On the use of the wave based method for the steady-state dynamic analysis of three-dimensional plate assemblies C. Vanmaele, W. Desmet, D. Vandepitte K.U.Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium e-mail:
[email protected]
Abstract Nowadays, the finite element method is the most commonly used prediction method for dynamic simulations of mechanical structures. A major disadvantage of this method is its practical frequency limitation at higher frequencies. A newly developed wave based prediction technique aims to relax this frequency limitation through an enhanced computational efficiency. This paper discusses the principles of the wave based method for the steady-state dynamic analysis of thin, flat plates of arbitrary geometry, coupled at an arbitrary angle. The beneficial convergence rate of the wave based method when compared with the finite element method is verified for two validation examples.
1
Introduction
In recent years, the vibrational and acoustic behaviour of a product became a criterion of growing importance in the product design process. This behaviour is often determined predominantly by the steady-state dynamic deformations of the mechanical structure. In view of supporting design decisions through virtual prototyping, the finite element method [1] is most commonly used for dynamic simulations of mechanical structures. This method is based on the discretization of the structure into small elemental domains. It expresses the dynamic field variables within each element in terms of local, non-exact shape functions. As a result, the number of elements and subsequently the size of a finite element model increases with increasing frequencies, such that the use of the finite element method is practically limited to low-frequency applications. For highfrequency applications the SEA method [2] can be used. This method divides the construction in a number of subdomains, for which only space- and frequency-averaged energy levels are predicted. The SEA method is, however, only applicable if all subdomains have a high modal overlap, which limits the use of the method to high-frequency applications. For mid-frequency applications, there exists still a lack of efficient prediction techniques. A new deterministic wave based prediction technique, based on the indirect Trefftz method [3], aims to overcome this mid-frequency issue. This method expresses the field variables in terms of global wave function expansions, which exactly satisfy the governing dynamic equations. This implies that there is only an approximation involved in the boundary conditions. Consequently, the system matrices are substantially smaller compared with the finite element method. The smaller system matrices result in a higher computational efficiency, which will allow the wave based method to become also applicable in the mid-frequency region. This paper discusses the development of the wave based method for the analysis of an assembly of flat plates coupled under arbitrary angles. The two considered validation examples confirm that the wave based method provides accurate results with substantially smaller models and computational efforts compared with the finite element method.
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Problem Definition
Figure 1: Two flat plates coupled at an angle α.
In its simplest form the problem consists of two flat plates coupled at an arbitrary angle α, as shown in figure 1. Each of both plates is excited with a harmonic normal point force, F1 and F2 , in respectively points (xF 1 , yF 1 ) and (xF 2 , yF 2 ). According to the thin plate theory of Kirchhoff [4], the steady-state out-of-plane displacements wzi (i = 1, 2) are governed by the following differential equation: 4 wzi (xi , yi ) = ∇4 wzi (xi , yi ) − kbi
with ∇4 = defined as,
∂4 ∂x4i
4
+ 2 ∂x∂2 ∂y2 + i
i
∂4 . ∂yi4
Fi δ(xF i , yF i ), Di
(1)
The plate bending wavenumber kbi and plate bending stiffness Di are s kbi = Di =
4
ρi h i ω 2 , Di
(2)
Ei h3i , 12(1 − νi2 )
with hi , Ei , νi and ρi respectively, the plate thickness, the elasticity modulus, the Poisson coefficient and the plate material density. The Kirchhoff theory remains valid as long as the bending wavelength λbi is approximately six times larger than the thickness hi of the plate.[5] When the coupling angle α is different from zero, the displacement field will also have a component in the plane of the plates. These in-plane displacements wxi and wyi are governed by the coupled dynamic Navier equations [6]: ∂ 2 wxi 1 − νi ∂ 2 wxi 1 + νi ∂ 2 wyi ρi (1 − νi2 )ω 2 + + + wxi = 0, 2 2 ∂xi ∂yi Ei ∂x2i ∂yi2 ∂ 2 wyi 1 − νi ∂ 2 wyi 1 + νi ∂ 2 wxi ρi (1 − νi2 )ω 2 + + + wyi = 0. 2 2 ∂xi ∂yi Ei ∂yi2 ∂x2i
(3) (4)
In order to simplify the solution of these equations, they can be transformed into two uncoupled equations. Two transformations are being considered for this case: a formulation in terms of dilatation and rotation
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and a formulation in terms of displacement potentials. Only the latter formulation in terms of displacement potentials will be further discussed. For more details concerning the formulation in terms of dilatation and rotation the reader is referred to a previous publication.[7] The transformation of the in-plane displacements wxi and wyi to the displacement potentials is based on the Helmholtz decomposition of a vector field.[8, 6] This theorem states that any vector field may be decomposed into an irrotational and a solenoidal part, provided that this vector field is piecewise differentiable. Applying the Helmholtz theorem to the in-plane displacement field results in the following decomposition, wxi = ∇φi + ∇ × ψi , (5) wyi with φi and ψi , respectively, an irrotational and solenoidal potential. By means of this transformation the coupled plate equations (3-4) are converted into the following uncoupled equations, ∇2 φi + kli2 φi = 0, 2
∇ ψi +
2 kti ψi
(6)
= 0,
(7)
where the in-plane longitudinal and shear wavenumbers are defined as, s s ρi (1 − νi2 ) 2ρi (1 + νi ) kli = ω and kti = ω . Ei Ei
(8)
To uniquely define the problem, four boundary conditions must be imposed along the entire boundary Γi \Γc . In case of clamped edges, the displacement field needs to fulfill the following four conditions: Rwi = wzi = 0, φi (i) Rwn i = Lwn = 0, ψi
(i)
Rθi = Lθ [wzi ] = 0, φi (i) Rws i = Lws = 0. ψi
(9)
Further explanation concerning simply supported and free edges can be found in [9]. Along the common interface Γc between the two plates, the force equilibrium requires: (2) Rmc = L(1) m [wz1 ] − Lm [wz2 ] = 0, (1)
(2)
(10) (2)
φ2 ψ2
RQc = LQ [wz1 ] + cos α LQ [wz2 ] − sin α LN = 0, φ1 φ2 (1) (2) (2) RN c = LN − cos α LN − sin α LQ [wz2 ] = 0, ψ1 ψ2 φ1 φ2 (1) (2) RT c = LT − LT = 0. ψ1 ψ2
(11) (12) (13)
The displacement compatibility along the coupling interface Γc leads to the following four conditions: φ1 (1) Rwc = wz2 − cos α wz1 + sin α Lwn = 0, (14) ψ1 Rθc Rwn c Rws c
(2)
(1)
= Lθ [wz2 ] + Lθ [wz1 ] = 0, φ2 φ1 (2) (1) = Lwn + cos α Lwn + sin α wz1 = 0, ψ2 ψ1 φ2 φ1 = L(2) + L(1) = 0. ws ws ψ2 ψ1
(15) (16) (17)
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The differential operators used in the boundary and interface conditions are defined as follows, normal angular displacement: normal bending moment: generalized shear force: in-plane normal displacement: in-plane tangential displacement: in-plane normal force: in-plane tangential force:
∂ ∂ni 2 ∂ ∂2 (i) Lm = −Di + νi 2 , ∂n2i ∂si 2 ∂ ∂ ∂2 (i) + (2 − νi ) 2 , LQ = −Di ∂ni ∂n2i ∂si ∂ ∂ L(i) , wn = ∂ni ∂si ∂ ∂ (i) − , Lws = ∂si ∂ni 2 E i hi ∂2 ∂2 ∂ E i hi (i) LN = , + νi 2 (1 + νi ) ∂ni ∂si (1 − νi2 ) ∂n2i ∂si 2 Ei hi ∂2 Ei hi ∂ ∂2 (i) LT = − , (1 + νi ) ∂ni ∂si 2(1 + νi ) ∂s2i ∂n2i (i)
Lθ = −
(18) (19) (20) (21) (22) (23) (24)
where ni and si are, respectively, the in-plane normal and in-plane tangential directions of the plate boundary or interface. The aforementioned coupling formalism gives an exact representation of the coupling. But, since the membrane stiffness is very large compared to the bending stiffness, the (low- and mid-frequency) coupling between the two plates can be approximated by only considering the bending energy. In fact, this implies that the interface is considered as a type of simply supported edge, as the displacement along the interface is zero a priori. The eight coupling conditions (10-17) are now replaced by the following four conditions, Rwc1 = wz1 = 0 , Rmc =
L(1) m [wz1 ]
(25) −
L(2) m [wz2 ]
+
(2) Lθ [wz2 ]
=0,
Rwc2 = wz2 = 0 , Rθc =
(1) Lθ [wz1 ]
(26) (27)
=0.
(28)
Although this simplified coupling is only an approximation of the physical reality, the prediction results show to be quite accurate compared with those of the exact coupling. Furthermore the model sizes and the subsequent computational times become significantly smaller. Both the prediction accuracy and the computational efficiency of both coupling methods are evaluated for two validation examples.
3
Basic principle of the wave based method
This section briefly discusses the basic principles of the wave based method for the example defined in the previous section. In contrast with the finite element method, the wave based method describes the field variables with an expansion of wave functions which exactly satisfies the governing dynamic equations. In this way, there is only an approximation error induced in the boundary and interface conditions. Minimizing this approximation error in an integral sense leads to the solution of the system.
3.1
Field variable expansions
In each subdomain (i) the out-of-plane displacement wzi is approximated as the following solution expansion w ˆzi , nbi X wzi (xi , yi ) ≈ w ˆzi (xi , yi ) = wbi Ψbi (xi , yi ) + w ˆF i (xi , yi ), (29) b=1
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in which each wave function Ψbi satisfies the homogeneous part of the dynamic equation (1), Ψbi (xi , yi ) = e−j(kxb,i xi +kyb,i yi )
2 2 4 with (kxb,i + kyb,i )2 = kbi .
(30)
The particular solution function w ˆF i satisfies the inhomogeneous part of the dynamic equation (1) arising from the external loading. For plate bending the displacement field of an infinite plate excited with a normal point force is selected, i jFi h (2) (2) H0 (kbi rF i ) − H0 (−jkbi rF i ) , w ˆF i (xi , yi ) = − 2 (31) 8kbi Di p with rF i = (xi − xF i )2 + (yi − yF i )2 , (2)
and where H0 is the zero-order Hankel function of the second kind. If included, the steady-state in-plane displacements wxi and wyi are approximated as expansions w ˆxi and w ˆyi , ) " (i) # ( Lwx φˆi (xi , yi ) wxi (xi , yi ) w ˆxi (xi , yi ) (32) ≈ = (i) wyi (xi , yi ) w ˆyi (xi , yi ) ψˆi (xi , yi ) Lwy " (i) # P nli Lwx φli Ψli (xi , yi ) l=1 P . = nti (i) Lwy t=1 ψti Ψti (xi , yi ) The irrotational (dilatational) and solenoidal (rotational) wave functions Ψli and Ψti satisfy their corresponding homogeneous differential equation (6) or (7),
3.2
Ψli (ri ) = e−j(kxl,i xi +kyl,i yi )
2 2 with kxl,i + kyl,i = kli2
(33)
Ψti (ri ) = e−j(kxt,i xi +kyt,i yi )
2 2 2 with kxt,i + kyt,i = kti .
(34)
Selection of wave functions
An infinite number of wave functions satisfies the homogeneous differential equations. A truncated set must be selected for each field variable. The selection of the wave functions for subdomain (i) is based on the dimensions of the smallest rectangular box circumscribing the subdomain, see figure 1. For plate bending, the wave functions with following wavenumbers are selected: q b1i π q 2 b1i π 2 2 2 (kxb,i , kyb,i ) = , ± kbi − kxb,i , , ±j kbi − kxb,i , Lxi Lxi q q b2i π b2i π 2 2 2 2 , ±j kbi − kyb,i , , (35) ± kbi − kyb,i , Lyi Lyi with (b1i , b2i ) = 0, ±1, · · · and with Lxi and Lyi the dimensions of the enclosing rectangular box. For the in-plane displacements, the following wave functions are included in the model: q l1i π q 2 l2i π 2 2 2 (kxl,i , kyl,i ) = , ± kli − kxl,i , ± kli − kyl,i , (36) Lxi Lyi (kxt,i , kyt,i ) =
q t1i π q 2 t2i π 2 2 2 , ± kti − kxt,i , ± kti − kyt,i , Lxi Lyi
(37)
with (l1i , l2i ) and (t1i , t2i ) = 0, ±1, · · · . Provided that all the subdomains are convex, it has been proven that this selection of wave functions enables the field variable expansions to converge towards the exact result.[9]
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The number of bending wave functions nbi that is included in expansion (29) is related to the excitation frequency and a characteristic length L of the enclosing rectangular box, nbi = T1
kb L , π
(38)
with T1 a parameter that is defined by the user. This way the largest wavenumber of the bending wave functions included in the model is at least T1 times the structural wavenumber at this frequency. To achieve a good coupling between the different displacement components, the number of in-plane wave functions nli and nti are chosen such that the largest in-plane wavenumbers along the coupling interface are of the same order of magnitude as the largest wavenumber of the bending waves.
3.3
Evaluation of boundary and interface conditions
The field variable expansions (29) and (32-33) exactly satisfy the governing dynamic equations (1) and (6-7), irrespective of the unknown wave function contribution factors wbi , φli and ψti . These contribution factors are determined by minimizing the approximation errors of the boundary and interface conditions through a weighted residual formulation. For the out-of-plane displacement, the approximation errors on boundary and interface conditions of the two subdomains are orthogonalized with respect to weighting functions w ˜z1 and w ˜z2 respectively, Z Z Z Z (1) (1) (1) LQ [w ˜z1 ]Rw1 dΓ + Lm [w ˜z1 ]Rθ1 dΓ − Lθ [w ˜z1 ]Rmc dΓ − w ˜z1 RQc dΓ = 0, (39) Γ1 \Γc
Z Γ2 \Γc
Γ1 \Γc
(2) LQ [w ˜z2 ]Rw2 dΓ +
Z Γ2 \Γc
L(2) ˜z2 ]Rθ2 dΓ + m [w
Γc
Z Γc
(2) LQ [w ˜z2 ]Rwc dΓ +
Γc
Z
L(2) ˜z2 ]Rθc dΓ = 0. (40) m [w
Γc
The approximation errors n˜ o n ˜ ofor the in-plane displacements are orthogonalized with respect to the weighting φ1 φ functions ψ˜ and ψ˜2 respectively, 1
2
"
Z Γ1 \Γc
Z Γ2 \Γc
(1) LN
" (2) LN
" # # Z ˜ φ˜1 (1) φ1 L R dΓ + Rws 1 dΓ w 1 n T ψ˜1 ψ˜1 Γ1 \Γc " # " # Z Z ˜1 ˜ φ (1) φ1 − L(1) R dΓ − L RT c dΓ = 0, N c wn ws ψ˜1 ψ˜1 Γc Γc
" # # Z ˜ φ˜2 (2) φ2 Rwn 2 dΓ + LT Rws 2 dΓ ˜ ψ˜2 ψ2 Γ2 \Γc " # " # Z Z ˜ ˜ (2) φ2 (2) φ2 + LT Rws c dΓ = 0. LN ˜ Rwn c dΓ + ψ2 ψ˜2 Γc Γc
(41)
(42)
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The weighting functions are chosen in a similar way as in the Galerkin weighting procedure, w ˜z1
= Ψb1 ,
w ˜ = Ψb2 , ( z2 ) ˜ φ1 Ψl1 = , 0 ψ˜1 0 = , Ψt1 ( ) φ˜2 Ψl2 = , 0 ψ˜2 0 = , Ψt2
with b = 1 · · · nb1 , with b = 1 · · · nb2 , with l = 1 · · · nl1 , with t = 1 · · · nt1 ,
(43)
with l = 1 · · · nl2 , with t = 1 · · · nt2 ,
with Ψb1 and Ψb2 the bending wave functions, Ψl1 and Ψl2 the dilatational wave functions and Ψt1 and Ψt2 the rotational wave functions. In this way, a square system of equations in the unknown contribution factors wb1 , wb2 , φl1 , φl2 , ψt1 and ψt2 is obtained. For the simplified coupling, i.e. without considering the in-plane displacements, the residuals (9) and (25-28) can be transformed into a weighted residual formulation in a similar way as for the full coupling. This will lead to a square system of equations in the unknowns wb1 and wb2 . Compared with the finite element method, the system matrices are substantially smaller. On the other hand, these matrices are fully populated, frequency dependent and contain complex elements. Nevertheless, the wave based method still exhibits a higher convergence rate than the finite element method as will be demonstrated for the validation examples.
4 Coupling of three rectangular plates
Figure 2: Coupling of three rectangular plates. A first validation example consists of three rectangular plates, coupled together as shown in figure 2. The first two plates are coupled under an angle of 30◦ , the second and third plate are coupled under an angle ◦ 9 2 3 of 45 . The plates are made of aluminium E = 70 · 10 N/m , ν = 0.3, ρ = 2790kg/m and have a
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thickness of 0.002m. All plate boundaries are clamped. A unit normal point force is applied to the first plate at position (xG , yG , zG ) = (0.3m, 0.35m, 0.1732m). The four response points in which the results will be calculated are indicated on figure 2 and have the following coordinates: w1 (0.2m, 0.1m, 0.1155m), w2 (0.8m, 0.4m, 0.35m), w3 (1.1m, 0.2m, 0.35m) and w4 (1.5m, 0.25m, 0.5438m).
Figure 3: Predicted displacement field at 100Hz.
Figure 4: Absolute error of prediction at 100Hz.
Figure 3 shows the predicted displacement field for the three plates when excited at 100Hz. This prediction was calculated with a wave model containing 228 bending wave functions, 114 dilatational wave functions and 138 rotational wave functions, leading to a total of 480 degrees of freedom. This figure indicates that both the boundary conditions and the displacement compatibility along the two interfaces are accurately represented. Remember that there is only an approximation induced on the boundary and interface conditions. This implies that the approximation error on boundaries and interfaces forms an indication of the overall prediction accuracy. The prediction error is represented in figure 4. This figure displays the amplitude of the absolute error of the wave based prediction. As reference the result of a finite element model consisting of 12 379 nodes and 61 895 degrees of freedom was used. The largest element size is 0.02m, where the structural wavelength is 0.4364m. Although the wave based model is substantially smaller than the finite element model, the achieved accuracy is very high. Note that the largest error is 2.7e−7 m and appears on one of the boundaries. Figure 5 compares the performance of the finite element method and the wave based method. The finite element model consists of 53 805 degrees of freedom (element size 0.014m) and the predictions are calculated using the direct solution method and a 4-noded quadrilateral shell discretization. The size of the wave based model increases with increasing frequencies. At 200Hz, for example, the model consists of 312 bending wave functions and 336 dilatational or rotational wave functions. As mentioned before, it is possible to approximate the coupling between two plates by discarding the in-plane displacements. This, of course, will lead to significantly smaller system matrices, but on the other hand it may also have some non-negligible implications on the prediction accuracy. In this case the model only includes the 312 bending wave functions. Figure 5 shows the predicted normal displacement wz calculated with the finite element model (- -) and the wave based model (--) for the four response points w1 − w4 . The figures on the top give the wave based result for the exact coupling; the figures on the bottom give the result when the in-plane displacements are neglected. As these figures indicate, the results of the wave based method, with and without in-plane displacements, agree very well with the finite element result. The only difference takes place at 157Hz, where the wave based method fails to predict the small peak for response points w2 and w3 , both with the inclusion of the in-plane movement or with its omission. At this frequency the structural wavelength equals almost exactly half of the length of the plates, namely 0.7m. As can be seen in formula (35), for a certain value of b1i , kxb,i will almost be the same as the structural wavenumber kbi which implies that kyb,i is close to zero. Consequently, the model includes bending wave functions which are almost linear dependent. This fact causes the slow convergence at this frequency. Further, it appears that the omission of the in-plane
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(a) response point 1
(b) response point 2
(c) response point 3
(d) response point 4
Figure 5: Frequency response functions (top: in-plane disp. included, bottom: in-plane disp. neglected).
Figure 6: Frequency response function with a reduced number of in-plane wave functions (response point 1).
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displacements does lead to good prediction results, in less than half of the computation time compared with the full coupling. When the in-plane displacements are taken into account, the predicted displacement for response point w1 , figure 5(a), exhibits a spurious resonance frequency at 195Hz. This is due to the very high number of dilatational and rotational wave functions. The number of in-plane wave functions is chosen such that the largest wavenumbers of both the bending wave functions and the in-plane wave functions are of the same order of magnitude, see section 3.2. But, due to the huge difference between the in-plane and bending wavenumbers, it is possible that this rule leads to an excessive number of in-plane wave functions. And indeed the result with only 156 in-plane wave functions no longer suffers from this spurious frequency, see figure 6. So apparently the current rule to determine the number of in-plane wave functions is not optimal. Consequently, a new guideline has to be derived in the near future.
5 Square duct
Figure 7: Four rectangular plates coupled to a tube. Figure 7 shows the second validation example, which consists of four rectangular plates coupled together to form a duct. The plates are made of aluminium E = 70 · 109 N/m2 , ν = 0.3, ρ = 2790kg/m3 and have a thickness of 0.0005m. The front plate lying in the (x, z)-plane is excited by a unit normal point force in the point F (xG , yG , zG ) = (0.2m, 0m, 0.02m). Again all plate boundaries at both ends of the duct are clamped. The four response points have the following coordinates: w1 (0.1m, 0m, 0.035m), w2 (0.3m, 0.025m, 0.05m), w3 (0.1m, 0.05m, 0.015m) and w4 (0.3m, 0.04m, 0m). The geometry of the structure is chosen such that the in-plane and out-of-plane displacements of adjacent plates are closely coupled.
Figure 8: Predicted displacement field at 800Hz. The real part of the predicted displacement field when excited at 800Hz is displayed in figure 8. This result was calculated with a wave based model containing 496 bending wave functions and 528 in-plane wave
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functions. Since the boundary conditions and displacement compatibility along the interfaces are accurately represented, the overall prediction accuracy is expected to be acceptable.
(a) response point 1
(b) response point 2
(c) response point 3
(d) response point 4
Figure 9: Frequency response functions (top: in-plane disp. included, bottom: in-plane disp. neglected).
Figure 10: Comparison of displacement inside the domain and on the interfaces. Figure 9 compares the predictions for the normal displacement wz obtained with the finite element method and the wave based method. The finite element model consists of a 4-noded quadrilateral shell discretization with 221 100 degrees of freedom (element size 0.0021m) and the predictions are calculated using the direct solution method. For the wave based method both the exact coupling formalism (1 024 degrees of freedom at 800Hz) and the simplified formalism (496 degrees of freedom at 800Hz) are included. The finite element results are plotted with a dashed line (- -), the wave based results are plotted with a full line (---). The fig-
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ures on the top give the wave based result for the exact coupling; the ones on the bottom for the simplified coupling. For the exact coupling the results correspond very well with the finite element results. For the simplified coupling on the other hand, there is a very big discrepancy between the results. Obviously the assumptions made by the simplified coupling are no longer valid in this frequency range. This is confirmed by figure 10. This figure plots the maximum amplitude of the displacement max(|w|) appearing on one of the 4 interfaces as a function of the excitation frequency (- -). The two other curves are the predicted normal displacement for response point 1 by the finite element and wave based method. It can be seen from this figure that in the frequency region where the prediction accuracy is very poor the actual displacements on the interfaces are quite large compared with the displacements inside the domain. Since with the simplified coupling the displacements along the interface are zero a priori, this figure indeed indicates that the assumptions made by the simplified coupling are no longer valid. This example demonstrates the possible danger of omitting the in-plane displacements from the model.
(a) response point 1
(b) response point 2
Figure 11: Frequency response functions in the mid-frequency range (top: in-plane disp. included, bottom: in-plane disp. neglected).
Figure 11 gives the predicted normal displacement between 800Hz and 1500Hz for response points 1 and 2. The models remain the same as before. The predictions resulting from the exact coupling are quite accurate but at some frequencies the method still suffers from convergence problems. These convergence problems arise from the poor numerical condition of the model matrix, which is typical for all Trefftz-based models. A similar problem occurs for the coupling of coplanar plates.[10] To overcome this, attempts will be made to increase the convergence rate for both coplanar and non-coplanar plate assemblies. For single domain problems, making the system matrices symmetric appeared an important improvement for the convergence rate. For multi-domain problems similar results are expected. Secondly will the coupling conditions be further investigated. Applying alternative coupling conditions such that the submatrices remain also non-singular for the subdomain resonance frequencies, could also enhance the convergence rate. Finally, the number of in-plane wave functions included in the model is most likely too high. Optimizing the rule that determines the number of in-plane wave functions will probably also increase the convergence rate. However, these three modifications still need further investigation. The predictions resulting from the simplified coupling do not suffer from these convergence problems, simply because the model includes a smaller number of wave functions. Where the prediction with this model was still very poor in the lower frequency range, it shows to be very accurate in this frequency range. Again it appears that this way of coupling can be very useful to obtain accurate results with very small system matrices. Nevertheless, this example also indicates the possible danger since the in-plane displacements can not always be neglected. As demonstrated with both validation examples, the wave based method achieves a good prediction accuracy with substantially smaller system matrices. To make a fair comparison between the wave based and finite element method figures 13 and 14 evaluate the computational efficiency of both methods. These figures
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Figure 12: Distribution of the response points over a surface.
Figure 13: Convergence curves for the out-of-plane displacement at 500Hz (λb = 0.0976m).
Figure 14: Convergence curves for the out-of-plane displacement at 700Hz (λb = 0.0825m).
plot the relative prediction error rel of the normal displacement averaged over 32 response points, 8 per surface, as a function of the CPU time at respectively 500Hz and 700Hz. As indicated in figure 12, the eight response points per surface are equally distributed over this surface. The averaged relative error hrel i is defined as follows, nrp 1 X hrel i = relj , (44) nrp j=1
with nrp the number of response points and relj the relative prediction error in response point j, relj =
ref kwzi (r j ) − wzi (r j )k ref kwzi (r j )k
.
(45)
ref The reference solution wzi (r j ) is calculated with a very fine finite element model with 820 480 degrees of freedom (element size 0.0011m). For the wave based method the indicated times include both the times needed for the construction of the model and for the solution of the model. Since the finite element models can be decomposed into frequency independent matrices, only the direct solution times are included in the indicated CPU times. All calculations are performed on a Windows 2000 system with a 2GHz processor and 1Gb RAM memory. Notice also that the finite element models are solved using the MSC/Nastran software while the wave based method is implemented in a C++ code. Both figures include the finite element method (- • -), the wave based method with the exact coupling (--4--) and the wave based method with the simplified coupling (--×--). The convergence curves at 500Hz, figure 13, clearly indicate the beneficial convergence rate of the wave based method compared with the finite element method. Comparing the two coupling formalisms, the higher convergence rate of the simplified coupling compared with the exact coupling is quite clear. The wave based method with simplified coupling reaches a quite accurate result with a relative error of 0.03 after less than half a second, where the finite element method needs over 30s. Also at 700Hz, figure 14, the convergence rate of the wave based method with the exact coupling is higher compared with this of the finite element method. As mentioned above, the simplified coupling does not converge to the correct solution
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since the assumptions are not valid at this frequency. Both figures indicate that the wave based models are not only substantially smaller compared with the finite element method, but also have an enhanced convergence rate.
6
Conclusions and future research
This paper reports on the development of the wave based method for the steady-state dynamic analysis of three-dimensional flat plate assemblies. It has been proposed that the coupling between two non-coplanar plates can be calculated not only including the in-plane displacements, but also using a simplified coupling formalism in which the in-plane displacements are neglected. Both validation examples prove that the simplified coupling can indeed lead to very accurate results and even has an enhanced convergence rate over the exact coupling. However, the second example also indicates the possible danger since the in-plane displacements are not always negligible. The validation examples clearly illustrate the potential of the wave based method to predict accurate results with substantially smaller prediction models compared with the finite element method. Nevertheless, at some frequencies the method still suffers from convergence problems. The same problem occurs for the coupling of coplanar plates. It is expected that the same modifications as for coplanar plates will increase the convergence rate. However these modifications still need further investigation. The second example also indicates the beneficial convergence rate of the wave based method over the finite element method. This enhanced computational efficiency will allow the wave based method to overcome the practical frequency limitation of the finite element method. The main drawback of the wave based method is its inability to deal with geometrically complex constructions. Since for convergence purposes all subdomains have to be convex, a complex geometry requires a high number of subdomains which will drastically decrease the convergence rate of the method. Therefore a hybrid approach [11] is being envisaged. In this approach, the wave based method will be coupled together with the finite element method to combine the advantages of both methods. The wave based method will be used to model the geometrically simple parts of the structures in a very efficient way. The finite element method, on the other hand, will be used to model the geometrically more complex parts of the structure.
Acknowledgements The research work of Caroline Vanmaele is financed by a scholarship of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).
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[6] J. Dominguez, Boundary Elements in Dynamics, Computational Mechanics Publications, Southampton, UK (1993). [7] C. Vanmaele, W. Desmet, D. Vandepitte, A wave based prediction technique for the steady-state dynamic analysis of three-dimensional plate structures, in Proceedings of The Tenth International Congress on Sound and Vibration, Stockholm, Sweden (2003), pp. 2817-2824. ¨ [8] H. L. F. Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, welch den Wirbelbewegungen entsprechen, 1858. [9] W. Desmet, A Wave based prediction technique for coupled vibro-acoustic analysis, Ph.D. dissertation, Katholieke Universiteit Leuven, Departement Werktuigkunde, Leuven (1998). [10] C. Vanmaele, W. Desmet, D. Vandepitte, A wave based prediction technique for the steady-state dynamic analysis of flat plate assemblies, in Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Eccomas 2004, Jyv¨askyl¨a, Finland (2004). [11] B. Van Hal, W. Desmet, D. Vandepitte and P. Sas, Coupled finite element - wave based approach for the steady-state dynamic analysis of acoustic systems, Journal of Computational Acoustics, Vol. 11, No. 2 (2003).
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