An enriched finite element method with interpolation cover functions for acoustic analysis in high frequencies
Introduction
As the increasingly requirements on the NVH (Noise, Vibration and Harshness) performance of vehicles (such as cars and airplanes), effective prediction and evaluation of acoustic performance is very essential in practical engineering applications. There are two types of numerical methods which are very prevalent for acoustic analysis, namely the classical finite element method (FEM) [1] and boundary element method (BEM) [2,3]. The BEM is more suitable for the calculation of exterior acoustic problems [4] (such as acoustic scattering and acoustic radiation) due to the automatically satisfaction of the Sommerfeld radiation condition, but for the interior acoustic problems, such as the acoustic analysis in automobile passenger compartments, researchers prefer to use the efficient FEM.
When the FEM is employed to analyze the acoustic problem which is governed by the Helmholtz equation, the troublesome numerical dispersion error always occurs and will become larger especially at high frequencies. In consequence, the obtained numerical solutions in high-frequency range are always inaccurate and unreliable [5]. Although improving the mesh quality (far beyond the so-called “rule of the thumb” [6]) can suppress the dispersion error to some extent, it will also require much more computational cost, especially when dealing with large-scale and complex 3D acoustic problems. To overcome this problem, researchers have proposed various numerical methods. One of them is the higher-order finite element method, including the p-version FEM [7] and the partition of unity method (PUM) [8]. However, the higher-order interpolation functions used in p-version FEM will lead to more computational efforts and the PUM may produce ill-conditioned system matrices. Another approach is the stabilized FEM, including the quasi-stabilized FEM [9], the Galerkin/least-squares FEM [10] and the residual-free bubbles method (RFB) [11], but these methods generally cannot completely eliminate dispersion error when they are applied to deal with the general acoustic problems. With the development of boundary-type and meshfree numerical techniques [12], [13], [14], [15], [16], the method of fundamental solutions (MFS) [17,18], singular boundary method [19], [20], [21], generalized finite difference method (GFDM) [22,23], the element-free Galerkin method (EFGM) [24], the point interpolation method (PIM) [25,26] have been developed for acoustic analysis. These methodologies have advantages over the standard FEM in terms of suppressing dispersion error. In addition, He et al. developed a mass-redistributed FEM [27], and Wang et al. proposed a gradient-weighted FEM [28]. These methods also show considerable potentials in reducing dispersion error.
In recent years, Liu and his coworkers [29] have combined the gradient smoothing techniques with the FEM to propose a series of smoothed finite element methods (S-FEMs). Some numerical researches have shown that the strain smoothing operations can effectively soften the stiffness of the discretized system, so more accurate results can be obtained than the standard FEM for solid mechanics. One of the S-FEMs is the node-based smoothed finite element method (NS-FEM) [30], but further researches showed that the NS-FEM exerts an excessive softening effect on the system stiffness which results in the temporal instability. Hence, NS-FEM is not suitable for acoustic analysis. Some improved methodologies, including the alpha finite element method (α-FEM) with a scale factor α [31] and the hybrid smoothed finite element method (HS-FEM) [32] have been developed to cure the drawbacks of NS-FEM. However, these methods also showed their own associated shortcomings. While it is found that the edge-based smoothed finite element method (ES-FEM) does not suffer from the temporal instability and can bring an appropriate softening effect to the discretized system, which means that ES-FEM can effectively achieve higher precision and suppress dispersion error at higher frequency range [33]. Unfortunately, the numerical dispersion error can only be suppressed by the ES-FEM at a very limited level and cannot be removed completely.
In this paper, an enriched finite element method (E-FEM) with interpolation cover functions is employed to analyze acoustic problems. In fact, this methodology has been proposed by Kim and Bathe for solid mechanics [34,35]. The main idea of this method is to enrich the standard finite element interpolation functions by using the additional interpolation cover functions to increase the accuracy and convergence rate of solutions. Kim [34] showed that this method is able to capture higher gradients of a field variable in the analysis of solid mechanics. In addition, this method has been applied to the analysis of shell elements [36,37]. Due to the good performance of this method in solid mechanics, in this work, we further extended it to solve acoustic problems, and the triangular elements and tetrahedral elements are still used because they are well adapted to complex geometric shapes. To clearly show the performance of the E-FEM in acoustic analysis, the corresponding numerical solutions from the FEM and ES-FEM are also provided for comparison. Several examples, including 2D and 3D acoustic problems, demonstrate that the E-FEM can provide better accuracy of numerical solutions and possess lower sensitivity to the distorted meshes without increasing the computational cost, and can effectively suppress dispersion error, especially for higher frequency range.
Below is the structure of the paper: Section 2 gives the basic equations. In Section 3, the formulation of the E-FEM for acoustic analysis is introduced in detail. In Section 4, a detailed dispersion analysis of the present E-FEM is performed. Section 5 summarizes the discretization error norm in computational acoustics. In Section 6, the performance of the E-FEM is evaluated by several numerical examples. Section 7 gives our conclusions.
Section snippets
Basic equations for acoustic problem
Assuming that the fluid medium in a finite domain Ω is in ideal conditions, the field acoustic pressure p′ in the medium can be described as followsin which, Δ denotes the Laplace operator, c is the acoustic velocity in the medium and t represents time, p′ is a harmonic perturbation and can be described aswhere , ω is the angular frequency, and the acoustic pressure p satisfies the following Helmholtz equationwhere the wavenumber is given by
The
Formulation of the E-FEM
In this sub-section, the discretized system equations for acoustic problem and the formulation of the present E-FEM are introduced detailedly. The standard weak formulation of Helmholtz equation is given by [33]where w is the test function. In the standard finite element formulation, the acoustic pressure p can be written using the following interpolationwhere pi and Ni denote the nodal field variable and the piecewise linear nodal
Dispersion analysis
As is known to all, when the standard FEM is applied to solve the Helmholtz equation, the numerical solutions always suffer from dispersion errors. Hence, it is important to investigate the dispersion properties of a numerical technique for acoustic analysis. In this sub-section, we will analyze and discuss the dispersion properties of the present E-FEM in detail based on the uniform mesh which is shown in Fig. 3, h and θ represent the average nodal distance and the angle between the wave
Discretization error
In the numerical analysis of acoustics, the control of pollution error is the focus of current research. According to the so-called “rule of the thumb” for computational acoustics [6], which describes that at least six elements should be required for one wavelength to ensure the reliability of the numerical results. However, the numerical error often increases significantly in the relatively high-frequency range even though the “rule of the thumb” is satisfied.
Generally, the gradient of
2D tube with Neumann boundary condition
As depicted in Fig. 5, a 2D numerical example is first investigated here. The dimension of the rectangular tube is 1.0m × 0.1m. The rectangular tube is filled with water and the acoustic velocity c in the medium is 1500m/s. Here we consider the Neumann boundary condition with vn = 10m/s, and the analytical solutions to this numerical example can be obtained bywhere ρ = 1000 kg/m3 is the density of the fluid medium.
When the 2D tube is bounded by
Conclusion
In this work, an enriched finite element method (E-FEM) in which the original linear nodal shape functions are enriched by the additional interpolation cover functions is proposed to analyze acoustic problems. We first give the fundamental formulations for the E-FEM in detail and then investigate its computational accuracy, convergence rate, computational efficiency and the sensitivity of meshes for acoustic analysis by using the benchmark case. Finally, we further evaluate the effectiveness of
Declaration of Competing Interest
None.
Acknowledgments
The first author wishes to thank the support of the Science and Technology Support Program Project of Guizhou Province (Grant No. [2021] General 341) and Fund of Equipment pre-research key laboratory (Grant No. 61420030404).
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