Stochastic finite element with material uncertainties

Finite Elements in Analysis and Design 64 (2013) 65–78

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Stochastic finite element with material uncertainties: Implementation in a general purpose simulation program Shen Shang, Gun Jin Yun n Department of Civil Engineering, The University of Akron, Akron, OH 44325, USA

a r t i c l e i n f o

abstract

Article history: Received 6 May 2012 Received in revised form 26 September 2012 Accepted 1 October 2012 Available online 30 October 2012

This paper presents a stochastic finite element (SFE) within a general purpose finite element analysis program ABAQUS to simulate the probabilistic structural response of stochastic materials. Discretization and quantification of random fields associated with material uncertainties are accomplished through Karhunen–Loe ve (KL) expansion in order to simulate the stochastic response of structures under material uncertainties. Although SFE is one of the most widely accepted approaches, its integrations into general-purpose finite element software are rare in literatures due to difficulties in its intrusive formulation and managing two different meshes for discretizing the physical and random field domains subjected to different meshing criteria. Therefore, issues on the separation of RF mesh from FE mesh have been addressed along with its efficient implementations. The proposed method can significantly reduce dimensionality of the stochastic domain and efficiently predict probability density functions of the structural response under material uncertainties through Monte Carlo simulations combined with the Latin hypercube sampling technique. & 2012 Elsevier B.V. All rights reserved.

Keywords: Stochastic finite element Abaqus uel Karhunen–Loe ve expansion Probabilistic analysis Material uncertainty Monte Carlo simulation

1. Introduction During last two decades, considerations of random nature of physical systems are gradually and widely recognized of importance by industry and researchers. As mentioned in [1] ‘‘During the next decade, probabilistic modeling of mechanical problems will be a topic of great importance and interest’’. Nowadays, the uncertainty assessment of engineering structures has increasingly become an important topic of interests in various engineering communities ranging from geo-physics to bio-engineering. Stochastic finite element method (SFEM) has been recognized as one of the primary computational techniques in probabilistic analysis of structures with uncertainties. Stefanou provided a state-of-the-art review of past and recent developments in the area of SFEM, indicating future directions as well as open issues to be studied in the future [2]. A vast amount of research has been conducted for tackling different problem definitions where uncertainties in geometry, material properties, loadings and dynamic systems were considered [3–25]. One of the most urgent needs for applications of SFEM in engineering problems is to develop a robust and efficient userfriendly framework that can interact with powerful third party codes. Several representative examples of structural reliability software can be found in literatures [26–31]. Noteworthy features

n

Corresponding author. Tel.: þ1 330 972 8489; fax: þ1 330 972 6020. E-mail addresses: [email protected] (S. Shang), [email protected] (G.J. Yun). 0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.10.001

and capability of these examples in interacting with the third party FE code have been studied in Ref. [29]. Two computer codes namely FERUM (Finite Element Reliability Using MATLAB) and OpenSees have been developed at the University of California Berkeley. FERUM is a MATLAB toolbox for structural reliability analysis and OpenSees, a widely used computational framework, contains reliability analysis tools in conjunction with static and dynamic nonlinear time history analysis. Numerical evaluation of stochastic structures under stress (NESSUS) is also a generalpurpose tool for computing the probabilistic response or reliability of complex systems [28]. The NESSUS is a highly interactive software that can be used in conjunction with commercial FE analysis programs such as ABAQUS, ANSYS, NASTRAN, etc. to solve problems in stochastic mechanics [29]. A stochastic FE library (StoFEL) was successfully coupled with ANSYS for approximating the structural response statistics and carrying out graphical postprocessing [32]. Finite element software ATENA and the stochastic package Feasible Reliability Engineering Tool(FReET) are integrated into software system SARA to carry out nonlinear analyses of reinforced concrete structures with uncertainties of structural input parameters [33]. On the commercial company side, ANSYS Inc. has integrated probabilistic design capabilities in its recent releases, namely the ANSYS Probabilistic Design System and the ANSYS DesignExplorer. More applications of the software for various industrial problems can be found in Ref. [34]. To the best of authors’ knowledge, there has been no attempt to integrate the SFE into ABAQUS/Standard. The SFE is able to account for uncertainties of mechanical properties as a homogeneous

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random field. To realize wide applications of SFE, two different meshes should be defined: 1) finite element (FE) mesh and 2) random field (RF) mesh. There are many random field discretization methods such as spatial average method [35], midpoint method [36–38], expansion optimal linear estimation method [39], Karhunen–Loe ve (KL)-expansion [40], etc. A comprehensive review and comparison of these methods can be found in [41]. The spatial averaging method assigns a value obtained as an average of stochastic field values over the finite element domain to the corresponding element in FE mesh[35]. The midpoint method assigns the random field value obtained at the location of the centroid of a certain FE element to the corresponding element in FE mesh [36–38]. The existing discretization methods for modeling random fields are fully reviewed and compared in [41]. In this paper, a stochastic finite element (SFE) is proposed in an intrusive way that employs Karhunen–Loe ve (KL) expansions for spectral decomposition of the covariance function of random fields. In particular, a general mapping–interpolation method has been proposed so that random field values could be efficiently mapped between the RF and FE meshes. Most importantly, the proposed methodology can easily be integrated with any working deterministic code, which is developed for handling very complex problem such as visco-elasto-plasticity. The paper is organized as follows. The basic concept and verification of numerical KL-expansion is presented in Section 2. Intrusive formulation of SFE is presented in Section 3. The general mapping method is introduced in Section 4. Section 5 describes the detailed implementation of SFE in ABAQUS-MATLAB software framework. Numerical studies are carried out to verify capabilities of SFE in Section 6. The paper ends with conclusions in Section 7.

2. Spectral decomposition of random field by  Karhunen–Loeve expansion

li ji ðxÞxi ðyÞ

x1 ,x2 A O

ð4Þ

where bc1 and bc2 are the correlation length parameters in different directions; LD1 and LD2 are the physical characteristic length in different directions; Lc1 ¼bc1Ld1 and Lc2 ¼bc2Ld2 are the correlation length in different directions; s is the standard deviation; bc and LD are the correlation length parameter and physical length of the symmetric CF kernel; Lc ¼bcLD is the correlation length. Since Eq. (3) is a separable integral kernel, the analytical solution of the integral function can be derived from Ref. [46]. However, Eq. (4) is an inseparable kernel [12]. Therefore, numerical solutions have to be provided for Eq. (4). 2.2. Galerkin finite element techniques for Karhunen–Loe ve expansion The analytical solution of the Fredholm integral equation does not always exist for most of covariance functions, especially in the case that the RF domain O takes irregular shapes, therefore, the Galerkin-based finite element approach is more preferred than analytical solutions. For computations, the truncated KL expansion is used in this paper M pffiffiffiffi X

li ji ðxÞxi ðyÞ

ð5Þ

i¼1

Uncertainties of material properties (e.g., elastic modulus, thickness, density) are often assumed to be Gaussian due to its simplicity and the lack of relevant experimental data although most material uncertainties appearing in engineering systems are non-Gaussian in nature according to [2,43,44]. In this paper, KL-expansion is employed for discretizing spatially varying random fields in the two-dimensional domain. KL expansion is able to use a few terms to capture the characteristic of the strongly correlated random fields. In this method, the 2D random fields with non-zero means are decomposed into a deterministic part and a stochastic part as follows: 1 pffiffiffiffi X

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 ðx1 x2 Þ2 þ y1 y2 A, C ðx1 ; x2 Þ ¼ s exp@ bc LD 2

Eðx, yÞ ¼ Eþ

2.1. Random field discretization by Karhunen–Loe ve expansion

Eðx, yÞ ¼ EðxÞ þ

Several covariance functions are available in different forms [12]. Choice of covariance functions is dependent on the nature of stochasticity of the random field. The covariance function (CF) for spatially varying random fields can also be constructed from field measurement data [45]. Two commonly used exponential forms are presented as follows:   9x1 x2 9 9y1 y2 9 C ðx1 ; x2 Þ ¼ s2 exp   ð3Þ , x1 ,x2 A O bc1 LD1 bc2 LD2

ð1Þ

i¼1

where E(x,y) is a random field; x is the position vector defined over the physical domain D; y represents primitive randomness that belongs to the space of random events; xi is the statistically uncorrelated Gaussian random variable with zero mean and unit standard deviation and EðxÞ is the mean value of E(x,y). However in this paper, EðxÞis defined as a constant and does not vary over the domain D. Therefore, EðxÞis simplified as E. li and ji (x) are the eigenvalue and the eigenfunction, which are the solution to the Fredholm integral equation of the second kind: Z C ðx1 ; x2 Þjk ðx2 ÞdA2e ¼ lk jk ðx1 Þ ð2Þ O2e

where C(x1,x2) is the covariance functions, A2e indicates the elemental domain (O2e) in terms of position vector x2 ¼(x2, y2), that is, dA2e ¼dx2dy2.

where M denotes the number of KL expansion terms. In Galerkin finite element approach, the eigenfunction is approximated by discretizing the problem domain. The approximation of the eigenfunction is expressed as

ji ðxÞ ¼

nnod XRF

  Lj ðxÞdij ¼ oLðxÞ 4 d

ð6Þ

j¼1

where dij is the jth nodal value of the ith eigenfunction of the covariance function; nnodRF is the number of nodes per RF element. Basis function Lj can be any complete set of 2D functions in the Hilbert space H. In this paper, shape functions of a ninenode quadratic quadrilateral finite element (Q9) (i.e. Lagrange interpolation function) have been selected as the basis function. Galerkin finite element procedures are carried out by substituting Eq. (6) into Eq. (2). For each RF element, Eq. (2) can be written in terms of a nodal vector {d}e and Q9 shape function oL(x)4 of the eigenfunction as follows: Z   C ðx1 ; x2 Þ o Lðx2 Þ 49Je 9dA2e d e ¼ li ji ðx1 Þ ð7Þ O2e

where 9Je9 is the Jacobian that maps from parametric coordinates to physical coordinates. For each RF element, a weighted residual of Eq. (7) by the same shape function oL(x1)4 can also be written as follows: Z Z

  2 C ðx1 ; x2 Þ oLðx1 Þ 4 T o Lðx2 Þ 49Je 9 dA2e dA1e d e O1e O2e Z   ¼ li oLðx1 Þ 4 T o Lðx1 Þ 4dA1e d e ð8Þ O1e

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67

After all RF elements are assembled, a generalized algebraic eigenvalue problem is obtained as follows: CD ¼ KBD

C¼

ð9Þ

N RF N RF Z X X 2e ¼ 1 1e ¼ 1

Z

2

O1e O2e

C ðx1 ; x2 Þ o Lðx1 Þ 4 T o Lðx2 Þ 4 9Je 9 dA2e dA1e ð10Þ

Kij ¼ dij li

B¼

ð11Þ

N RF Z X e¼1

Oe

oLðxÞ 4 T o LðxÞ 49Je 9dAe

ð12Þ

where values in the jth column of D matrix are jth eigenfunction values at nodal points; C, D, K, and B are ndofRF by ndofRF dimensional matrices, where ndofRF is the total DOFs of the RF mesh; dij is the Kronecker delta. Elemental matrices Ce and Be have to be computed for each RF element and assembled into the global matrices C and B. Detailed information for computing the elemental matrices can be found in [47]. The property of symmetry holds for matrices B and Be if the covariance function is symmetric. However, only if both of the covariance function and RF domain O are symmetric, the global matrix C will be symmetric. After the general eigenvalue problem is solved, eigensolutions of the Fredholm integral equation are obtained. The size of finite elements should be selected to adequately capture the essential features of the stochastic spatial variability of the material properties. The strategy used to select the number or the size of the element used in decomposition is comprehensively studied in [47]. For the KL expansion method, it is recommended to choose the number of elements satisfying the following condition: NRF Z

LD Lc

ð13Þ

where LD is the length of the physical domain D and Lc is the correlation length. This inequality is not only suitable for onedimension domain but can also be applied to 2D domain. Following Eq. (13), a reasonable random field mesh can be selected for the KL expansion to accurately capture the random field fluctuations. All the numerical examples presented in this paper satisfy this condition. It can be observed that if the correlation length decreases, NRF has to be increased. Verification of Galerkin FE techniques for KL expansion: A unit square domain is presented in the following example to show the numerically reproduced covariance function and the decomposition error between the numerical covariance function and the analytical covariance function. The covariance function presented in Eq. (4) is used. The physical area AD ¼1, correlation length Lc ¼bcLd ¼1, and standard deviation s ¼1. 36 square elements are used for the RF mesh. The selected eigenvalues and eigenfunctions are depicted in Figs. 1 and 2. Numerically reproduced covariance function and analytical covariance function referring to a point (x2 ¼0.5 and y2 ¼0.5) are plotted in Fig. 3(a) and (b). To calculate the decomposition error, the global error estimator shown in Eq. (14) was evaluated: R R

ed ¼

D D 9C ðx1 ; x2 Þ

Fig. 1. First 50 eigenvalues of the covariance function (Eq. (3)).

PM

i¼1









li jRF x1 , y1 jRF x2 , y2 9dAdA i i A2D s2

ð14Þ

In this example, ndofRF ¼169 and the calculated decomposition error ed equals to 3.1934  10  4. This negligible amount of errors implies that KL expansion shows sufficient accuracy in discretizing the analytical covariance function.

Fig. 2. Eigenfunctions of covariance function (Eq. (3)) with NRF ¼9, LD1 ¼ LD2 ¼1, bc1 ¼bc2 ¼ 1 and s ¼ 1.

3. Intrusive formulation of stochastic finite element The ABAQUS/Standard UEL subroutine named SFEQ8 (stochastic Q8 finite element) is coded in FORTRAN language. As an 8-node plane stress finite element, the element totally contains eight nodes with 16 DOFs (translation on 1 and 2 directions at each node). Labels of nodes, faces and integration points for output are shown in Fig. 4. SFEQ8 is based on the isoparametric formulation. Shape functions used are the same as the one for Q8 finite element [48]. The elemental stiffness matrix of the stochastic finite element is decomposed into deterministic and stochastic parts as follows: ½Ke  ¼ ½Ke d þ½Ke s

ð15Þ

The deterministic element stiffness matrix [Ke]d can be calculated by a standard method in finite element procedures. Full

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Fig. 3. (a) Numerically reproduced covariance function by using 6 by 6 square elements in RF mesh referring to x2 ¼ 0.5, y2 ¼ 0.5; (b) analytical covariance function referring to x2 ¼0.5, y2 ¼0.5.

Fig. 4. Configuration of UEL SFEQ8.

integration scheme in Gaussian quadrature is used as in Eq. (16). ZZ Z 1Z 1 ½BT ½D½Bt9J9dxdZ ½Ke d ¼ ½BT ½D½Bdxdy ¼ 1

¼

3 X 3 X

1

T

½B ½D½Bt9J9W i W j

ð16Þ

i¼1j¼1

where [B] is the strain-displacement matrix; 9J9 is the determinant of Jacobian matrix; W is the weight factor of Gaussian quadrature; t is the thickness, and [D] is the material constitutive matrix where the mean value of Young’s modulus is used. In the case of the plane stress condition, the stochastic elemental stiffness matrix is formulated by substituting the second term of Eq. (5) into Young’s modulus in the material constitutive matrix: ZZ ^ xi , Z ; yÞ½Btdxdy ½Ke s ¼ ½BT ½Dð j Z

1

Z

1

¼ 1

¼

1

3 X 3 X

^ x , Z ; yÞ½Bt9J9dxdZ ½BT ½Dð i j ^ x , Z ; yÞ½Bt9J9W i W j ½BT ½Dð i j

i¼1j¼1

2 1 ^EG ðx , Z ; yÞ 6 i j ^ xi , Z ; yÞ ¼ 6m ½Dð j 1m2 4 0

E^ G ðxi , Zj ; yÞ ¼

8 X i¼1

m 1 0

N i ðxi , Zj ÞEFE i ðx, yÞ

0

3

0 7 7 5 ð1mÞ

ð17Þ

2

ð18Þ

where [Ke]s is the stochastic element stiffness matrix; x is the position vector of FE nodal points, and m is the Poisson’s ratio; EFE i (x, y) is the randomly distributed elastic modulus at FE nodal points which is calculated by using the general mapping– interpolation method introduced in Section 4; E^ G ðxi , Zj ; yÞ is the interpolated Young’s modulus at Gauss points. As shown in Eq. (18), Young’s modulus values at Gauss points are interpolated from nodal Young’s modulus values using eight-node shape functions. It is worth noting that instead of using a constant value of elastic modulus over the entire elements, every Gauss point has different elastic modulus realized based on the stochastic parameters of the random field. This numerical integration technique allows us to simulate stochastic material behavior. ABAQUS assembles element stiffness matrices into a global stiffness matrix and delivers the matrix to the numerical solver with prescribed boundary conditions. The detailed implementation of SFE in the MATLAB-ABAQUS combined framework will be introduced in Section 5.

4. General mapping–interpolation method between random field and finite element meshes 4.1. Separation of random field mesh from finite element mesh In this Section, a general mapping and interpolation method is presented to obtain the random field in FE mesh. Refinement of the FE mesh generally depends on geometrical features or stress states by the given loadings. However, distribution of random material properties is not necessarily identical to the FE mesh because inherent material heterogeneity is mainly related to manufacturing process or post-event internal defects. Complete separation of the RF mesh from the FE mesh is beneficial because RF and FE meshes are subjected to different criteria in terms of mesh density for reasonable accuracy. Therefore, two different kinds of meshes: (1) RF element mesh and (2) FE mesh will be generated. Two different meshes are assumed to have identical boundary shape and size but can have different number of elements. Since the RF mesh is different from the FE mesh, a method is needed to transfer the spatial distributed material properties from RF mesh to FE mesh. From a computational point of view, the discretized value of the random field has to be finally transferred to the Gaussian points in FE mesh, because the Gaussian integration of the elemental stiffness matrix is carried out by using the material properties defined at Gaussian points. For this purpose, eigenfunction values at RF nodal points, which are obtained from KL expansion, are transferred to FE nodal

S. Shang, G.J. Yun / Finite Elements in Analysis and Design 64 (2013) 65–78

points. Then, the discretized value of the random field at FE nodal points is calculated based on the eigenfunctions at FE nodal point. At last, the random field is interpolated to the location of the Gaussian points in FE mesh by using the eight-node shape function. An example that shows the mapping and interpolation error of the eigenfunctions based on a global error estimator is presented. 4.2. General mapping–interpolation between random field mesh and finite element mesh In order to deal with large-scale and stochastically complex engineering problems, a general mapping–interpolation method needs to be developed. First, let’s assume the eigenvalues and eigenfunctions values at all RF nodal points have been already determined through KL expansion. Because two meshes are independent, FE nodal points that are located inside or on the edge of each RF element must be identified. Within each RF element, RF mesh isoparametric coordinate values corresponding to the FE nodal points are calculated for subsequent interpolations. For a certain RF element, a nonlinear equation that relates coordinates ðxji , yji Þ of FE nodes with coordinates ðX jm , Y jm Þ of RF element nodes can be formulated as follows: 9 X

69 j

isoparametric coordinates xi and Zi j . By using the trust region method (‘fsolve’ built-in function of MATLAB), the nonlinear equation is solved as an unconstrained nonlinear optimization problem. An initial guess of every isoparametric coordinate is set j as ½xi , Zji ¼[0.0,0.0]. According to numerical examples presented in this paper, solutions are always converged within less than three optimization iterations with a terminating criterion that the objective function value is smaller than a preset tolerance (10  7). One demonstration is shown in Fig. 6. j After solving all the optimization problems, xi and Zji are substituted to Eq. (20) to calculate the eigenfunction value at the location of ith FE nodal point within jth RF element: k

jjFE ¼ i

9 X

j

N m ðxi , Zji Þ k jjRF m

ð20Þ

m¼1

j

Nm ðxi , Zji ÞX jm xji ¼ 0

m¼1 9 X

j

N m ðxi , Zji ÞY jm yji ¼ 0

ð19Þ

m¼1

where xji and yji are the physical coordinates of ith nodal point of current finite element which is located within jth RF element; X jm and Y jm is the physical coordinates of mth RF nodal point that j belongs to jth RF element. xi and Zi j are the isoparametric coordinates attached to jth RF element, which correspond to ith FE nodal point. Fig. 5 shows the coordinate transformation from physical coordinates to isoparametric coordinates within a certain RF element. Since both of the RF and FE physical nodal coordinates are known, the system of nonlinear equations can be solved for the

Fig. 6. A demonstration of the coordinate transformation between two fully independent RF mesh and FE mesh on non-rectangular domain.

Fig. 5. Illustration of the transformation of coordinates within a certain RF element (Green dotted lines are FE mesh). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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where k jjRF m is the value of kth eigenfunctions at mth nodal point of the jth RF element; k jjRF is the value of kth eigenfunctions i value at ith nodal point in FE mesh that is located within jth RF element. The Young’s modulus distribution at FE nodal points EFE can be calculated based on the obtained jFE using Eq. (21). EFE ðx, yÞ ¼ E þ

M pffiffiffiffi X

li jFE i ðxÞxi ðyÞ

ð21Þ

i¼1

It is worth noting that the optimization problem needs to be solved only one time as long as the RF and FE meshes do not j change during simulations. The values of xi and Zi j will be saved and retrieved later at the time of the numerical integration of the stochastic part of elemental stiffness matrices. In the case of

Monte Carlo Simulation (MCS), each realization requires reevaluation of Young’s modulus at the nodal points of FE mesh. Estimation of errors in the general mapping–interpolation method: The proposed method mapped eigenfunctions obtained from the KL expansion RF mesh to the FE mesh. After that, the Young’s modulus is generated by using eigenvalues and eigenfunctions obtained from KL expansion in FE mesh. To evaluate the mapping and interpolation error, a local error estimator for the reproduced covariance function is defined as 



eI x1 , y1 ; x2 , y2 ¼

M X

        9li jRF x1 , y1 jRF x2 , y2 li jFE x1 , y1 jFE x2 , y2 9 i i i i

i¼1

ð22Þ

Fig. 7. Schematic of Monte Carlo simulation using SFE and post-processing with UEL framework interfaced with MATLAB codes (RF, Random Field; KL, Karhunen–Loe ve).

Fig. 8. The 225 Q8 stochastic finite element mesh overlapped with nine Q9 random field elements of a square plate.

S. Shang, G.J. Yun / Finite Elements in Analysis and Design 64 (2013) 65–78 RE where jFE i and ji are eigenfunction values evaluated on FE and RF meshes, respectively. Since locations of nodal points in the RF and FE mesh mismatch, a global error estimator has to be used as

PM

eI ¼

i¼1

R R

D D 9li















71

estimator eI was found to 3.8  10  8. This small error estimator means the mapping and interpolation error of eigenfunctions between the RF and FE mesh is very small and negligible.



jRF x1 , y1 jRF x2 , y2 li jFE x1 , y1 jFE x2 , y2 9dAdA i i i i 2 2 AD s ð23Þ

To demonstrate the mapping and interpolation error evaluation, the same physical domain used in Section 2.2 is discretized into a RF mesh with nine elements and the FE mesh with 100 elements. It is assumed that the physical area AD ¼ 1, correlation length Lc ¼1, and standard deviation s ¼1. The global error

5. Implementation of stochastic finite elements for Monte Carlo Simulation In this section, detailed implementations for Monte Carlo Simulation (MCS) using the SFE are presented to predict probabilistic structural response caused by material uncertainties. A flow chart showing the MCS is depicted in Fig. 7. ABAQUS is one of the most widely used sophisticated finite element analysis tools and MATLAB is also a commonly used tool for numerical simulations. Therefore, SFE is implemented in user-defined element (UEL) in ABAQUS and a stochastic decomposition tool developed in MATLAB codes is interfaced with the UEL. 5.1. Monte Carlo Simulation with SFE and KL expansion

Fig. 9. The locations of the FE nodal points used to evaluate coefficients of correlation (the number is the label of the nodes).

Table 1 Coefficients of correlation before and after the nonlinear transformation. Paired points

rN rL 100(9rL  rN9)/rL(%)

(21,338) 0.3375 0.3354 0.6261

(36,93) 0.5937 0.5930 0.1180

(49,73) 0.8200 0.8206 0.0731

(31,41) 0.9094 0.9092 0.0220

Monte Carlo simulation (MCS) is used to simulate the stochastic structural response under material uncertainty. Even though it is expensive computationally, Monte Carlo simulation is the only currently available universal methodology for accurately solving problems in stochastic mechanics involving strong nonlinearities and large variations of non-Gaussian uncertain system parameters in Reliability-based design [42]. For MCS that considers the material uncertainty, zero-mean and unit-variance independent Gaussian random variables ðxi Þ in Eq. (5) need to be generated. In MCS, a set of these random variables is described as realizations, observations, or samples. Therefore, the stochastic problem is converted to a large number of deterministic problems. If computations for solving the deterministic problem are computationally costly, the MCS will be very time-consuming because the simple random sampling scheme needs a large number of samples to draw statistically meaningful results. In order to reduce computational times in MCS, an efficient sampling approach has to be   applied. In jth Gaussian realization, xndof RF j is generated by using Latin hypercube sampling (LHS) technique that preserves marginal probability distributions for each variable simulated. This LHS method allows for a small number of samples maintaining feasible representations of statistical characteristics of random fields [49]. The spatial random distribution of the elastic modulus on FE nodal points EFE(x,y) is calculated by using Eq. (5) combining with the general mapping–interpolation method. The user subroutine UEXTERNALDB is used to import the EFE(x,y) at the beginning of every elemental calculation carried out within ABAQUS. For each element, the code in UEL searches

93 36 93 Fig. 10. Normalized (a) E36 N vs. EN and (b) EN vs. EN (5000 samples).

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Fig. 11. Randomly distributed elastic modulus for four different cases: (a) Lc1 ¼ 1 m, sL1 ¼ 1 GPa; (b) Lc2 ¼ 1 m, sL2 ¼0.1 GPa; (c) Lc3 ¼0.5 m, sL3 ¼ 1 GPa; (d) Lc4 ¼0.5 m, sL4 ¼0.1 GPa.

Fig. 12. Representations of the approximated RF by using different number of KL expansion terms (M): (a) M ¼5; (b) M ¼ 20; (c) M ¼ 35; (d) M ¼49 (Lc ¼1, sL ¼ 1 GPa).

S. Shang, G.J. Yun / Finite Elements in Analysis and Design 64 (2013) 65–78

for EFE(x,y) at corresponding eight nodal points of SFE. After that, Eqs. (16), (17) are used to compute the elemental stiffness matrix.

6. Numerical examples for effects of material uncertainties on probabilistic structural response 6.1. A square plate with randomly distributed Young’s modulus A physical model of a thin square plate with 1 m length shown in Fig. 8 is selected as one of the numerical examples. Boundary edges indicated by the surfaces G1 and G2 are restrained in the direction normal to the edge. The edge G3 is subjected to a uniformly distributed loading of magnitude 1800 kN. Random field domain O is discretized into nine Q9 Lagrange elements (NRF ¼9), thus the RF mesh has a total of ndofRF ¼49 DOFs. Physical domain D has a different mesh that consists of 225 Q8 stochastic elements (NFE ¼225) with a total of ndofFE ¼736 DOFs.

73

6.1.1. Distortion check of covariance structure by transformation from normal to lognormal distribution In order to have non-negative elastic modulus values in realizations, the lognormal distribution was assumed. Mean value   EL and standard deviation (sL) of Young’s modulus that follows the lognormal distribution are first assumed. Using Eq. (24), statistical parameters in lognormal distribution are converted to those in the normal distribution. Because the KL expansion assumes that the random field  follows Gaussian distribution,  variance s2N and mean values EN in normal distribution are used as in Eq. (1) and Eq. (4) for generating multiple realizations of spatially varying Young’s modulus through KL expansion. Realizations of the random field in the normal distribution are converted to realizations in the lognormal distribution by Eq. (25). The transformation technique was used in Monte Carlo Simulation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # u " 2 u sL 1 sN ¼ tln þ 1 ; EN ¼ lnEL  s2N ð24Þ 2 EL

Fig. 13. (a) random distribution of elastic modulus; (b) stress in X direction; (c) stress in Y direction; (d) in-plane shear stress; (e) displacement in X direction; (f) displacement in Y direction.

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" # M pffiffiffiffi X ^EL ðx, yÞ ¼ exp EN ðx, yÞ þ li ji ðxÞxi ðyÞ

ð25Þ

i¼1

Although such a mapping guarantees the proper probability distribution, it may distort the correlation structure. Methods for correcting this problem can be found in [36]. As mentioned in [42,50], if the coefficient of the variance (CV) is small, the distortion of the correlation structure caused by the nonlinear transformation will be small. To ensure no significant distortion of the correlation structure, an example is presented. Eq. (4) is used as the covariance function. EL ¼ 30 GPa, the standard deviation sL ¼3 GPa and the correlation length Lc ¼1. Four pairs of arbitrary locations have been selected as shown in Fig. 9. Coefficients of correlation between two points for the four pairs are calculated before and after the transformation based on the MCS with 5000 samples and shown in Table 1. Fig. 10 shows the normalized Young’s modulus sampled from nodal point 36 versus the one from nodal point 93 before and after the transformation.

It can be concluded that, if the CV is small, the nonlinear transformation used in Eq. (25) will change the correlation structure very slightly. Therefore, the largest CV used is equal to 0.1 and no correction method is used in this paper. 6.1.2. Results of static stochastic finite element analysis In this example, materials are assumed to be high-strength concrete which has a lognormal mean value of elastic modulus EL ¼ 30 Gpa (4351.13 ksi). The Poisson’s ratio is set to be m ¼0.15 as a deterministic parameter. The exponential CF in Eq. (4) is adopted in this example. Four combinations of different correlation lengths and standard deviations of the random field were assumed: (1) Lc1 ¼ 1.0 m, sL1 ¼1.0 GPa; (2) Lc2 ¼1.0 m, sL2 ¼ 0.1 GPa; (3) Lc3 ¼0.5 m, sL3 ¼1.0 GPa; (4) Lc4 ¼0.5 m, sL4 ¼0.1 GPa. Fig. 11 shows realizations of spatially varying Young’s modulus for each of four different statistical distributions. One important observation in the discretization of the random field by KL expansion is the effect of correlation length (Lc) on the distribution.

Fig. 14. Stochastic responses to material uncertainty (a) displacement with mSR ¼1.201 mm, sSR ¼0.0283 mm; (b) strain with mSR ¼ 1.201  10  3, sSR ¼ 2.68  10  5; and (c) stress with mSR ¼ 36 MPa, sSR ¼ 3.64  10  3 MPa in x direction at Point A by using sL ¼1 GPa and Lc ¼ 1 m.

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As the correlation length decreases, the level of fluctuation of spatial variation apparently increases based on comparisons between Fig. 11(a) and (d). Comparing Fig. 11(a) and (c) and (b) and (d), variance effects could also be observed. Young’s modulus ranges from 30.82 GPa to 29.39 GPa in the case of sL ¼1.0 GPa (Fig. 11(a)–(c)) whereas it ranges from 30.02 GPa to 29.92 GPa in the case of sL ¼0.1 GPa (Fig. 11(b)–(d)) as expected. For any physical domain that is discretized into ndofRF, representations of the spatially varying Young’s modulus by using the different number of terms in the truncated KL expansion are shown in Fig. 12. As more terms are included in the truncated KL expansion, discretization errors in the approximate realization of random distribution tend to decrease as usual in Galerkin-based finite element approaches. Comparing random distribution of Young’s modulus with the different number of terms, it can be observed that by using more terms in computation, the approximated random field becomes more detailed. Stochastic finite element analysis results from one realization with Lc ¼0.5 m and sL ¼1.0 GPa are shown in Fig. 13. Results of the stochastic finite element analysis were verified through equilibrium and compatibility conditions. Although the structure is under uniformly distributed loading, it does not show constant stress states due to material uncertainties. Fig. 13(a)–(d) indicates such non-uniform stress states. 6.1.3. Monte Carlo Simulation for analysis of statistical material response In order to demonstrate the capability of SFE for simulating the stochastic material response, MCS is performed. In this example, 1000 samples are generated by using LHS and assuming Gaussian distribution. Eq. (4) is used as the covariance function with sL ¼1 GPa and Lc ¼ 1 m. The PDFs and CDFs of the material response: (a) displacement (Ux); (b) strain (exx); (c) stress (sxx) at Point A are plotted in Fig. 14. mSR and sSR in lognormal distribution are calculated from the simulated responses (SR) under the realized material uncertainty. It can be seen that the computed analytical PDFs are reasonably comparable with the simulated PDFs. The stresses and strains at Point A in X direction are plotted in Fig. 15. Due to the spatially varying material uncertainty, stresses and strains appear to be uncorrelated in Fig. 15. However, as shown in Fig. 16, two displacement components Ux and Uy are strongly correlated due to displacement compatibility conditions. The possible locations of Point A after deformation are plotted in Fig. 16.

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Fig. 15. Probabilistic distribution of stresses and strains at the final load step and at Point A.

Fig. 16. Probabilistic distribution of the position of Point A at the final loading step.

6.2. A quarter circular tube with randomly distributed Young’s modulus In order to show the generality of the proposed SFE, a quarter circular tube model was used in this example. As shown in Fig. 17, it is notable that RF mesh intersects the FE mesh. Fig. 17 shows an illustration of FE mesh intersected with RF mesh for the quarter circular tube that has inner radius of 2 m and outer radius of three meters. Displacements along the edge G1 and G2 are restrained in their normal directions. An external concentrated load of magnitude 100 N is applied at Point B in horizontal direction. The RF domain O is discretized into NRF ¼10 RF elements, resulting in a total of ndofRF ¼55 DOFs. The physical domain D is meshed into NFE ¼210 SFEs, resulting in a total of ndofFE ¼705 DOFs. The exponential CF in Eq. (4) is used for the second-order statistical feature of the random field. Lognormal distribution described in Section 6.1 is used for generating multiple realizations of Young’s modulus distribution. Young’s modulus of the material are assumed to have EL ¼ 0:26 MPa (lognormal mean), sL ¼0.026 MPa (standard deviation), Lc ¼ 1 m and m ¼0.3 (Poisson ratio). Young’s modulus distribution from one realization is shown in Fig. 18. One

Fig. 17. Finite element mesh intersected with random field mesh for the quarter circular tube model.

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thousand samples are generated for MCS to obtain the stochastic material response. The stochastic displacement in X direction at Point A is plotted in Fig. 19.

6.3. A retaining wall with randomly distributed Young’s modulus To demonstrate potentials of the proposed approach in dealing with large scale complex engineering problems, a retaining wall is

used as a more realistic example shown in Fig. 20. The geometry, boundary condition, RF mesh, and FE mesh are presented. The RF domain O is discretized into NRF ¼184 RF elements, resulting in a total of ndofRF ¼811 DOFs. The physical domain D is meshed into NFE ¼2962 SFEs, resulting in a total of ndofFE ¼9179 DOFs. As shown in Fig. 20, both of the RF and FE mesh use the free mesh control. Displacements along edge G1 is restrained in X and Y directions. The lateral earth pressure is applied on edge G2 with the magnitude of 60 MPa. The exponential CF in Eq. (4) is still applied here. Young’s modulus of the material are assumed to be EL ¼ 30 GPa (lognormal mean), sL ¼3 GPa (standard deviation), Lc ¼2 m and m ¼0.15 (Poisson ratio). Fig. 21 shows the deformation of the right edge predicted from MCS with 2000 samples. The red dash line presents the predicted deformation when the Young’s modulus on every material point has the same value as the assumed mean.

7. Conclusions

Fig. 18. The lognormal Young’s modulus spatial distribution from one realization.

In this paper, a stochastic finite element (SFE) was presented along with implementations into a general-purpose finite element analysis program ABAQUS in order to simulate the probabilistic response of structures with material uncertainties. Discretization and quantification of random fields associated with material uncertainties are accomplished through the Karhunen–Loe ve (KL) expansion. In particular, issues on the separation of RF mesh from FE mesh have been addressed by suggesting a general mapping– interpolation method for stochastic finite element simulations.

Fig. 19. Stochastic displacement response from simulation at Point A in X direction with mSR ¼  0.0715 m, sSR ¼0.006 m by using sL ¼ 0.026 MPa and Lc ¼ 1 m.

Fig. 20. The problem definition of a retaining wall with randomly distributed Young’s modulus.

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References

Fig. 21. Deformation of the right edge AB predicted from Monte Carlo simulation with 2000 samples (the deformation is magnified by six times).

Further applications of the presented SFE computation techniques in ABAQUS/standard element library will significantly impact research and practices in the field of probabilistic computational mechanics. In the numerical demonstrations, Young’s modulus in twodimensional structures is considered to be a spatially varying random field. Galerkin finite element technique was used to solve the homogeneous Fredholm integral equation. After the material distribution at RF nodal points is obtained, the general mapping– interpolation method is employed to calculate the material property distribution at FE nodal points. Numerical examples that include a square plate, a quarter circular tube, and a retaining wall were presented to show the flexibility and fidelity of the SFE analysis. The probabilistic analyses were also carried out by using MCS and Latin hypercube sampling technique. Through numerical demonstrations, we could draw several conclusions as follows: (1) the proposed general mapping–interpolation method completely separates the RF mesh from the FE mesh. By using this method, the discretized random field, which is distributed at the nodal points in RF mesh, is seamlessly and accurately transformed to the Gaussian points in FE mesh. Therefore, different criteria for these two different meshes can flexibly be applied; (2) varying the number of truncated terms in KL expansion, random fields could be efficiently approximated; (3) MCS showed that material uncertainties resulted in reasonable probabilistic distributions of the structural response. Although problems presented in this paper are defined in an isotropic linear elastic regime, various potential applications of SFE are expected. It is able to be extended to simulate stochastic responses under uncertainties related to plastic behavior, damage, and fatigue mechanisms of anisotropic materials on a basis of different formulations in the UEL. Furthermore, it provides a promising model to be used in stochastic inverse analysis that is capable of identifying and characterizing the uncertainty in material properties on a basis of experimental tests.

Acknowledgment This research is supported by the New Faculty Startup Fund from the University of Akron. The authors are grateful for this support.

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