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Fast Variation Method for Elastic Strip Calculation Sergey V. Biryukov, Member, IEEE Abstract—A new, fast, variation method (FVM) for determining an elastic strip response to stresses arbitrarily distributed on the flat side of the strip is proposed. The remaining surface of the strip may have an arbitrary form, and it is free of stresses. The FVM, as well as the wellknown finite element method (FEM), starts with the variational principle. However, it does not use the meshing of the strip. A comparison of FVM results with the exact analytical solution in the special case of shear stresses and a rectangular strip demonstrates an excellent agreement.
I. Introduction he metallic electrode is one of the main elements responsible for the frequency properties of surface acoustic wave (SAW) interdigital transducers (IDT). Because of a finite ratio between the electrode thickness h and the SAW wavelength, and the tendency to exploit ever-increasing frequencies, the elastic properties of the electrode and its form should be taken into account. With increasing of the thickness ratio, however, the simple perturbation methods break down, and the only efficient method for electrode—or, in other words, elastic strip calculations—is currently the finite element method (FEM) [1]–[4]. This numerical method is used mainly for calculating an elastic displacement response to given mechanical forces or stresses applied to the flat surface of the strip bottom [3], [4]. Such a relationship is needed for precise analysis of IDTs and a new method of its derivation is considered in the present paper. The FEM was created many years ago for calculations of complex mechanical constructions such as dams or bridges. Its main idea is to divide the geometric region of the displacement field into small meshes and to interpolate the field in the resulting meshes by simple basis functions. Each mesh with its own basis functions, attached to some node points inside the mesh, is so-called a finite element. The field amplitudes at the node points are found with the aid of the variational principle. In spite of its simple model, the FEM is cumbersome for practical implementation due to the necessity to create a meshing algorithm. So, the creation of an individual, effective computer program demands a long time. However, IDT strip forms seem to be too simple to be analyzed only by the FEM in the general formulation.
Fig. 1. An elastic strip cross section.
T
Manuscript received March 7, 2001; accepted December 3, 2001. The author is with the Institute of Solid State and Materials Research Dresden, Dresden, D-01171, Germany (e-mail:
[email protected]). Permanent address: Mints Radiotechnical Institute, Moscow, 125083, Russia.
In this paper we propose a new, fast method to calculate the elastic strip, which is here referred to as the fast variation method (FVM). The main idea of the FVM is to treat the strip cross section as one mesh to avoid dividing. Instead, to be optimized by variational principle, some simple representation for the two-dimensional field of displacements in the form of a finite sum everywhere over the strip cross section is suggested. The reliability of the method is confirmed by comparing the numerical results with the exact analytical solution in the case of shear stresses and a rectangular strip. The developed FVM is simpler for computer simulations as compared with the FEM. The results of the present paper were reported partially at IEEE conferences [5], [6].
II. The Background Consider an elastic isotropic strip of an arbitrary form with the flat bottom and with width 2a and thickness h, and the Cartesian coordinate system x = x1 , y = x2 , z = x3 (Fig. 1). Let ρ, µ, and λ be the strip mass density and Lame constants, respectively. Suppose that the strip’s upper surface is defined by a function f ≥ 0 as: z = hf (x/a),
(1)
and this surface is free of stresses. The relationship between the exciting normal stresses and displacements on the flat strip surface z = 0, −a ≤ x ≤ a is required. Let the dependence of fields on time t and frequency ω be determined by the factor e−iωt ; this factor is omitted below. The displacement vector ui (i =
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1, 2, 3) in the strip volume is related to the stress tensor σij by the well-known expression: ∂ui ∂uj ∂uk +µ + σij = λδij , (2) ∂xk ∂xj ∂xi where i, j, k = 1, 2, 3, δij is the Kronecker symbol, and the Einstein’s summation convention for repeating indexes is used. The strip Lagrangian L having regard to the above boundary conditions can be written in the following form [7]: 1 L= 2
∂ui 1 dS σij − ρω 2 ∂xj 2
S
−a
III. Shear Horizontal Polarization dx σi3 ui , (3)
where S is the strip cross section area and the independence of all fields of the coordinate y is assumed. The last term in (3) contains the normal stress distribution σi3 = σi3 (x) on the strip bottom, which may be prescribed arbitrarily. We are going to find the displacement field ui (x, z) excited by this distribution by using the wellknown variational principle δL = 0. In other words, the actual field ui (x, z) is regarded as a stationary point of the Lagrangian L. In such a manner, an arbitrary displacement field variation δui at this point should comply with: 1 ∂ui ∂(δui ) δL = dS δσij + σij 2 ∂xj ∂xj S
− ρω
2
dx σi3 (x)δui = 0, −a
S
If the exciting stress distribution consists only of the nonzero component σ23 (x), the only component of the displacement field to be generated is u2 . In this case, the system of linear equations outlined above takes the following form: (2) (2) Kmnkl − ω 2 Mmnkl Akl = Fmn , (8) k,l
where
where the variation δσij is expressed in terms of δui in accordance with (2): ∂(δuk ) ∂(δui ) ∂(δuj ) δσij = λδij +µ + . (5) ∂xk ∂xj ∂xi We propose now a simple representation for the field of displacements ui (x, z) everywhere over the strip cross section as follows: ui (x, z) = A(i) (6) mn Φm (x)Ψn (z). m,n
Here Φm (x) and Ψn (z) are basis functions specified in advance; their number and properties may be chosen by “hand” in order to obtain a good approximation of the dis(i) placement field. Amn are unknown coefficients that should be found from the variational principle (4). To do this, it is necessary only in (4) to replace ui with its representation (6) and the variation δui with: δui (x, z) = δA(i) (7) mn Φm (x)Ψn (z),
Kmnkl = µ
hf(x/a)
dx
−a
0
+ Φm (x)Ψn (z)Φk (x)Ψl (z) , a Mmnkl = ρ
(9)
dz Φm (x)Ψn (z)Φk (x)Ψl (z) , (10)
hf(x/a)
dx
−a
0
a
(2) Fmn = −Ψn (0)
dx σ23 (x)Φm (x),
(11)
−a
and a prime means a derivative. In order to produce calculations, simple basis functions were chosen as follows: Φm (x) = (x/a)m , m = 0, . . . , M, Ψn (z) = (z/h)n , n = 0, . . . , N,
(12) (13)
where (M +1) and (N +1) are the numbers of the involved basis functions. Substitution of the functions (12) and (13) into the coefficients (9)–(11) gives: h mk Kmnkl = µ Jm+k−2,n+l+1 a n+l+1
a nl (14) + Jm+k,n+l−1 , h n+l−1 ρha Jm+k,n+l+1 , n+l+1 a = −δn0 dx σ23 (x)(x/a)m ,
Mmnkl =
m,n
and take into account the relations (2) and (5). Because (i) (4) is true for arbitrary variations δAmn of the coefficients
dz Φm (x)Ψn (z)Φk (x)Ψl (z)
a
(4)
a dS ui δui +
(i)
a dS ui ui +
S
(i)
Amn , each expression in front of any variation δAmn in (4) should be set equal to zero. As a result, the FVM sys(i) tem of linear algebraic equations for the coefficients Amn is obtained. The number of these equations is equal to the number of the coefficients involved in the field represen(i) tation (6). After finding the coefficients Amn the desired displacement distribution (6) will be defined. Let us consider now two independent cases of shear horizontal- and sagittal-displacement polarization.
(2) Fmn
−a
(15) (16)
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where the only factor depending on the strip form is:
Jmn
a 1 dx(x/a)m [f (x/a)]n ; m ≥ 0, n > 0 . = a −a (17) 0; m < 0 or n ≤ 0
Once the system (8) of (M +1)×(N +1) equations is solved by any standard method, the most important displacement distribution at the strip bottom is:
u2 (x) ≡ u2 (x, 0) =
M
(2)
Am0 (x/a)m ,
(18)
m=0
as follows from (6), (12), and (13) at z = 0. The simple integral in (17) can be calculated numerically or analytically for any strip form function f . For example, in the important case of the rectangular strip f (x/a) = 1, −a < x < a [Fig. 2(a)], this integral does not depend on the index n and its analytical expression is:
Jmn =
1 + (−1)m , m+1
m ≥ 0, n > 0.
(19)
In a more general case of a trapezoidal form [Fig. 2(b)], 1 + (−1)m m+1 m p p+1 m+1 p (−1) ∆ + (m + 1) Cm (1 − ∆) , p+n+1 (20) p=0
Fig. 2. Elastic strip forms: (a) rectangular, (b) trapezoidal, and (c) triangular.
Jmn =
m ≥ 0, n > 0, p where Cm = m!/[(m − p)!p!] is the binomial coefficient, ∆ = (a−d)/a, and d is the half-width of the strip flat upper surface. In the partial case of a triangular form [Fig. 2(c)], ∆ = 1 in (20).
where
a (11) Kmnkl
=
hf(x/a)
dz (λ + 2µ) Φm (x)Ψn (z)Φk (x)Ψl (z)
dx −a
0
+ µΦm (x)Ψn (z)Φk (x)Ψl (z) ,
IV. Sagittal Polarization
a (13) Kmnkl
=
hf(x/a)
−a
If the exciting stress distribution consists of the nonzero components σ13 (xa), or σ33 (x), or both, two displacement components u1 and u3 are generated in each case. The FVM system of linear equations can be written by analogy with (8) in the following form: (11) (1) (13) (3) (1) Kmnkl − ω 2 Mmnkl Akl + Kmnkl Akl = Fmn , k,l
(31) (1) (33) (3) (3) Kmnkl Akl + Kmnkl − ω 2 Mmnkl Akl = Fmn ,(21) k,l
dz λΦm (x)Ψn (z)Φk (x)Ψl (z)
dx 0
+ µΦm (x)Ψn (z)Φk (x)Ψl (z) , a
(31) Kmnkl
=
hf(x/a)
dz λΦm (x)Ψn (z)Φk (x)Ψl (z)
dx −a
0
a
hf(x/a)
(22)
+ µΦm (x)Ψn (z)Φk (x)Ψl (z) ,
(33) Kmnkl
=
dx −a
+
dz (λ + 2µ) Φm (x)Ψn (z)Φk (x)Ψl (z)
0 µΦm (x)Ψn (z)Φk (x)Ψl (z) ,
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the coefficients Mmnkl are defined by (10) and
V. The Exact Solution
a (1) Fmn
= −Ψn (0)
dx σ13 (x)Φm (x),
(23)
dx σ33 (x)Φm (x).
(24)
−a a (3) = −Ψn (0) Fmn −a
In the particular case of the basis functions (12) and (13), the coefficients of the system (21) take an explicit form: (11) Kmnkl
(13)
Kmnkl (31)
Kmnkl (33)
Kmnkl
h mk = (λ + 2µ) Jm+k−2,n+l+1 a n+l+1
a nl Jm+k,n+l−1 , +µ h n+l−1 λml + µnk = Jm+k−1,n+l , n+l λnk + µml (25) = Jm+k−1,n+l , n+l h mk =µ Jm+k−2,n+l+1 a n+l+1
a nl + (λ + 2µ) Jm+k,n+l−1 , h n+l−1
In the case of a shear horizontal polarization and a rectangular strip [Fig. 2(a)], the exact solution was derived by the author in [8], and it was analyzed and used for investigation of some shear wave scattering problems [8]– [10]. This solution has a simple coordinate representation. If u2 (x) is the y component of elastic displacements and σ23 (x) is the normal component of the stress tensor on the surface z = 0, the following coordinate representation of the integral impedance relation between these components takes place [9], [10]: a σ23 (x) =
a = −δn0
dx σ13 (x)(x/a)m ,
(26)
∞ µ νn βn tanh(hβn ) a n=0 πn(x + a) πn(x + a) cos cos , 2a 2a
ζ(x, x ) = −
βn2 =
u2 (x) =
M
(1)
Am0 (x/a)m ,
(28)
m=0
u3 (x) ≡ u3 (x, 0) =
2a
−
ρω 2 µ
(32)
dx Y (x, x )σ23 (x ),
−a ≤ x ≤ a.
−a
−a
u1 (x) ≡ u1 (x, 0) =
πn 2
For comparison with the FVM, the inverse relation of (30) is required:
(27)
After solving of 2(M + 1) × (N + 1) linear equations (21), the bottom displacement distributions can be found by analogy with (18):
M
(3) Am0 (x/a)m .
(29)
m=0
By this means the response of elastic displacements u1 (x), u2 (x), and u3 (x) to given arbitrary mechanical stresses σ13 (x), σ23 (x), and σ33 (x), applied to the bottom strip surface, is found by (18), (28), (29) and (16), (26), (27). To check the reliability of the FVM, the known exact analytical solution in the special case of shear stresses and a rectangular strip [8] is used below. However, it needs to be transformed to the form that is convenient for comparison.
(31)
where ν0 = 1/2, νn = 1, n ≥ 1,
a dx σ33 (x)(x/a)m .
(30)
The impedance ζ(x, x ) exactly takes into account the boundary conditions on the strip stress free surface, and it is given by:
−a a (3) Fmn = −δn0
−a ≤ x ≤ a.
−a
the coefficients Mmnkl and the factor Jmn are calculated in (15) and (17), respectively, and
(1) Fmn
dx ζ(x, x )u2 (x ),
(33)
In other words, the admittance Y (x, x ) should be calculated. To do this, one should multiply the relation (30) by cos [πm(x + a)/(2a)] with integer m and take into account the following orthogonality condition: νn a
a −a
πm(x + a) πn(x + a) dx cos cos = δmn . 2a 2a (34)
After simple manipulations, the impedance relation (30) is reduced to the inverse admittance relation (33) with the following expression for the admittance: ∞ 1 coth(hβn ) νn µa n=0 βn πn(x + a) πn(x + a) cos cos . 2a 2a
Y (x, x ) = −
(35)
In order to use (33) and (35) for practical calculations of elastic displacements as a function of arbitrary distributed
biryukov: elastic strip calculation method
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shear horizontal stresses, it is necessary at first to calculate the integral then make the summation [5]: u2 (x) = −
∞ 1 coth(hβn ) πn(x + a) νn cos µa n=0 βn 2a a
dx cos
−a
πn(x + a) σ23 (x ). (36) 2a
This exact analytical expression is used for checking the FVM results below. In practice, the number of summands is taken limited, certainly. This number depends on the desired accuracy. It follows from (36) that u2 (x) → ∞ under the resonance conditions βn = 0 and coth(hβn ) = ∞. Taking into account (32), the corresponding resonance frequencies can be found: πn µ ω= , n = 1, 2, 3 . . . (37) 2a ρ and
µ πm 2 πn 2 + , ω= ρ h 2a n = 0, 1, 2, . . . , m = 1, 2, 3 . . . ,
Fig. 3. The convergence of FVM results to the exact solution in the case of delta-like shear stresses and the rectangular strip for the frequency away from the first resonance (h/a = 0.2, Ω = 0.7, g = 10, x0 /a = 0.3).
(38)
respectively. They can be recognized, respectively, as corresponding to the longitudinal and transversal resonances related with the strip modes excited by the strip bottom stresses and reflected from strip edges x = ±a and from both upper z = h and bottom z = 0 sides.
VI. Calculations To check the reliability of the FVM by the exact solution (36) in the case of rectangular strip three functions for modeling, the exciting stress distributions on the strip bottom were chosen for analysis. One of them is the deltalike function: 2 2 2 g σd (x) = √ e−g (x−x0 ) a , (39) π where a parameter g specifies the width of this function with the maximum at the point x = x0 . The second and third ones are the functions with square root singularities [4] at the edge points x = ±a: cos(πγx/a) σs (x) = , 1 − (x/a)2
sin(πγx/a) σa (x) = , 1 − (x/a)2 (40)
which contain either the symmetric or antisymmetric oscillating multiplier with a parameter γ. The dimension coefficient N/m2 in (39) and (40) is implied and omitted everywhere. Throughout this item, the strip material parameters for aluminum (ρ = 2695 kg/m3 , µ = 2.5×1010 N/m2 ,
λ = 6.1 × 1010 N/m2 ) are used. The normalized dimensionless frequency: ρ Ω =ωa (41) µ is also introduced below for convenience. The displacement responses u2 (x) to the delta-like shear stress distribution σ23 (x) = σd (x) are shown in Figs. 3 and 4 for the strip thickness h/a = 0.2 and for the frequencies Ω = 0.7 and Ω = 1.5. In other words, these frequencies are away from the first resonance, Ω = π/2 [see (41) and (37) with n = 1], and nearby, respectively. It is seen from Figs. 3 and 4 that FVM results converge rapidly to the exact solution (36) as the numbers M and N of involved basis functions (12) and (13) increase. As usual, the proximity of resonance gives rise to the growth of the displacement field magnitude. The displacement responses to the shear stresses (40) with the parameter γ = 2 for the symmetric, σ23 (x) = σs (x), (Fig. 5) and antisymmetric, σ23 (x) = σa (x), (Fig. 6) stresses calculated by the FVM, also demonstrate an excellent agreement with the exact solution (36). It should be noted that the displacement field is limited everywhere in spite of the square root singularities of the exciting stresses (40). This is due to the convergence of the integrals in (11) and (36). After checking of the FVM, more complicated cases of strip forms and the general case of polarization can be considered. The results of FVM simulation for different strip forms (Fig. 2) and symmetric shear stresses σ23 (x) = σs (x) are shown in Fig. 7 for the strip thickness h/a = 0.2, the frequency Ω = 0.7, and the oscillation parameter γ = 3.
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Fig. 6. The convergence of FVM results to the exact solution in the case of antisymmetric shear stresses and the rectangular strip (h/a = 0.2, Ω = 0.7, γ = 2). Fig. 4. The convergence of FVM results to the exact solution in the case of delta-like shear stresses and the rectangular strip for the frequency nearby the first resonance (h/a = 0.2, Ω = 1.5, g = 10, x0 /a = 0.3).
The parameter d = a/2 in the trapezium case. It is interesting to note that the displacement distributions are smoothed out in contrast to the stress distribution. In Figs. 8 and 9, the results of FVM simulation in the most complicated case of sagittal polarization and different strip forms are shown for the longitudinal component of the exciting stresses σ13 (x) = σs (x) and for the normal component σ33 (x) = σs (x), respectively. It should be noted from a comparison of Figs. 8 and 9 that the displacements generated by the third component of stress are considerably greater than the displacements generated by the first component. Moreover, the displacement distributions in Fig. 9 depend strongly on the strip form. As for the convergence of results in Figs. 7–9, their changes with the parameters M and N over M = 11 and N = 2 are negligible.
VII. Conclusions
Fig. 5. The convergence of FVM results to the exact solution in the case of symmetric shear stresses and the rectangular strip (h/a = 0.2, Ω = 0.7, γ = 2).
A new FVM for determining the elastic strip response to stresses arbitrarily distributed on the flat side of the strip has been developed. The FVM has been derived from the variational principle, and it does not use the meshing of the strip in contrast to the FEM. The reliability of the method has been checked by comparing FVM results with the exact analytical solution in the special case of shear stresses and a rectangular strip. The transformation of this solution to the form convenient for comparison has been done. An excellent agreement has been demonstrated. The method is simpler for computer simulations as compared to the well-known FEM. The application of FVM to problems of SAW propagation in periodic electrode structures recently was reported [11].
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Fig. 9. The influence of the strip form on the displacement distributions in the case of the normal stress component (h/a = 0.2, Ω = 0.7, γ = 3, d = a/2, M = 11, N = 2). Fig. 7. The influence of the strip form on the displacement distribution in the case of the shear stress component (h/a = 0.2, Ω = 0.7, γ = 3, d = a/2, M = 11, N = 2).
Acknowledgments The author is grateful to M. Weihnacht for discussions and to Vectron International (VI) for support of this work.
References
Fig. 8. The influence of the strip form on the displacement distributions in the case of the longitudinal stress component (h/a = 0.2, Ω = 0.7, γ = 3, d = a/2, M = 11, N = 2).
[1] K. Hasegawa and M. Koshiba, “Finite-element solution of Rayleigh-wave scattering from reflective gratings on a piezoelectric substrate,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 37, pp. 99–105, Mar. 1990. [2] P. Bauerscmidt, R. Lerch, J. Machui, W. Ruile, and G. Visintini, “Reflection and transmission coefficients of SAW in a periodic grating computed by finite element method,” in Proc. IEEE Ultrason. Symp., 1990, pp. 421–423. [3] H. P. Reichinger and A. R. Baghai-Wadji, “Dynamic 2D analysis of SAW-devices including massloading,” in Proc. IEEE Ultrason. Symp., 1992, pp. 7–10. [4] P. Ventura, J. M. Hod´e, and M. Solal, “A new efficient combined FEM and periodic Green’s function formalism for the analysis of periodic SAW structures,” in Proc. IEEE Ultrason. Symp., 1995, pp. 263–268. [5] S. V. Biryukov, “Analytical expression for a rectangular strip response to arbitrary surface shear stresses,” in Proc. IEEE Int. Forum Wave Electron. Its Applicat., 2000, pp. 290–293. [6] S. V. Biryukov, “Fast variation method for elastic strip calculation,” in Proc. IEEE Ultrason. Symp., 2000, pp. 155–158. [7] E. P. EerNisse and R. Holland, “On variational techniques for piezoelectric device analysis,” Proc. IEEE, vol. 55, pp. 1524– 1525, Aug. 1967. [8] S. V. Biryukov, “Quadratic effects in the scattering of horizontally polarized surface acoustic waves by a rectangular projection,” Sov. Phys.—Acoust., vol. 33, no. 1, pp. 7–10, 1987. [9] S. V. Biryukov, “Shear surface waves in a periodic grating of low rectangular corrugations,” Sov. Phys.—Acoust., vol. 38, no. 4, pp. 328–331, 1992. [10] S. V. Biryukov, V. G. Polevoi, and M. Weihnacht, “Exact expression for the impedance of a rectangular strip and second-order effects in Bleustein-Gulyaev wave scattering,” J. Appl. Phys., vol. 84, pp. 6698–6707, Dec. 15, 1998.
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[11] S. V. Biryukov and M. Weihnacht, “Elastic electrode polarization in a spatial harmonic field and the natural boundary element method,” in Proc. IEEE Ultrason. Symp., 2001, to be published.
Sergey V. Biryukov (M’98) was born in Vladimir region, Russia, in 1952. He received the M.Sc. degree in physics from the Moscow Institute of Physics and Technology (MIPT) in 1975, the Ph.D. degree in physics and mathematics from the same institute in 1978, and the D.Sc. degree in physics and mathematics in 1989 from the Institute of General Physics of the Russian Academy of Sciences, Moscow. Since 1978 he has been with the Radiophysics Research Laboratory (Theoretical
Department), Mints Radiotechnical Institute, Moscow, Russia; at present he is a head scientist. During 1990–1992 he also worked as a part-time professor of the General Physics Chair at the MIPT. Since 1993 he has been collaborating with the Institute of Solid State and Materials Research Dresden, Germany; and since 1999 he has been working there as a visiting scientist. His researches are concentrated mainly on a theory of SAW interaction with irregularities of a solidstate boundary and on the application of the theory to the analysis of SAW devices. His present scientific interests include development of the surface impedance method to the problems of acoustic and electromagnetic waves propagation in inhomogeneous media. He is one of the authors of the monograph Surface Acoustic Waves in Inhomogeneous Media (Moscow: Nauka, 1991; Berlin, Heidelberg: Springer-Verlag, 1995). D.Sc. Biryukov is a member of IEEE. He was awarded the former USSR Highest Young Scientist Award in Science and Technology in 1984.
