is very similar to it. By minimizing the discretization error, an optimal parameter equipped in this novel SNS-FEM is formulated. Both theoretical analysis and numerical examples demonstrate that the SNS-FEM with the optimal parameter reduces the dispersion error significantly compared with the FEM and the well performed GLS, especially for mid-and high-wave number problems. AMS subject classifications: 35J05, 65C20, 65N30, 65N22, 68U20, 81U30 Key words: Acoustic, numerical method, the stable node-based smoothed finite element method (SNS-FEM), dispersion error, hexagonal patches.

show abstract

Section: Introductionmentioning

confidence: 99%

Dispersion Error Analysis of Stable Node-Based Finite Element Method for the Helmholtz Equation

Hu¹,

Cui²,

Zhang³

et al. 2018

CICP

View full text Add to dashboard Cite

show abstract

“…They are, (i) the stabilized FEM, such as the Galerkin/leastsquares finite element method (GLS) [3,4], the quasistabilized finite element method (QSFEM) [5]. (ii) higher order methods, such as generalized high order approximations ( p-version) [6,7], the partition of unity method (PUM) [8,9] and the discontinuous enrichment method (DEM) [10,11] (iii) meshless method, such as element-free Galerkin method (EFGM) [12,13]. They all can give improved solutions compared to the standard FEM, however, properly "softened" stiffness for the discrete model is much more effective and direct to the root of the numerical pollution error [14].…”

Section: Introductionmentioning

confidence: 99%

Dispersion free analysis of acoustic problems using the alpha finite element method

Liu

Zhong

et al. 2010

Comput Mech

View full text Add to dashboard Cite

The classical finite element method (FEM) fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known "pollution error" caused by the numerical dispersion, i.e. the numerical wave number is always smaller than the exact one. This dispersion error is essentially rooted at the "overly-stiff" feature of the FEM model. In this paper, an alpha finite element method (α-FEM) is then formulated for the acoustic problems by combining the "smaller wave number" model of FEM and the "larger wave number" model of NS-FEM through a scaling factor a ∈ [0, 1]. The motivation for this combined approach is essentially from the features of "overly-stiff" FEM model and "overly-soft" NS-FEM model, and accurate solutions can be obtained by tuning the α-FEM model. A technique is proposed to determine a particular alpha with which the α-FEM model can possess a very "close-to-exact" stiffness, which can effectively reduce the dispersion error leading to dispersion free solutions for acoustic problems. Theoretical and numerical studies shall demonstrate the excellent properties of the present α-FEM.

show abstract

“…、边界元法(Boundary element method ， BEM) [5][6] 、无网格法 [7][8][9][10] 和光滑有限元法(Smoothed finite element method, SFEM) [11][12][13][14] 等，其中有限元法是最常用的声学数值方法。有限元法对各种声学问题具有良好的适应性，计算简单，效率较高。但有限元系统的刚度偏硬，在声学问题上表现为数值波数与实际波数存在相位误差，从而导致色散误差，并且随波数的增大而增大。为了使有限元法获得可月 2016 年 8 月崔向阳等：二维声学数值计算的梯度最小二乘加权 53 靠的精度，人们通常采用增加网格剖分密度或提高插值阶次 [15] 的方法。DERAEMAEKER 等 [16] 研究发现，为保证可靠的计算精度，单位波长内的节点数要在 10 个以上，这在中高频问题上很难实施。通过增加网格剖分密度或提高插值阶次的方法来降低色散误差，提高求解精度的方法，需要更多的存储空间和计算时间。近年来，有许多学者通过其他新的方法来控制色散误差，取得了很大的进展。 BOUILLARD 等 [7] 将无单元法 (Element free Galerkin, EFG)应用到声学数值计算中，提高了计算精度，然而，基于移动最小二乘构造的无单元法的形函数不满足 Kronecker delta 性质，边界条件施加困难 [17] ，为数值实现带来了不便。姚等将分区光滑径向点插值法 (Cell-based smoothed radial point interpolation method, CS-RPIM) [18] 推广到声学计算中，使模型刚度得到软化，有效地控制了色散误差；也有学者将混合有限元-无网格法 [19][20] 推广到声学计算中，在一定程度上提高了计算精度。然而，无网格法计算效率普遍偏低，很难用来解决计算规模庞大的实际工程问题。针对 FEM 和无网格法的优缺点，将移动最小二乘(Moving least-squares, MLS)权函数 [21] 应用到…”

unclassified

Gradient Weighted by Moving Least-squares for Two Dimension Acoustic Numerical Computation

Cui¹

2016

JME

View full text Add to dashboard Cite

Abstract：It is well known that one key issue of analyzing acoustic problems using finite element method (FEM) is "numerical dispersion error" due to the "overly stiff" nature of the FEM. This will cause the accuracy deterioration in the solution when it comes to high wave number or irregular meshes. To overcome this problem, a gradient weighted by moving least-squares (GW-MLS) is presented for analyzing acoustic problem. The gradient of acoustic pressure is reconstructed by the weight function of moving least-squares (MLS). And this makes the GW-MLS model much softer than the "overly stiff" FEM model. In acoustic GW-MLS, the acoustic mass matrix and the vectors of boundary integrals are constructed by the standard FEM to insure the integral accuracy of the mass matrix and the boundary conditions applied on region boundary accurately. Numerical examples including a 2D tube model and car cavity model are presented. The results demonstrate that the GW-MLS reduces the numerical dispersion error effectively. And because of this, the GW-MLS achieves higher accuracy as compared with FEM when meshes are seriously distorted especially in calculating high wave number problems.

show abstract

Higher order finite and infinite elements for the solution of Helmholtz problems

Cited by 26 publications

References 43 publications

Dispersion Error Analysis of Stable Node-Based Finite Element Method for the Helmholtz Equation

Dispersion Error Analysis of Stable Node-Based Finite Element Method for the Helmholtz Equation

Dispersion free analysis of acoustic problems using the alpha finite element method

Gradient Weighted by Moving Least-squares for Two Dimension Acoustic Numerical Computation

Contact Info

Product

Resources

About