International Journal for Multiscale Computational Engineering, 6(5)435–449(2008)
Employing the Discrete Fourier Transform in the Analysis of Multiscale Problems Michael Ryvkin School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
ABSTRACT The idea of employing the discrete Fourier transform casted as the representative cell method for the solution of multiscale problems is illustrated. Its application in combination with analytical (structural mechanics methods, Wiener-Hopf method, integral transform methods) and numerical (finite element method, higher-order theory) methods is demonstrated. Both cases of 1-D and 2-D translational symmetry are addressed. In particular, the problems for layered, cellular, and perforated materials with and without flaws (cracks) are considered. The method is shown to be a convenient and universal analysis tool. Its numerical efficiency allowed us to solve optimization problems characterized by multiple reanalysis.
KEYWORDS discrete Fourier transform, periodic microstructure, representative cell, flaw, crack
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1. INTRODUCTION In many cases, the multiscale problems arise in the analysis of elastic systems possessing periodicity in geometric and elastic properties. Typical examples of such systems on a macroscale are large space structures or two-dimensional grids [1]. On a much smaller scale, there is a large family of composite materials with periodic microstructure, including fiber-reinforced and cellular materials [2]. It appears that in this situation, it is possible to avoid the homogenization procedure and to get the result using the Discrete Fourier Transform (DFT). The employing of this transform in the analysis of the periodic systems has long been known. A rather comprehensive review of the works on this subject is given in [3]. It is found that most of the works are either from the field of dynamics or related to the buckling analysis in the framework of the Bloch wave approach. Among the recent publications, it is pertinent to note the works on the fracture in lattices (e.g., [4]), the Bloch wave study of pattern transformations in periodic materials [5], and the analysis of finite repetitive structures [6]. In the present article, the application of the representative cell method suggested in [7] for the solution of multiscale problems is illustrated. This method is based on the DFT and is found to be a very convenient and universal mathematical tool that can be easily employed for the analysis of periodic structures in combination with analytical as well as numerical methods. In the next section, the idea of the method is illustrated by means of a trivial example. In Section 3, the applications for the structures and the layered materials posessing 1D translation symmetry are considered, whereas in Section 4, the case of 2-D symmetry is addressed. In both cases, special attention is given to the flaw existence violating the periodicity. In the final section, several concluding remarks are drawn. 2. ILLUSTRATIVE EXAMPLE It is convenient to present the idea of the representative cell approach by means of the known structural mechanics problem on bending of an infinite periodically supported beam (Fig. 1). The beam has uniform bending stiffness EJ and, in the global coordinate system X, Y , occupies the region ∞ < X < ∞. The system periodicity in the geometric and elas-
MICHAEL RYVKIN
tic properties is characterized by the vector b of 1D translational symmetry defining the local scale of the problem. The beam is loaded by the two transverse forces Q, S acting in two neighboring spans at the distances tQ and tS from the left support, respectively. Note that since the problem is linear, it was sufficient to consider only one force and then obtain the multiple force solution by superposition. The two forces are considered in the example simultaneously to emphasize the fact that the loading in the representative cell problem, which will be obtained later, is complex valued. The infinite domain is considered as an assemblage of an infinite number of repetitive cells k = 0, ±1, ±2, ... and local systems of coordinates −L ≤ x ≤ L are introduced in each cell in an identical manner as shown in Fig. 1(b). Consequently, the problem of deriving the bending deflections Y (X) in the global coordinates ∞ < X < ∞ is replaced by the problem of displacements yk (x), k = 0, ±1, ±2, ... in local coordinate systems, which is formulated as follows. (i) Within the cell, the deflections obey the fourthorder differential equation EIykIV = Pk ,
k = 0, ±1, ±2, ...
(1)
Pk = Qδk0 δ(x − tQ ) + Sδk1 δ(x − tS )
(2)
where
Here δij denotes the Kronecker symbol and δ(x) is the Delta function. (ii) At the support point x = 0, which can be referred to as the inner boundary Γ0 , the boundary conditions are yk (±0) = 0 θk (+0) = θk (−0) , Mk (+0) = Mk (−0)
(3) k = 0, ±1, ±2, ... (4) (5)
where θk (x) = yk0 (x) is the rotation angle and Mk (x) = −EJyk00 (x) is the bending moment. (iii) At the outer boundaries Γ± : x = ±L, the continuity conditions between the neighboring cells must be fulfilled. Defining the generalized displacement and force vectors as uk = {yk , θk }
International Journal for Multiscale Computational Engineering
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EMPLOYING THE DISCRETE FOURIER TRANSFORM IN THE ANALYSIS OF MULTISCALE PROBLEMS
Q
S
tQ
tS
(a) 11111 00000 00000 11111 00000 11111
11111 00000 00000 11111 00000 11111
00000 00000 11111 X11111 00000 11111
11111 00000 00000 11111 00000 11111
2L Y
(b) 11111 00000 00000 11111 00000 11111
x
y k=−1
k=−2
11111 00000 00000 11111 00000 11111
x
y
f_ * − u *
11111 00000 00000 11111 00000 11111
x
y k=1
k=0
(c)
11111 00000 00000 11111 00000 11111
P * 11111 00000 00000 11111 00000 11111
x
y k=2
k=3
f+ * x u+ *
y FIGURE 1. (a) Infinite periodically supported beam subjected to local loading, (b) the representative cell scheme, and (c) the representative cell problem
Fig. 1abc
and fk = {Mk , Vk }, respectively, where Vk = −EJyk000 (x) is the shear force, we have u+ k fk+
= u− k+1 =
− fk+1
,
(6) k = 0, ±1, ±2, ... (7)
The superscript (±) denotes the corresponding boundary values, for example, u+ k ≡ {yk (L), θ(L)} and u− ≡ {y (−L), θ(−L)}. k k The formulated problem for the infinite beam is reduced to the problem for a finite beam of the length 2L being the representative cell of the problem (see Fig. 1(c)) by means of the DFT:
Volume 6, Number 5, 2008
g∗ (ϕ)=
∞ X
gk exp(iϕk)
(8)
k=−∞
gk =
1 2π
Z
π
g∗ exp(−iϕk)dϕ,
−π
