generalized neumann expansion and its application in...
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 325025, 13 pageshttp://dx.doi.org/10.1155/2013/325025
Research ArticleGeneralized Neumann Expansion and Its Application inStochastic Finite Element Methods
Xiangyu Wang,1 Song Cen,1,2,3 and Chenfeng Li4
1 Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China2High Performance Computing Center, School of Aerospace, Tsinghua University, Beijing 100084, China3 Key Laboratory of Applied Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China4College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK
Correspondence should be addressed to Song Cen; [email protected]
Received 18 July 2013; Accepted 13 August 2013
Academic Editor: Zhiqiang Hu
Copyright ยฉ 2013 Xiangyu Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertainsystems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easyto implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator isproposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solutionis sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationshipbetween the GNEmethod, the perturbationmethod, and the standard Neumann expansionmethod is also discussed. Based on thelinks between these three methods, quantitative error estimations for the perturbationmethod and the standard Neumannmethodare obtained for the first time in the probability context.
1. Introduction
Over the past few decades, the stochastic finite elementmethod (SFEM) has attracted increasing attention from bothacademia and industries. The objective of SFEM is to analyzeuncertainty propagation in practical engineering systems thatare affected by various random factors, for example, hetero-geneous materials and seismic loading. The current researchof SFEM has two main focuses: probabilistic modeling ofrandom media and efficient numerical solution of stochasticequations. Properties of heterogeneous materials (e.g., rocksand composites) vary randomly through the medium andare generally modeled with stochastic fields. Various proba-bilistic methods [1โ7] have been developed to quantitativelydescribe the random material properties, which includethe spectral representation method [1, 2], the Karhunen-Loeฬve expansion method [3, 4], and the Fourier-Karhunen-Loeฬve expansion method [5, 6], to name a few. In thesemethods, the random material properties are represented bya linear expansion such that they can be directly insertedinto the governing equations of the physical system, which
facilitates further numerical analysis.The resulting equationsare stochastic equations, whose solution is the second mainfocus of SFEM research and also the focus of this work.
For static and steady-state problems, the resulting SFEMequations have the following form [8โ13]:
(A0+ A1๐ผ1+ A2๐ผ2+ โ โ โ + A
๐๐ผ๐) u (๐) = f (๐) , (1)
where A1, . . . ,A
๐are ๐ ร ๐ symmetric and deterministic
matrices, u(๐) the unknown stochastic nodal vector, and f(๐)the nodal force vector. If the material properties are modeledas Gaussian fields, then ๐ผ
1, ๐ผ2, . . . , ๐ผ
๐are independent Gaus-
sian random variables.The uncertainties from the right-handside of (1) are relatively easy to handle because they are notdirectly coupled with the random matrix from the left-handside.
A number of methods have been developed for thesolution of (1), including Monte Carlo methods, the per-turbation method, the Neumann expansion method, andvarious projection methods. The Monte Carlo methods [14โ19] first generate a number of samples of the randommaterialproperties, then solve each sample with the deterministic
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2 Mathematical Problems in Engineering
FEM solver, and finally evaluate the statistics from thesedeterministic solutions. The Monte Carlo methods are sim-ple, flexible, friendly to parallel computing, and not restrictedby the number of random variables [20]. The computationalcost of Monte Carlo methods is generally high especiallyfor large-scale problems, although there exist a number ofefficient sampling techniques such as important sampling[14, 19], subset simulation [15, 17], and line sampling [18, 19].Owning to its generosity and accuracy when sufficient sam-pling can be ensured, the Monte Carlo method is often takenas the reference method to verify other SFEM techniques.The perturbation method [11, 21โ25] expresses the stiffnessmatrix, the load vector, and the unknown nodal displacementvector in terms of Taylor expansion with respect to theinput randomvariables, and the Taylor expansion coefficientscorresponding to the unknown nodal displacement are thensolved by equating terms of the same order from both sidesof (1). Its accuracy decreases when the random fluctuationincreases, and there has not been a quantitative conclusionfor the relationship between the solution error and the levelof perturbation, making it difficult to confidently deploy theperturbation method in practical problems. The Neumannexpansion method [9, 26, 27] is an iterative process for thesolution of linear algebraic equations. Because the spectralradius of the iterative matrix must be smaller than one,it is often considered not suitable for problems with largerandom variations. However, the fluctuation range allowedby the Neumann expansion method is generally higher thanthe perturbation methods, because it is easier to adopthigher order expansions in the former one. It should benoted that the stationary iterative procedure employed inNeumann expansion is not as efficient as the precondi-tioned conjugate gradient procedure employed in MonteCarlo methods. Therefore, the computational efficiency ofthe Neumann expansion method is inferior to direct MonteCarlo simulations. In the projection methods [3, 4, 12, 28โ35], the unknown nodal displacements are projected ontoa set of orthogonal polynomials or other orthogonal bases,after which the Galerkin approach is employed to solvethe projection coefficients. However, the projection methodstypically result in an equation system much larger than thedeterministic counterpart, and thus these methods are oftenlimited to cases with a small set of base functions and afew independent random variables. In order to handle prob-lems with more random variables, the stochastic collocationmethods [36โ42] have been developed, in which a reducedrandom basis is used in the projection operation and theprojection coefficients are determined by approximating theprecomputedMonte Carlo solutions. However, the stochasticcollocationmethods are believed to be less rigorous and theiraccuracy is poorer than the full projectionmethods using theGalerkin approach. Other variants of the projection proce-dures can be found in [13, 43, 44]. The fatal drawback of theprojection types of solvers is their low efficiency [45], becausethe size of the base function set grows exponentially with thenumber of randomvariables and the order of the polynomialsused. Moreover, the accuracy of the projection methods iscriticized by [46, 47]. For linear stochastic problems, a jointdiagonalization solution strategy was proposed in [8โ10], in
which the stochastic solution is obtained in a closed form byjointly diagonalizing the matrix family in (1). This method isnot sensitive to the number of randomvariables in the system,while there has not been a mathematically rigorous proof forits accuracy.
We propose a generalized Neumann expansion methodfor the solution of (1). The new GNE method is many timesmore efficient than the standard Neumann method, which iscompromised by increased memory demand. In the contextof probability, a mathematically rigorous error estimation isestablished for the new GNE scheme. It is also proved thatfor (1), the GNE scheme, the standard Neumann expansionmethod, and the perturbation method are equivalent. Hence,the stochastic error estimation obtained here also shed alight on the standard Neumannmethod and the perturbationmethod, which has remained as an outstanding challenge inthe literature.The rest of the paper is organized as follows.TheGNE scheme and its error estimation are derived in Section2, after which it is validated and examined in Section 3 via acomprehensive numerical test. Concluding remarks are givenin Section 4.
2. Generalized Neumann Expansion Method
Consider a linear elastic statics system with Youngโs modulusmodeled as a random field. By using the Karhunen-Loeฬveexpansion [3, 4] or the Fourier-Karhunen-Loeฬve expansion[5, 6], random Youngโs modulus can be expressed as
๐ธ (x, ๐) = ๐ธ0(x) + โ
๐
โ๐๐๐ผ๐(๐) ๐๐(x) , (2)
where ๐ธ(x, ๐) denotes random Youngโs modulus at point x,๐ธ0(x) is the expectation of ๐ธ(x, ๐), ๐ผ
๐(๐) are uncorrelated
random variables, and ๐๐and ๐
๐(x) are deterministic eigen
values and eigen functions corresponding to the covariancefunction of the source field ๐ธ(x, ๐). If ๐ธ(x, ๐) is modeled as aGaussian field, ๐ผ
๐(๐) become independent standard Gaussian
random variables. For the sake of simplicity, we drop therandom symbol๐ and for the rest of the paper use๐ผ
๐to denote
the random variables in (2).Following the standard finite element (FE) discretization
procedure, a stochastic system of linear algebraic equationscan be built from (2)
(A0+โ๐
๐ผ๐A๐) u = f . (3)
In the above equation, A0is the conventional FE stiffness
matrix, which is symmetric, positive definite, and sparsewhile A
๐are symmetric and sparse matrices. The determin-
istic matrices A๐share the same entry pattern as the stiffness
matrix A0, but they are usually not positive definite.
2.1. RelatedMethods. TheMonteCarlomethods first generatea set of samples of๐ผ
๐in (3), and each sample of๐ผ
๐corresponds
to a deterministic equation, which can be solved by thestandard FEMmethod.
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Mathematical Problems in Engineering 3
In the perturbation procedure, the unknown randomvector u is expanded as [21, 25, 26]
u = u0+โ๐
๐ผ๐u๐+โ๐
โ๐
๐ผ๐๐ผ๐u๐๐+ โ โ โ . (4)
Substituting (4) into (3) yields
(A0+โ๐
๐ผ๐A๐)(u
0+โ๐
๐ผ๐u๐+โ๐
โ๐
๐ผ๐๐ผ๐u๐๐+ โ โ โ ) = f ,
(5)
from which the components u0, u๐, u๐๐of u are solved as
follows:
u0= Aโ10f ,
u๐= โAโ10A๐u0,
u๐๐= Aโ10A๐Aโ10A๐u0.
(6)
It should be noted that, in the stochastic context, there has notbeen a rigorous convergence criterion for the perturbationscheme described above.
In the Neumann expansion method, the solution of (3) iswritten as
u = (A0+ ฮA)โ1f = (I + B)โ1Aโ1
0f , (7)
where ฮA = โ๐๐ผ๐A๐, B = Aโ1
0ฮA and I is the identity matrix.
According to Neumann expansion
(I + B)โ1 = I โ B + B2 โ B3 + โ โ โ , (8)
solution (7) can be simplified as
u = u0โ Bu0+ B2u
0โ B3u
0+ โ โ โ , (9)
where u0= Aโ10f . The spectral radius of matrix B must be
smaller than 1 to ensure the convergence of (9).
2.2. The Generalized Neumann Expansion Method. TheNeu-mann expansion (8) can be generalized to the inverse ofmultiple matrices:
(I +โ๐
๐ผ๐B๐)
โ1
= I โโ๐
๐ผ๐B๐+โ๐
โ๐
๐ผ๐๐ผ๐B๐B๐โ โ โ โ . (10)
The generalized Neumann expansion (GNE) in (10) can beproved as follows:
(I +โ๐
๐ผ๐B๐)(I +โ
๐
๐ผ๐B๐)
โ1
= I โโ๐
๐ผ๐B๐+โ๐
โ๐
๐ผ๐๐ผ๐B๐B๐โ โ โ โ
+ ๐ผ1B1โโ๐
๐ผ1๐ผ๐B1B๐+ โ โ โ
+ ๐ผ2B2โโ๐
๐ผ2๐ผ๐B2B๐+ โ โ โ
...
+ ๐ผ๐B๐โโ๐
๐ผ๐๐ผ๐B๐B๐+ โ โ โ
= I,
(11)
where ๐ is the total number of B๐.
The Neumann expansion (8) and the generalized Neu-mann expansion (10) are equivalent in mathematics and onlydiffer in organizing thematrix operations.This difference canlead to a significant acceleration in computing the solution of(3). Specifically, let B
๐= Aโ10A๐, and applying the GNE in (10)
to (3) gives
u = (A0+โ๐
๐ผ๐A๐)
โ1
f = (I +โ๐
๐ผ๐Aโ10A๐)
โ1
Aโ10f
= u0โโ๐
๐ผ๐Aโ10A๐u0+โ๐
โ๐
๐ผ๐๐ผ๐Aโ10A๐Aโ10A๐u0โ โ โ โ
= u0โโ๐
๐ผ๐u๐+โ๐
โ๐
๐ผ๐๐ผ๐u๐๐โ โ โ โ
(12)
in which
u๐= Aโ10A๐u0,
u๐๐= Aโ10A๐Aโ10A๐u0.
(13)
Equation (12) is the GNE solution.
2.3. Convergence Analysis and Error Estimation. The conver-gence condition of the GNE (10) is essentially the same as theNeumann expansion (8), that is, ๐(โ๐ผ
๐B๐) < 1, where ๐(โ )
denotes the spectral radius of a matrix. Hence, for the GNEsolution in (12), the convergence condition is
๐ (โ๐ผ๐ฮโ1
0A๐) < 1. (14)
It should be noted that
๐ (โ๐ผ๐ฮโ1
0A๐) = ๐ (โ๐ผ
๐Cโ1A๐Cโ1) = โ๐ผ๐C
โ1A๐Cโ1๐
2
,
(15)
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4 Mathematical Problems in Engineering
where โ โ โ๐2
denotes the spectral norm of a matrix, and thesymmetric and positive definite matrix C is defined as
A0= C2. (16)
As a type ofmatrix norm, the spectral norm has the followingproperties:
โP +Qโ๐2
โค โPโ๐2
+ โQโ๐2
,
โPQโ๐2
โค โPโ๐2โQโ๐
2
,
โโPโ๐2
= โโPโ๐2
,
(17)
where P and Q are general matrices and โ is a positiveconstant. Hence,โ๐ผ๐C
โ1A๐Cโ1๐
2
โค๐ผ1Cโ1A1Cโ1๐
2
+๐ผ2Cโ1A2Cโ1๐
2
+ โ โ โ ๐ผ๐Cโ1A๐Cโ1๐
2
,
๐ (โ๐ผ๐Cโ1A๐Cโ1)
โค ๐ (๐ผ1Cโ1A1Cโ1) + ๐ (๐ผ
2Cโ1A2Cโ1)
+ โ โ โ ๐ (๐ผ๐Cโ1A๐Cโ1) ,
๐ (โ๐ผ๐ฮโ1
0A๐)
โค ๐ (๐ผ1ฮโ1
0A1) + ๐ (๐ผ
2ฮโ1
0A2) + โ โ โ ๐ (๐ผ
๐ฮโ1
0A๐)
โค ๐max (๐ผ๐)max (๐ (ฮโ1
0A๐)) .
(18)
Based on the above relation, a sufficient but not necessarycondition for convergence is
๐max (๐ผ๐)max (๐ (ฮโ1
0A๐)) < 1. (19)
The maximum spectral radius ๐max = max(๐(ฮโ1
0A๐)) can
be obtained from A0and A
๐. Equation (19) can be used
to verify the convergence of GNE in advance. When ๐ผ๐
are independent standard Gaussian random variables, theirabsolute values are smaller than 3 with the probability of99.7%. In this case, (19) can be further simplified:
๐max (๐ผ๐)max (๐ (ฮโ1
0A๐)) โ 3๐๐max < 1,
๐max <1
3๐.
(20)
Considering the ๐th-order GNE (10) or Neumann expansion(8), their error can be estimated as follows:
(I + B)โ1 โ (I โ B + B2 + โ โ โ + (โ1)๐B๐)๐
2
=B๐+1(I + B)โ1๐
2
โคB๐+1๐
2
(I + B)โ1๐
2
=๐๐+1๐ต
1 โ ๐๐ต
,
(21)
where๐๐ตis the spectral normofB, which is assumed to satisfy
๐๐ต< 1. (22)
The exact solution of (3) is
u = (I +โ๐
๐ผ๐Aโ10A๐)
โ1
Aโ10f = (I + B)โ1Aโ1
0f ,
Cu = (I +โ๐
๐ผ๐Cโ1A๐Cโ1)โ1
Cโ1f
= (I +D)โ1Cโ1f ,
(23)
where B = โ๐๐ผ๐Aโ10A๐, D = โ
๐๐ผ๐Cโ1A๐Cโ1. The ๐th-order
GNE or Neumann expansion solution is given by
uฬ = (I โ B + B2 + โ โ โ + (โ1)๐B๐)Aโ10f ,
Cuฬ = (I โD +D2 + โ โ โ + (โ1)๐D๐)Cโ1f .(24)
The norm of the solution error is defined asโCu โ Cuฬโ2
=[(I +D)
โ1 โ (I โD +D2 + โ โ โ + (โ1)๐D๐)]Cโ1f2
=D๐+1(I +D)โ1Cโ1f2 =
D๐+1Cu2
โคD๐+1๐
2
โCuโ2 = ๐๐+1
๐ทโCuโ2 = ๐
๐+1
๐ตโCuโ2,
(25)
where ๐๐ตis the spectral radius of B.
The relative error of the solution is defined as
RE =โ(u โ uฬ)๐A
0(u โ uฬ)
โu๐A0u
=โ(u โ uฬ)๐C2 (u โ uฬ)
โu๐C2u=โCu โ Cuฬโ2โCuโ2
โค ๐๐+1๐ต.
(26)
Comparing (21) and (26), it can be seen that the accuracy forthe solution vector u is even better than the accuracy for thematrix inverse (I + B)โ1.
2.4. Solutions forMore General Stochastic Algebraic Equations.For the solution of (3), the GNE solution in (12) sharesthe same form as the perturbation solution in (5). Hence,for solving (3), the GNE method, the perturbation method,and the standard Neumann expansion method are identical.The error estimation obtained in (26) also holds for theperturbation approach.
However, for the following more general stochastic alge-braic equations [26]:
(A0+โ๐
๐ผ๐A๐+โ๐
โ๐
๐ผ๐๐ผ๐A๐๐+ โ โ โ ) u
= f0+โ๐
๐พ๐f๐+โ๐
โ๐
๐พ๐๐พ๐f๐๐+ โ โ โ ,
(27)
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Mathematical Problems in Engineering 5
the three methods are not equivalent. Specifically, ap-plying the perturbation method to the above equationyields
(A0+โ๐
๐ผ๐A๐+โ๐
โ๐
๐ผ๐๐ผ๐A๐๐+ โ โ โ )
ร (u0+โ๐
๐ฝ๐u๐+โ๐
โ๐
๐ฝ๐๐ฝ๐u๐๐+ โ โ โ )
= f0+โ๐
๐พ๐f๐+โ๐
โ๐
๐พ๐๐พ๐f๐๐+ โ โ โ .
(28)
Equation (28) can be reorganized as
A0u0
+โ๐
๐ผ๐A๐u0
+โ๐
โ๐
๐ผ๐๐ผ๐A๐๐u0
โ โ โ
+โ๐
A0๐ฝ๐u๐
+โ๐
โ๐
๐ผ๐A๐๐ฝ๐u๐
+โ๐
โ๐
โ๐
๐ผ๐๐ผ๐A๐๐๐ฝ๐u๐
โ โ โ
+โ๐
โ๐
A0๐ฝ๐๐ฝ๐u๐๐+โ๐
โ๐
โ๐
๐ผ๐A๐๐ฝ๐๐ฝ๐u๐๐+โ๐
โ๐
โ๐
โ๐
๐ผ๐๐ผ๐A๐๐๐ฝ๐๐ฝ๐u๐๐โ โ โ
โ โ โ โ โ โ โ โ โ d
= f0+โ๐
๐พ๐f๐+โ๐
โ๐
๐พ๐๐พ๐f๐๐+ โ โ โ .
(29)
If ๐ผ๐= ๐ฝ๐= ๐พ๐, then the perturbation components u
0, u๐, u๐๐,
. . . , u๐1๐2โ โ โ ๐๐
of u can be solved as
A0u0= f0,
A0u๐+ A๐u0= f๐,
A0u๐๐+ A๐u๐+ A๐๐u0= f๐๐,
...
A0u๐1๐2โ โ โ ๐๐
+ A๐1
u๐2๐3โ โ โ ๐๐
+ A๐1๐2
u๐3๐4โ โ โ ๐๐
+ A๐1๐2โ โ โ ๐๐
u0= f๐1๐2โ โ โ ๐๐
.
(30)
Solving (27) with the Neumann expansion gives
Aโ1 = (A0+ ฮA)โ1
= (I + B)โ1Aโ10= (I โ B + B2 โ B3 + โ โ โ )Aโ1
0,
(31)
where B = Aโ10ฮA, ฮA = โ
๐๐ผ๐A๐+ โ๐โ๐๐ผ๐๐ผ๐A๐๐+ โ โ โ . Then
u = (I โ B + B2 โ B3 + โ โ โ )Aโ10(f0+ ฮf) , (32)
in which
ฮf = โ๐
๐พ๐f๐+โ๐
โ๐
๐พ๐๐พ๐f๐๐+ โ โ โ . (33)
Because higher order entries are included in B, the termBAโ10(f0+ ฮf) (the first-order term of the Neumann solution
in (32)) does not correspond to the first-order entries of u.Therefore, theNeumann solution in (32) and the perturbationsolution in (30) are different for the general stochasticequation (27).
Solving (27) with the GNE approach gives
u = (A0+โ๐๐
๐ผ๐1
A๐1
+โ๐2
โ๐1
๐ผ๐1
๐ผ๐2
A๐1๐2
+ โ โ โ
+โ๐๐
โ โ โ โ๐2
โ๐1
๐ผ๐1๐2โ โ โ ๐๐
A๐1๐2โ โ โ ๐๐
)
โ1
f
= (I +โ๐๐
๐ผ๐1
B๐1
+โ๐2
โ๐1
๐ผ๐1
๐ผ๐2
B๐1๐2
+ โ โ โ
+โ๐๐
โ โ โ โ๐2
โ๐1
๐ผ๐1๐2โ โ โ ๐๐
B๐1๐2โ โ โ ๐๐
)
โ1
Aโ10f
= (I +โ๐
๐ฝ๐C๐)
โ1
Aโ10f
= (I โโ๐
๐ฝ๐C๐+โ๐
โ๐
๐ฝ๐๐ฝ๐C๐C๐
โโ๐
โ๐
โ๐
๐ฝ๐๐ฝ๐๐ฝ๐C๐C๐C๐+ โ โ โ )Aโ1
0f ,
(34)
where f = f0+ โ๐๐พ๐f๐+ โ๐โ๐๐พ๐๐พ๐f๐๐+ โ โ โ , B
๐= Aโ10A๐,
B๐๐= Aโ10A๐๐, and other B terms can be similarly formed.
Matrices C๐are determined by realignment of B
๐, B๐๐, . . .
and random variables ๐ฝ๐are formed by rearrangement of
๐ผ๐1
, ๐ผ๐1๐2
, . . . , ๐ผ๐1๐2โ โ โ ๐๐
.In summary, when solving (27), the GNEmethod and the
Neumannmethod are identical, while they are not equivalentto the perturbation procedure. However, if the higher orderterms in the GNE or Neumann expansion solutions areomitted, they degenerate into the perturbation solution.
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6 Mathematical Problems in Engineering
x
y
p
Figure 1: Cross section of the random material sample.
Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method.
2.4.1. Example I. Only first-order entries are included in bothsides of (27) such that the equation becomes
(A0+โ๐
๐ผ๐A๐) u = f
0+โ๐
๐ผ๐f๐. (35)
The first-order perturbation solution is
u0= Aโ10f0,
u๐= Aโ10f๐โ Aโ10A๐u0,
u = u0+โ๐
๐ผ๐u๐
= Aโ10f0+โ๐
๐ผ๐Aโ10f๐โโ๐
๐ผ๐Aโ10A๐u0,
(36)
and the first-order GNE solution is
u = (Aโ10โโ๐
๐ผ๐Aโ10A๐Aโ10)(f0+โ๐
๐ผ๐f๐)
= Aโ10f0โโ๐
๐ผ๐Aโ10A๐u0+โ๐
๐ผ๐Aโ10f๐
โโ๐
โ๐
๐ผ๐๐ผ๐Aโ10A๐f๐.
(37)
Comparing (36) and (37), it can be seen that theGNE solutioncontains an extra second-order term โโ
๐โ๐๐ผ๐๐ผ๐Aโ10A๐f๐.
2.4.2. Example II. Up to second-order terms are included inthe left-hand side of (27), while only the constant term isincluded in the right-hand side. In this case, the equationbecomes
(A0+โ๐
๐ผ๐A๐+โ๐
โ๐
๐ผ๐๐ผ๐A๐๐) u = f
0. (38)
Denote u0= Aโ10f0, B๐= Aโ10A๐, and B
๐๐= Aโ10A๐๐; then the
first-order perturbation solution can be expressed as
u0= Aโ10f0,
u๐= โโ๐
Aโ10A๐u0= โโ๐
B๐u0,
u = u0+โ๐
๐ผ๐u๐= u0โโ๐
๐ผ๐B๐u0,
(39)
while the first-order GNE solution is
u = (I +โ๐
๐ผ๐Aโ10A๐+โ๐
โ๐
๐ผ๐๐ผ๐Aโ10A๐๐)
โ1
u0
= (I +โ๐
๐ผ๐B๐+โ๐
โ๐
๐ผ๐๐ผ๐B๐๐)
โ1
u0
= (I โโ๐
๐ผ๐B๐โโ๐
โ๐
๐ผ๐๐ผ๐B๐๐) u0.
(40)
Comparing (39) and (40), the second-order termsโโ๐โ๐๐ผ๐๐ผ๐Aโ10A๐๐u0in the GNE solution do not appear in
the perturbation solution (39).
3. Numerical Examples
3.1. Example Description. As shown in Figure 1, a plain strainproblem with an orthohexagonal cross section is consideredhere. The length of each edge of the hexagon is 0.4m. ThePoisson ratio of thematerial is set as 0.2, andYoungโsmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as
๐ (๐1, ๐2) = ๐1๐โ๐12/๐2โ๐2
2/๐2 , (41)
in which ๐1= ๐ฅ1โ ๐ฅ2and ๐2= ๐ฆ1โ ๐ฆ2are the coordinate
differences and ๐1and ๐2are adjustable constants. As shown in
Figure 2, the cross section is discretized with triangular linearelements, where each side is divided into 8 equal parts. Thereare in total 384 elements and 217 nodes.
-
Mathematical Problems in Engineering 7
Figure 2: Finite element mesh.
The relative error is defined as
RE =CuGNE โ CuMC
2CuMC
2ร 100%, (42)
inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution. Unlessstated explicitly, the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41).
The convergence condition of the GNE method has beenrigorously proved. The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters, forwhich a number of numericaltests are performed in the following subsections.
3.2. Error Estimation. The third-order GNE solution isadopted.The constants ๐
1and ๐2in (41) are varied to examine
the error estimator in (26). The constant ๐1is the variance of
the Gaussian field, and the constant ๐2controls the shape of
the covariance function. The effective correlation length ๐ฟeffof the covariance function ๐ (๐) is defined by
๐ (๐ฟeff)
๐ (0)= 0.01. (43)
For any two points in the medium, if the distance betweenthem is larger than ๐ฟeff, their material properties are onlyweakly correlated. The effective correlation length of theGaussian field defined in (41) is
๐ (๐)
๐ (0)= ๐โ๐
2/๐2 = 0.01 โ ๐ฟeff = 2.146โ๐2. (44)
In the first set of tests, the constant ๐1is varied to change the
variance of the Gaussian field. The parameter settings are
๐1= 0.01, ๐
2= 3.2๐ธ โ 2, ๐ฟeff = 0.38,
๐1= 0.04, ๐
2= 3.2๐ธ โ 2, ๐ฟeff = 0.38.
(45)
In the second set of tests, the constant ๐2is varied to change
the shape of the covariance function. The parameter settingsare
๐1= 0.01, ๐
2= 0.8๐ธ โ 2, ๐ฟeff = 0.19,
๐1= 0.01, ๐
2= 2.4๐ธ โ 2, ๐ฟeff = 0.33,
๐1= 0.01, ๐
2= 4.0๐ธ โ 2, ๐ฟeff = 0.43.
(46)
3.2.1. The Influence of ๐1. Corresponding to ๐
1= 0.01 and
๐1= 0.04, the standard deviation ๐ of the stochastic field
is 0.1 and 0.2, respectively. A total number of 4340 samplesare generated for each case. For each realization, both theMonte Carlo solution and the third-order GNE solution arecomputed. The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4, in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation. It can been seen that for both cases, the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by ๐
1.
3.2.2. The Influence of ๐2. The constant ๐
2is adjusted to take
three different values 0.8 ร 10โ2, 2.4 ร 10โ2, and 4.0 ร 10โ2,while the constant ๐
1remains unchanged. Following (41),
the effective correlation length grows with the increase of๐2. Again, 4340 samples are generated for each stochastic
field, and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed. Figure 5plots the relationship between spectral radius and the relativeerror, which verifies that, for all cases, the theoretical errorestimation in (26) holds. The changing of ๐
2, which changes
the variation frequency of the stochastic field, does not affectthe error estimation.
As indicated in (44), the effective correlation length ๐ฟeffis directly associated with the parameter ๐
2. To investigate
the influence of ๐ฟeff on the relative error, the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different ๐ฟeff cases. Comparing thethree ECDF curves at different effective correlation lengths,it is found that with the decrease of the effective correlationlength, the accuracy of the GNE solution improves.
3.3. The Relative Error for Other Types of Stochastic Fields.In the last two subsections, the error estimation analysis hastaken the Gaussian assumption; that is, all random variables๐ผ๐in (3) are assumed to be independent Gaussian random
variables. In this subsection, we investigate the relative errorof the GNE method for other types of stochastic fields.Specifically, the random variables ๐ผ
๐in (3) are assumed to
have the two-point distribution, the binomial distribution,and the ๐ฝ distribution, respectively. For the case of two-pointdistribution, values {โ0.3๐ธ
0, 0.3๐ธ0} are assigned with equal
probability, that is, 0.5. For the case of binomial distribution,ten times Bernoulli trials with the probability 0.5 are adopted,
-
8 Mathematical Problems in Engineering
โ14 โ12 โ10 โ8 โ6 โ4โ14
โ12
โ10
โ8
โ6
โ4
Spectral radius โผ
Rela
tive e
rror
โผlo
g (RE
)
Solution with c1 = 0.01log (RE) = 4 log (๐B)
4 log (๐B)
Figure 3: The relation between the relative error and the spectral radius.
โ11 โ10 โ9 โ8 โ7 โ6 โ5 โ4 โ3 โ2 โ1 0โ11
โ10
โ9
โ8
โ7
โ6
โ5
โ4
โ3
โ2
โ1
0
Solution with c1 = 0.04log (RE) = 4 log (๐B)
Spectral radius โผ
Rela
tive e
rror
โผlo
g (RE
)
4 log (๐B)
Figure 4: The relation between the relative error and the spectral radius.
and the integer values between 0 and 10 are linearly projectedonto [โ0.3๐ธ
0, 0.3๐ธ0]. For the case of ๐ฝ distribution, the
probability density function is defined as
๐ (๐ฅ | ๐, ๐) =1
๐ต (๐, ๐)๐ฅ๐โ1(1 โ ๐ฅ)๐โ1๐ผ
(0,1)(๐ฅ) , (47)
where ๐ต(๐, ๐) is a ๐ฝ function, and the filter function ๐ผ(0,1)(๐ฅ)
is defined as
๐ผ(0,1) (๐ฅ) = {
1 ๐ฅ โ (0, 1)
0 ๐ฅ โ (0, 1) .(48)
Three kinds of ๐ฝ distributions are considered, with ๐ = ๐ =0.75, ๐ = ๐ = 1, and ๐ = ๐ = 4, respectively.The interval (0, 1)is linearly mapped to [โ0.3๐ธ
0, 0.3๐ธ0].
Corresponding to these five distributions, the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7. Comparing the ECDF curves at differenteffective correlation lengths, the same trend is observed asthe Gaussian case. That is, the relative error decreases withthe drop of effective correlation length.
3.4. Efficiency Analysis. Using the same computing platform,the Monte Carlo method, the Neumann expansion method,
-
Mathematical Problems in Engineering 9
โ14 โ12 โ10 โ8 โ6 โ4
โ14
โ12
โ10
โ8
โ6
โ4
Solution with Leff = 0.19Solution with Leff = 0.33
Solution with Leff = 0.43log (RE) = 4 log (๐B)
Spectral radius โผ
Rela
tive e
rror
โผlo
g (RE
)
4 log (๐B)
Figure 5: The relation between the relative error and the spectral radius.
0 1 2 3 4 5 6 7 8 90
0.8
0.6
0.4
0.2
1
Relative error
ECD
F
Leff = 0.19
Leff = 0.33Leff = 0.43
ร10โ3
Figure 6: ECDF of the relative errors.
and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples), and thecomputation time is shown in Table 1.The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length. A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 ร 434.
If the mesh density is doubled, then the dimensionof each matrix is 1634 ร 1634. The number of randomvariables remains the same in each case. The correspondingcomputation time is shown in Table 2.
The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures. However, the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease.
4. Conclusions
Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations. The new method,
-
10 Mathematical Problems in Engineering
0 0.5 1 1.5 2 2.5Relative error
ECD
F
0
0.8
0.6
0.4
0.2
1
ร10โ5
(a) Two-point distribution
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Relative error
ECD
F
0
0.8
0.6
0.4
0.2
1
ร10โ7
(b) Binomial distribution
0 0.5 1 1.5 2 2.5 3 3.5 4
ECD
F
Relative error ร10โ60
0.8
0.6
0.4
0.2
1
(c) ๐ฝ distribution with ๐ = ๐ = 0.75
0 0.5 1 1.5 2 2.5 3 3.5
ECD
F
Relative error
0
0.8
0.6
0.4
0.2
1
ร10โ6
(d) ๐ฝ distribution with ๐ = ๐ = 1
0 1 2 3 4 5 6Relative error
ECD
F
0
0.8
0.6
0.4
0.2
1
ร10โ7
Leff = 0.19
Leff = 0.33Leff = 0.43
(e) ๐ฝ distribution with ๐ = ๐ = 4
Figure 7: ECDF of the relative error.
-
Mathematical Problems in Engineering 11
Table 1: Computation time for 4340 samples (unit: s).
Number of random variables 10 11 16 21 39
Monte Carlo 40.78 43.26 54.02 65.55 107.13
GNE
First order 0.44 0.44 0.48 0.53 0.69Second order 0.79 0.83 1.05 1.28 2.21Third order 1.30 1.43 2.29 3.58 13.33
Neumann
First order 26.74 28.96 40.43 52.23 93.90Second order 28.63 30.88 42.28 53.70 95.22Third order 30.28 32.57 44.05 55.49 97.08
Table 2: Computation time for 16340 samples (unit: s).
Number of random variables 11 16 21
Monte Carlo 6560.01 7871.31 9118.39
GNEM
First order 13.31 14.85 16.39Second order 16.06 19.23 22.50Third order 24.55 41.86 70.97
Neumann
First order 3325.30 4851.50 6017.91Second order 3639.38 5010.79 6314.89Third order 3881.43 5473.71 6609.92
termed generalized Neumann expansion, generalizes Neu-mann expansion from one matrix to multiple matrices.The GNE method is equivalent to Neumann expansion forgeneral stochastic equations, and for stochastic linear equa-tions, both methods are also equivalent to the perturbationmethod. According to this equivalence relation, a rigorouserror estimation is obtained, which is applicable to all thethree methods. Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations.
Conflict of Interests
The authors do not have any conflict of interests with thecontent of the paper.
Acknowledgments
The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181), the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080),and the National Basic Research Program of China (Projectno. 2010CB832701). The authors would also like to thankthe support from British Council and Chinese Scholarship
Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies.
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Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawi Publishing Corporation http://www.hindawi.com Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2013
ISRN Discrete Mathematics
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Differential EquationsInternational Journal of
Volume 2013