Spectral methods for fuzzy structural dynamics: modal vs direct - PowerPoint PPT Presentation

Spectral methods for fuzzy structural dynamics: modal vs direct approach S Adhikari Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Wales, UK IUTAM Symposium - Dynamical Analysis of Multibody Systems with Design Uncertainties, Stuttgart, Germany

Outline of the talk Introduction 1 Brief overview of fuzzy variables 2 Legendre polynomials for fuzzy uncertainty propagation 3 Problem description Legendre polynomials Projection in the basis of Legendre polynomials Fuzzy finite element analysis 4 Formulation of the problem Galerkin approach Fuzzy eigen analysis using spectral projection 5 Numerical example: axial vibration of a rod 6 Summary and conclusion 7

Fuzzy variables The consideration of uncertainties in numerical models to obtain the variability of response is becoming more common for finite element models arising in practical problems. When substantial statistical information exists, the theory of probability and stochastic processes offer a rich mathematical framework to represent such uncertainties. In a probabilistic setting, uncertainty associated with the system parameters, such as the geometric properties and constitutive relations (i.e. Young’s modulus, mass density, Poisson’s ratio, damping coefficients), can be modeled as random variables or stochastic processes. These uncertainties can be quantified and propagated, for example, using the stochastic finite element method

Fuzzy variables The reliable application of probabilistic approaches requires information to construct the probability density functions of uncertain parameters. This information may not be easily available for many complex practical problems. In such situations, non-probabilistic approaches such as interval algebra, convex modelsand fuzzy set based methods can be used. Fuzzy finite element analysis aims to combine the power of finite element method and uncertainty modelling capability of fuzzy variables. In the context of computational mechanics, the aim of a fuzzy finite element analysis is to obtain the range of certain output quantities (such as displacement, acceleration and stress) given the membership of data in the set of input variables. This problem, known as the uncertainty propagation problem, has taken the centre stage in recent research activities in the field.

Membership functions of a fuzzy variable Membership functions of a fuzzy variable; (a) symmetric triangular membership function; (b) asymmetric triangular membership function; (c) general membership function. The value corresponding to α = 1 is the crisp (or deterministic) value. The range corresponding to α = 0 is the widest interval. Any intermediate value of 0 < α < 1, yields an interval variable with a finite lower and upper bound.

Uncertainty propagation We are interested in the propagation of a m -dimensional vector of Fuzzy variables, which can be expressed as y = � f ( x ) ∈ R n (1) f ( • ) : R m → R n is a smooth nonlinear function of the input fuzzy where � vector x . The fuzzy uncertainty propagation problem can be formally defined as the solution of the following set of optimisation problems � �  �  y jk min = min f j ( x α k ) � �  ∀ j = 1 , 2 , . . . , n ; k = 1 , 2 , . . . , r (2) � y jk max = max f j ( x α k ) Two ways the solution can be made efficient: (1) evaluate � f ( x ) in Eq (1) fast, and/or (2) do a better job with the optimisation in Eq (2). The main idea proposed here is based on efficient and accurate construction of a (hopefully cheaper!) ‘response surface’ for every α -cut followed by an optimization approach.

Transformation of a fuzzy variable Transformation of a Fuzzy variable x to ζ ∈ [ − 1 , 1 ] for different α -cuts. The � � x ( α ) , x ( α ) transformation ϕ ( • ) maps the interval → [ − 1 , 1 ] . l h

Transformation of a fuzzy variable � � x ( α ) , x ( α ) Suppose the transformation ϕ ( • ) maps the interval → [ − 1 , 1 ] . i l i h Here x ( α ) and x ( α ) denotes the lower and upper bound of the Fuzzy i l i h variable x i for a given α -cut. Denote ζ i as the variable bounded between [ − 1 , 1 ] . � � Considering the variable x ( α ) x ( α ) , x ( α ) lies between , the linear i i l i h transformation ϕ ( • ) can be identified as � �   x ( α ) − x ( α )   � − 1 i i l ζ i = ϕ ( x ( α ) � ) = 2 (3) i   x ( α ) − x ( α ) 2 i h i l The inverse transformation of (3) can be obtained as � � � � x ( α ) + x ( α ) x ( α ) − x ( α ) i h i l x ( α ) i h i l = ζ i + (4) i 2 2

Transformation of a fuzzy variable Substituting x ( α ) in Eq. (1) for all α and i , we can can formally express i the uncertainty propagation problem in terms of the vector valued variable ζ = { ζ i } ∀ i = 1 , ··· , m ∈ [ − 1 , 1 ] m as y ( α ) = f ( α ) ( ζ ) ∈ R n (5) Now we utilise the orthogonal property of Legendre polynomials in [ − 1 , 1 ] for this uncertainty propagation problem 1 . 1 Adhikari, S. and Khodaparast, H. H., ”A spectral approach for fuzzy uncertainty propagation in finite element analysis”, Fuzzy Sets and Systems, 243[1] (2014), pp. 1-24.

Univariate Legendre polynomials In [ − 1 , 1 ] , Legendre polynomials are orthogonal with respect to the L 2 inner product norm � 1 2 L j ( ζ ) L k ( ζ ) d ζ = 2 k + 1 δ jk (6) − 1 δ jk denotes the Kronecker delta (equal to 1 if j = k and to 0 otherwise) and L j ( ζ ) is the j th order Legendre polynomial. Each Legendre polynomial L j ( ζ ) is an k th-degree polynomial and can be expressed using Rodrigues’ formula as � ( 1 − ζ 2 ) k � d k 1 L k ( ζ ) = ; k = 0 , 1 , 2 , · · · (7) 2 k k ! d ζ k

Univariate Legendre polynomials Table : One dimensional Legendre Polynomials upto order 10 k Legendre Polynomials of order k : L k ( ζ ) ; ζ ∈ [ − 1 , 1 ] 0 1 1 ζ � � 3 ζ 2 − 1 1 2 � � 2 5 ζ 3 − 3 ζ 1 3 � � 2 35 ζ 4 − 15 ζ 2 + 3 1 4 � � 8 63 ζ 5 − 70 ζ 3 + 15 ζ 1 5 � � 8 231 ζ 6 − 315 ζ 4 + 105 ζ 2 − 5 1 6 � � 16 429 ζ 7 − 693 ζ 5 + 325 ζ 3 − 35 ζ 1 7 � � 16 6435 ζ 8 − 12012 ζ 6 + 6930 ζ 4 − 1260 ζ 2 + 35 1 8 � � 128 12155 ζ 9 − 25740 ζ 7 + 18018 ζ 5 − 4620 ζ 3 + 315 ζ 1 9 � � 128 46189 ζ 10 − 109395 ζ 8 + 90090 ζ 6 − 30030 ζ 4 + 3465 ζ 2 − 63 1 10 256

Multivariate Legendre polynomials Multivariate Legendre polynomials can be defined as products of univariate Legendre polynomials, similar to that of Hermite polynomials. This can be constructed by considering products of different orders. We consider following sets of integers i = { i 1 , i 2 , · · · i p } , i k ≥ 1 (8) β = { β 1 , β 2 , · · · β p } , β k ≥ 0 (9) The multivariate Legendre polynomial associated with sequence ( i , β ) as the products of univariate Legendre polynomials p � L i , β ( ζ ) = L β k ( ζ k ) (10) k = 1 The multivariate Legendre polynomials also satisfy the orthogonality condition with respect to the L 2 inner product norm in the respective dimension.

Projection approach We follow the idea of homogeneous chaos proposed by Wienner involving Hermite polynomials. If a function f ( ζ ) is square integrable, it can be expanded in Homogeneous Chaos as � ∞ f ( ζ ) = ˆ ˆ y i 0 h 0 + y i 1 Γ 1 ( ζ i 1 ) i 1 = 1 i 1 i 1 i 2 � ∞ � � ∞ � � ˆ ˆ + y i 1 , i 2 Γ 2 ( ζ i 1 , ζ i 2 ) + y i 1 i 2 i 3 Γ 3 ( ζ i 1 , ζ i 2 , ζ i 3 ) (11) i 1 = 1 i 2 = 1 i 1 = 1 i 2 = 1 i 3 = 1 i 3 ∞ i 1 i 2 � � � � ˆ + y i 1 i 2 i 3 i 4 Γ 4 ( ζ i 1 , ζ i 2 , ζ i 3 , ζ i 4 ) + . . . , i 1 = 1 i 2 = 1 i 3 = 1 i 4 = 1 Γ p ( ζ i 1 , ζ i 2 , · · · ζ i m ) is m -dimensional homogeneous chaos of order p (obtained by products of one-dimensional Legendre polynomials).

Projection approach We can concisely write P − 1 � f ( ζ ) = y j Ψ j ( ζ ) (12) j = 0 where the constant y j and functions Ψ j ( • ) are effectively constants ˆ y k and functions Γ k ( • ) for corresponding indices. Equation (12) can be viewed as the projection in the basis functions Ψ j ( ζ ) with corresponding ‘coordinates’ y j . The number of terms P in Eq. (12) depends on the number of variables m and maximum order of polynomials p as � m + p � p � ( m + j − 1 )! P = = (13) j !( m − 1 )! p j = 0

Univariate Legendre polynomials Table : Two dimensional Legendre polynomial based homogeneous chaos basis up to 4 th order j p Construction of Ψ j Ψ j 0 p = 0 L 0 1 1 p = 1 L 1 ( ζ 1 ) ζ 1 2 L 1 ( ζ 2 ) ζ 2 3 / 2 ζ 12 − 1 / 2 3 L 2 ( ζ 1 ) 4 p = 2 L 1 ( ζ 1 ) L 1 ( ζ 2 ) ζ 1 ζ 2 3 / 2 ζ 22 − 1 / 2 5 L 2 ( ζ 2 ) 5 / 2 ζ 13 − 3 / 2 ζ 1 6 L 3 ( ζ 1 ) � 3 / 2 ζ 12 − 1 / 2 � 7 p = 3 L 2 ( ζ 1 ) L 1 ( ζ 2 ) ζ 2 � 3 / 2 ζ 22 − 1 / 2 � 8 L 1 ( ζ 1 ) L 2 ( ζ 2 ) ζ 1 5 / 2 ζ 23 − 3 / 2 ζ 2 9 L 3 ( ζ 2 ) 8 ζ 14 − 15 4 ζ 12 + 3 / 8 35 10 L 4 ( ζ 1 ) � 5 / 2 ζ 13 − 3 / 2 ζ 1 � 11 L 3 ( ζ 1 ) L 1 ( ζ 2 ) ζ 2 � 3 / 2 ζ 12 − 1 / 2 � � 3 / 2 ζ 22 − 1 / 2 � 12 p = 4 L 2 ( ζ 1 ) L 2 ( ζ 2 ) � 5 / 2 ζ 23 − 3 / 2 ζ 2 � 13 L 1 ( ζ 1 ) L 3 ( ζ 2 ) ζ 1 35 8 ζ 24 − 15 4 ζ 22 + 3 / 8 14 L 4 ( ζ 2 )