332 Computational Mechanics of Composite Materials A1=A的 (7.38) 威=赋+军+贷广戏 (7.39) Likewise,the recurrence relations for the forcing terms simplify to Pia =p (7.40) =破-及At+p时 (7.41) Since the term A"is unchanged by reduction,it is clear that A=A.Similarly, p"is unchanged by reduction,so p"=p.The situation for B and g"is more complicated.We solve for them analytically using the solution of(7.32). Consider the case p=0.Clearly,then,it is the case that =o.The solution of(7.32)is therefore given by x()=-exp(At)g (7.42) where A=(+BA",=(+B+).The average of this solution must also solve (7.31)since it is the equation for the average value of the solution by definition.The average value of x(r)in (7.32)on the interval [0,1]is given by -onah月-t-en同arg (7.43) The solution to(7.31)is given by x6=-1+B6-A(q。+入) (7.44) The right hand sides of(7.43)and (7.44)are demonstrated to be equal for all A; setting A=0 and solving for B"yields the solution given in the statement of the proposition.The case when p0 proceeds similarly. Solutions of (7.32)have the same "average"or coarse-scale behaviour as solutions of(7.12).The main point is that this homogenisation procedure allows for coefficients to vary on arbitrarily many intermediate scales,which is in contrast to the classical homogenisation examples,which did not allow for intermediate scales
332 Computational Mechanics of Composite Materials h k h Ak+1 = A (7.38) ( ) h k h k h k h k k h Bk B A I B A 1 2 1 4 − + = + + δ (7.39) Likewise, the recurrence relations for the forcing terms simplify to h k h k p = p +1 (7.40) ( ) h k h k h k h k k h k q q A I B p 1 1 2 − + = − + δ (7.41) Since the term h A is unchanged by reduction, it is clear that ( ) 0 −∞ A = A h . Similarly, h p is unchanged by reduction, so ( ) 0 −∞ p = p h . The situation for h B and h q is more complicated. We solve for them analytically using the solution of (7.32). Consider the case 0 ( ) 0 = −∞ p . Clearly, then, it is the case that ( ) 0 −∞ q = q h . The solution of (7.32) is therefore given by x t (At)q ~ ~ ( ) = −exp (7.42) where ( ) h h A I B A ~ −1 = + , = ( )( ) + + λ − h h q I B q ~ 1 . The average of this solution must also solve (7.31) since it is the equation for the average value of the solution by definition. The average value of x(t) in (7.32) on the interval [0,1] is given by x ( ) () At dt q ( ) I A A q ~ ~ ~ exp ~ ~ exp 1 1 0 − = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − ∫ (7.43) The solution to (7.31) is given by = −( )( ) + − + λ −∞ −∞ −∞ − (−∞) 0 1 ( ) 2 0 ( ) 1 0 ( ) x0 I B A q (7.44) The right hand sides of (7.43) and (7.44) are demonstrated to be equal for all λ; setting λ=0 and solving for h B yields the solution given in the statement of the proposition. The case when 0 ( ) 0 ≠ −∞ p proceeds similarly. Solutions of (7.32) have the same “average” or coarse-scale behaviour as solutions of (7.12). The main point is that this homogenisation procedure allows for coefficients to vary on arbitrarily many intermediate scales, which is in contrast to the classical homogenisation examples, which did not allow for intermediate scales
Multiresolutional Analysis 333 As formulated above,the multiresolution approach to homogenisation requires the computation of A and B,i.e.a limit over infinitely many scales.The typical practice is to compute successive A and B terms until finer approximations vary by less than some specified tolerance,and use these matrices as approximations to A and B. Besides establishing the general framework for multiresolution reduction and homogenisation,it is observed that for systems of linear ordinary differential equations,using the Haar basis (or a multiwavelet basis)provides a technical advantage.Since the functions of the Haar basis on a fixed scale do not have overlapping supports,the recurrence relations for the operators and forcing terms in the equation may be written as local relations and solved explicitly.Thus,for ODEs,an explicit local reduction and homogenisation procedure is possible. Let us consider for illustration (7.1)with initial conditions at x=0.It may be rewritten as the coupled first-order system d v(x)=-f(x) dx (7.45) u(x)=e(x)v(x) dx By writing in an integral form one can obtain 8880-oA (7.46 g(t)=0 as well as p(t)= 0 -f0 Using the reduction procedure in the Haar basis for a system of linear differential equations,the goal is to find constants B,A",p","such that 6+圈}*a-48} (7.47) after reduction to the scale Vo will be the same as (7.46)reduced to that scale.This is accomplished by solving the recursion relations between the operators in the reduced equations explicitly,element-by-element in each matrix.This is possible to do because of the non-overlapping supports of the Haar basis functions on a fixed scale.The result for the first two coefficients is
Multiresolutional Analysis 333 As formulated above, the multiresolution approach to homogenisation requires the computation of ( ) 0 −∞ A and ( ) 0 −∞ B , i.e. a limit over infinitely many scales. The typical practice is to compute successive ( ) 0 J A − and ( ) 0 J B − terms until finer approximations vary by less than some specified tolerance, and use these matrices as approximations to ( ) 0 −∞ A and ( ) 0 −∞ B . Besides establishing the general framework for multiresolution reduction and homogenisation, it is observed that for systems of linear ordinary differential equations, using the Haar basis (or a multiwavelet basis) provides a technical advantage. Since the functions of the Haar basis on a fixed scale do not have overlapping supports, the recurrence relations for the operators and forcing terms in the equation may be written as local relations and solved explicitly. Thus, for ODEs, an explicit local reduction and homogenisation procedure is possible. Let us consider for illustration (7.1) with initial conditions at x=0. It may be rewritten as the coupled first-order system ⎪ ⎩ ⎪ ⎨ ⎧ = = − − ( ) ( ) ( ) ( ) ( ) 1 u x e x v x dx d v x f x dx d (7.45) By writing in an integral form one can obtain ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ x − dt v t f t e t u t v u v x u x 0 1 ( ) 0 ( ) ( ) 0 0 0 ( ) (0) (0) ( ) ( ) (7.46) Thus, in the notation of (7.11), B(x)=0, ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − 0 0 0 ( ) ( ) 1 e x A x , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = (0) (0) v u λ and q(t)=0 as well as ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ( ) 0 ( ) f t p t . Using the reduction procedure in the Haar basis for a system of linear differential equations, the goal is to find constants h h h h B , such that A , p , q ( ) ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + x h h h h p dt v t u t q A v x u x I B 0 ( ) ( ) ( ) ( ) λ (7.47) after reduction to the scale V0 will be the same as (7.46) reduced to that scale. This is accomplished by solving the recursion relations between the operators in the reduced equations explicitly, element-by-element in each matrix. This is possible to do because of the non-overlapping supports of the Haar basis functions on a fixed scale. The result for the first two coefficients is
334 Computational Mechanics of Composite Materials (7.48) where (7.49) Similar expressions for p and g"can be found.Note that we have p==0 if fx)=0 identically.Furthermore,in general B",A"do not depend on p and q.As a first-order system of ordinary differential equations,the homogenised equation yields ()=f( dx 床u)=M1-2M2) (7.50) what is somewhat different from the classical result.This difference results from the fact that the multiresolution homogenisation procedure allows the coefficients e(x)to vary on arbitrarily many scales,whereas the classical approach presented before allows only for coefficients of the form ex/E).In the multiresolution context this amounts to restricting the coefficients to an asymptotically fine scale. Let us apply the same limit in the preceding section to the coefficients appearing in the multiresolution approach.We start with the coefficients of the form ex/). Applying this homogenisation scheme to the elliptic equation with these coefficients yields two terms,M (e)and M (e).If we take the limit as g->0,it is found that limM (E)=M (7.51) E-0 and limM,()=0 (7.52) Thus,the factor M2 is present in the multiresolution context but does not appear in the classical approach,and it is zero when the limit found in the classical method is applied to the result of the multiresolution methodology
334 Computational Mechanics of Composite Materials ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 0 0 0 2 , 0 0 0 0 M1 M 2 B A h h (7.48) where = ∫ 1 0 1 ( ) 1 dt e t M , ∫ − = 1 0 2 1 2 ( ) dt e t t M (7.49) Similar expressions for h p and h q can be found. Note that we have 0 = = h h p q if f(x)=0 identically. Furthermore, in general h h B , do not depend on A p and q. As a first-order system of ordinary differential equations, the homogenised equation yields ⎪ ( ) ⎩ ⎪ ⎨ ⎧ = − = ( ) 2 ( ) ( ) ( ) 1 2 u x M M v x dx d v x f x dx d h (7.50) what is somewhat different from the classical result. This difference results from the fact that the multiresolution homogenisation procedure allows the coefficients e(x) to vary on arbitrarily many scales, whereas the classical approach presented before allows only for coefficients of the form e( ) x / ε . In the multiresolution context this amounts to restricting the coefficients to an asymptotically fine scale. Let us apply the same limit in the preceding section to the coefficients appearing in the multiresolution approach. We start with the coefficients of the form e( ) x / ε . Applying this homogenisation scheme to the elliptic equation with these coefficients yields two terms, ( ) ε 1 M and ( ) ε 2 M . If we take the limit as 0 ε → , it is found that 1 1 0 lim M ( ) = M → ε ε (7.51) and lim 2 ( ) 0 0 = → ε ε M (7.52) Thus, the factor M2 is present in the multiresolution context but does not appear in the classical approach, and it is zero when the limit found in the classical method is applied to the result of the multiresolution methodology
Multiresolutional Analysis 335 Let us note that this formula of the homogenised parameter e introduces new,closer bounds on the wavelet function defining material parameters than is done by classical formulation:integralsand must be of real values and e(x)must be positive defined to assure homogenisability of the problem.The counter-example is the family of sinusoidal wavelets of the form e(x)=eo+asin(),where eo.a,Le.Taking for example L=10,eo=20 and c0.1,MAPLE symbolic integration returns j0=-299242-19.07199i.The classical small parameter homogenisation method should be applied in that case; otherwise another wavelet decomposition of the real composite is to be performed. 7.3 Multiscale Homogenisation for the Wave Propagation Equation For illustration,let us consider the following ordinary differential equation (ODE)corresponding to unidirectional acoustic wave propagation in a multiscale medium with uniaxial distribution of nonhomogeneities [71,188]: (7.53) u(x)=ioM(x)u(x);xE[0,1] d where physical coefficients M(x)for both composite layers are defined by M(x)= Mo,0≤x< lM1,2≤x≤1 (7.54) These equations are solved using the methods typical for a deterministic problem and are derived for equal volume ratios of both layers.Otherwise,they should be complemented with the ratios cI and c2.The corresponding homogenised equation can be rewritten for the deterministic system as )=Kwu田 (7.55) dx It can be demonstrated [71]that the homogenised coefficientK is equal to K=log+0+B6-支A-A-） (7.56
Multiresolutional Analysis 335 Let us note that this formula of the homogenised parameter e(eff) introduces new, closer bounds on the wavelet function defining material parameters than is done by classical formulation: integrals ∫ L e x dx 0 ( ) and ∫ L e x xdx 0 ( ) must be of real values and e(x) must be positive defined to assure homogenisability of the problem. The counter-example is the family of sinusoidal wavelets of the form ( ) L x e x e π ( ) = 0 +α sin , where e0 ,α, L∈ℜ . Taking for example L=10, e0=20 and α=0.1, MAPLE symbolic integration returns . . i L e( x ) xdx 2 99242 19 07199 0 ∫ = − − . The classical small parameter homogenisation method should be applied in that case; otherwise another wavelet decomposition of the real composite is to be performed. 7.3 Multiscale Homogenisation for the Wave Propagation Equation For illustration, let us consider the following ordinary differential equation (ODE) corresponding to unidirectional acoustic wave propagation in a multiscale medium with uniaxial distribution of nonhomogeneities [71,188]: u(x) i M (x)u(x) dx d = ω ; ] x ∈[0,1 (7.53) where physical coefficients M(x) for both composite layers are defined by ⎩ ⎨ ⎧ ≤ ≤ ≤ < = , 1 , 0 ( ) 2 1 1 2 1 0 M x M x M x (7.54) These equations are solved using the methods typical for a deterministic problem and are derived for equal volume ratios of both layers. Otherwise, they should be complemented with the ratios c1 and c2. The corresponding homogenised equation can be rewritten for the deterministic system as ( ) ( ) ( ) u x K u x dx d eff = (7.55) It can be demonstrated [71] that the homogenised coefficient K(eff) is equal to ( ( ) ) ( ) 0 1 ( ) 2 0 ( ) 1 0 ( ) log −∞ −∞ − −∞ K = I + I + B − A A eff (7.56)
336 Computational Mechanics of Composite Materials for BO=S3-D-D8-iAED+S) (7.57) and AS=SA-DF-Ds+) (7.58) The right hand side coefficients denote SA=4+(4),DA=4-(4) (7.59) Sg=B6+(B月)，Dg=B6-(B) (7.60) F=I+Sg+÷D% (7.61) where A=lim A.B=lim B) (7.62) After some algebra it is found that =a时}. 7.63) where the following extension is used: (7.64) =1+i0M-o M0) 2j8j48 Taking into account that the coefficients B and A in(7.63)represent physical properties of the composite components with the total number of various scales tending to infinity,it is possible to determine an analogous definition of the homogenised coefficient for a composite with some finite scale number. Furthermore,using the stochastic second order perturbation second probabilistic moment methodology,it is relatively easy to determine the first two probabilistic moments of the homogenised coefficient defined by (7.63)
336 Computational Mechanics of Composite Materials for ( ) ( ) B S B DA DB S A F DB S A = ′ − ′ − ′ − ′ ′ + ′ −∞ − 4 1 1 4 1 4 ( ) 1 0 (7.57) and ( ) A S A DA F DB S A = ′ − ′ ′ + ′ −∞ − 4 ( ) 1 1 0 (7.58) The right hand side coefficients denote ( ) ( )( )1 ( ) 0 1 ( ) 2 1 1 −∞ −∞ S A ′ = A + A , ( ) ( )( )1 ( ) 0 1 ( ) 2 1 1 −∞ −∞ D′ A = A − A (7.59) ( ) ( )( )1 ( ) 0 1 ( ) 2 1 1 −∞ −∞ S B ′ = B + B , ( ) ( )( )1 ( ) 0 1 ( ) 2 1 1 −∞ −∞ DB ′ = B − B (7.60) S B DA F = I + ′ + ′ 4 1 (7.61) where ( ) (1 ) lim J j J Aj A − →∞ −∞ = , ( ) (1 ) lim J j J Bj B − →∞ −∞ = (7.62) After some algebra it is found that ( ) A j = Aj (−∞) 1 , ( ) I I I A A B j j j − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − −∞ 2 1 2 exp 2 1 ( ) 1 , j=1,2 (7.63) where the following extension is used: M ... O( ) i M M i I M i exp A exp n j j j j j + + ω ω + ω− ω = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ω = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 3 3 2 2 2 8 48 2 2 (7.64) Taking into account that the coefficients B and A in (7.63) represent physical properties of the composite components with the total number of various scales tending to infinity, it is possible to determine an analogous definition of the homogenised coefficient for a composite with some finite scale number. Furthermore, using the stochastic second order perturbation second probabilistic moment methodology, it is relatively easy to determine the first two probabilistic moments of the homogenised coefficient defined by (7.63)