A stochastic computational method based on goal ...

ENGEO-04349; No of Pages 11 Engineering Geology xxx (2016) xxx–xxx

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A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials S.Sh. Ghorashi d, T. Lahmer d, A.S. Bagherzadeh d, G. Zi c, T. Rabczuk a,b,d,⁎ a

Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam School of Civil, Environmental and Architectural Engineering, Korea University, Republic of Korea d Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar, Germany b c

a r t i c l e

i n f o

Article history: Received 3 June 2016 Received in revised form 29 July 2016 Accepted 30 July 2016 Available online xxxx Keywords: Dual-weighted residual Error estimation Quantity of interest Mesh adaptivity Goal-oriented Random field Geological materials

a b s t r a c t Computational modeling of geological materials is challenging. Firstly, they are heterogeneous with numerous uncertainties in the input parameters and secondly, the computational cost of modeling geological structures is time consuming due to the large and different length scales involved. In this article, we propose an efficient computational method for heterogeneous geological materials based on goal oriented error estimation and adaptive mesh refinement. Instead of estimating the error in a specific norm, the proposed novel error estimation approach which is called dual-weighted residual error estimation, approximates the error with respect to the quantity of interest. The dual-weighted residual error estimation is a dual-based scheme which requires an adjoint problem. The adjoint or dual problem is described by defining the quantity of interest in a functional form. Then by solving the primal and dual problems, errors in terms of the specified quantities are calculated. In many applications in engineering geology, the material is heterogeneous. In such cases, the material properties can be regarded as random fields. This variety of material properties leads to non-uniform distributions of the solution gradients, e.g. stresses. Therefore, it is vital to apply a reliable error estimation approach to be able to do efficiently the mesh-adaptivity procedure with regard to varying material parameters with pre-defined correlation lengths. Hence, the proposed error estimator is extended by accounting for a random field model to describe the material properties. Local estimated errors are exploited in order to accomplish the mesh adaptivity procedure. The goal-oriented mesh adaptivity controls the local errors in terms of the prescribed quantities. Both refinement and coarsening processes are applied to raise the efficiency. The performance of the proposed computational approach is demonstrated for several examples. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Nowadays complicated engineering applications can be analyzed by numerical methods running on available computers. One of the most robust and reliable computational approaches is Finite Element Method (FEM). Although, many other numerical methods (Chen et al., 2012; Chau-Dinh et al., 2012; Ren et al., 2012; Vu-Bac et al., 2016; Quayum et al., 2015; Zhu et al., 2016) such as meshfree methods (Nguyen et al., 2008; Rabczuk and Belytschko, 2004; Ghorashi et al., 2011; Khazal et al., 2015; Amiri et al., 2014a, b; Rabczuk et al., 2010; Zhuang et al., 2012, 2014), isogeometric analysis method and its extensions (Hughes et al., 2005; Ghorashi et al., 2012a, b, 2015; Jia et al., 2014; Anitescu et al., 2015; Valizadeh et al., 2015; Ghasemi et al., 2015), efficient remeshing techniques (Areias et al., 2013a, b, 2014, 2015, 2016; Areias ⁎ Corresponding author at: Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail address: [email protected] (T. Rabczuk).

and Rabczuk, 2013; Nguyen-Xuan et al., 2013) or multiscale methods (Yang et al., 2015; Budarapu et al., 2014a, b; Talebi et al., 2013, 2014, 2015) have been introduced and successfully applied in different fields, FEM still plays an important role in computational mechanics field. For the special case of randomly varying material properties, the FEM can be enhanced to describe the true physical behavior of the material even more precisely by the use of random fields, a technique in particular of high importance in the field of reliability analysis. In the FEM, mesh discretization highly affects the solution accuracy and obviously the computational effort. Therefore, it is of great importance to be able to minimize the computational cost while the expected solution accuracy is gained. Mesh adaptation is a profitable approach to achieve this goal. As a criterion for mesh configuration and updating, estimation of discretization error is required. A good error estimator plays a very important role to implement an efficient refinement procedure in numerical methods. An error estimate should be performed to locate the situations of error distribution in the problem domain. The error estimation methods based on classical

http://dx.doi.org/10.1016/j.enggeo.2016.07.012 0013-7952/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: Ghorashi, S.S., et al., A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials, Eng. Geol. (2016), http://dx.doi.org/10.1016/j.enggeo.2016.07.012

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S.S. Ghorashi et al. / Engineering Geology xxx (2016) xxx–xxx

Fig. 3. Initial discretization of the plate with hole subjected to tension. Fig. 1. The plate with hole subjected to far-field uni-directional tension.

energy norm are categorized into two classes namely the residualsbased (Babuška and Rheinboldt, 1978; Amani et al., 2012, 2014) and the recovery-based (Zienkiewicz and Zhu, 1987) methods. In a residual-based error estimator, the residuals of a governing differential equation and its boundary conditions are considered as an error criteria while, the gradient of solutions is utilized in recovery-based methods (Zienkiewicz and Zhu, 1992a, b). It is practically important to be able to estimate the error in the socalled quantity of interest (QoI) rather than the global energy norm. A new type of error estimation procedure called goal-oriented error estimation (GOEE) has been proposed to estimate the error with respect to the QoI (Becker and Rannacher, 1996, 1998; Bangerth and Rannacher, 2003; Prudhomme and Oden, 1999; Stein et al., 2007; Zaccardi et al., 2013; González-Estrada et al., 2014). It results in quantifying the effect of local errors on the accuracy of the solution with respect to the specific quantities. Therefore, this methodology is so beneficial for adaptivity schemes and quality assessment in engineering applications. In this paper, the dual-weighted residual error estimation besides the conventional residual-based error estimation is applied in a threedimensional elasticity problem. Local estimated errors are exploited in order to accomplish the local mesh adaptivity, refinement/coarsening, considering hanging nodes. The simulations are carried out by using the open source FEM library, deal.II (Bangerth et al., 2007). The goal-oriented mesh adaptivity controls the local errors in terms of the prescribed quantity. The convergence rates are plotted to illustrate the superiorities of the goal-oriented mesh adaptivity methodology over the traditional methods. The rest of the paper is organized as follows: In Section 2, the goaloriented error estimation based on dual-weighted residual is described. Section 3 describes one approach to generate random fields based on triangular decompositions of the covariance matrix. In Section 4, a

numerical example considering both homogeneity and heterogeneity is investigated by applying different adaptivity strategies. Finally, some concluding remarks are outlined in Section 5. 2. Goal-oriented error estimation In engineering applications the entire solution of the problem may not be interested, but rather some certain aspects of it. This is the main idea of applying the goal-oriented error estimation (GOEE). For example, in an elasticity problem one might want to know about values of the stress at certain points to predict whether maximal load values of joints are safe. In GOEE procedure, solution of a dual/auxiliary/adjoint problem is also required. 2.1. Primal problem Let us consider the following elastic equation: −∂ j σ ij ¼ f i ;

i; j; k; l ¼ 1; 2; 3

ð1Þ

where the stress is defined as σij = cijkl ∂kul. The displacement and traction boundary conditions can be written as ui ¼ ui ;

i ¼ 1; 2; 3;

(a)

on Γd :

ð2Þ

(b)

(c) Fig. 2. Geometry and boundary conditions of the plate with hole subjected to tension.

Fig. 4. Meshes at the 4-th adaptivity step of the plate with hole subjected to tension by applying: (a) Kelly refinement, (b) residual-based adaptivity and (c) DWR adaptivity.


3


Fig. 5. Exact relative errors (%) of σav xx |S versus degrees of freedom for the plate with hole subjected to tension.

t i ¼ cijkl ∂k ul n j ;

i; j; k; l ¼ 1; 2; 3;

on Γn ;

ð3Þ Fig. 7. Geometry and boundary conditions of the square-shape problem.

where the u and t are prescribed displacement and traction imposed on the boundaries Γd and Γn, respectively. The values cijkl are the stiffness coefficients. In isotropic case, by introduction of the Lamé's parameters, λ and μ, the coefficient tensor is defined as, cijkl ¼ λδij δkl þ μ δik δjl þ δil δjk :

ð4Þ

Then, the elastic equation can be rewritten in the following form: −∂i λ∂ j u j −∂ j μ∂i u j −∂ j μ∂ j ui ¼ f i ;

i; j ¼ 1; 2; 3

ð5Þ

and its corresponding variational form of this problem is to find u ∈ V such that aðu; vÞ ¼ lðvÞ

u∈V

(a)

(b)

(c)

(d)

(e)

(f)

ð6Þ

where aðu; vÞ ¼

X

ðλ∂l ul ; ∂k vk ÞΩ þ

k;l

lðvÞ ¼

X

X k;l

ðμ∂k ul ; ∂k vl ÞΩ þ

X

ðμ∂k ul ; ∂l vk ÞΩ ð7Þ

k;l

ð f l ; vl ÞΩ

ð8Þ

l

and (a, b)Ω denotes ∫ Ω abdΩ and V is the Hilbert space n o V ¼ v∈H 1 ; v ¼ 0; on Γd :

ð9Þ

Fig. 6. Exact relative errors (%) of σav xx |S versus computational times (s) for the plate with hole subjected to tension.

Fig. 8. DWR-based mesh adaptivity of the square-shape sample under pressure and shear in homogeneous case: (a) Step 1 with 1694 DoFs, (b) Step 2 with 3274 DoFs, (c) Step 3 with 6162 DoFs, (d) Step 4 with 11,712 DoFs, (e) Step 5 with 21,984 DoFs, (f) Step 6 with 41,606 DoFs.


4


(b)

(a)

Fig. 11. Geometry and boundary conditions of the 2-material layered problem.

(c) Fig. 9. Meshes at the 4-th adaptivity step of the square-shape sample under pressure and shear in homogeneous case by applying: (a) Kelly refinement, (b) residual-based adaptivity and (c) DWR adaptivity.

2.2. Dual problem A Quantity of Interest (QoI) can be characterized as a continuous linear functional on the space of admissible functions. If the QoI functional is non-linear, it may be linearized and then be used (Zaccardi et al., 2013). Also, sometimes the linear functional may not be continuous. For instance, when we are interested in the error in the solution at a particular point in the domain. In GOEE, the evaluation of a functional of the solution, J(u), is of interest rather than the solution values, u, everywhere. Since the exact solution, u, is not available, but only its numerical approximation uh, it is sensible to check if the computed value J(uh) is within certain limits of the exact value J(u). The aim would be attaining the bounds on the error, J(e) = J(u)− J(uh). Errors in QoI are calculated by means of solving a dual problem. Let us denote the solution of the following auxiliary problem by z, aðv; zÞ ¼ J ðvÞ

∀v∈V

(a)

(b)

(c)

(d)

(e)

(f)

ð10Þ

where a(⋅, ⋅) is the bi-linear form associated with the differential equation. Then, by considering v = e = u − uh the error, we have J ðeÞ ¼ aðe; zÞ

ð11Þ

Fig. 10. Relative errors (%) of uA in homogeneous case.

Fig. 12. DWR-based mesh adaptivity of the square-shape sample under pressure and shear in 2-material layered case: (a) Step 1 with 1742 DoFs, (b) Step 2 with 3444 DoFs, (c) Step 3 with 6916 DoFs, (d) Step 4 with 13,642 DoFs, (e) Step 5 with 27,204 DoFs, (f) Step 6 with 53,592 DoFs.


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S.S. Ghorashi et al. / Engineering Geology xxx (2016) xxx–xxx 1 0.9 1.2 0.8 0.7 1.1 0.6 0.5

(a)

1

0.4

(b)

0.9 0.3 0.2 0.8 0.1 0

0.7 0

0.2

0.4

0.6

0.8

1

Fig. 15. Random field in the range [0.7, 1.3] for the square-shape problem.

(c)

The edge residual, rh, is obtained by exchanging half of the edge integral of cell K with the neighbor element K′ and considering the

Fig. 13. Meshes at the 4-th adaptivity step of the square-shape sample under pressure and shear in 2-material layered case by applying: (a) Kelly refinement, (b) residual-based adaptivity and (c) DWR adaptivity.

which it can be rewritten as follow using the Galerkin orthogonality, J ðeÞ ¼ a e; z−zh ¼ a e; eQ

ð12Þ

where zh ∈ Vh is an approximation of z which is here considered as point interpolation of the dual solution, zh =Ihz. For elasticity, the error identity reads J ðeÞ ¼

X X λ∂l el ; ∂k eQk þ μ∂k el ; ∂k eQl Ω

k;l

k;l

(b)

(c)

(d)

(e)

(f)

Ω

X μ∂k el ; ∂l eQk : þ

ð13Þ

Ω

k;l

(a)

By splitting the scalar products into terms on all elements and integrating by parts on each of them, we have J ðeÞ ¼

o X n h R ; eQ þ rh ; eQ K

K

ð14Þ

∂K

with the cell residual Rhi ¼ f i þ ∂i λ∂ j uhj þ ∂ j μ∂i uhj þ ∂ j μ∂ j uhi ;

i; j ¼ 1; 2; 3:

ð15Þ

Fig. 14. Relative errors (%) of uA in 2-material layered case (homogeneous case).

Fig. 16. DWR-based mesh adaptivity of the square-shape sample under pressure and shear in heterogeneous case: (a) Step 1 with 1722 DoFs, (b) Step 2 with 3374 DoFs, (c) Step 3 with 6472 DoFs, (d) Step 4 with 12,488 DoFs, (e) Step 5 with 23,766 DoFs, (f) Step 6 with 45,612 DoFs.


6


opposite sign of their normal vectors 8 > 1 > > < n j cijkl ∂k uhl jK −cijkl ∂k uhl jK 0 h 2 r i jΓ ¼ > 0 > > : ti − cijkl ∂k uhl n j

if Γ ⊂ ∂K j ∂Ω if Γ ⊂ Γd if Γ ⊂ Γn :

ð16Þ

As a result of the above equations, the error of the finite element discretization with respect to arbitrary (linear) functionals J(⋅) is represented. This GOEE is a weighted form of a residual estimator, where eQ = z − zh are weights indicating how important the residuals on a certain cell is for the evaluation of the given functional. Since it is an element-wise quantity, it can be used as a mesh refinement criterion. However, the GOEE requires knowledge of the dual solution z, which carries the information about the quantity we want to know to best accuracy. For this purpose, we compute the dual solution numerically, and approximate z by some numerically obtained ~z. It is noted that it is not sufficient to compute this approximation ~z using the same method as used for the primal solution uh, since then ~z−I h ~z ¼ 0, and the overall error estimate would be zero. Rather, the approximation ~z has to be from a larger space than the primal finite element space. There are various ways to obtain such an approximation. In this paper, we compute it one higher order finite element space.

Fig. 18. Relative errors (%) of uA in 2-material layered case (heterogeneous case).

2.3.2. Average stress on a surface The other quantity which we look for with the best accuracy is the mean value of a component of stress on a specified surface S, σav ij |S(i, j = 1, 2, 3). The corresponding functional, which is substituted in the right-hand side of Eq. (10), is defined as J ðvÞ ¼

1 jΓS j

Z

σ ij ðvÞdΓ:

ð18Þ

ΓS

2.3. Quantity of interest functional 3. Random fields The dual problem, Eq. (10), is defined by the functional corresponding to the quantity of interest. In this contribution two quantities including the point displacement and average value of stress on a specific surface are of interest. 2.3.1. Displacement at a point Firstly, consider displacement at a point x0, u(x0), as a quantity of interest. Therefore, the corresponding functional is defined as. J ðvÞ ¼ vðx0 Þ ¼

Z

vδðx−x0 ÞdΩ

ð17Þ

Ω

where δ is the Dirac delta function.

It has become state of the art to model the spatial variability of a material by random fields. The generated fields have in common that indeed the material properties at any point are random, however a certain correlation between neighboring points is accounted for to avoid naturally non-realistic rapid changes. There are different strategies to generate random fields, e.g. for their use in Finite Element Modeling, see (Liu et al., 1986). The technique applied in this work bases on realizations of Gaussian random fields depending on a prescribed correlation structure. We are assuming stationary Gaussian fields with given mean μ and standard deviation σ. For modeling the variability of the field, we are using a covariance matrix approach including a Gaussian model. For a two dimensional situation the correlation function is linked to the covariance matrix Cov(X,X′) by ρðτ1 ; τ 2 Þ ¼

Cov X; X 0 σ X σ X0

(b)

(a)

(c) Fig. 17. Meshes at the 4-th adaptivity step of the square-shape sample under pressure and shear in heterogeneous case by applying: (a) Kelly refinement, (b) residual-based adaptivity and (c) DWR adaptivity.

Fig. 19. Geometry and boundary conditions of concrete under pressure.


7


Fig. 20. Initial mesh considered for the problem of concrete under pressure. Fig. 22. Relative errors (%) of σav yy |S in homogeneous concrete under pressure.

where σX and σX′ are the standard deviations of at the points X and X′, which obtain, however, the same value due to the assumption of stationarity. According to a Gaussian model, the correlation function is given by

2jτ1 j 2jτ2 j ρðτ1 ; τ2 Þ ¼ exp − − θ1 θ2

which shares the Markovian assumption in each spatial direction. The parameters θ1 and θ2 are the correlation lengths and decide about the variability of the field. The computation of the field is achieved by a Choleski Decomposition of the covariance matrix. If Cov is a positive definite covariance matrix generated from a correlation function for any discretized field, then a zero mean field F can be computed by F ¼ LU where L stems from an Choleski decomposition of Cov, i.e. Cov ¼ LLT and U is a vector of independent zero mean normally distributed

1.3

0.5

(a)

(b)

(c)

0.4 1.2 0.3 0.2 1.1 0.1 0

1

−0.1 0.9 −0.2 −0.3 0.8 −0.4

(a)

(d)

(e)

−0.5 −0.5

0.7 0

0.5

(f)

Fig. 21. DWR-based mesh configuration based adaptivity of the sample with a hole under pressure in homogeneous case: (a) Step 1 with 654 DoFs, (b) Step 2 with 1312 DoFs, (c) Step 3 with 2556 DoFs, (d) Step 4 with 4854 DoFs, (e) Step 5 with 9390 DoFs, (f) Step 6 with 17,758 DoFs.

(b) Fig. 23. Random field in the range [0.7, 1.3] produced by the Gauss correlation variables: (a) [0.03, 0.01] and (b) [0.05, 0.04].


8


direction (see Fig. 1). The analytical solution of stress is given by, a2 3 3 a4 cos2θ þ cos4θ þ 4 cos4θ 2 2r r 2 a2 1 3 a4 cos2θ− cos4θ − 4 cos4θ σ y ðr; θÞ ¼ − 2 2r r 2 a2 1 3 a4 sin2θ þ sin4θ þ 4 sin4θ σ xy ðr; θÞ ¼ − 2 2r r 2 σ x ðr; θÞ ¼ 1−

Fig. 24. Relative errors (%) of σav yy |S in heterogeneous concrete (produced by random field 1) under pressure.

Regarding the problem symmetry, just a quarter of the plate with finite dimensions, as illustrated in Fig. 2, is modeled where the top and right edges are subjected to the tractions obtained from analytical stresses, as follow t ¼ σn

random variables with standard deviation of one. F is then an autocorrelated realization of the random field at a discrete set of points with covariance matrix Cov. An appropriate Matlab Package is written by (Constantine, 2010) extending the work of (Davis, 1987) which was used to generate the fields as sketched in Figs. 15 and 23. 4. Numerical examples In order to verify and show the efficiency of the proposed goal-oriented mesh adaptivity procedure and the effect of heterogeneity on the adaptivity process, several numerical examples considering homogeneity and heterogeneity are investigated. In all examples, it is assumed that the material remains in linear elastic under the imposed loading. In spite of the goal-oriented mesh adaptivity process, mesh adaptivity based on global refinement and conventional recovery- and residualbased error estimations are also performed to be able to compare the results. In the global refinement process, all elements are subdivided by 4 elements at each refinement step. In order to do h-adaptivity, a recovery-based error estimation developed by Kelly et al. (1983) and described residual-based and dual weighted residual error estimation schemes have been applied. The elements are sorted according to the error magnitude. The elements which belong to the 30% of the elements with higher errors are selected for refinement and the elements which are within the 3% of the elements with lowest errors are coarsened in the next adaptive step, i.e. four elements are replaced by a 4-times bigger element. 4.1. Plate with a hole under far-field uni-directional tension In order to verify the proposed approach and compare it with other introduced techniques, we consider firstly a plate with a centered circular hole subjected to far-field uni-directional tension, σ∞, along the x-

ð19Þ

ð20Þ

where σ is the stress tensor and n is the unit vector normal to the corresponding surface. The following parameters are considered for modeling the problem with steel material: Far-field stress σ∞ = 100 MPa, Young's modulus E = 200 GPa and Poisson's ratio ν =0.3. The average value of stress σav xx |S on the surface S which is 1/8 of the curved surface (see Fig. 2) is considered as the quantity of interest. The analytical solution is σav xx |S = 291.19 MPa. Firstly, 32 elements are used for discretization (see Fig. 7). Different strategies including global refinement and adaptivity based on estimated errors by using Kelly, residual and dual-weighted residual (goal-oriented error estimation) have been applied and investigated. Fig. 3 shows the initial configuration before adaptive mesh refinement. Each methodology leads to different element errors and therefore, the resulting meshes are different. Fig. 4 illustrates the resulting meshes at the 4-th adaptivity step of different schemes. It is seen that in the DWR adaptivity mesh concentration is around the surface S where the quantity of interest has been defined. The relative errors of the σav xx |S are depicted in Fig. 7. It is seen that the goal-oriented error estimation leads to an adaptivity process with much better convergence rate. Since for estimating the error in dual-weighted residual methodology, an auxiliary/dual problem needs to be solved, for the same degrees of freedom, more computational effort is required especially that for the current case dual problem is solved by adopting one higher order finite elements. Fig. 5 illustrates the exact relative error of σav xx versus the degrees of freedom. In order to evaluate the efficiency of the proposed goal-oriented adaptivity process, Fig. 6 depicts the exact relative error (%) of σav xx |S versus the consumed computational time (s). It is shown that for achieving very precise solution, the goal-oriented adaptivity is the most efficient method. For the rest of examples studied in this paper, the analytical solutions are not available. Therefore, the quantity of interest value calculated in the finest mesh obtained in the goal-oriented mesh adaptivity (by using dual-weighted residual error estimation) is considered as the reference solution and is applied for estimating the error of other simulations. 4.2. Point displacement evaluation of a square-shape sample under pressure and shear Consider a plane strain problem with the loading and boundary conditions shown in Fig. 7 and the average material properties of clay: Modulus of elasticity Eav ¼ 30 MPa;

Fig. 25. Relative errors (%) of σav yy |S in heterogeneous concrete (produced by random field 2) under pressure.

Poisson0 s ratio νav ¼ 0:2:

The uniform distributed 21 × 21 nodes are considered as the initial discretization. Displacement at the Point A is the quantity of interest.



(a)

(b)

(b)

(c)

(a)

4.2.1. Homogeneous case In the first investigation, material is assumed homogeneous. Displacement at the Point A in the finest mesh obtained in the goal-oriented mesh adaptivity with 532,572 DoFs is considered as the reference solution (uA = 2.99199 mm) and is applied for estimating the error of quantity of interest in other simulations. Fig. 8 shows the first six mesh configurations obtained in the goaloriented adaptive process. Fig. 9 demonstrates the mesh configuration at the 4-th adaptivity step of different strategies. It is seen that the meshes resulted by the estimated errors are so different. The DWRbased adaptivity leads to more mesh concentration around the Point A whose displacement is of interest. Relative errors (in percentage) of Point A displacement for global refinement and different mesh adaptivity strategies are illustrated in Fig. 10. Superiority of the Goal-oriented mesh adaptivity is shown compared to the other conventional mesh-adaptivity schemes. 4.2.2. 2-Materials layered case In this section, for better demonstrating the effect of material change on the mesh-adaptivity procedure, the two-material layered as shown in Fig. 11 is investigated. Mesh configurations of the first six adaptive steps obtained by applying DWR error estimation are shown in Figs. 12 and 13 depicts the resulted meshes of 4-th adaptivity step of different strategies. It is seen that dense elements are formed along the material change line and the adaptivity procedures capture it well. Among the applied methodologies, DWR-adaptivity results in mesh concentration around the Point A, too. The reference uA = 3.47691 mm is taken from the finest solution of the DWR-based adaptivity with 414,656 degrees of freedom. The relative errors are plotted in Fig. 14.

(c)

(a)

9

(b)

4.2.3. Heterogeneous case In this section, a heterogeneous material is considered. The material heterogeneity is modeled by producing a random field consisting of 101 × 101 uniformly distributed points by the correlation variables 0.95 and 0.01 in x and y directions, respectively and scaling them to [0.7, 1.3]. Random variable inside the intervals (squares) is obtained by averaging the random variables at the square vertices. Therefore, at each arbitrary point, a random value is defined. The resulting random field is shown in Fig. 15. The material properties, Eav and νav, are multiplied by this random field. The reference uA = 3.16901 mm is taken from the fine mesh, with 681,284 DoFs, obtained in the 10-th adaptivity step by using DWR error estimation. Mesh configurations of different DWR-based adaptivity steps are illustrated in Fig. 16. Fig. 17 presents the resulted meshes in the 4-th adaptivity step of the applied methodologies. It is seen that mesh obtained from the residual-based adaptivity captures the highly material changes more in comparison with the DWR adaptivity. Because the local effect of the elements around the point A on the error of the QoI is dominant in this problem. The resulting relative errors are depicted in Fig. 18. It shows the superiority of the proposed adaptivity approach over other conventional approaches when a specific quantity is of interest. 4.3. Mean stress evaluation on a surface in a square-shape sample with a hole under pressure Consider a plane strain problem with the geometry and boundary conditions shown in Fig. 19 made out of concrete with the following material properties.

(c)

Fig. 26. Meshes at the 4-th adaptivity step of the sample with a hole under pressure in homogeneous and heterogeneous cases by applying: (a) Kelly refinement, (b) residualbased adaptivity and (c) DWR adaptivity.


10


Modulus of elasticity Eav = 14 GPa, Poisson's ratio νav = 0.15, pressure P =3 MPa and density ρ =2400 kg/m. The average stress on the curved surface S, σav yy |S, is of interest. Fig. 20 illustrates the initial discretization. 4.3.1. Homogeneous case Firstly, the concrete material is assumed homogeneous. The average stress on the curved face S, σav yy |S = − 8 , 730,730 Pa, at the finest mesh (with 232,038 degrees of freedom) obtained by DWR adaptivity is considered as reference solution and is used for approximating the error of σav yy |S in other simulations. Discretization of the first six adaptivity steps applied in the DWR additivity is shown in Fig. 21. It is seen that mesh concentration is more around the surface S. The resulting relative error (in percentage) of σav yy |S for all the simulations is plotted in Fig. 22. The average convergence rate of global, Kelly-, residual-based and DWR adaptivities are 0.36, 0.40, 0.50 and 1.05, respectively. It is shown that the convergence rate of DWR adaptivity is much higher than other applied methods. 4.3.2. Heterogeneous case Since the concrete is not fully homogeneous, its heterogeneity is assumed by two random fields in the range [0.7, 1.3] (see Fig. 23) which is multiplied by Eav and νav considered in the homogeneous case. The random fields are produced by applying uniformly distributed points with 1 the horizontal and vertical 110 m distances by applying the Gauss correlation variables [0.03, 0.01] and [0.05, 0.04], respectively. At each arbitrary points, the random variable is considered as the average value of those on the square vertices surrounding it. Like before, the reference solution is obtained by calculating the average of stress on the surface S at the finest mesh obtained by DWR adaptivity. The reference solutions for random fields 1 and 2 are σav yy |S = − 8 , 597,010 and σav yy |S = − 9 , 137, 850, obtained by applying 236,466 and 236,200 degrees of freedom, respectively. The relative errors of σav yy |S for all the simulations are depicted in Figs. 24 and 25 for random fields 1 and 2, respectively. The convergence rate of adaptivity based on dual-weighted residual error estimation tends is the highest among other conventional approaches. In order to demonstrate the effect of different heterogeneities on the produced meshes applying different adaptivity approaches, mesh discretization of the fourth adaptivity step of all the simulations are presented in Fig. 26. 5. Conclusions In this paper, a goal-oriented error estimation approach called dualweighted residual has been applied in adaptive mesh refinement procedure to quantify and control the local error in quantities of interest in a homogeneous and heterogeneous media. Heterogeneity has been produced by a random-field model. Several numerical examples have been investigated and the goal-oriented adaptivity has been applied considering the quantity of point displacement and mean value of a component of stress on a specific surface. By comparing the results with the conventional recovery-based and residual-based error estimations, the efficiency and advantages of the proposed approach have been illustrated. The results show that when a specific quantity is of interest, the goal-oriented adaptivity is a competitive computational approach to calculate it with high accuracy. Material heterogeneity affects the errors and subsequently the refined meshes, but the degree of its influence depends on the problem and the quantity of interest. Acknowledgments Ghorashi, Lahmer and Rabczuk gratefully acknowledge the financial support of the German Research Foundation (DFG) through Research

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Please cite this article as: Ghorashi, S.S., et al., A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials, Eng. Geol. (2016), http://dx.doi.org/10.1016/j.enggeo.2016.07.012 Powered by TCPDF (www.tcpdf.org)

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