Collocation based high dimensional model representation for stochastic partial differential equations

Transcription

1 Collocation based high dimensional model representation for stochastic partial differential equations S Adhikari 1 1 Swansea University, UK ECCM 2010: IV European Conference on Computational Mechanics, Paris Adhikari (SU) Reduced methods for SPDE 20 May / 52

2 Nanoscale Energy Harvesting: ZnO nanowires ZnO materials have attracted extensive attention due to their excellent performance in electronic, ferroelectric and piezoelectric applications. Nano-scale ZnO is an important material for the nanoscale energy harvesting and scavenging. Investigation and understanding of the bending of ZnO nanowires are valuable for their potential application. For example, ZnO nanowires are bend by rubbing against each other for energy scavenging. Adhikari (SU) Reduced methods for SPDE 20 May / 52

3 Rubbing the right way When ambient vibrations move a microfibre covered with zinc oxide nanowires (blue) back and forth with respect to a similar fibre that has been coated with gold (orange), electrical energy is produced because ZnO is a piezoelectric material; Nature Nanotechnology, Vol 3, March 2008, pp 123. Adhikari (SU) Reduced methods for SPDE 20 May / 52

4 Power shirt Adhikari (SU) Reduced methods for SPDE 20 May / 52

5 Collection of ZnO A collection of vertically grown ZnO NWs. This can be viewed as the sample space for the application of stochastic finite element method. Adhikari (SU) Reduced methods for SPDE 20 May / 52

6 Stochastic nanomechanics: computational challenges When applying the continuum stochastic mechanics at the nanoscale, the following points need to be considered: The finite element discretization should be very small to take account of nanoscale spatial resolution (large n). Due to the small length-scale, the uncertainties are relatively large (as can be seen in the SEM images (large σ). The correlation length, which governs the statistical correlation between two points in the space is generally very small. This is because the interaction between the atoms reduces significantly with distance. This can be understood, for example, by looking at the Lennard-Jones potential V (r) = 4ɛ [ ( rmin r ) 12 ( rmin r ) ] 6 (large M). Since the standard deviation σ, the degrees-of-freedom n and the number of random variables M are all expected to be large, stochastic nanomechanics is particularity challenging as the computational cost can be significantly higher. Adhikari (SU) Reduced methods for SPDE 20 May / 52

7 Outline of the talk 1 Introduction Stochastic elliptic PDEs 2 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Properties of the spectral functions 3 Error minimization in the Hilbert space The Galerkin approach Computational method 4 Numerical illustration ZnO nanowires Results for larger correlation length Results for smaller correlation length 5 Conclusions 6 Acknowledgements Adhikari (SU) Reduced methods for SPDE 20 May / 52

8 Introduction Stochastic elliptic PDEs Stochastic elliptic PDE We consider the stochastic elliptic partial differential equation (PDE) [a(r, ω) u(r, ω)] = p(r); r in D (1) with the associated boundary condition u(r, ω) = 0; r on D (2) Here a : R d Ω R is a random field, which can be viewed as a set of random variables indexed by r R d. We assume the random field a(r, ω) to be stationary and square integrable. Based on the physical problem the random field a(r, ω) can be used to model different physical quantities. Adhikari (SU) Reduced methods for SPDE 20 May / 52

9 Introduction Stochastic elliptic PDEs Discretized Stochastic PDE The random process a(r, ω) can be expressed in a generalized fourier type of series known as the Karhunen-Loève expansion a(r, ω) = a 0 (r) + νi ξ i (ω)ϕ i (r) (3) Here a 0 (r) is the mean function, ξ i (ω) are uncorrelated standard Gaussian random variables, ν i and ϕ i (r) are eigenvalues and eigenfunctions satisfying the integral equation D C a(r 1, r 2 )ϕ j (r 1 )dr 1 = ν j ϕ j (r 2 ), j = 1, 2,. Truncating the series (3) upto the M-th term, substituting a(r, ω) in the governing PDE (1) and applying the boundary conditions, the discretized equation can be written as [ ] M A 0 + ξ i (ω)a i u(ω) = f (4) i=1 i=1 Adhikari (SU) Reduced methods for SPDE 20 May / 52

10 Introduction Stochastic elliptic PDEs Polynomial Chaos expansion Using the Polynomial Chaos expansion, the solution (a vector valued function) can be expressed as u(ω) = u i0 h 0 + u i1 h 1 (ξ i1 (ω)) + i 1 i 1 =1 i 2 =1 + i 1 i 1 =1 i 1 =1 i 2 =1 i 3 =1 + u i1,i 2 h 2 (ξ i1 (ω), ξ i2 (ω)) i 2 i 1 i 2 i 3 i 1 =1 i 2 =1 i 3 =1 i 4 =1 u i1 i 2 i 3 h 3 (ξ i1 (ω), ξ i2 (ω), ξ i3 (ω)) u i1 i 2 i 3 i 4 h 4 (ξ i1 (ω), ξ i2 (ω), ξ i3 (ω), ξ i4 (ω)) +..., Here u i1,...,i p R n are deterministic vectors to be determined. Adhikari (SU) Reduced methods for SPDE 20 May / 52

11 Introduction Stochastic elliptic PDEs Polynomial Chaos expansion After the finite truncation, concisely, the polynomial chaos expansion can be written as û(ω) = P H k (ξ(ω))u k (5) k=1 where H k (ξ(ω)) are the polynomial chaoses. The value of the number of terms P depends on the number of basic random variables M and the order of the PC expansion r as P = r j=0 (M + j 1)! j!(m 1)! (6) Adhikari (SU) Reduced methods for SPDE 20 May / 52

12 Introduction Stochastic elliptic PDEs Some basics of linear algebra Definition (Linearly independent vectors) A set of vectors {φ 1, φ 2,..., φ n } is linearly independent if the expression n k=1 α kφ k = 0 if and only if α k = 0 for all k = 1, 2,..., n. Remark (The spanning property) Suppose {φ 1, φ 2,..., φ n } is a complete basis in the Hilbert space H. Then for every nonzero u H, it is possible to choose α 1, α 2,..., α n 0 uniquely such that u = α 1 φ 1 + α 2 φ α n φ n. Adhikari (SU) Reduced methods for SPDE 20 May / 52

13 Introduction Stochastic elliptic PDEs Polynomial Chaos expansion We can split the Polynomial Chaos type of expansions as û(ω) = n H k (ξ(ω))u k + k=1 P k=n+1 H k (ξ(ω))u k (7) According to the spanning property of a complete basis in R n it is always possible to project û(ω) in a finite dimensional vector basis for any ω Ω. Therefore, in a vector polynomial chaos expansion (7), all u k for k > n must be linearly dependent. This is the motivation behind seeking a finite dimensional expansion. Adhikari (SU) Reduced methods for SPDE 20 May / 52

14 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space Theorem There exist a finite set of functions Γ k : (R m Ω) (R Ω) and an orthonormal basis φ k R n for k = 1, 2,..., n such that the series û(ω) = n Γ k (ξ(ω))φ k (8) k=1 converges to the exact solution of the discretized stochastic finite element equation (4) with probability 1. Outline of proof: The first step is to generate a complete orthonormal basis. We use the eigenvectors φ k R n of the matrix A 0 such that A 0 φ k = λ 0k φ k ; k = 1, 2,... n (9) Adhikari (SU) Reduced methods for SPDE 20 May / 52

15 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space We define the matrix of eigenvalues and eigenvectors Λ 0 = diag [λ 01, λ 02,..., λ 0n ] R n n ; Φ = [φ 1, φ 2,..., φ n ] R n n (10) Eigenvalues are ordered in the ascending order: λ 01 < λ 02 <... < λ 0n. Since Φ is an orthogonal matrix we have Φ 1 = Φ T so that: Φ T A 0 Φ = Λ 0 ; A 0 = Φ T Λ 0 Φ 1 and A 1 0 = ΦΛ 1 0 ΦT (11) We also introduce the transformations à i = Φ T A i Φ R n n ; i = 0, 1, 2,..., M (12) Note that Ã0 = Λ 0, a diagonal matrix and A i = Φ T à i Φ 1 R n n ; i = 1, 2,..., M (13) Adhikari (SU) Reduced methods for SPDE 20 May / 52

16 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space Suppose the solution of Eq. (4) is given by [ ] 1 M û(ω) = A 0 + ξ i (ω)a i f (14) Using Eqs. (10) (13) and the orthonormality of Φ one has [ ] 1 M û(ω) = Φ T Λ 0 Φ 1 + ξ i (ω)φ T Ã i Φ 1 f = ΦΨ (ξ(ω)) Φ T f i=1 i=1 where [ Ψ (ξ(ω)) = Λ 0 + (15) 1 M ξ i (ω)ãi] (16) and the M-dimensional random vector ξ(ω) = {ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)} T (17) i=1 Adhikari (SU) Reduced methods for SPDE 20 May / 52

17 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space Now we separate the diagonal and off-diagonal terms of the Ãi matrices as à i = Λ i + i, i = 1, 2,..., M (18) Here the diagonal matrix ] Λ i = diag [à = diag [ ] λ i1, λ i2,..., λ in R n n (19) and i = Ãi Λ i is an off-diagonal only matrix. M M Ψ (ξ(ω)) = Λ 0 + ξ i (ω)λ i + ξ i (ω) i i=1 i=1 }{{}}{{} Λ(ξ(ω)) (ξ(ω)) 1 (20) where Λ (ξ(ω)) R n n is a diagonal matrix and (ξ(ω)) is an off-diagonal only matrix. Adhikari (SU) Reduced methods for SPDE 20 May / 52

18 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space We rewrite Eq. (20) as Ψ (ξ(ω)) = [ [ 1 Λ (ξ(ω)) I n + Λ 1 (ξ(ω)) (ξ(ω))]] (21) The above expression can be represented using a Neumann type of matrix series as Ψ (ξ(ω)) = [ s ( 1) s Λ 1 (ξ(ω)) (ξ(ω))] Λ 1 (ξ(ω)) (22) s=0 Adhikari (SU) Reduced methods for SPDE 20 May / 52

19 Spectral decomposition in a vector space Projection in a finite dimensional vector-space Polynomial Chaos expansion Taking an arbitrary r-th element of û(ω), Eq. (15) can be rearranged to have n n ( f) û r (ω) = Ψ kj (ξ(ω)) φ T j (23) Defining k=1 Φ rk Γ k (ξ(ω)) = j=1 n j=1 ( ) Ψ kj (ξ(ω)) φ T j f and collecting all the elements in Eq. (23) for r = 1, 2,..., n one has û(ω) = (24) n Γ k (ξ(ω)) φ k (25) k=1 Adhikari (SU) Reduced methods for SPDE 20 May / 52

20 Spectral decomposition in a vector space Properties of the spectral functions Spectral functions Definition The functions Γ k (ξ(ω)), k = 1, 2,... n are called the spectral functions as they are expressed in terms of the spectral properties of the coefficient matrices of the governing discretized equation. The main difficulty in applying this result is that each of the spectral functions Γ k (ξ(ω)) contain infinite number of terms and they are highly nonlinear functions of the random variables ξ i (ω). For computational purposes, it is necessary to truncate the series after certain number of terms. Different order of spectral functions can be obtained by using truncation in the expression of Γ k (ξ(ω)) Adhikari (SU) Reduced methods for SPDE 20 May / 52

21 Spectral decomposition in a vector space Properties of the spectral functions First-order spectral functions Definition The first-order spectral functions Γ (1) k (ξ(ω)), k = 1, 2,..., n are obtained by retaining one term in the series (22). Retaining one term in (22) we have Ψ (1) (ξ(ω)) = Λ 1 (ξ(ω)) or Ψ (1) kj (ξ(ω)) = δ kj λ 0k + M i=1 ξ i(ω)λ ik(26) Using the definition of the spectral function in Eq. (24), the first-order spectral functions can be explicitly obtained as n ( ) Γ (1) k (ξ(ω)) = Ψ (1) kj (ξ(ω)) φ T φ T k j f = f λ 0k + M i=1 ξ (27) i(ω)λ ik j=1 From this expression it is clear that Γ (1) k (ξ(ω)) are non-gaussian random variables even if ξ i (ω) are Gaussian random variables. Adhikari (SU) Reduced methods for SPDE 20 May / 52

22 Spectral decomposition in a vector space Properties of the spectral functions Second-order spectral functions Definition The second-order spectral functions Γ (2) k (ξ(ω)), k = 1, 2,..., n are obtained by retaining two terms in the series (22). Retaining two terms in (22) we have Ψ (2) (ξ(ω)) = Λ 1 (ξ(ω)) Λ 1 (ξ(ω)) (ξ(ω)) Λ 1 (ξ(ω)) (28) Using the definition of the spectral function in Eq. (24), the second-order spectral functions can be obtained in closed-form as Γ (2) k (ξ(ω)) = φ T k f λ 0k + M n j=1 i=1 ξ i(ω)λ ik ( φ T j f) M i=1 ξ i(ω) ikj ( λ 0k + M i=1 ξ i(ω)λ ik ) ( λ 0j + M i=1 ξ i(ω)λ ij ) (29) Adhikari (SU) Reduced methods for SPDE 20 May / 52

23 Spectral decomposition in a vector space Properties of the spectral functions Analysis of spectral functions The spectral basis functions are not simple polynomials, but ratio of polynomials in ξ(ω). We now look into the functional nature of the solution u(ω) in terms of the random variables ξ i (ω). Theorem If all A i R n n are matrices of rank n, then the elements of u(ω) are the ratio of polynomials of the form p (n 1) (ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)) p (n) (ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)) (30) where p (n) (ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)) is an n-th order complete multivariate polynomial of variables ξ 1 (ω), ξ 2 (ω),..., ξ M (ω). Adhikari (SU) Reduced methods for SPDE 20 May / 52

24 Spectral decomposition in a vector space Properties of the spectral functions Analysis of spectral functions Suppose we denote so that A(ω) = [ A 0 + ] M ξ i (ω)a i R n n (31) i=1 From the definition of the matrix inverse we have u(ω) = A 1 (ω)f (32) A 1 = Adj(A) det (A) = CT a det (A) (33) where C a is the matrix of cofactors. The determinant of A contains a maximum of n number of products of A kj and their linear combinations. Note from Eq. (31) that M A kj (ω) = A 0kj + ξ i (ω)a ikj (34) i=1 Adhikari (SU) Reduced methods for SPDE 20 May / 52

25 Spectral decomposition in a vector space Properties of the spectral functions Analysis of spectral functions Since all the matrices are of full rank, the determinant contains a maximum of n number of products of linear combination of random variables in Eq. (34). On the other hand, each entries of the matrix of cofactors, contains a maximum of (n 1) number of products of linear combination of random variables in Eq. (34). From Eqs. (32) and (33) it follows that u(ω) = CT a f det (A) (35) Therefore, the numerator of each element of the solution vector contains linear combinations of the elements of the cofactor matrix, which are complete polynomials of order (n 1). The result derived in this theorem is important because the solution methods proposed for stochastic finite element analysis essentially aim to approximate the ratio of the polynomials given in Eq. (30). Adhikari (SU) Reduced methods for SPDE 20 May / 52

26 Spectral decomposition in a vector space Properties of the spectral functions Analysis of spectral functions Theorem The linear combination of the spectral functions has the same functional form in (ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)) as the elements of the solution vector, that is, û r (ω) p(n 1) r (ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)) p (n), r = 1, 2,..., n (36) r (ξ 1 (ω), ξ 2 (ω),..., ξ M (ω)) When first-order spectral functions (27) are considered, we have û (1) r (ω) = n k=1 Γ (1) k (ξ(ω)) φ rk = n k=1 φ T k f λ 0k + M i=1 ξ i(ω)λ ik φ rk (37) All (λ 0k + M i=1 ξ i(ω)λ ik ) are different for different k because it is assumed that all eigenvalues λ 0k are distinct. Adhikari (SU) Reduced methods for SPDE 20 May / 52

27 Spectral decomposition in a vector space Properties of the spectral functions Analysis of spectral functions Carrying out the above summation one has n number of products of (λ 0k + M i=1 ξ i(ω)λ ik ) in the denominator and n sums of (n 1) number of products of (λ 0k + M i=1 ξ i(ω)λ ik ) in the numerator, that is, û (1) r (ω) = ( j=1 k λ 0j + ) M i=1 ξ i(ω)λ ij ( n 1 k=1 λ 0j + ) (38) M i=1 ξ i(ω)λ ij n k=1 (φt k f)φ rk n 1 Adhikari (SU) Reduced methods for SPDE 20 May / 52

28 Error minimization in the Hilbert space The Galerkin approach The Galerkin approach There exist a set of finite functions Γ k : (R m Ω) (R Ω), constants c k R and orthonormal vectors φ k R n for k = 1, 2,..., n such that the series n û(ω) = c k Γk (ξ(ω))φ k (39) k=1 converges to the exact solution of the discretized stochastic finite element equation (4) in the mean-square sense provided the vector c = {c 1, c 2,..., c n } T satisfies the n n algebraic equations S c = b with M S jk = Ã ijk D ijk ; j, k = 1, 2,..., n; Ãi jk = φ T j A i φ k, (40) i=0 [ ] D ijk = E ξ i (ω) Γ j (ξ(ω)) Γ k (ξ(ω)) ] ( ) and b j = E [ Γj (ξ(ω)) φ T j f. (41) Adhikari (SU) Reduced methods for SPDE 20 May / 52

29 Error minimization in the Hilbert space The Galerkin approach The Galerkin approach The error vector can be obtained as ( M ) ( n ) ε(ω) = A i ξ i (ω) c k Γk (ξ(ω))φ k f R n (42) i=0 k=1 } The solution is viewed as a projection where { Γk (ξ(ω))φ k R n are the basis functions and c k are the unknown constants to be determined. We wish to obtain the coefficients c k such that the error norm χ 2 = ε(ω), ε(ω) is minimum. This can be achieved using the Galerkin approach so that the error is made orthogonal to the basis functions, that is, mathematically ) ε(ω) ( Γj (ξ(ω))φ j or Γj (ξ(ω))φ j, ε(ω) = 0 j = 1, 2,..., n (43) Adhikari (SU) Reduced methods for SPDE 20 May / 52

30 Error minimization in the Hilbert space The Galerkin approach The Galerkin approach Imposing the orthogonality condition and using the expression of the error one has [ ( M ) ( n ) ] E Γj (ξ(ω))φ T j A i ξ i (ω) c k Γk (ξ(ω))φ k Γ j (ξ(ω))φ T j f = i=0 k=1 (44) Interchanging the E [ ] and summation operations, this can be simplified to ( n M k=1 i=0 ) [ (φ T j A i φ k E ξ i (ω) Γ j (ξ(ω)) Γ k (ξ(ω))] ) c k = ] ( ) E [ Γj (ξ(ω)) φ T j f (45) or ( n M ) Ã ijk D ijk c k = b j (46) k=1 i=0 Adhikari (SU) Reduced methods for SPDE 20 May / 52

31 Error minimization in the Hilbert space Computational method Computational method The mean vector can be obtained as ū = E [û(ω)] = n k=1 ] c k E [ Γk (ξ(ω)) φ k (47) The covariance of the solution vector can be expressed as [ ] Σ u = E (û(ω) ū) (û(ω) ū) T = n k=1 j=1 n c k c j Σ Γkj φ k φ T j (48) where the elements of the covariance matrix of the spectral functions are given by ])] Σ Γkj = E [( Γk (ξ(ω)) E [ Γk (ξ(ω))]) ( Γj (ξ(ω)) E [ Γj (ξ(ω)) (49) Adhikari (SU) Reduced methods for SPDE 20 May / 52

32 Error minimization in the Hilbert space Computational method Summary of the computational method 1 Solve the eigenvalue problem associated with the mean matrix A 0 to generate the orthonormal basis vectors: A 0 Φ = Λ 0 Φ 2 Select a number of samples, say N samp. Generate the samples of basic random variables ξ i (ω), i = 1, 2,..., M. 3 Calculate the spectral basis functions (for example, first-order): φ T k f Γ k (ξ(ω)) = λ 0k + M i=1 ξ i (ω)λ ik 4 Obtain the coefficient vector: c = S 1 b R n, where b = f Γ, S = Λ 0 D 0 + M [ i=1 Ãi ] D i and D i = E Γ(ω)ξ i (ω)γ T (ω), i = 0, 1, 2,..., M 5 Obtain the samples of the response from the spectral series: û(ω) = n k=1 c kγ k (ξ(ω))φ k Adhikari (SU) Reduced methods for SPDE 20 May / 52

33 Error minimization in the Hilbert space Computational method Computational complexity The spectral functions Γ k (ξ(ω)) are highly non-gaussian in nature and do not in general enjoy any orthogonality properties like the Hermite polynomials or any other orthogonal polynomials with respect to the underlying probability measure. The coefficient matrix S and the vector b should be obtained numerically using the Monte Carlo simulation or other numerical integration technique. The simulated spectral functions can also be recycled to obtain the statistics and probability density function (pdf) of the solution. The main computational cost of the proposed method depends on (a) the solution of the matrix eigenvalue problem, (b) the generation of the coefficient matrices D i, and (c) the calculation of the coefficient vector by solving linear matrix equation. Adhikari (SU) Reduced methods for SPDE 20 May / 52

34 Error minimization in the Hilbert space Computational method Computational complexity Both the linear matrix algebraic and the matrix eigenvalue problem scales in O(n 3 ) in the worse case. The calculation of the coefficient matrices scales linearly with M and n(n + 1)/2 with n. Therefore, this cost scales with O((M + 1) n(n + 1)/2). The overall cost is 2O(n 3 ) + O((M + 1) n(n + 1)/2). For large M and n, asymptotically the computational cost becomes C s = O(Mn 2 ) + O(n 3 ). The important point to note here that the proposed approach scales linearly with the number of random variables M. For comparison, in the classical PC expansion one needs to solve a matrix equation of dimension Pn, which in the worse case scales with (O(Pn) 3 ). Since P M, we have O(P 3 n 3 ) O(Mn 2 ) + O(n 3 ). Adhikari (SU) Reduced methods for SPDE 20 May / 52

35 Numerical illustration ZnO nanowires Collection of ZnO: Close up Uncertainties in ZnO NWs in the close up view. The uncertain parameter include geometric parameters such as the length and the cross sectional area along the length, boundary condition and material properties. Adhikari (SU) Reduced methods for SPDE 20 May / 52

36 Numerical illustration ZnO nanowires ZnO nanowires For the future nano energy scavenging devices several thousands of ZnO NWs will be used simultaneously. This gives a natural framework for the application of stochastic finite element method due a large sample space. ZnO NWs have the nano piezoelastic property so that the electric charge generated is a function of the deformation. It is therefore vitally important to look into the ensemble behavior of the deformation of ZnO NW for the reliable estimate of mechanical deformation and consequently the charge generation. For the nano-scale application this is especially crucial as the margin of error is very small. Here we study the deformation of a cantilevered ZnO NW with stochastic properties under the Atomic Force Microscope (AFM) tip. Adhikari (SU) Reduced methods for SPDE 20 May / 52

37 Numerical illustration ZnO nanowires ZnO nanowires (a) The SEM image of a collection of ZnO NW showing hexagonal cross sectional area. (b) The atomic structure of the cross section of a ZnO NW (the red is O 2 and the grey is Zn atom) Adhikari (SU) Reduced methods for SPDE 20 May / 52

38 Numerical illustration ZnO nanowires ZnO nanowires (c) The atomistic model of a ZnO NW (d) The continuum idealization of a cantilevered ZnO NW under an AFM tip. grown from a ZnO crystal in the (0, 0, 0, 1) direction. Adhikari (SU) Reduced methods for SPDE 20 May / 52

39 Numerical illustration ZnO nanowires Problem details We study the deflection of ZnO NW under the AFM tip considering stochastically varying bending modulus. The variability of the deflection is particularly important as the harvested energy from the bending depends on it. We assume that the bending modulus of the ZnO NW is a homogeneous stationary Gaussian random field of the form EI(x, ω) = EI 0 (1 + a(x, ω)) (50) where x is the coordinate along the length of ZnO NW, EI 0 is the estimate of the mean bending modulus, a(x, ω) is a zero mean stationary Gaussian random field. The autocorrelation function of this random field is assumed to be C a (x 1, x 2 ) = σ 2 ae ( x 1 x 2 )/µ a (51) where µ a is the correlation length and σ a is the standard deviation. Adhikari (SU) Reduced methods for SPDE 20 May / 52

40 Numerical illustration ZnO nanowires Problem details We consider a long nanowire where the continuum model has been validated. We use the baseline parameters for the ZnO NW from Gao and Wang (Nano Letters 7 (8) (2007), ) as the length L = 600nm, diameter d = 50nm and the lateral point force at the tip f T = 80nN. Using these data, the baseline deflection can be obtained as δ 0 = 145nm. We normalize our results with this baseline value for convenience. Two correlation lengths are considered in the numerical studies: µ a = L/3 and µ a = L/10. The number of terms M in the KL expansion becomes 24 and 67 (95% capture). The nanowire is divided into 50 beam elements of equal length. The number of degrees of freedom of the model n = 100 (standard beam element). Adhikari (SU) Reduced methods for SPDE 20 May / 52

41 Numerical illustration Results for larger correlation length Moments: larger correlation length (e) Mean of the normalized deflection. (f) Standard deviation of the normalized deflection. Figure: The number of random variable used: M = 24. The number of degrees of freedom: n = 100. Adhikari (SU) Reduced methods for SPDE 20 May / 52

42 Numerical illustration Results for larger correlation length Error in moments: larger correlation length Statistics Methods σ a = 0.05 σ a = 0.10 σ a = 0.15 σ a = 0.20 Mean 1st order Galerkin 2nd order Galerkin Standard 1st order Galerkin deviation 2nd order Galerkin Percentage error in the mean and standard deviation of the deflection of the ZnO NW under the AFM tip when correlation length is µ a = L/3. For n = 100 and M = 24, if the second-order PC was used, one would need to solve a linear system of equation of size The results shown here are obtained by solving a linear system of equation of size 100 using the proposed Galerkin approach. Adhikari (SU) Reduced methods for SPDE 20 May / 52

43 Numerical illustration Results for larger correlation length Pdf: larger correlation length (a) Probability density function for σ a = (b) Probability density function for σ a = 0.1. The probability density function of the normalized deflection δ/δ 0 of the ZnO NW under the AFM tip (δ 0 = 145nm). Adhikari (SU) Reduced methods for SPDE 20 May / 52

44 Numerical illustration Results for larger correlation length Pdf: larger correlation length (c) Probability density function for σ a = (d) Probability density function for σ a = 0.2. The probability density function of the normalized deflection δ/δ 0 of the ZnO NW under the AFM tip (δ 0 = 145nm). Adhikari (SU) Reduced methods for SPDE 20 May / 52

45 Numerical illustration Results for smaller correlation length Moments: smaller correlation length (e) Mean of the normalized deflection. (f) Standard deviation of the normalized deflection. Figure: The number of random variable used: M = 67. The number of degrees of freedom: n = 100. Adhikari (SU) Reduced methods for SPDE 20 May / 52

46 Numerical illustration Results for smaller correlation length Error in moments: smaller correlation length Statistics Methods σ a = 0.05 σ a = 0.10 σ a = 0.15 σ a = 0.20 Mean 1st order Galerkin 2nd order Galerkin Standard 1st order Galerkin deviation 2nd order Galerkin Percentage error in the mean and standard deviation of the deflection of the ZnO NW under the AFM tip when correlation length is µ a = L/3. For n = 100 and M = 67, if the second-order PC was used, one would need to solve a linear system of equation of size 234,500. The results shown here are obtained by solving a linear system of equation of size 100 using the proposed Galerkin approach. Adhikari (SU) Reduced methods for SPDE 20 May / 52

47 Numerical illustration Results for smaller correlation length Pdf: smaller correlation length (a) Probability density function for σ a = (b) Probability density function for σ a = 0.1. The probability density function of the normalized deflection δ/δ 0 of the ZnO NW under the AFM tip (δ 0 = 145nm). Adhikari (SU) Reduced methods for SPDE 20 May / 52

48 Numerical illustration Results for smaller correlation length Pdf: smaller correlation length (c) Probability density function for σ a = (d) Probability density function for σ a = 0.2. The probability density function of the normalized deflection δ/δ 0 of the ZnO NW under the AFM tip (δ 0 = 145nm). Adhikari (SU) Reduced methods for SPDE 20 May / 52

49 Conclusions Conclusions 1 We consider discretised stochastic elliptic partial differential equations. 2 The solution is projected into a finite dimensional complete orthonormal vector basis and the associated coefficient functions are obtained. 3 The coefficient functions, called as the spectral functions, are expressed in terms of the spectral properties of the system matrices. 4 If n is the size of the discretized matrices and M is the number of random variables, then the computational complexity grows in O(Mn 2 ) + O(n 3 ) for large M and n in the worse case. 5 We consider a problem with 24 and 67 random variables and n = 100 degrees of freedom. A second-order PC would require the solution of equations of dimension 32,400 and 234,500 respectively. In comparison, the proposed Galerkin approach requires the solution of algebraic equations of dimension n only. Adhikari (SU) Reduced methods for SPDE 20 May / 52

50 Conclusions Conclusions (e) Basic building blocks. (f) Possible solution. An analogy of PC based solution. Adhikari (SU) Reduced methods for SPDE 20 May / 52

51 Conclusions Conclusions (g) Basic building blocks. (h) Possible solution. An analogy of the proposed spectral function based Galerkin solution. Adhikari (SU) Reduced methods for SPDE 20 May / 52

52 Acknowledgements Acknowledgements Adhikari (SU) Reduced methods for SPDE 20 May / 52