Application of a Stochastic Finite Element Procedure to Reliability Analysis Bruno Sudret & Marc Berveiller Electricit´e de France, R&D Division, Site des Renardi`eres, F-77818 Moret-sur-Loing
Maurice Lemaire Institut Franc¸ais de M´ecanique Avanc´ee, LaRAMA, Campus des C´ezeaux, BP265, F-63175 Aubi`ere
Keywords: Stochastic Finite Element, Polynomial Chaos, Structural Reliability, FORM/SORM, Parametric Study
ABSTRACT: The stochastic finite element procedure presented in this paper consists in representing in a probabilistic form the response of a system whose material properties and loading are random. Each random variable is expanded into Hermite polynomial series. The response is expanded onto the so-called polynomial chaos. The coefficients of the expansion of the response (e.g. the nodal displacement vector) are obtained by a Galerkin-type method in the space of probability. This stochastic finite element procedure is applied to the parametric study of the reliability of a foundation on a two-layer elastic soil mass. The method reveals accurate and more efficient than a direct coupling between a finite element code and a probabilistic code.
1 INTRODUCTION The Spectral Stochastic Finite Element Method (SSFEM) was introduced in the early 90’s for solving stochastic boundary value problems in which the spatial variability of a material property (e.g. Young’s modulus) was modeled as a random field (Ghanem and Spanos, 1991). The approach, originally developed for a linear elastic problem using Gaussian random fields, was later extended to heat transfer problems (Ghanem, 1999b; Xiu and Karniadakis, 2002) and lognormal fields (Ghanem, 1999b). Recently, applications in fluid mechanics (Le Maˆıtre et al., 2001), modal analysis (Van Den Nieuwenhof, 2003), soil structure interaction (Ghiocel and Ghanem, 2002; Clouteau and Lafargue, 2003) have also been proposed. In most papers, the end result is the calculation of the two first moments of the response (i.e. mean and standard deviation). Application to structural reliability as a postprocessing of SSFEM has been demonstrated in Sudret and Der Kiureghian (2000); Sudret and Der Kiureghian (2002). In all the above references, the basic assumption is the description of the system randomness in terms of one or two random fields, which reduces the spectrum of applications. The aim of this paper is to develop a stochastic finite element procedure for problems involving any number of random variables of any type (i.e. non necessarily Gaussian and lognormal) describing uncertainties in the material properties as well as the loading. The proposed stochastic finite element method (SFEM) relies upon three ingredients : 1
– expansion of the input random variables in terms of Hermite polynomials of standard normal variables, – expansion of the response onto the so-called polynomial chaos, – post-processing of the results for reliability analysis. The method is applied to study the reliability of a foundation on a two-layer elastic soil mass. Comparisons with a classical finite element reliability analysis (direct coupling between the FORM method and a finite element code) are given in terms of accuracy and efficiency. 2 EXPANSION OF RANDOM VARIABLES ONTO THE POLYNOMIAL CHAOS Let us denote by L2 (Θ, F, P ) the Hilbert space of random variables with finite variance. Let us consider a random variable X with prescribed probability density function fX (x). Classical results Malliavin (1997) allow to expand X as a Hermite polynomial series expansion : X=
∞ X
ai Hi (ξ)
(1)
i=0
where ξ denotes a standard normal variable, {Hi , i = 0, · · · , ∞} are Hermite polynomials defined by : 2 2 i x2 d − x2 Hi (x) = (−1) e e (2) dxi and {ai , i = 0, · · · , ∞ } are coefficients to be evaluated. Two methods are now presented for this purpose. 2.1
Projection method This method was used by Puig et al. (2002); Xiu and Karniadakis (2002). Due to the orthogonality of the Hermite polynomials with respect to the Gaussian measure, it comes from Eq.(1) : E[XHi (ξ)] = ai E[Hi2 (ξ)]
(3)
where E[Hi2 (ξ)] = i!. Let FX denote the cumulative distribution function (CDF) of the random variable to approximate and Φ the standard normal CDF. By using the transformation to the standard normal space X → ξ : FX (X) = Φ(ξ), one can write : X(ξ) = FX−1 (Φ(ξ)) Thus : ai =
1 1 E[X(ξ)Hi (ξ)] = i! i!
Z
R
FX−1 (Φ(t))Hi (t)ϕ(t)dt
(4)
(5)
where ϕ is the standard normal probability density function (PDF). When X is a normal or lognormal random variable, coefficients {ai , i = 0, · · · , ∞} can be evaluated analytically : X ≡ N(µ, σ) a0 = µ, a1 = σ, ai = 0 for i ≥ 2 X ≡ LN(λ, ζ)
ai =
ζi i!
exp [λ + 12 ζ 2 ] for i ≥ 0
(6)
For other types of distribution, the quadrature method may be used for evaluating the integral in Eq.(5) (Berveiller and Sudret, 2003).
2
2.2
Collocation method This method was introduced by Isukapalli (1999). It is based on a least square minimization of the ˜: discrepancy between the input variable X and its truncated approximation X ˜= X
p X
ai Hi (ξ)
(7)
i=0
Let {ξ (1) , · · · , ξ (n) } be n outcomes of ξ. From Eq.(4), we obtain n outcomes {X (1) , · · · , X (n) }. The least square method consists in minimizing the following quantity with respect to {ai , i = 0, · · · , p} : 2 p n n X X X ˜ (i) )2 = X (i) − (X (i) − X aj Hj (ξ (i) ) (8) ∆X = i=1
i=1
j=0
This leads to the following linear system yielding the expansion coefficients {ai , i = 0, · · · , p} : n n n X X X (i) (i) (i) (i) (i) (i) H0 (ξ )H0 (ξ ) · · · H0 (ξ )Hp (ξ ) X H0 (ξ ) a0 i=1 i=1 i=1 .. .. .. .. .. = (9) . . . . . n n n X X X ap (i) (i) (i) (i) (i) (i) Hp (ξ )H0 (ξ ) · · · Hp (ξ )Hp (ξ ) X Hp (ξ ) i=1
i=1
i=1
Both methods are illustrated in Figure 1 in case of a lognormal distribution LN(0.6501, 0.2936) with mean 2 and standard deviation 0.6. 0.8 theoretical projection collocation 0.7
0.6
0.5
Method Projection Collocation
0.4
0.3
a0 2.0000 1.9986
a1 0.5871 0.5869
a2 0.0862 0.0872
a3 0.0084 0.0085
0.2
0.1
0 0
0.5
1
1.5
2
2.5
3
3.5
4
F IG . 1: Theoretical and approximated PDF for a lognormal distribution LN(0.6501, 0.2936) - Coefficients of the expansion (p = 3)
3 STOCHASTIC FINITE ELEMENT METHOD IN LINEAR ELASTICITY Using classical notations (Zienkiewicz and Taylor, 2000), the finite element method for static problems in linear elasticity yields a linear system of size Nddl × Nddl where Nddl denotes the number of degrees of freedom of the structure : K ·U = F (10) where K is the global stiffness matrix, U is the basic response quantity (vector of nodal displacements) and F is the vector of nodal forces. 3
In SFEM, due to the introduction of input random variables, the basic response quantity is a random vector of nodal displacements U (θ). Each component is a random variable expanded onto the so-called polynomial chaos : ∞ X U (θ) = U j Ψj ({ξk (θ)}M (11) k=1 ) j=0
where {ξk (θ)}M k=1 denotes the set of standard normal variables appearing in the discretization of all input random variables and {Ψj ({ξk }M k=1 )} are multidimensional Hermite polynomials that form 2 an orthogonal basis of L (Θ, F, P ). As the polynomial chaos contains the one-dimensional Hermite polynomials in each random variable, the expressions of each input random variable may be injected in the polynomial chaos expression : Xi =
∞ X
xij Hj (ξi ) ≡
j=0
∞ X
x ˜ij Ψj ({ξk }M k=1 )
(12)
j=0
In the sequel, the dependency of Ψj in {ξk }M k=1 will be omitted for the sake of clarity. 3.1
Taking into account randomness in material properties In the deterministic case, the global stiffness matrix reads : [ [Z K = ke = B T · D · BdΩe e
(13)
Ωe
e
where B is the matrix that [relates the components of strain to the element nodal displacement, D is the elasticity matrix and is the assembly procedure over all elements. When material properties e
are described by means of random variables, the elasticity matrix hence the global stiffness matrix become random. The latter may be expanded onto the polynomial chaos as follows : K=
∞ X
K j Ψj
(14)
B T · E[DΨj ] · BdΩe
(15)
j=0
where K j = E[KΨj ] =
[Z e
Ωe
Note that B is a deterministic matrix while D is random. In case of an isotropic elastic material with random independent Young’s modulus E and Poisson’s ratio ν, the latter may be written as (plane strain problems) : ˜ D = E(λ(ν)D µ(ν)D2 ) (16) 1 + 2˜ where ˜ λ(ν) = and
1 1 D1 = 1 0
ν (1 + ν)(1 − 2ν) 1 1 1 0
1 1 1 0
0 0 0 0
,
4
,
µ ˜(ν) =
1 0 D2 = 0 0
1 2(1 + ν) 0 1 0 0
0 0 1 0
(17) 0 0 0 1
(18)
Random Young’s modulus E is expanded as in Eq.(12). Functions of the random Poisson’s ratio ˜ {λ(ν), µ ˜(ν)} may be expanded in the same way, using either the projection or the collocation method : E=
∞ X
˜ λ(ν) =
ei Hi (ξE )
i=0
∞ X
λj Hj (ξν )
µ ˜(ν) =
j=0
∞ X
µj Hj (ξν )
(19)
j=0
˜ Note that the same standard normal variable ξν is used to expand both functions λ(ν) and µ ˜(ν). By substituting for Eq.(19) in (16), one finally gets : D=
∞ X ∞ X
ei λj Hi (ξE )Hj (ξν )D1 +
i=0 j=0
∞ X ∞ X
ei µj Hi (ξE )Hj (ξν )D2
(20)
i=0 j=0
Products Hi (ξE )Hj (ξν ) may be injected into the polynomial chaos Ψk (ξE , ξν ), finally yielding : D=
∞ X (αk D1 + βk D2 )Ψk (ξE , ξν )
(21)
k=0
If the structure under consideration is made of several materials, the above procedure is applied using different random variables in each element group having the same material properties. 3.2
Taking into account randomness in loading The vector of nodal forces may be written : F=
Nq X
qiF i
(22)
i=1
N
q denote random loading parameters and F i ”load where Nq is the number of load cases, {q i }i=1 pattern” vectors corresponding to a unit value of q i . Note that this formulation equally applies to pinpoint forces, pressure or initial stresses. Coefficients q i can be expanded onto the polynomial chaos (see Eq.(12)) : ∞ ∞ X X i i q = qj Hj (ξi ) = q˜ji Ψj (23)
j=0
j=0
Finally the random vector of nodal forces reads : F=
Nq ∞ X X
q˜ji Ψj F i =
i=1 j=0
3.3
∞ X j=0
e Ψj F j
Global linear system By using (11), (14) and (24), the discretized stochastic equilibrium equation reads : ! ∞ ∞ ∞ X X X e j Ψj F K i Ψi · U j Ψj = i=0
j=0
(24)
(25)
j=0
After a truncature of the series appearing in Eq.(25) at order P , the residual in the equilibrium equation is : ! P −1 P −1 P −1 X X X e j Ψj P = K i Ψi · U j Ψj − F (26) i=0
j=0
5
j=0
Coefficients {U 0 , · · · , U P −1 } are computed using the Galerkin method minimizing the residual defined above, which is equivalent to requiring that this residual be orthogonal to the space spanned by −1 {Ψj }Pj=0 (Ghanem and Spanos, 1991) : E[P Ψk ] = 0 ,
k = {0, · · · , P − 1}
(27)
This leads to the linear system : where K j,k =
P −1 X
K 0,0 K 1,0 .. .
··· ···
K P −1,0 · · ·
K 0,P −1 K 1,P −1 .. . K P −1,P −1
·
U0 U1 .. . U P −1
K i dijk and :
=
e0 F e F 1 .. . e F P −1
(28)
i=0
dijk = E[Ψi Ψj Ψk ] =
0
i!j!k! i+j−k j+k−i k+i−j ! ! ! 2 2 2
,
if (i + j + k) even and k ∈ [|i − j|, i + j]
otherwise
(29) This system can be solve directly (Ghanem and Kruger, 1996) or using a hierarchical approach (Ghanem, 1999a, 2000) after a direct resolution at a lower order. 4 APPLICATION TO RELIABILITY ANALYSIS Solving the linear system (28) yields the expansion coefficients of the vector of nodal displaceP −1 X U j Ψj . It is easy to show that any response quantity (e.g. strain or stress component) ments U = j=0
may be also expanded onto the polynomial chaos (Berveiller and Sudret, 2003). Thus the mechanical response of the system S (i.e. the set of all nodal displacements, strain or stress components) may be P −1 X written as S = S j Ψj . j=0
In reliability analysis, the failure criterion of a structure is defined in terms of a limit state function g(X, S(X)) which may depend both on basic random variables X and response quantities S(X). After performing a SFEM analysis as described above, it is clear that any limit state function is analytical and defined in terms of standard normal variables : P −1 X g(X, S(X)) ≡ g {ξk }M S j Ψj ({ξk }M (30) k=1 , k=1 ) j=0
Thus the reliability problem, which is already formulated in the standard normal space, may be solved by any available method including Monte-Carlo Simulation, FORM/SORM, Importance Sampling, etc. (Ditlevsen and Madsen, 1996). Application of this approach (SFEM+FORM) in the context of spatial variability in an elastic mechanical problem can be found in Sudret and Der Kiureghian (2002). Application to one (resp. two)-layer elastic foundation with random elastic properties and loading can be found in Sudret et al. (2003). Both references show the accuracy of the approach for reasonable 6
reliability indices up to β = 4 − 5 (less than 5% discrepancy in β with respect to the reference solution). In this paper we further develop the approach to carry out efficient parametric studies. Suppose the reliability of an elastic structure is investigated as a function of a load parameter λ, for instance : g(X, λ) = S¯ − λS(X)
(31)
where S¯ is an admissible threshold and S is a response quantity (e.g. nodal displacement) depending on the basic random variable X and the load parameter λ. In order to get the probability of failure with respect to λ (fragility curve), two strategies may be adopted : – for each value of λ, a finite element reliability analysis is carried out by coupling a finite element code and a reliability code (Der Kiureghian and Ke, 1988; Lemaire and Mohamed, 2000). A parametric study with respect to λ may be carried out, which enhances the efficiency in terms of numbers of calls to the finite element code. – a single SFEM analysis corresponding to λ = 1 is performed. The limit state function (31) rewrites : P −1 X Sj Ψj ({ξk }M (32) gSF EM (X, λ) = S¯ − λ k=1 ) j=0
The successive reliability analysis corresponding to Eq.(32) are then almost costless since this limit state function is analytical. Both approach are now compared on an application example in geotechnical engineering. 5 APPLICATION EXAMPLE : SERVICEABILITY OF A FOUNDATION Let us consider an elastic soil mass made of two layers of different isotropic linear elastic materials lying on a rigid substratum. A foundation on this soil mass is modeled as a uniform pressure λ applied over a length 2B of the free surface (figure 5). Due to the symmetry, half of the structure is modeled by finite element. The mesh of half of the foundation (due to symmetry) comprises 99 nodes and 80 4-node elements which allows a 1%-accurate evaluation of the maximal settlement compared to a reference solution. The model parameters are listed in Table 1. There are four random variables, namely the Young’s modulus and Poisson’s ratio of each layer. The reliability of the foundation with respect to the maximum admissible settlement u ¯ is investigated. Hence the parametric limit state function : g(X, λ) = u ¯ − λ uA (E1 , ν1 , E2 , ν2 ) (33) where uA (E1 , ν1 , E2 , ν2 ) is the maximal displacement obtained for a unit load λ = 1. The reliability index is computed by FORM for different values of λ using the two strategies mentioned in Section 4, namely : – a direct coupling between the finite element code Code Asterr [http ://www.code-aster.org] and the probabilistic code PROBAN (parametric study) – a single SFEM analysis followed by a parametric FORM reliability analysis from Eq.(33). These tools are implemented in a Matlab package. Results are reported in Table 3 and plotted in Figure 3. Column #2 corresponds to the direct coupling reference solution, columns #3-5 corresponds to different strategies of resolution in the SFEM analysis, namely a complete resolution at order 2 or 3 and a hierarchical resolution at order 3 with pre-resolution at order 2 (denoted by p = 3(2)). It appears that for reliability indices lower than 5, we have excellent results for the three different methods used. For reliability indices greater than 5, the direct method with an order 2 gives poor results, the two other results are equivalent. More 7
F IG . 2: Mesh of the foundation TAB . 1: Parameters of the foundation Parameter Notation Type Mean Upper layer Young’s Modulus E1 Lognormal 50 MPa Lower layer Young’s Modulus E2 Lognormal 70 MPa Beta [0, 0.5] 0.3 Upper layer Poisson’s Ratio ν1 Lower layer Poisson’s Ratio ν2 Beta [0, 0.5] 0.3 Width 2B Deterministic 10 m Mesh width L Deterministic 60 m Deterministic 7.75 m Upper layer soil thickness t1 Lower layer soil thickness t2 Deterministic 22.25 m Threshold u ¯ Deterministic 0.1 m
Coef. of Var. 20% 20% 20% 20% -
generally, other results not reported in this paper show that the hierarchical resolution at order 3 with a pre-resolution at order 2 is the best compromise between accuracy and efficiency. Table 2 reports the computer processing time for the complete parametric study in each case, where the unit time corresponds to one single deterministic analysis (i.e. p = 0 in the SFEM context). It appears that the direct coupling is about as computationally expansive as the SFEM with p = 3. However the same accuracy is obtained for the SFEM (p = 3(2)) hierarchical solution at about one tenth of the cost. TAB . 2: Computer processing time required by the direct coupling and SFEM at various orders Direct Coupling SFEM p = 0 Coupling p = 0 p = 2 p = 3 p = 3(2) Time 1 1280 1 56 1291 105
14
1 direct p=2 direct p=3 hierarchical p=3(2) Coupling
12
0.9
direct p=2 direct p=3 hierarchical p=3(2) Coupling
0.8
10
0.7 8 0.6 6 0.5 4 0.4 2 0.3 0
0.2
−2
−4 0.1
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0 0.1
0.6
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
F IG . 3: Left : Reliability index β vs. load P - Right : Failure probability Pf vs. P (fragility curve).
8
TAB . 3: Reliability index for the settlement of a foundation with different loads λ λ (MPa) Direct Coupling SFEM βDirectCoupling − βSF EM p=3 p=2 p=3 p = 3(2) 0.100 9.4605 12.0038 10.7559 10.7350 -1.2954 0.150 6.9900 7.9454 7.6122 7.6175 -0.6222 0.200 5.1135 5.4536 5.4004 5.4047 -0.2869 0.250 3.6212 3.7013 3.7412 3.7228 -0.1200 0.300 2.3963 2.3674 2.4322 2.4024 -0.0359 0.350 1.3637 1.2971 1.3622 1.3377 0.0015 0.400 0.4732 0.4054 0.4612 0.4536 0.0120 0.450 -0.3084 -0.3589 -0.3183 -0.3037 0.0099 0.500 -1.0022 -1.0285 -1.0091 -0.9723 0.0069 0.550 -1.6268 -1.6257 -1.6351 -1.5799 0.0083 0.600 -2.1942 -2.2657 -2.2133 -2.1463 0.0191 6 CONCLUSION This paper presented a stochastic finite element method able to solve linear mechanical problems with random materials properties and loading. Basic random variables can be in any number and of any type. The paper focuses on the application of SFEM to solving parametric reliability problems. The approach appears accurate for reliability indices up to β = 5, at a cost which is one order of magnitude smaller than the direct coupling in the application example. Note that the SFEM also provides straightforwardly the statistical moments of the response (Sudret et al., 2003) and that any other reliability analysis (e.g. corresponding to other limit state functions) is costless since the complete probabilistic content of the response has been characterized once for all in Eq.(11). Work is currently in progress to obtain the expansion coefficients in Eq.(11) in a different way, which will open SFEM to non linear problems. R´ef´erences Berveiller, M. and B. Sudret (2003). Th`ese sur les e´ l´ements finis stochastiques. Rapport d’avancement 1. Technical Report HT-26/03/039/A, EDF R&D internal report. (in French). Clouteau, D. and R. Lafargue (2003). An iterative solver for stochastic soil-structure interaction. In P. Spanos and G. Deodatis (Eds.), Computational Stochastic Mechanics (CSM-4), pp. 119–124. Der Kiureghian, A. and J.-B. Ke (1988). The stochastic finite element method in structural reliability. Prob. Eng. Mech. 3(2), 83–91. Ditlevsen, O. and H. Madsen (1996). Structural reliability methods. J. Wiley and Sons, Chichester. Ghanem, R.-G. (1999a). Ingredients for a general purpose stochastic finite elements implementation. Comp. Meth. Appl. Mech. Eng. 168, 19–34. Ghanem, R.-G. (1999b). Stochastic finite elements with multiple random non-Gaussian properties. J. Eng. Mech., ASCE 125(1), 26–40. Ghanem, R.-G. (2000). Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Advances in Engineering Software 31, 607–616. Ghanem, R.-G. and R. Kruger (1996). Numerical solution of spectral stochastic finite element systems. Comp. Meth. Appl. Mech. Eng. 129(3), 289–303. 9
Ghanem, R.-G. and P.-D. Spanos (1991). Stochastic finite elements - A spectral approach. Springer Verlag. Ghiocel, D. and R. Ghanem (2002). Stochastic finite element analysis of seismic soil-structure interaction. J. Eng. Mech. 128, 66–77. Isukapalli, S. S. (1999). Uncertainty Analysis of Transport-Transformation Models. Ph. D. thesis, The State University of New Jersey. Le Maˆıtre, O., O. M. Knio, H. N. Najm, and R. Ghanem (2001). A stochastic projection method for fluid flow. J. Comp. Phys. 173, 481–511. Lemaire, M. and A. Mohamed (2000). Finite element and reliability : A happy marriage ? In Reliability and Optimization of Structural Systems, Keynote Lecture at the 9th IFIP WG 7.5 Working Conference, pp. 3–14. Nowak, A. and Szerszen, M. (Eds.). Malliavin, P. (1997). Stochastic Analysis. Springer. Puig, B., F. Poirion, and C. Soize (2002). Non-Gaussian simulation using Hermite polynomial expansion : convergences. Prob. Eng. Mech. 17, 253–264. Sudret, B., M. Berveiller, and M. Lemaire (2003). El´ements finis stochastiques spectraux, nouvelles perspectives. In Proc. 16`eme Congr`es Franc¸ais de M´ecanique - Nice. (in French). Sudret, B. and A. Der Kiureghian (2000). Stochastic finite elements and reliability : A state-of-the-art report. Technical Report no UCB/SEMM-2000/08, University of California, Berkeley. 173 pages. Sudret, B. and A. Der Kiureghian (2002). Comparison of finite element reliability methods. Prob. Eng. Mech. 17, 337–348. Van Den Nieuwenhof, B. (2003). Stochastic Finite Elements for elastodynamics : random fields and shape uncertainty modeling using direct and modal perturbation-based aproaches. Ph. D. thesis, Universit´e Catholique de Louvain. Xiu, D. and G. E. Karniadakis (2002). The Wiener-Askey polynomial chaos for stochastic differentiel equations. J. Sci. Comput. 24(2), 619–644. Zienkiewicz, O.-C. and R.-L. Taylor (2000). The finite element method. Butterworth Heinemann, 5th edition.
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