Isogeometric analysis for acoustic fluid-structure interaction problems
Graphical abstract
Introduction
Vibration leading to noise in the environment is becoming one of the major concerns in our daily usage of industrial products. Products having quieter operation are given higher priority e.g., in passenger cabins, machinery etc. In fact, most automobile, aerospace industries are considering acoustically comfortable designs at the development stage itself. In order to design the products, it is very much required to characterize the dynamic behavior of the structures. Structures also have different behavior under the influence of the surrounding fluid medium and necessitates a coupled analysis [1]. Often their dynamic behavior changes with geometry, the material properties of the solid as well as the fluid medium in contact with. Analytical solutions which are normally based on the a priori modal solutions restricts the problem domain to very simple geometries. Tanaka et al [2] reported that the modal coupling theorem is valid under the assumption that the fluid medium is non-dense and the cavity walls are not thin. Thus, numerical solutions are often sought to study the behavior of the coupled system. Of the many numerical techniques, the commonly used numerical techniques are Finite Difference Method (FDM), Finite Element Method (FEM), Boundary Element Method (BEM), Finite Volume Method (FVM), Fast Multipole Method (FMM) etc. FEM is one of the widely accepted numerical techniques. FEM basically requires meshing of the domain. In conventional finite element analysis, a Computer Aided Design (CAD) model is first converted to an Analysis Suitable Geometry (ASG) by meshing the domain and the analysis is carried over the meshed structure/domain. Geometrical approximations also exist while converting the CAD geometry to a meshed domain. This meshing process consumes more time in a product design life cycle [3]. As product design process is an iterative process and design changes happen quite often in its design life cycle, the changes has to be updated in the model and then re-mesh the model to analyze the system behavior. Not only the tedious task of meshing exists in analyzing the model using FEM, the numerical solutions with FEM often suffers from numerical dispersions even in non-dispersive mediums. This leads to requirement of minimum 10 nodes per wavelength [4]. Thus, for coupled problems, the required degrees of freedom to achieve a substantially accurate result is increased due to the numerical dispersion error.
In 2005, Hughes et al [5] proposed Isogeometric Analysis, which basically incorporates the same bases used to model the geometry for analysis as well. In doing so, a tight coupling between CAD and analysis is obtained which eases the process of model updation in the product cycle. In CAD, NURBS has become a very standard tool for modeling complex geometries like free form surfaces, conic sections etc. Also, the NURBS has certain properties like partition of unity, compact support, non-negative or positive bases, higher continuity, affine transformation etc., which makes it more suitable as bases for the analysis as well. In IGA, there is no need of converting the CAD geometry into a meshed structure and thus, no geometry approximation is considered. A higher regularity is also obtained in using NURBS bases in IGA. IGA has been implemented to study the behavior of systems in many areas of engineering viz., structural mechanics [6], [7], fluid mechanics [8], [9], acoustics [10], electromagnetics [11], [12] etc.
Cotrell et al [13] studied IGA for structural vibrations and reported that no optical branching is observed in the frequency spectra when higher regularity NURBS bases are used. They reported that NURBS bases give lesser dispersion error compared to the conventional Lagrange bases used in FEM. Reali et al [14] studied the effect of increase in the order of approximation spaces with Lagrangian and NURBS bases for one-dimensional structural vibration and wave propagation problems. They demonstrated that with the increase in the order of approximation space, Lagrangian bases showed diverging behavior in the higher modes whereas NURBS bases showed higher convergence behavior for all the range of modes.
Simpson et al [10] proposed isogeometric based boundary element method for acoustics and reported superior convergence compared to Lagrangian based discretization. Peake et al [15], [16] proposed Extended Boundary Element Method (XIBEM) in two-dimensional and three-dimensional acoustic wave scattering problems and observed better convergence compared to conventional BEM. Idesman et al [17] studied elastic wave propagation using IGA and observed lesser dispersion error compared to conventional FEM. Using IGA, Nørtoft et al [18] analyzed sound propagation through laminar flow in 2-dimensional ducts. Wu et al [19] used IGA in modeling interior acoustics and observed better performances over -FEM. Venås [20] reported that IGA showed better convergence for exterior acoustic scattering and acoustic fluid-structure interaction with Infinite Elements (IE). Coox et al.et al [21] studied the performance of IGA for interior acoustics in two-dimensional problems and reported that the errors observed in the eigen frequencies and the eigen vectors with IGA are lesser than the conventional Lagrangian FEM.
Several other new methods are also gaining importance in Acoustic fluid-structure interaction problems. Chen et al [22] have considered Chebyshev polynomial expansions for approximating the field variables and have shown its applicability for an arbitrarily restrained rectangular plate coupled to a cavity with general wall impedance. Xie et al [23] have also considered Chebyshev polynomial expansions for approximating the field variables. They proposed a variational formulation for vibro-acoustic analysis of a panel backed by an irregularly-bounded cavity using domain partitioning technique. Li et al [24] proposed Smooth-FEM for coupled structural-acoustic problems. They reported that the gradient smoothing over the edge-based and face-based smoothing domains has an softening effect compared to the overly stiff models in conventional FEM and leads to significant improvements in the accuracy of the solution for the coupled system. He et al [25] proposed α-FEM acoustic problems which makes use of a combination of overly-stiff and overly-soft models. They reported that by choosing a proper value of α, dispersion error can be reduced leading to better accurate solutions. Gong et al [26] have also proposed cell-based smoothed three-node Mindlin plate element for coupled structural acoustic problems. They reported that the solution accuracy of the cell based SFEM coupled to face based SFEM (FS-FEM) improves the solution accuracy compared to conventional FEM due to the smoothing effects.
On the other hand, there are several formulations available in the literature depending on the primary variable of the fluid medium. Some of them are based on displacement [27], displacement potential [28], velocity potential [29] and mixed formulation [30], [31] etc. Of all the formulations, even though displacement-pressure formulation leads to non-symmetric system matrices, they are still widely accepted as it does not increase the degrees of freedom and the bandedness of the system matrices while other formulations based on velocity leads to increase in degrees of freedom as well as the appearance of spurious rotational modes [27] in the numerical solution. Bermúdez et al [32] also showed the different formulations in acoustic fluid structure interaction problems and reported that displacement-pressure formulation showed lowest eigen frequency errors compared to all other formulations.
From the literature, it is observed that the IGA may be a promising analysis tool for acoustic-fluid structure interaction problems. In the present study, IGA discretization based on variational formulation is presented for acoustic fluid-structure interaction problems. In Section 2, the problem statement and the variational formulations are presented. In Section 3, formulation of Isogeometric analysis for acoustic fluid structure problem is presented. In Section 4, two benchmark problems viz., coupling of i) a beam and a rectangle cavity in two dimension; and ii) a flexible plate coupled to a cuboidal cavity in three dimension with weak and strong couplings are considered. Also, a circular plate coupled to a doubly curved barrel shaped cavity filled with water is also considered. The efficiency of IGA is shown by comparing the IGA solutions to classical FEM solutions. Finally, the applicability of IGA is illustrated through an example of a two dimensional car cavity coupled to a beam which is considered to act like a sunroof of a car.
Section snippets
Governing equations of an acoustic fluid-structure interaction problem
In acoustic fluid-structure interaction problems, the primary physical variables of interest are the structural displacement () and the fluid pressure (pf). The governing equation of structural part is the momentum balance while the governing equation in fluid part is the wave equation. The solid/structure as well as the fluid medium are assumed to be linear systems. The schematic of the acoustic fluid-structure interaction problem is shown in Fig. 1. Let Ωs and Ωf are domains
Isogeometric analysis of acoustic fluid-structure interaction
This section describes the application of isogeometric analysis for acoustic fluid-structure interaction problems. In subsection 3.1, a brief introduction to NURBS is described. Readers are directed to Piegl and Tiller [33] for a complete description on NURBS. In subsection 3.2, NURBS based discrete form of governing equations for the coupled problem is presented.
Numerical Results
Two benchmark problems viz., a beam coupled to rectangular cavity in two dimension (2D) and a plate coupled to a cuboidal cavity in three dimension (3D) are considered. To study the performance, both weak and strong coupling are considered with the acoustic medium as air and water respectively. Also, a circular plate coupled to a doubly curved barrel shaped cavity filled with water is also considered. Finally, a car cavity filled with air coupled to a beam acting as the sunroof is considered.
Conclusion
Four problems viz. a beam coupled to rectangular fluid cavity, a Kirchhoff plate coupled to a cuboidal cavity, a circular plate coupled to a doubly curved barrel shaped cavity and a car interior cavity in two dimension coupled to a beam acting as the sunroof are solved using Isogeometric Analysis and Finite Element Method. The performance of IGA is studied for the various problems considered.
In the two dimension, FEM results showed higher errors which becomes more prominent for strong coupling
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2021, Composite StructuresCitation Excerpt :The dimensions and material properties of the cavity & fluid studied in Ref. [21] are Lx = 10 m; Ly = 4 m; E = 210 GPa; second moment of area of the beam = 1.59 × 10-4 m4, mass per unit length = 50 kg/m; density of air = 1.2 kg/m3; sonic velocity of air = 340 m/s. Based on progressive mesh refinement, 30x9 is found to be adequate to model the cavity size considered here. Structural frequencies of the beam, frequencies of the fluid cavity and coupled frequencies calculated using the present work match well with the results reported in Ref. [21] as given in Table 1. The SPL obtained at the center of the fluid cavity for a steady-state mechanical harmonic point load (1 N) excited at the center of the beam is also compared with the response presented in Ref. [21] as shown in Fig. 4 to validate the response calculation.