Structural reliability using non-intrusive stochastic finite elements Marc Berveiller(a,b) , Bruno Sudret(a) & Maurice Lemaire(b) (a)

Electricit´e de France, R&D Division, Site des Renardi`eres, F-77818 Moret-sur-Loing, France LaMI EA 3867 - FR CNRS 2856, Universit´e Blaise Pascal & Institut Francais de M´ecanique Avanc´ee, F-61375 Aubi`ere France (b)

ABSTRACT: The stochastic finite element method allows to solve stochastic boundary value problems where material properties and loads are random. The method is based on the expansion of the stochastic mechanical response onto the so-called polynomial chaos. In this paper, a non intrusive method based on a projection method is presented. This method is illustrated by the analysis of a cracked pipe in the context of non linear fracture mechanics.

1 INTRODUCTION The finite element method is currently used for evaluating the behavior of mechanical systems. This method produces accurate results, but we have to consider that input parameters (material properties, loading parameters) are deterministic. The new step in the numerical simulation is to take into account the randomness of input parameters. Then reliability analysis can be performed with classical methods like Monte-Carlo simulation or FORM/SORM. These methods requires running numerous deterministic finite element analysis to obtain one result (moments of the response, or probability of failure). However, the engineer is often interested not only in a single value of the probability of failure but also in its evolution as a function of a parameter or also having the moments of the response. The stochastic finite element method (Ghanem and Spanos 1991), developed in the early 90’s, allows to take in account randomness in material properties and in loading parameters. The result is the expansion of the response quantities onto the socalled polynomial chaos. New methods are developed for computing the expansion coefficients of a response quantity with deterministic finite element analysis and analytical computations. These methods are called non intrusive methods. When using the properties of orthogonality of the polynomial chaos, these coefficients may alternatively be cast as an expectation formula, which reduces to computing an integral. This method is called the projection method (Berveiller, Sudret, and Lemaire 2004). This integral may be evaluated by Monte Carlo simulation or by a quadrature method. The projection method is illustrated by the analysis of a crack in a pipe weld in the context of fracture mechanics. The obtained expansion onto the polynomial chaos is interpreted as a stochastic response surface cast in the standard normal space. Then usual reliability methods such as FORM/SORM combined with importance sampling can be used. 2 ”HISTORICAL” STOCHASTIC FINITE ELEMENT METHOD 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

(1)

where K is the global stiffness matrix, U is the basic response quantity (vector of nodal displacement) and F is the vector of nodal forces. 1

In the stochastic finite element method (Ghanem and Spanos 1991), due to the introduction of input random variables for material properties and loading, the matrix K and the vector F become random. Thus the basic response quantity becomes a random vector of nodal displacements U (θ). Each component is a random variable expanded onto the so-called polynomial chaos : U (θ) =

P −1 X

U j Ψj {ξk (θ)}M k=1

j=0




(2)

where {ξk (θ)}M variables appearing in the discretization k=1 denotes the set of standard normal
} are multidimensional Hermite polynomials of all input random variables and {Ψj {ξk (θ)}M k=1 2 that form an orthogonal basis of L (Θ, F, P ), which is the Hilbert space of random variables with finite variance.In the original work by Ghanem et al., the coefficients {U 0 , · · · , U P −1 } are computed using the Galerkin method. Note that the following relationship holds: P =

(M + p)! M ! p!

(3)

where P is the number of coefficients to be computed to represent each scalar response quantity, M is the number of input random variables (or number of random variables used to discretize input random fields) and p is the maximum degree of the polynomial approximation. This resolution implies that the equilibrium equation must be modified with a specific implementation into the computer code. New methods have been recently developed for computing these coefficients using deterministic finite element analysis and analytical computations. They are called in the sequel non-intrusive methods. 3 NON INTRUSIVE METHOD 3.1 Principle The principle of the non intrusive method (Field, Red-Horse, and Paez 2000; Field 2002; Berveiller, Sudret, and Lemaire 2004) is to compute the coefficients U j in Eq.(2) from a set of deterministic analysis, which can be performed by any classical finite element code. By using the orthogonality of the polynomial chaos basis, Eq.(2) yields: Uj =

E[U Ψj ] E[Ψ2j ]

,

, j = {0, · · · , P − 1}

(4)

The denominator is easy to compute (Sudret and Der Kiureghian 2000). The numerator can be cast as an integral (Eq.(5)), which can be evaluated by various numerical methods: such as simulation methods or quadrature schemes. Z U (x)Ψj (x)ϕM (x)dx E[U Ψj ] = (5) RM

1

2

where ϕM (x) = (2π)−M/2 e− 2 kxk . 3.2 Simulation Method Monte Carlo simulation The Monte-Carlo simulation (Rubinstein 1981) is a classical method to estimate the integral of a function. The main disavantage of the Monte-Carlo simulation is that it require a large number of samples to get a good estimation of the integral. Stratified simulation, like the Latin Hypercube simulation, can be used for decreasing the number of samples. Latin Hypercube simulation The Latin Hypercube simulation (MacKay, Beckman, and Conover 1979; Olsson, Sandberg, and Dahlblom 2003) is a simulation method which forces the samples to cover more uniformly the space of parameters (see figures 1(a) and 1(b)). 2

4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −4

−3

−2

−1

0

1

2

3

4

−4 −4

−3

−2

−1

0

1

2

3

4

(a) Repartition of 100 samples for the Monte Carlo simula-(b) Repartition of 100 samples for the Latin Hypercube simtion ulation

3.3 Quadrature Method The integral in Eq. (5) can be evaluated by using a quadrature scheme (Field 2002; Matthies and Keese 2004; Berveiller, Sudret, and Lemaire 2004). The fundamental theorem of Gaussian quadrature states that given an integer q, one can find a set of weights wj and a set of points ξj that satisfy: Z

RM

U (x)Ψj (x)ϕM (x)dx ≈

q X

i1 =1

···

q X

wi1 · · · wiM U (X(ξi1 ) · · · X(ξiM ))Ψj (ξi1 · · · ξiM ) (6)

iM =1

In each dimension, integration points ξj for a q-order quadrature scheme are roots of the Hermite polynomial of degree q + 1. The method consists in the following steps: 1. choose an order p for the polynomial chaos;, 2. compute the polynomial chaos basis, 3. compute the integration points ξj and weights wj associated to order q; 4. using the isoprobabilistic transformation ξ → X, compute all possible combinations for the input parameters X(ξ) for the finite element code; 5. evaluate Eq.(6); 6. evaluate Eq.(4). 4 APPLICATION TO RELIABILITY ANALYSIS It is easy to show that any response quantity (e.g. strain or stress component) may be also expanded onto the polynomial chaos. Thus the mechanical response of the system S (i.e. the set of all nodal displacements, strain or stress components) may be written as: S=

P −1 X

S j Ψj

(7)

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).

3

When using the polynomial representation of the response (Eq.(7)), it is clear that any limit state function is analytical and defined in terms of standard normal variables :   P −1 X  S j Ψj {ξk }M (8) g(X, S(X)) ≡ g {ξk }M k=1 , k=1 j=0

Thus the reliability problem, which is already formulated in the standard normal space, may be solved by any method including Monte-Carlo simulation, FORM/SORM, Importance Sampling, etc. (Ditlevsen and Madsen 1996). The probability density function of the response can be computed by various methods including Monte-Carlo simulation and parametric FORM analysis (Sudret and Der Kiureghian 2002). 5 APPLICATION EXAMPLE 5.1 Problem statement In nuclear power plants, pipes undergo thermal and mechanical cycles that can lead to initiation and propagation of cracks. When a crack is observed, it is important to know whether the structure has to be repaired or if it can be stated that an accident will not occur. Therefore, reliability analysis can provide the probability of failure knowing that there is a crack and that the load can reach accidental values defined in a particular range. Figure 1 shows an axisymmetrically cracked pipe submitted to internal pressure and axial tension. Due to the boundary conditions at the pipe ends, the applied hydraulic pressure induces, besides the radial pressure, longitudinal tension forces. L -   σt + σ0 66666666666666666666666 σt + σ0 P circumferential crack ???????????  ????????????   -

σt + σ0

σt + σ0

Figure 1. Axisymmetrically cracked pipe

The input parameters are described as follows: • the crack length a = 15mm • the pipe length L = 1m • the internal pressure P = 15.5M P a • the inner radius Ri = 393.5mm • the thickness t = 393.5mm • the applied tensile stress σt (varying from 0 up to 200 MPa). This load is taken as a deterministic parameter in the reliability analysis. Indeed we are interested in obtaining the failure probability as a function of the tensile stress in order to be able to decide if pipe repairing has to be done for a given crack length and loading effect. • end effect stress σ0 given by σ0 = P

Ri2 (Ri + t)2 − Ri2

The material is steel, whose constitutive law is described by a Ramberg-Osgood law. The strainstress relationship reads:   σy σ n σ (9) ε = +α E E σy 4

where E denotes the Young’s modulus, σy the yield strength, n the strain hardening exponent and α the Ramberg-Osgood coefficient. We are interested in the evaluation of the crack driving force J as a function of the applied tensile stress σt , which can be computed by a finite element analysis. Figure 2 shows the mesh used for solving the problem.

Figure 2. Mesh of the cracked pipe

250

200

σ (MPa)

150

100

50

0 0

0.02

0.04

0.06

0.08

0.1

ε

Figure 3. Tensile test curve of the steel

In this problem, four random variables are considered (see Table 1), namely the Young’s modulus E, the yield strength σy , the strain hardening exponent n and the Ramberg-Osgood coefficient α. These random variables are assumed to be statistically independent. Table 1. Description of the input random variables

Parameter Young’s modulus Ramberg-Osgood coefficient Strain hardening exponent Yield strength

Notation E α n σy

Distribution Lognormal Normal Normal Lognormal

Mean 175 500 MPa 1.15 3.5 259.5 MPa

Coef. of Var. 5% 13% 3% 3.8%

5.2 Reliability analysis In order to evaluate the structural integrity, the probability that the crack driving force J exceeds the ductile tearing resistance J0.2 is to be computed : Pf = Prob(J ≥ J0.2 )

(10)

g = J0.2 − J(E, σy , n, α)

(11)

The limit state function is written as:

Consider that J0.2 is modeled as a lognormal random variable with mean value 52 MPa.mm and coefficient of variation 18.27%. A reference solution of (10) is obtained by a direct coupling between the finite element code Code Asterr1 and the reliability code Proban (Det Norske Veritas 1

http://www.code-aster.org

5

2000) using the First Order Reliability Method (FORM). P On the other hand, the stochastic finite element approximation J˜ = Pk=1 Jk Ψk (ξ) is computed. A polynomial chaos of third degree is used. Eq.(3) shows that P = 35 response coefficients are to be computed. Once the coefficients are known, Eq.(11) is replaced by an approximation of the limit state function: 35 X Jk Ψk (ξ) g˜ = J0.2 − (12) k=1

FORM is first applied (as explained above, the approximation limit state function in this case is polynomial in standard normal variable making this solving scheme inexpensive). Then importance sampling aroung the design point is used in order to get accurate values of Pf (a coefficient of variation of 5% is obtained using 1000 simulations). Note that is inexpensive since Eq.(12) is analytical. Figures (4) and (5) shows the failure probability vs. the tensile stress and the log of the failure probability obtained with the non-intrusive methods (Latin hypercube simulation and quadrature) respectively. It is important to note that each latin hypercube simlation requires a full non linear finite element analysis. For the quadrature scheme, 3 (resp. 4) integration points are used in each dimension and for the Latin hypercube simulation, 500, 1000 and 1500 samples are used respectively. We can note that for the Latin Hypercube simulation, 1500 samples are not enough to get an accurate estimation of the failure probability. Using the quadrature method, 3 or 4 of the integration points provide the same results and they are very accurate compared to the reference solution. Table 2 gives the number of calls to the finite element code required by the various methods. The quadrature method is more than 6 times faster than the reference simulation. It should be noted that the direct coupling is started from scratch at each value of σt , since the runs used for FORM analysis at a lower value of σt are of no use for a new analysis. 1 Reference 0.9 quadrature : 3 points quadrature : 4 points 0.8 LHS : 500 points LHS : 1000 points 0.7 LHS : 1500 points 0.6 0.5 0.4 0.3 0.2 0.1 0 130

140

150

160 170 Tensile stress (MPa)

180

190

200

Figure 4. Failure probability vs. tensile stress

Table 2. Computer processing time required by the direct coupling and the quadrature method

Parameter Number of Points Time

Direct Coupling 542

Quadrature 3 4 81 256

500 500

LHS 1000 1500 1000 1500

Figure 6 presents the PDF for σt = 200 MPa of J computed with 3 integration points. It is 6

0 −1 −2 −3 −4 −5

Reference quadrature : 3 points quadrature : 4 points LHS : 500 points LHS : 1000 points LHS : 1500 points

−6 −7 −8 130

140

150

160 170 Tensile stress (MPa) Figure 5. Log of the failure probability vs. tensile stress obtained

180

190

200

rather close to a lognormal PDF with mean 71.64 and standard deviation 7.81. 0.06 0.05 0.04 0.03 0.02 0.01 0 30

40

50

60

70

80

90 100 110 120

Figure 6. PDF of J for σt = 200 MPa

6 CONCLUSIONS The paper presents a non intrusive method for computing coefficients of the polynomial series expansion of the response in non linear stochastic finite element analysis. This method is based on a projection of the solution onto the so-called polynomial chaos. The method leads to evaluating a multidimensional integral. This can be carried out by a quadrature scheme or by a simulation method (Monte-Carlo or Latin Hypercube). The method is applied to solve a reliability problem as a post-processing. It is illustrated by the analysis of a crack in a pipe in the context of non linear fracture mechanics. A parametric study is carried out as a function of the applied axial stress σt . The influence of the number of integration points used in the quadrature scheme and the number of the samples in the Latin Hypercube simulation is investigated. Based on the selection scheme, it appears that a number of integration points equal to 3 provides excellent accuracy in the tail (probability of exceedance of a threshold). In comparison the Latin hypercube simulation method provides poor result for a much larger computational effort. The great advantage of the non intrusive approach compared to the classical Galerkin approach is that only deterministic finite 7

element models are run. Hence the full non linear capabilities of the finite element code may be used without additional implementation. Note that if several output quantities are of interest, the marginal cost to estimate the response coefficients is low. Indeed the deterministic finite element analysis are usually the expensive part of the evaluation. REFERENCES Berveiller, M., B. Sudret, and M. Lemaire (2004). Comparison of methods for computing the response coefficients in stochastic finite element analysis. In Proc. AsRANET 2, Barcelona, Spain. Det Norske Veritas (2000). PROBAN user’s manual, V.4.3. Ditlevsen, O. and H. Madsen (1996). Structural reliability methods. J. Wiley and Sons, Chichester. Field, R. (2002). Numerical methods to estimate the coefficients of the polynomial chaos expansion. In 15th ASCE Engineering Mechanics Conference. Field, R., J. Red-Horse, and T. Paez (2000). A non deterministic shock and vibration application using polynomial chaos expansion. In 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability. Ghanem, R.-G. and P.-D. Spanos (1991). Stochastic finite elements - A spectral approach. Springer Verlag. MacKay, M. D., R. J. Beckman, and W. J. Conover (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 2, 239–245. Matthies, H. and A. Keese (2004). Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comp. Meth. Appl. Mech. Eng. in press. Olsson, A., G. Sandberg, and O. Dahlblom (2003). On latin hypercube sampling for structural reliability analysis. Struct. Saf. 25, 47–68. Rubinstein, R.-Y. (1981). Simulation and the Monte Carlo methods. John Wiley & Sons. Sudret, B. and A. Der Kiureghian (2000). Stochastic finite elements and reliability : A state-ofthe-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. Zienkiewicz, O.-C. and R.-L. Taylor (2000). The finite element method. Butterworth Heinemann, 5th edition.

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