Abstract
Wavebased method (WBM), which is derived from indirect Trefftz method, is a new novel deterministic prediction technique for analyzing structural dynamicresponse problems in the midfrequency range. WBM has geometrical limitations, which can be overcome through complementgraph conception by introducing the multilevel modeling theory. To illustrate the feasibility and efficiency of WBM, the dynamicresponse process is implemented using a thinplate structure as a numerical example. The numerical result extends the ability of WBM to deal with the dynamic problem in concave domain.
1. Introduction
Structuralacoustic radiation problems are research hotspot issue in engineering design and manufacturing. The problem over the entire frequency range of interest can be categorized into three groups, i.e., lowfrequency, midfrequency and highfrequency ranges. The distinction among these different frequency ranges is based on the ratio between the freefield wavelength ($\lambda $) and the characteristic dimension ($L$) of design problem. Numerical calculation method is used as an efficient tool to analyze structuralacoustic problems. The most commonly used numerical prediction techniques for structuralacoustic problems are the deterministic element based methods, such as finite element method (FEM) and boundary element method (BEM). These element based methods use locally supported simple polynomials as shape functions within the elements. Given that these methods have no restrictions regarding the geometric complexity of a problem, they are efficient for analyzing a generally shaped model. However, the apparent drawback of these element based methods is the increase in number of elements and computational efforts to maintain reasonable prediction accuracy within highfrequency ranges and at short wavelengths. In addition, prediction results within the highfrequency range are substantially affected by interpolation and pollution errors. Thus, the practical applicability of element based methods is limited only to the lowfrequency range. Statistical energy analysis (SEA) has many advantages for predicting system response within the highfrequency range. However, SEA application is limited only to the highfrequency range and is not valid for midlow frequency range dynamicresponse problems because of the modal density insufficiency of SEA for obtaining reliable results. A socalled midfrequency gap exists between the application range of lowfrequency and highfrequencies. This gap has no available mature and adequate prediction techniques, thereby leading to a very high computational loading for elementbased methods and to a lack of randomness to make SEA applicable.
A novel and indirect Trefftz functionbased deterministic prediction technique has been introduced to improve midfrequency range computational efficiency and computational accuracy. The Trefftz method is a meshless numerical method for solving boundary value problems where approximate solutions are expressed as a linear combination of functions that automatically satisfy governing equations [1, 2]. The wavebased method (WBM) that is derived from the indirect Trefftz approach presents many advantages, such as relatively small system matrices, absence of pollution errors, and capacity to relax the frequency limitation [3]. WBM is suitable for coping with the midfrequency dynamic problem through an enhanced computational efficiency. WBM divides the problem domain into a limited number of large subdomains instead of dividing the problem domain into a large number of elemental domains. In addition, WBM facilitates description of the dynamicresponse variables by a set of wave function i.e., the exact solutions of the governing differential equations. This property of WBM implies that only an approximation is involved in the boundary conditions. Consequently, the system matrices are substantially smaller compared with the elementbased methods. Furthermore, WBM degrees of freedom (DOFs) are the contributory factors of the wave functions in the field variable expansion that leads to a substantial reduction in DOFs. The resulting smaller system matrices result in a higher computational efficiency that makes WBM capable of predicting the dynamicresponse problem in both lowfrequency range and midfrequency range.
Over the past 16 years, many WBMresults studies have dealt with several dynamic problems, such as platestructural dynamic problems and interior/exterior acousticradiation problems [4, 5]. The basic principles of WBM for convexplate problems have been discussed, and the beneficial convergence rate of WBM compared with that of either FEM or BEM has been validated for various twodimensional (2D) problem examples [6, 7]. WBM for the steadystate dynamicresponse problem of thin, flat plates coupled at an arbitrary angle, including both outofplane and inplane plate dynamic behaviors, has been discussed [8]. The purpose of the hybrid deterministicstatistical approach, in which both WBM and SEA are applied to deal with the midfrequency problem, has been presented [9, 10]. WBM has also been applied to analyze the acousticradiation problem that involves uncoupled interior and exterior acoustic radiation in the acoustic cavity [11, 12]. Furthermore, many studies aimed at extending the application of WBM have been published. The structuredesign sensitivity formulation of a semicoupled structuralacoustic problem using the WBM has been implemented [13]. Moreover, the thinplate vibration power flow problem with Kirchhoff plate bending theory has been analyzed with WBM theory, and results show that WBM is theoretically accurate [14].
However, one of the apparent disadvantages of WBM method is that, a sufficient condition for the WBM approximations to converge towards the exact solution is convexity domain of the considered problem. The tendency of the structuralacoustic domain to be nonconvex is one of the general structuralacoustic problems. Thus, the main drawback of the WBM is its inability to deal with geometrically complex constructions. To maximize the computational efficiency of WBM, its application was limited to moderate geometrical complexity system [15]. Nonconvex problems are partitioned into many convex subdomains. However, increased numbers of subdomains results in increased numbers of interfaces, it is a multidomain problem. This relationship shows an example of a multidomain problem, which leads to an additional computational loading during the course of WBM use. Moreover, some geometrical shapes such as plate with circular hole or inclusions cannot be divided into convex subdomains. To alleviate this convexitycondition problem, two WBM extension tactics, including hybrid WBMFEM and multilevel WBM, have been recently proposed. The hybrid WBMFEM involves a combination of the geometrical flexibility of the structural FEM and the indispensable convergence properties of the WBM; computational efficiency is also assured [16, 17]. Consequently, to overcome the geometrical restrictions, Bspline function is used to describe the curved edges, and the numerical integration procedure is used to evaluate the weighted residual formulation of WBM, respectively [18]. In response to the unbounded problem, a significantly enhanced WBM multilevel conception version has been introduced to efficiently analyze 2D acoustic scattering and inclusions problem [19, 20]. The basic theory of multilevel conception is to decompose the multiple scattering problems into a set of linked singlescattering problem; the multilevel theory has extended the application of WBM in the dynamicresponse problem of plate structure with holes or inclusions [21]. The bounded Helmholtz problem with multiple inclusions has been demonstrated using an efficient and flexible numerical strategy called multilevel wavebased numerical modeling [22].
The present study aims to develop a new novel approach to utilizing WBM toward a concave domain structure by introducing multilevel theory and complementgraph conception. Theory of complementgraph conception is investigated to alleviate or remove some of the geometrical constraints upon WBM application. The rest of this paper is organized as follows. The basic formulation of WBM is reviewed in Section 2. The complementgraph conception is performed in Section 3. The validity and numerical efficiency of the proposed method, as well as concavedomain numerical implementation, are illustrated in Section 4. Finally, the conclusions are drawn in Section 5.
2. Basic formulation of WBM
According to the indirect Trefftz principle, the wave functions accurately satisfy the governing dynamic equations. The general WBM modeling procedure consists of four steps: 1) partitioning of the problem domain into a number of convex subdomains, 2) wave function set selection of each convex subdomain, 3) WBM system matrix construction by using a weighted residual formulation of the boundary and interface conditions, 4) solution formulation of the system of equations to generate the wave function contribution factors and post processing of the dynamicresponse variables.
2.1. Problem description
Basic principles of WBM for a general convex plate design domain are briefly discussed. As shown in the Cartesian coordinate system in Fig. 1, a thinplate structure exists and is excited by a harmonic normal point force $F$ at position ${r}_{F}({x}_{F},{y}_{F})$. In Fig. 1, ${\mathrm{\Omega}}_{S}$ is the structure design domain, and ${\mathrm{\Gamma}}_{w\theta}$, ${\mathrm{\Gamma}}_{wm}$ and ${\mathrm{\Gamma}}_{wQ}$ are the structure boundary conditions.
Fig. 1Convex plate domain
Two types of platebending theories exist, namely, Kirchhoff platebending theory and ReissnerMindlin platebending theory. WBM as described in this research is based on the Kirchhoff platebending theory. In Kirchhoff theory, the rotational inertia and shear deformation effects are negligible. Considering the thinplate structure problem, the Kirchhoff platebending theory could yields accurate results in low and mid frequency range. According to the Kirchhoff platebending theory, the time harmonic outofplane displacements function $w(x,y)$ in the problem domain ${\mathrm{\Omega}}_{S}$ is governed by the following fourthorder partial differential equation:
where ${\nabla}^{4}={\partial}^{4}/\partial {x}^{4}+2{\partial}^{4}/\partial {x}^{2}\partial {y}^{2}+{\partial}^{4}/\partial {y}^{4}$. $\delta $ is the 2D Diracdelta function. The bending wave number ${k}_{b}$ and the plate structure bending stiffness $D$ are defined as: ${k}_{b}=\sqrt[4]{\rho t{\omega}^{2}/D}$ and $D=E(1+i\eta ){t}^{3}/12(1{\mu}^{2})$, respectively. In Eq. (1), symbols $t$, $E$, $\mu $, $\omega $, $\eta $, and $\rho $ represent the plate thickness, material elasticity modulus, material Poisson’s ratio, circular frequency of bending wave, material loss factor, and plate material density, respectively. In addition, symbols $i$ is the unit imaginary number. Kirchhoff bending theory is tested to be correct when the bending wave length $\lambda $ is six times greater the plate thickness $t$.
In platestructure dynamicresponse analysis, Kirchhoff platebending theory is governed by a fourthorder equation differential equation that uniquely defines the displacement function $w(x,y)$. The boundary conditions are specified for each evaluated point in the platestructure boundary. In addition, the boundary conditions of the considered plate domain ${\mathrm{\Omega}}_{S}$ has three parts (Fig. 1): 1) the kinematic boundary condition ${\mathrm{\Gamma}}_{w\theta}$ where the outofplane translational and the normal rotational displacements are imposed, 2) the mixed boundary conditions ${\mathrm{\Gamma}}_{wm}$ that impose the outofplane translational displacements and the normal bending moments, and 3) the mechanical boundary conditions ${\mathrm{\Gamma}}_{mQ}$ that impose the normal bending moments and the equivalent shear forces.
The differential operators for the outofplate displacement, then the rotational displacement, bending moment and generalized shear force are defined as:
where ${\gamma}_{n}$ and ${\gamma}_{S}$ are the two arc coordinates in the normal and tangential directions of the platestructure boundary, respectively.
2.2. The field variable expansion
For WBM application to bounded problems, the convexity of the considered problem domain is a sufficient condition for the theoretical convergence of WBM. This application approximated the field variables as an expansion based from functions that exactly satisfy the governing dynamic equations. In addition, an approximation error is induced in the boundary and interface conditions. Integrally minimizing this approximation error leads to the solution of the system.
In the considered design domain, the time harmonic outofplane displacement function $w(x,y)$ is approximated by the following field variable expansion:
where each wave function ${\mathrm{\Psi}}_{b}(x,y)$ satisfies the homogeneous part of the dynamic function shown by:
where $({k}_{b,x}^{2}+{k}_{b,y}^{2}{)}^{2}={k}_{b}^{4}$ and ${w}_{b}$ represent the unknown participation factors of wave functions; ${\mathrm{\Psi}}_{b}(x,y)$, ${k}_{b,x}^{}$ and ${k}_{b,y}^{}$ represent the $x$ and $y$ component of the wave vector, respectively; and ${n}_{b}$ is the number of wave functions. Function ${\widehat{w}}_{F}(x,y)$ is a particular solution function to consider the inhomogeneous portion of the dynamic equation that arises from the external loading applied to the plate structure. In addition, the wave functions are restricted to the interval [−1, 1] in the application region to ensure that the amplitudes of the wave functions are less than 1.
The particular solution function ${\widehat{w}}_{F}(x,y)$ satisfies the inhomogeneous part of the dynamic Eq. (1) arising from the external loading. Only an excitation by normal point forces is considered in this study, and the particular solution function for a point force loading of an infinite plate is given as:
where ${r}_{sF}=\sqrt{(x{x}_{F}{)}^{2}+(y{y}_{F}{)}^{2}}$, ${H}_{0}^{\left(2\right)}$ is the zerothorder Hankel function of the second kind. In Eq. (5), ${H}_{0}^{\left(2\right)}\left(x\right)={J}_{0}\left(x\right)i{Y}_{0}\left(x\right)$, ${J}_{0}\left(x\right)$ and ${Y}_{0}\left(x\right)$ are the Bessel functions of the first and second kind, respectively.
2.3. Weighted residual formulation
The field variable Eq. (3) fits exactly with the governing dynamic Eq. (1), irrespective of the unknown wave function contribution factors ${w}_{\mathrm{b}}$. These contribution factors are determined by minimizing the approximation errors of the boundary conditions and the continuity condition through the weighted residual formulation. The weighted residual formulation ${\widehat{w}}_{F}(x,y)$of the boundary conditions is defined as:
Substitution of the field variable expansion and the weighting function expansion into weighted residual formulation leads to a typical WBM system solution.
2.4. Wave function selection
In general, an infinite number of wave functions can satisfy the homogeneous differential dynamic Eq. (1). To apply WBM in numerical analysis, wave functions expansion must be truncated to a finite set. In WBM, the maximum wave number ${k}_{b}$, i.e., ${k}_{b}^{max}$, depends on the analysis frequency range and the dimensions of considered problem. Wave numbers ${k}_{b}$ (${k}_{b,x}$ or ${k}_{b,y}$) are selected based on the outline dimensions (${L}_{x}$×${L}_{y}$) of a (preferably the smallest) rectangular circumscribing the plate domain ${\mathrm{\Omega}}_{S}$.
Among the first types of wave number, ${k}_{b,x}$ and ${k}_{b,y}$ are chosen such that an integer number of half wavelengths equals the length of the rectangular box in the corresponding direction. The other component of the wave number is calculated from the wave number ${k}_{b}$ of the considered frequency. Vanmaele [6] has proposed two sets of wave functions, shown in Eq. (7). The first set of wave functions is cosine functionrelated whereas the second set is sine functionrelated. The two types of wave function are written as:
Desmet [4] proved that the first group of wave functions is theoretically sufficient for the convergence of the WBM in the convexconsidered domain. In this study, the cosineassociated wave functions are chosen, and wave numbers can be expressed as:
Among the plate structure domain, the number of bending wave functions ${n}_{b}$ that are included in the field variable expansion in Eq. (3), is related to the excited force frequency and the dimensions of the enclosing rectangular box: ${n}_{b}=4({n}_{b1}+1)+4({n}_{b2}+1)$.
However, finite wave functions can only be used in practical calculations. An infinite number of wave functions should satisfy the homogeneous differential equations. Therefore, the set of wave functions is truncated by applying a frequencydependent truncation rule. The truncation rule is stated as:
where $T$ ($\ge $2) is the truncation coefficient; a truncation coefficient of $T=$ 10 is adopted in this study. The integer truncation value ${n}_{b1}$ and ${n}_{b2}$. ${L}_{x}$ and ${L}_{\mathrm{y}}$ are the length parameter and width parameter of the smallest rectangle around the outline domain, respectively. Through this formula, the largest wave number ${k}_{b}^{max}$ of the bending wave functions included in the plate structure is obtained within the considered frequency. As a result, the number of wave functions increases approximately linearly with excitation frequency.
3. Multilevel modeling analysis
This section presents an efficient extension of multilevel modeling approach to the concave domain by using the complementgraph conception.
3.1. Basic conception of multilevel modeling
A sufficient condition to ensure the convergence of WBM is that the design domain should be convexity domain. This is the main drawback of WBM. In response, the nonconvex problem geometries must be decomposed into many convexity subdomains. As a result, when the number of subdomains increases, the integration length also increases. This further leads to an additional computational loading. On the other hand, some geometrical shapes, such as problem domain with circular and elliptic shapes, cannot be divided into convex subdomains. To deal with this problem, two kinds of approaches aim to overcome these limitations were introduced recently. The first method involves use of multilevel modeling framework to alleviate the problems considered domain decomposition and geometrical requirement of structural problems with inclusions. The second approach is utilizing a hybrid FEWBM formulation, i.e., FEM is applied in complex domain, whereas WBM is applied in convex domain. The multilevel modeling approach is also presented in this paper.
The theoretical basis for the multilevel conception is the superposition principle and the multiplescattering theories. In the multilevel theory of WBM, the original problem can be decomposed into a single bounded subproblem and one or more unbounded subproblems (also termed levels). The bounded subproblem boundary matched the boundary of original problem. In addition, the unbounded subproblems boundary matches as well as the original problem boundary. Subsequently, the original problem is decomposed into the following levels with corresponding wave function sets, boundary residuals, and weighting functions. In the multilevel theory, the dynamicresponse of various levels is superposed to form the dynamic response of the original problem.
Considering that the original problem converted bounded levels to unbounded levels, the former contains no holes or inclusions, and WBM tools become usable. In the multilevel modeling, the bounded levels ${\mathrm{\Omega}}_{b}$ describe the interior dynamic behavior of bounded subproblems. Meanwhile, the unbounded levels ${\mathrm{\Omega}}_{ub}$ consider the scattering behavior of an inclusion in an infinite homogeneous medium to model one hole or inclusion in an infinite domain. Consequently, this technique facilitates obtaining efficient and flexible numerical tactics for analyzing the midfrequency dynamics of complex geometrical domain.
To model these unbounded parts, new wave function sets are defined for unbounded plate problems. The solution function $w(x,y)$ is written as the superposition of several field variable expansions ${w}_{b}(x,y)$ and ${w}_{ub}(x,y)$, which need to satisfy the governing equation. The wave function ${w}_{b}(x,y)$ of bounded fields should fit the nonreflecting boundary conditions imposed at ${\mathrm{\Gamma}}_{\mathrm{\infty}}$. Meanwhile, the wave function ${w}_{ub}(x,y)$ of unbounded fields should satisfy the nonreflecting boundary conditions imposed at ${\mathrm{\Gamma}}_{b}$. Modeling each of the subproblems using WBM and coupling them using the superposition principle help obtain the consequent response fields of the original problem.
A multilevel modeling approach is developed for unbounded acoustic multiplescattering problems. The procedure for this approach involves four steps: 1) division of the original problem into various levels containing unbounded and bounded subdomains; 2) selection of wave functions for each level containing a subdomain design i.e., a wave function that can represent any arbitrary field on that boundary; 3) construction of the system of equations, and the weighted residual formulation; and 4) solution formulation and postprocessing showing each point representing combined levels. Through this modeling technique, the dynamicresponse problem of domain with concave boundary is investigated.
3.2. Basic conception of complement graph
The discussion on multilevel modeling in the previous section illustrates the feasibility and efficiency to cope with the 2D domain problem through multiple circular holes or square holes [19]. However, some difficulties are encountered, such as the concavedomain problem.
To overcome these geometrical limitations in WBM, a complementgraph conception is proposed in this study. The multilevel modeling theory and superposition principle constitute the basic foundation of complement graphconception. As shown in Fig. 2(a), a concave domain $\mathrm{\Omega}$ consists of two kinds of boundary characteristics i.e., convex curve ${\mathrm{\Gamma}}_{1}$, and concave curve ${\mathrm{\Gamma}}_{2}$. Considering the curve ${\mathrm{\Gamma}}_{2}$, the concave domain $\mathrm{\Omega}$ could not be divided into many kinds of convex subdomains, and the traditional multilevel modeling could also not solve this problem. To deal with these concavedomain problem limitation, the addition a convex complement graph domain ${\mathrm{\Omega}}_{C}$ on to the concave boundary curve ${\mathrm{\Gamma}}_{2}$, as shown in Fig. 2(b), results in a new convex domain ${\mathrm{\Omega}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\mathrm{\Omega}\cup {\mathrm{\Omega}}_{C}$. In this study, the added convexdomain ${\mathrm{\Omega}}_{C}$ is named the complement graph.
As shown in the Fig. 2, the concave domain ${\mathrm{\Omega}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ has two convex boundaries, ${\mathrm{\Gamma}}_{1}\cup {\mathrm{\Gamma}}_{3}$. The concave domain ${\mathrm{\Omega}}_{C}$ also has two convex boundaries, ${\mathrm{\Gamma}}_{2}\cup {\mathrm{\Gamma}}_{3}$. Defining the convex domains of ${\mathrm{\Omega}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and ${\mathrm{\Omega}}_{C}$ as levels renders multilevel modeling theory applicable. Apparently, the convex design domains ${\mathrm{\Omega}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and ${\mathrm{\Omega}}_{C}$, are easy to deal with through WBM theory. In the next section, the feasibility and efficiency of complementgraph conception is illustrated through numerical verification studies where methodology is applied for Kirchhoff platebending problems.
Fig. 2Complementgraph conception
a)
b)
3.3. An application of complement graph
The concavedomain problem is discussed in this section. Combination complementgraph conception and multilevel modeling theory are considered, and the application of the problem is shown in Fig. 3.
Fig. 3Complement graph method
According to traditional WBM and multilevel modeling, accurately modeling this plate structure accurately (Fig. 3) is possible. Dividing the concave problem into convex subdomains is also impossible. The quarter circle boundary ${\mathrm{\Gamma}}_{i,1}$ could be approximated by a large number of straight edges, but this step leads to a rough approximation and a large number of subdomains. Consequently, the computational efficiency of WBM is jeopardized
Through the multilevel approach and complementgraph conception, the concave domain ${\mathrm{\Omega}}_{S}$ problem is decomposed into two levels: one bounded level and one unbounded level that are both coupled at the boundaries. The bounded levels ${\mathrm{\Omega}}_{b}^{}$ describe the interior dynamic response behavior of the bounded subproblems with boundaries ${\mathrm{\Gamma}}_{O}\cup {\mathrm{\Gamma}}_{\mathrm{i},2}\cup {\mathrm{\Gamma}}_{i,3}$. The unbounded levels ${\mathrm{\Omega}}_{ub}^{}$ consider the scattering behavior of an inclusion in an infinite homogeneous medium, with inner boundaries ${\mathrm{\Gamma}}_{i,1}\cup {\mathrm{\Gamma}}_{i,2}\cup {\mathrm{\Gamma}}_{i,3}$. The solution field function $w(x,y)$ is written as the superposition of several fields, ${w}_{b}$ and ${w}_{ub}$, which is required to satisfy the governing Kirchhoff platebending equation. Furthermore, the unbounded fields should also satisfy nonreflecting boundary conditions imposed at ${\mathrm{\Gamma}}_{\mathrm{\infty}}$.
3.4. Wave functions of complement graph
After identifying the field variable expansions for the different levels, they combined with the common domain ${\mathrm{\Omega}}_{S}$ by using the superposition principle. In the bounded domain ${\mathrm{\Omega}}_{b}$, the bounded wave functions should be defined based on the indirect Trefftz principle that is shown by Eq. (4).
In a similar manner, the unbounded wave functions should be defined based on the indirect Trefftz principle; the wave function can be written as: $\left({\nabla}^{2}+{k}_{b}^{2}\right)\left({\nabla}^{2}{k}_{b}^{2}\right){w}_{ub}(x,y)=0$, where no external force is applied within the considered unbounded level.
To satisfy the Kirchhoff platebending theory, a sufficient condition is achieved when unbounded wave functions satisfy either $({\nabla}^{2}+{k}_{b}^{2}){w}_{ub}(x,y)=0$ or $({\nabla}^{2}{k}_{b}^{2}){w}_{ub}(x,y)=0$ and the nonreflecting boundary conditions imposed at boundary ${\mathrm{\Gamma}}_{\mathrm{\infty}}$. Considering the purely outgoing nature of the subdomain, extra caution is unnecessary. Furthermore, the subdomian is implicitly represented in the wave function set [22].
After defining the wave function of the unbounded domain and bounded domain, the field variable expansions of bounded and unbounded levels are combined using the superposition principle. Consequently, the multilevel field variable expansion can be stated as:
where ${\sum}_{b=1}^{{n}_{b}}{w}_{b}{\mathrm{\Psi}}_{b}(x,y)+{\widehat{w}}_{bF}(x,y)$ is the solution of the bounded domain problem, and ${\sum}_{ub=1}^{{n}_{ub}}\u200d{w}_{ub}{\mathrm{\Psi}}_{ub}(x,y)$ is the solution of the unbounded domain problem. Consequently, the composite multilevel function set is defined, and the wave model can be constructed by enforcing the boundary conditions through the application of a weighted residual formulation. This formulation is analogous to the weighted residual formulation for bounded problems Eq. (6) introduced in Section 2, but residuals are now expressed in terms of the new, composite wave function set. Using the weighted residual formulation, the proposed multilevel displacement expansion and weighting function expansion leads to a linear system of algebraic equations. These equations could be solved to obtain the wave function contribution factors of the displacement of the considered problem domain $\mathrm{\Omega}$.
4. Numerical results and discussions
To demonstrate the efficiency and feasibility of the nonconvex domain complementgraph conception and multilevel modeling, a plate structure under the external harmonic excitation in the air is presented. The convergence of the WBM is verified, and WBM predictions are compared with those obtained with the FEM.
4.1. Model description
Taking for example a rectangle plate with one elliptic cutout in the leftbottom corner, the calculation of the vibration by the WBM is carried out. Fig. 4 shows the geometry of the considered problem.
Fig. 4Rectangle plate
A rectangle plate containing one elliptic cutout in the leftbottom corner is excited by a harmonic mechanical loading with prescribed under excitation frequency, amplitude, and direction. The boundary of the plate is simply supported. The semimajor axis and semiminor axes of the ellipse are 0.4 m and 0.3 m, respectively. A unit point timeharmonic force with $z$ direction is applied at point $F$ (m, m), as shown in Fig. 4. Two evaluation points location point locations exist, i.e., $F$ (m, m) and $W$ (0.6 m, m).
For rectangle plate structure, the rectangular Cartesian coordinate system is used, with the coordinate origin set at the ellipse center. The length ${L}_{x}$ of plate is 0.8 m, the width ${L}_{y}$ is 0.6 m, and thickness is 0.001 m. The plate is steel based, and the mechanical properties of this steel are shown in Table 1.
Table 1Mechanical properties of materials
Material  Elasticity modulus (MPa)  Poisson ration  Loss factor  Density (kg·m^{3}) 
Steel  $E=$ 210 000  0.3  $\eta =$ 0  7800 
4.2. Calculation and comparison
Fig. 4 shows the impossibility to divide the problem into convex subdomains, and traditional WBM could not to deal with this problem. Through the complementgraph conception and the multilevel modeling theory, the multilevel WBM is introduced. In the numerical calculation, the variations of vibrations are with the frequency ranges 0800 Hz, and the frequency step size is 2 Hz. to compare the computational efficiency of multilevel WBM and FEM in the low and mid frequency range, convergence analysis of structural dynamic response is performed.
In the dynamicresponse analysis, the truncation coefficient $T$ is defined as 10 to illustrate that the multilevel extension of WBM accurately describes the dynamic response behavior. In addition, multilevel WB model contains only 250 basis functions in the displacement expansion (150 bounded and 100 unbounded wave functions). Furthermore, WBM is implemented in the MATLAB/R2007 environment.
Considering that the business software MSC.Patran/Nastran is a proven and widely used numerical tool for structural dynamicresponse analysis, all FEM predictions are calculated using MSC/Nastran 2005. The finest 8noded Nastran model is defined as a reference for convergence curves. To ensure high numerical precision, the model is fine meshed and consisted of 78 200 quadratic elements within the plate structure.
Fig. 5 shows a contour of the structural displacement at 500 Hz obtained using both MSC/Nastran 2005 and a multilevel WBM. Results show that multilevel WBM can accurately describe the spatial distribution of the dynamic displacement field by using complementgraph conception. Fig. 5 shows that using WBM can achieve accuracy similar to FEM but with less DOFs and fast convergence. In the other frequencies, the displacement contour plots are considerable correlate.
Fig. 5Comparison of predictions in the displacement response of WBM and FEM
a) WBM, 250 DOFs
b) FEM, 196167 DOFs
Fig. 6 compares the predicted displacement response for evaluation point W within (0800 Hz) frequency and the technique includes both FEM and multilevel WBM. Fig. 7 compares the predicted displacement response for evaluation point $F$ within (0800 Hz) frequency and the technique also includes both FEM and multilevel WBM.
According to Figs. 6 and 7, multilevel WBM predictions considerable agree with conventional prediction techniques such as FEM throughout the entire frequency range of interest. Compared with FEM, WBM has less computational loading and more efficient convergence properties.
Fig. 6Displacement response at point W
Fig. 7Displacement response at point F
Comparison of data in Figs. 57 reveals that the numerical result of Patran/Nastran and WBM are similar, whereas WBM has less DOFs than that using MSC/Nastran soft code. Generally, these examples show that the multilevel WBM for platebending problems perform better than FEM within both low and mid frequency ranges.
Numerical results exhibit considerable improvement in terms of efficiency in concave problem by using the complementgraph conception and multilevel modeling theory. Numerical results show that the multilevel multilevel modeling theory of WBM can accurately describe the spatial distribution of the dynamic displacement field combination through complementgraph conception. Similar, displacement contour plots are considerably correlated within different frequency ranges.
4.3. Convergence analysis
The accuracy of FEM analysis depends on the number of elements, selected shape functions. While, the accuracy of WBM analysis depends on the number of wave function. Fig. 8 plots the variation in relative errors and the degrees of freedom (DOFs) for the displacement prediction amplitudes of point $W$ (m, m) at 500 Hz frequency.
Fig. 8 shows that WBM has better convergence than element based methods. The high computational efficiency is likely to become even more definite.
Fig. 8Comparison of the convergence of FEM and WBM
5. Conclusions
WBM is applied to platebending vibration, with special focus on the concavedomain problem. A new approach is also introduced for the nonconvex domain problem. Complementgraph conception effectiveness for concavedomain problem is investigated through a plate structure as a numerical example. The potential of enhanced WBM is demonstrated through validation examples. The application of more complex concave problem such as threedimensional vibroacoustic concavedomain problem requires further studies.
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This work was partially supported by the Youth Fund of State Key Laboratory of Ocean Engineering (Grant No. GKZD01005922) of Shanghai Jiaotong University. This support is gratefully acknowledged by the authors.