Collocation based high dimensional model representation for - PowerPoint PPT Presentation

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 2010 1 / 52

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 2010 2 / 52

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 2010 3 / 52

Power shirt Adhikari (SU) Reduced methods for SPDE 20 May 2010 4 / 52

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 2010 5 / 52

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 �� r min � 6 � � 12 − � r min the Lennard-Jones potential V ( r ) = 4 ǫ (large r r 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 2010 6 / 52

Outline of the talk Introduction 1 Stochastic elliptic PDEs Spectral decomposition in a vector space 2 Projection in a finite dimensional vector-space Properties of the spectral functions Error minimization in the Hilbert space 3 The Galerkin approach Computational method Numerical illustration 4 ZnO nanowires Results for larger correlation length Results for smaller correlation length Conclusions 5 Acknowledgements 6 Adhikari (SU) Reduced methods for SPDE 20 May 2010 7 / 52

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 2010 8 / 52

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-Lo` eve expansion ∞ � √ ν i ξ i ( ω ) ϕ i ( r ) a ( r , ω ) = a 0 ( r ) + (3) i = 1 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 ) d r 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 Adhikari (SU) Reduced methods for SPDE 20 May 2010 9 / 52

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 i 0 h 0 + u i 1 h 1 ( ξ i 1 ( ω )) i 1 = 1 ∞ i 1 � � + u i 1 , i 2 h 2 ( ξ i 1 ( ω ) , ξ i 2 ( ω )) i 1 = 1 i 2 = 1 ∞ i 1 i 2 � � � + u i 1 i 2 i 3 h 3 ( ξ i 1 ( ω ) , ξ i 2 ( ω ) , ξ i 3 ( ω )) i 1 = 1 i 2 = 1 i 3 = 1 i 1 i 2 i 3 ∞ � � � � + u i 1 i 2 i 3 i 4 h 4 ( ξ i 1 ( ω ) , ξ i 2 ( ω ) , ξ i 3 ( ω ) , ξ i 4 ( ω )) + . . . , i 1 = 1 i 2 = 1 i 3 = 1 i 4 = 1 Here u i 1 ,..., i p ∈ R n are deterministic vectors to be determined. Adhikari (SU) Reduced methods for SPDE 20 May 2010 10 / 52

Introduction Stochastic elliptic PDEs Polynomial Chaos expansion After the finite truncation, concisely, the polynomial chaos expansion can be written as P � ˆ u ( ω ) = 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 r � ( M + j − 1 )! P = (6) j !( M − 1 )! j = 0 Adhikari (SU) Reduced methods for SPDE 20 May 2010 11 / 52

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 k φ 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 2 + . . . α n φ n . Adhikari (SU) Reduced methods for SPDE 20 May 2010 12 / 52

Introduction Stochastic elliptic PDEs Polynomial Chaos expansion We can ‘split’ the Polynomial Chaos type of expansions as n P � � ˆ u ( ω ) = H k ( ξ ( ω )) u k + H k ( ξ ( ω )) u k (7) k = 1 k = n + 1 According to the spanning property of a complete basis in R n it is always possible to project ˆ u ( 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 2010 13 / 52

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 n � ˆ u ( ω ) = 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 0 k φ k ; k = 1 , 2 , . . . n (9) Adhikari (SU) Reduced methods for SPDE 20 May 2010 14 / 52

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 0 1 , λ 0 2 , . . . , λ 0 n ] ∈ R n × n ; Φ = [ φ 1 , φ 2 , . . . , φ n ] ∈ R n × n (10) Eigenvalues are ordered in the ascending order: λ 0 1 < λ 0 2 < . . . < λ 0 n . Since Φ is an orthogonal matrix we have Φ − 1 = Φ T so that: A − 1 Φ T A 0 Φ = Λ 0 ; A 0 = Φ − T Λ 0 Φ − 1 = ΦΛ − 1 0 Φ T and (11) 0 We also introduce the transformations � A i = Φ T A i Φ ∈ R n × n ; i = 0 , 1 , 2 , . . . , M (12) Note that � A 0 = Λ 0 , a diagonal matrix and A i Φ − 1 ∈ R n × n ; i = 1 , 2 , . . . , M A i = Φ − T � (13) Adhikari (SU) Reduced methods for SPDE 20 May 2010 15 / 52

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 � ˆ u ( ω ) = A 0 + ξ i ( ω ) A i f (14) i = 1 Using Eqs. (10)–(13) and the orthonormality of Φ one has � � − 1 M � Φ − T Λ 0 Φ − 1 + ξ i ( ω ) Φ − T � A i Φ − 1 f = ΦΨ ( ξ ( ω )) Φ T f ˆ u ( ω ) = i = 1 (15) where � � − 1 M � ξ i ( ω ) � Ψ ( ξ ( ω )) = Λ 0 + A i (16) i = 1 and the M -dimensional random vector ξ ( ω ) = { ξ 1 ( ω ) , ξ 2 ( ω ) , . . . , ξ M ( ω ) } T (17) Adhikari (SU) Reduced methods for SPDE 20 May 2010 16 / 52