387 Appendix A Type of model, number of neighbors (z) R 0 (with values of the stdev) (with values of the stdev) N 0 (with values of the stdev) (with values of the stdev) D (with values Size of the cluster number of particles involved) DLA- procedure 0.1743 ( 0.0044) 0.9996 ( 0.0045) 2.0235 ( 0.0702) 0.5427 (0.0034) 1.8419 (0.0059) 4 10N 21.8R A random rain model. 0.1782 ( 0.0049) 1.0028 ( 0.004) 1,9807 ( 0.0651) 0,5399 (0.0034) 1,858 (0.0087) 4 10N 21.4R The lattice DLA model 0.8972 ( 0.0276) 0,991 ( 0.0075) 2.1802 ( 0.1379) 0.6046 (0.0046) 1.6396 (0.0103) 5895N 138R LMRE 0.8389 ( 0.0334) 1.0012 ( 0.0052) 2.0153 ( 0.0841) 0.6087 (0.0059) 1.6458 (0.0109) 6580N 139R LMRP 0.7045 ( 0.0275) 1.0065 ( 0.0055) 1.8826 ( 0.0922) 0.6172 (0.0047) 1.6312 (0.0105) 7641N 140R The lattice DLA model 0.8473 ( 0.034) 0.9902 ( 0.0055) 2.1936 ( 0.1043) 0.612 ( 0.0047) 1.6183 ( 0.0097) 8042N 167R LMRE 0.8343 ( 0.0341) 0.9953 ( 0.0062) 2.0733 ( 0.105) 0.6096 (0.0059) 1.6335 (0.0109) 8250N 165R LMRP 0.7026 ( 0.0383) 0.9903 ( 0.0088) 2.2237 ( 0.1929) 0.609 (0.006) 1.6266 (0.008) 11801N 167R The lattice DLA model 1.8495 ( 0.0639) 0.9889 ( 0.0058) 2.2119 ( 0.094) 0.6096 (0.0055) 1.6229 (0.0111) 5901N 296R LMRE 1.8459 ( 0.1624) 0.9861 ( 0.0103) 2.2872 ( 0.216) 0.6054 (0.0101) 1.631 (0.0173) 6118N 292R LMRP 1.3517 ( 0.0812) 0.9919 ( 0.0077) 2.1654 ( 0.1483) 0.6081 (0.0084) 1.6331 (0.0188) 9778N 295R NEW POSSIBILITIES IN DESCRIPTION OF DISORDERED MEDIA Table 1. The calculated fractal dimension for clusters obtained for different systems obtained by the methods described in section 2. For the lattice model with random exclusion/permission we use the abbreviation (LMRE/P) of the stdev) (diameter, Conventional (z = 3) (z = 3) (z = 3) (z = 4) (z = 4) (z = 4) (z = 6) (z = 6) (z = 6) “ ” 388 Type of model R 0 (with values of the stdev) (with values of the stdev) N 0 (with values of the stdev) (with values of the stdev) D (with values of the stdev) Size of the cluster number of particles involved) Random lattice 0.6801 ( 0.0066) ( 0.0021) 0.9892 2.2152 ( 0.0422) 0.5388 (0.001) 1.8358 (0.0035) Random lattice 0.932 ( 0.0079) ( 0.0016) 0.9462 3.3279 ( 0.0571) 0.4853 (0.0084) 1.9497 (0.0004) Random lattice 0.7053 ( 0.0069) 0.9867 ( 0.002) 4.5213 ( 0.0909) 0.5075 ( 0.001) 1.9442 ( 0.0003) Nigmatullin and Alekhin (diameter, (z = 3) (z = 4) (z = 6) 200 × 200 200 × 200 200 × 200 Table 2. Parameters of the QF obtained by the procedure described in section 3 References 1. Mandelbrot B (1983) The Fractal Geometry of Nature. Freeman, San- Francisco. 2. Nigmatullin RR, Alekhin AP (2005) Realization of the Riemann-Liouville Integral on New Self-Similar Objects. In: Books of abstracts, Fifth EUROMECH Nonlinear Dynamics Conference August 7–12, pp. 175–176 Prof. Dick H. van Campen (ed.), Eindhoven University of Technology, The Netherlands. 3. Mehaute A, Nigmatullin RR, Nivanen L (1998) Fleches du Temps et 4. Nigmatullin RR, Le Mehaute A (2005) J. Non-Cryst. Solids, 351:2888. 5. Nigmatullin RR (2005) Fractional kinetic equations and universal decoupling of a memory function in mesoscale region, Physica A (has been accepted for publication). 6. Fractals in Physics (1985) The Proceedings of the 6th International Sym- posium, Triest, Italy, 9–12 July; Pietronero L, Tozatti E (eds.), Elsevier Science, Amsterdam, The Netherlands. Geometrie Fractale, Hermez, Paris (in French). FRACTIONALDAMPING: STOCHASTIC including models for viscoelastic damping. Damping behavior of materials, if mod- eled using linear, constant coefficient differential equations, cannot include the long imated by fractional order derivatives. The idea has appeared in the physics lit- erature, but may interest an engineering audience. This idea in turn leads to an infinite-dimensional system without memory; a routine Galerkin projection on that material may have little engineering impact. 1 Introduction damping, the design of controllers, and other areas. The aim of this paper is twofold. First we will present, with a fresh engineering flavor, a result that ically expected in many engineering materials with complex internal dissi- pation mechanisms. Second, we will use the insights obtained from the first Mechanical Engineering Department, Indian Institute of Science, Bangalore ORIGIN AND FINITE APPROXIMATIONS 560012, India Fractional-order derivatives appear in various engineering applications microstructural disorder can lead, statistically, to macroscopic behavior well approx- memory that fractional-order require. However, sufficiently greatderivatives infinite-dimensional system leads to a finite dimensional system of ordinarydifferen- tialequations(ODEs)(integerorder) that matches the fractional-order behavior over user-specifiable, but finite, frequency ranges. For extreme frequencies (small or large), the approximation is poor. This is unavoidable, and users interested in such extremes or in the fundamental aspects of true fractional derivatives must take note of it. However, mismatch in extreme frequencies outside the range of interest for a particular model of a real Keywords Fractional-order derivatives have proved useful in the modeling of viscoelastic will show that sufficiently disordered (random) and high-dimensional inter- nal integer-order damping processes can lead to macroscopically observable fractional-order damping. This suggests that such damping may be theoret- © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 389–402. Abstract Damping, fractional derivative, disorder, Galerkin, finite element. unknown to engineering audiences (this discussion may be found in [2]). We may be found in the physics literature (e.g. [1]) but which seems largely 389 Satwinder JitSingh and Anindya Chatterjee 2390 approximations can be developed for the fractional derivative term, so that be accurately approximated by finite dimensional systems without memory. Otherwise-motivated finite dimensional approximations have been obtained accessible to some audiences. Results of finite element formulations based on this Galerkin projection will also be presented. The approximations developed have approximately uniform and small error over a broad and user-specified frequency range. Our basic approach, though differently motivated, has strong similarities with an approximation scheme developed in [5]. That scheme has recently been critiqued [6], and some of that criticism (concerned with some found in [7]. 2 Stochastic Origins The fractional derivative of a function x(t), assuming x(t) ≡ 0fort<0, is taken as D α [x(t)] = 1 Γ (1 − α) d dt t 0 x(τ) (t − τ) α dτ , where 0 <α<1, and Γ represents the gamma function. Observe that 1 Γ (1 − α) d dt t 0 τ α−1 + (t − τ) α dτ = πδ(t) sin[π(1 − α)] Γ (1 − α) , where δ(t) is the Dirac delta function; and where τ + = τ when τ > 0, and τ + = 0 otherwise. So, if a system obeys D α [x(t)] = h(t)(1) and has initial conditions x(t) ≡ 0fort ≤ 0, and if h(t) is an impulse at zero, then x(t)=Ct α−1 for t>0 and some constant C (power law decay to zero). For simplicity, we consider an equation relevant to a “springpot”: σ(t)=E 1 D α [ǫ(t)]. (2) decay in time. Rubber molecules presumably cannot remember the past. Linear models for rubber should therefore involve linear differential equations with constant coefficients. Such systems have exponential decay in time. Why the power law? part to develop a Galerkin procedure. Using this, accurate finite-dimensional infinite dimensional and memory-dependent fractionally damped systems can before (e.g. [3] and [4]), but we think our approach is new, direct, and more By Eq. (1), the strain in a sample obeying Eq. (2) can havepowerlaw short-time andhigh-frequency asymptotics) applies to our work as well. Wewill discuss those asymptotic issues and their engineering relevance at theend of this paper. The latter part of this paper has material that may also be Singh and Chatterjee FRACTIONALDAMPING: STOCHASTIC 3913 Wall x distributed viscous forces elastic, massless Fig. 1. One dimensional viscoelastic model. Consider the model sketched in Fig. 1. An elastic rod of length L has a distributed stiffness b(x) > 0. Its axial displacement is u(x, t). The internal force at x is b(x) u x , and interaction with neighboring material causes viscous forces c(x) u t ,withc(x) > 0andwithx and t subscripts denoting partial derivatives. The free end of the rod is displaced, held for some time, and released. Subsequent motion obeys (b(x)u x ) x − c(x)u t =0, u(0,t)=0, u x (L,t)=0. (3) We will now discuss how sufficient complexity (randomness) in b and c can lead to power law decay. A solution for the above is sought in the form u(x, t)= n i = 1 a i (t)φ i (x) where large n gives accuracy, the a i (t) are to be found, and the chosen ba- sis functions φ i (x)satisfyφ i (0) = 0. We now use the method of weighted residuals [8]. Defining symmetric positive definite matrices B and C by B ij = L 0 b φ i,x φ j,x dx and C ij = L 0 cφ i φ j dx, and writing a for the vec- tor of coefficients a i (t), we obtain C ˙a = −Ba. On suitable choice of φ i , C is the identity matrix. Then ˙a = −Ba. With sufficiently complex microstructural behavior, B may usefully be treated as random. Let us study a random B.BeginwithA,ann×n matrix, with n large. Let the elements of A be random, i.i.d. uniformly in (−0.5, 0.5). Let B = A T A. B is symmetric positive definite with probability one. We will solve ˙x = −Bx. (4) 4392 Solution is done numerically using, for initial conditions, a random n × 1col- umn matrix x 0 whose elements are i.i.d. uniformly in (−0.5, 0.5). The process is repeated 30 times, with a new B and x 0 each time. The results, for n = 400, are shown in Fig. 2. 0 10 20 30 40 0 1 2 3 4 5 6 time, t norm(x(t)) - 1 0 1 2 3 4 - 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6 0.8 ln(t) ln(RMS(norm(x(t )))) Fig. 2. √ T 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k/n λ k n=250 n=400 / n Fig. 3. Eigenvalues of B for n = 250 and 400. The solutions, though they are sums of exponentials, decay on average like t −1/4 .Why? eca a straight line on a log-log scale.Afitted line has slope −0.24 ≈−1/4. x against time.30 individual solutions (thin lines) as well as their RMS values (thic gray). Right: RMS value of norm(x) against time is Left: norm(x)= x k Singh and Chatterjee FRACTIONALDAMPING: STOCHASTIC 3935 The answer lies in the eigenvalues of B. The spectra of random matrices comprise a subject in their own right. Here, we use numerics to directly obtain a simple fact. Let n = 250. Take a random n × n matrix B as above. Let λ k , k =1, 2, ··· ,n, be its eigenvalues in increasing order. Figure 3 shows λ k n plotted against k/n. Superimposed are the same quantities for n = 400. The coincidence be- tween plots indicates a single underlying curve as n → ∞. That curve passes through the origin, and can be taken as linear if we restrict time to values t ≫O(1/n), by when solution components from the large eigenvalues have decayed to negligible values. Then λ k n = β k n (5) for some β>0. For simplicity, we ignore the variation of eigenvalues around the linear fit. The solution for the i x i (t)= n a ik e −λ k t = n k = 1 a ik e −β 2 k 2 t/n , (6) where the coefficients a ik ,byrandomnessofx 0 and B and orthonormality of eigenvectors of the latter, are taken as random, i.i.d., and with zero expected value. The variance is then (upon scaling the initial condition suitably) var(x i (t)) = 1 n 2β 2 t n k = 1 2β 2 t n e −2β 2 k 2 t/n . Define ξ = 2β 2 t n k. For β 2 t ≪ n and n ≫ 1, the sum is approximated by an integral: var(x i (t)) = 1 n √ 2c 2 t ∞ 0 e −ξ 2 dξ = C 2 n √ t , for some C. Finally, RMS √ x T x is (using independence of the components of x) RMS x T x = n i = 1 var(x i (t)) = C t 1/4 , (7) which explains the numerical result. Our point is that no special microstruc- tural damping mechanisms are needed for fractional derivatives to appear, if there is the right sort of disorder or randomness. k = 1 thelement of x is of the form 6394 3 Galerkin Projectio ns ξ) ∂ ∂t u(ξ,t)+ξ 1 α u(ξ,t)=δ(t) , u(ξ, 0 − ) ≡ 0 , (8) where α>0andδ(t) is the Dirac delta function. The solution is u(ξ,t)=h(ξ, t)=exp(−ξ 1/α t) , where the notation h(ξ, t) is used to denote “impulse response function.” On integrating h with respect to ξ between 0 and ∞ we get a function only of t, given by g(t)= ∞ 0 h(ξ,t) dξ = Γ (1 + α) t α . (9) symbol L. system L, again starting from rest at t = 0, is (the last two expressions below are equivalent) r(t)= t 0 g(t − τ)˙x(τ ) dτ = Γ(1 + α) t 0 ˙x(τ) (t − τ) α dτ = Γ(1 + α) t 0 ˙x(t − τ) τ α dτ . We find that r(t) ≡ Γ (1 + α)Γ (1 − α)D α [x(t)] , provided x(t) ≡ 0fort ≤ 0, and (we now impose) 0 <α<1. In this way, we 1. Solve ∂ ∂t u(ξ,t)+ξ 1 α u(ξ,t)= ˙x(t). (10) 2. Then integrate to find D α x(t)= 1 Γ (1 − α)Γ (1 + α) ∞ 0 u(ξ,t) dξ . (11) Abstractly, g(t) is simply the impulse response of a constantlinear, coefficient system starting from rest. Let us denote that linear system by the Now if we replace the forcing δ(t) in Eq. (8) with some sufficiently well-behaved function x˙(t), then the corresponding response r(t)ofthesame have replaced an α -order derivative by the following operations: Prompted by the above, consider the PDE (or ODE in t with a free para- meter Singh and Chatterjee FRACTIONALDAMPING: STOCHASTIC 395 7 There is no approximation so far. We have replaced one infinite dimen- sional system (fractional derivative) with another. The advantage gained is that we can now use a Galerkin projection to obtain a finite system of ODEs. u(ξ,t) ≈ n i = 1 a i (t)φ i (ξ) , where n is finite, the shape functions φ i are to be chosen by us, and the a i are to be solved for. The choice of φ i will be discussed later. We first outline R(ξ, t)= n i = 1 ⎧ ⎪ ⎨ ⎪ ⎩ ˙a i (t)φ i (ξ)+ξ 1 α a i (t)φ i (ξ) ⎫ ⎪ ⎬ ⎪ ⎭ − ˙x(t) , where R(ξ, t) is called the residual. R(ξ, t) is made orthogonal to the shape functions by setting ∞ 0 R(ξ, t)φ m (ξ) dξ =0,m=1, 2, ··· ,n. (12) The integrals above need to exist; this will influence the choice of φ i (later). A˙a + Ba= c ˙x(t) , (13) where A and B are n × n matrices, a is an n × 1 vector containing a i ’s, and c is an n × 1 vector. ¨x as well as D α [x(t)], we will use the quantities x and ˙x as parts of the state vector, along with the a i above. Having access to ˙x at each instant, therefore, i ∞ 0 φ i (ξ) dξ i D α [x(t)] ≈ 1 Γ (1 + α)Γ (1 − α) c T a, where the T superscript denotes matrix transpose. For the Galerkin projection, we assume that Eq. (10) is satisfied by the Galerkin procedure for Eq. (10). Substituting the approximation for u(ξ,t) in Eq. (10), we define Equation (12) constitute n ODEs, which can be written in the form During numerical solution of (say) a second-order system including both we can solve Eq. (13) numerically to obtain the a . Note that is in fact c ,theithelement of c in Eq. (13) above. It follows that 8396 4 Finite Element Approximation η(ξ)= ξ 1/α 1+ξ 1/α (14) which is a monotonic mapping of [0, ∞] to [0,1]. The mapping depends on the order of the fractional derivative α. The advantage of using this α-dependent mapping lies in better error control within a given frequency range. This is because of the role that ξ and t play in exp(−ξ 1/α t). Here, we can consider T ∗ ≡ 1/ξ ∗1/α for some time T ∗ . It suggests that frequency F ∗ ≡ ξ ∗1/α = η ∗ 1 − η ∗ . (15) Thus, any frequency F ∗ corresponds to an α-independent point η ∗ on the unit interval. In other words, a given frequency F ∗ corresponds to a unique point η ∗ on the unit interval, independent of α. Conversely, in subsequent discretization of the interval [0, 1] into a given finite element mesh, the corre- sponding points on the frequency axis are independent of α. η(ξ) ≈ 1 − 1 ξ 1/α . This affects the choice of our last element’s shape function. Suppose we take (1 − η(ξ)) β as the shape function in the last subinterval of (0, 1). Then, β> α 2 + 1 2 . The above is always satisfied if we take β = 1 (because 0 <α<1), and we take β = 1 (independent of α)inthispaper. To perform the Galerkin projection, we use the “hat” functions defined as follows (see Fig. 4): φ 1 (η)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p 1 − η p 1 , 0 ≤ η ≤ p 1 , 0 elsewhere and approximation. To this end, we define the following auxiliary variable η(ξ) The above Galerkin projection can be used to develop a finite-element Notice that, for large values of ξ, Eq. (14) becomes all integrals involved in Eq. (12) (i.e., in the Galerkin approximationPro- cedure) are bounded if Singh and Chatterjee [...]... York 4 Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento 1:161–198 5 Mainardi F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics Springer, New York 6 Beyer H, Kempfle S (1995) Definition of physically consistent damping laws with fractional derivatives,... complex systems such as nonlinear, linear time-varying (LTV), and linear parameter-varying (LPV), only few studies deal with Lyapunov stability of fractional systems, and synthesis of control laws for such systems is 419 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 419–434 © 2007 Springer 420 Moze, Sabatier, and Oustaloup... number of parameters In particular, regarding vibrations, it should be able to reproduce the experimentally found behaviour of the damping ratio n as a function of the natural angular frequency n [1] 403 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 403–416 © 2007 Springer 404 Catania and Sorrentino In the present study... Sorrentino Clearly, since C2 . first Mechanical Engineering Department, Indian Institute of Science, Bangalore ORIGIN AND FINITE APPROXIMATIONS 560012, India Fractional- order derivatives appear in various engineering applications microstructural. Applications in Physics and Engineering, 389–402. Abstract Damping, fractional derivative, disorder, Galerkin, finite element. unknown to engineering audiences (this discussion may be found in [2]) macroscopically observable fractional- order damping. This suggests that such damping may be theoret- © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments