A Novel SFEM Algorithm Using the Probabilistic Transformation Method Abdo AbouJaoude***, Seifedine Kadry**, Khaled EL-Tawil* ***PhD student, Lebanese University, Faculty of Sciences, Beirut, Lebanon.
[email protected] """Professor, AUL university, Beirut, Lebanon,
[email protected] "'Associate Professor, Lebanese University, Faculty of Engineering, Beirut, Lebanon. Khaled_tawil@yahoo. com Abstract. The probabilistic transformation method with the Finite Element Analysis is a new technique to solve random differential equation. The advantage of this technique is finding the "exact" expression of the Probability Density Function of the solution when the Probability Density Function of the input is known. However the disadvantage is due to the characteristics (geometries and materials) of the analyzed structure included in the random differential equation. In this paper, a developed formula is used to generalize this technique by obtaining the "exact" joint probability density function of the solution in any situations, as well as the proposed technique for the non-linear case. Keywords: Probabilistic transformation method, FEA, random matrix, stochastic differential equation. PACS: 02.50.-r
INTRODUCTION For several decades, the theory of probability has been used in the theory of random differential equations to model the random properties. The solution of a Stochastic Differential Equation is obtained when evaluating the statistical characteristic of the solution process like the mean, standard deviation, high order moments and the most important characteristic "the probability density function" (p.d.f.). The probabilistic transformation method together with the finite element analysis is a new technique for evaluating the p.d.f of the random differential equation. As opposed to the stochastic finite element method or perturbation based methods, no series expansion is involved in this expression. The power of this technique is demonstrated in the context of Stochastic Differential Equation [1-2], in the reliability analysis [3] and in the random eigenvalue problem [4]. The main problem of this technique is limited and applicable only for some special cases such as non-linearity, complexity and the heterogeneous in the materials of the structure. In this article, the idea is to evaluate in exact form the joint density function of the solution of random differential equation for any complex structure, based on a developed formula that gives the p.d.f of the inverse of stiffness matrix. A technique also was proposed in the non-linear case for the classic technique.
PTM-FEM TECHNIQUE The Probabilistic Transformation Methods (PTM) evaluate the Probability Density Function (PDF) of the system output by multiplying the input PDF by the Jacobean of the inverse function. The idea of PTM is based on the following formulae[5].
fu{^)=fp{p)-VnU =fp{p)
d(p ^(M) du
Where/) is the input parameter, u is the response (solution) and (p~^{u) is the inverse transformation, which can be determined either analytically or numerically. The PTM-FEM technique is a combination between the finite element method and the probabilistic transformation method.
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NON-LINEARITY TECHNIQUE In the complex mechanical structure, we always face the problem of non linearity in the stiffness matrix. The difficulty then arises when the non-linear element is random. The method proposed in this article is to linearize the element of stiffness matrix by using the Variable Changes Technique which in turn allows us to apply the PTM-FEM. Some times the non-linearity present in the numerator of the element of stiffness matrix; for example k- =
; here the non linearity is in inertia moment /
which is supposed as a random variable and its probability density function (p.d.f) is known. To linearize / we suppose M = / ' , t h e n we apply the PTM Technique to get the p.d.f of M. we then EM apply easily the PTM to the stiffness element k- = which is linear in terms of random variable M. some times the non-linearity appears in the denominator of the term of stiffness matrix; for example EI
, , ^ . Here the random variable is the lengths (/j, /j ) of the analyzed structure. We apply
k- = —. —-I- —
VM
'2 J
the same technique and suppose for example Ki = — and N = — . Then we apply PTM to find the p.d.f of M and N again we suppose L = M + N and we find the p.d.f of L and finally we suppose T = — and we evaluate its p.d.f. In this case, the term of stiffness become linear in terms of the L random variable k- = EIT then easily we can apply the PTM to find the p.d.f of k-. GENERALIZATION OF PTM-FEM The PTM-FEM is a new powerful technique that has been developed for calculating the most important factor in probability which is the Probability Density Function as all the other probabilistic characteristics are calculated from p.d.f; if we look at the literature we won't find any method to evaluate the p.d.f in a deterministic form. For example, the SFEM, which is the famous method in this domain, can give us only the first and second moment and in some cases the moment order 3 and 4 only. Usually the inverse of stiffness matrix is non-linear specially for a complex matrix. As we see, though the PTM-FEM technique is powerful and good yet it's very difficult to generalize due to the non homogenize of the structure of stiffness matrix which in turn should be well studied due to the non homogenize of mechanical structure. The application of PTM-FEM should be studied problem by problem, i.e. each structure has its own PTM-FEM. For this reason, to generalize this technique, we should work on the whole stiffness matrix with the p.d.f of the stiffness matrix and not for the element of stiffness matrix as before. In this section, we developed a new formula which is very important in the calculation in close form of the p.d.f of stiffness matrix. In linear elastic static, the displacement formulation, by using FEM, gives the structural equilibrium in the form:
MM=M
(1)
where [.^J is the structure stiffness matrix, JL'^j is the vector of Nodal Displacements and | F | is the vector of Applied Nodal Forces. Suppose the matrix probability density function of the non-singular stiffness matrix K is given by
p^{K):IR""
-^IR.
We are interested in the p.d.f of H = K-^ e IR"'", (2) because in general from (1), U = K F and the main problem is to find the p.d.f of K . This implies that we want to obtain the joint probability density functions of all the elements of H. The elements of H are complicated non-linear functions of the elements of K. Therefore, even if the elements of K have Gaussian Distribution, the joint distribution of the elements of H will be nonGaussian. Also note that H may not have any banded structure even if K is of banded nature. The key 1506 Downloaded 09 Mar 2010 to 194.146.153.202. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/proceedings/cpcr.jsp
step to obtain the p.d.f of the inverse of a random matrix is the calculation of the Jacobean of the nonliner matrix transformation of the above equation. Now, our proposed formula is: J = \K\
-(n+l)
(3)
This expression of the Jacobean of the inverse matrix transformation plays the central role in the determination of the response of a linear stochastic system. Thereafter, we proof this formula for n= 1 and 2 then we proof it for all n. For a single-degree-of-freedom system (« = 1), Eq. (1) reduces to ku =f where k, u, f ^ -'-f^ . Suppose the probability density function (may be non-Gaussian) of the random variable k is given by Pk(k) and we are interested in deriving the p.d.f of The Jacobean of the above transformation
dh dk Using the Jacobean, the probability density function of h can be obtained using PTM technique, as follows: 11-2
II
(I
Phih) = \J\Pkik)-^ Phih) = \h\
pA-
The final step is to multiply the p.d.f of h=k"^ by the load vector which is supposed to be deterministic in this case to obtain the p.d.f of u. le- Pu(t*) = Ph(h)xfFor n=2, suppose the 2 x 2 stiffness symmetric matrix is given by
a b
K
Here a, b and c are real scalar variables which can be random. From the direct matrix inversion we get
1 ac-b"
H =K
We observe that the structure of the inverse of stiffness matrix represents complex coupled nonlinear transformations of the original (random) variables in the stiffness matrix. This is why it is difficult to say that the PTM of univariate case is always applicable so the best and the one deterministic solution to work on is the probability of the stiffness matrix itself Differentiating the upper triangular part of the A matrix, with respect to the independent elements of the K matrix one, has
J
dH dK
(ac-b^f 1
{ac-b^y 1 {ac-by
1 (ac-b^)
be -ac-b ab
-c 2bc
-ac-b^ ab
lab -a'
-b' 2ab -a'
2
2bc
be {ac-b^f
lab -a
{ac-b^f
[ - a V -a^b^c^ +2a^b^c^ +2a^b^c^ -2ab'c-2ab'c (ac-b^J
={ac-b^y^
={ac-b^y-(2+1)
2bc -b'
-ac-b ab
+ ab'c + b']
K
General Proof: The matrix differential of (2)
dH = -K'dKK'
(4)
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It is sometimes convenient when we work with matrix differentiation to rearrange the elements of an mxn matrix A={aij}, in the form on an ww-dimensional column vector. The conventional way of doing so is to successively stack the first, second,..., nth columns a-^,a2,•••,Cl„ of A under each other, giving the ww-dimensional column vector: [a^ flj... . j , the [(7 -l)n + ifh and [(/' - 1 ) « + jfh rows of G„ equal the [(7 -1)(« - j/l) + i\h row of
In{n+\)l2' ^^^
i^' ^^y
^^^ equal to the /«f«+7j/27-dimensional
row vector whose
[(7 — lj(« — 7/2)+/y/z element is 1 and whose remaining elements are 0. and (for / > 7 ), the [(7—lj(«— 7/2)+/y/z column
of
G„
is
an n^ dimensional
column
vector
whose
[(7 — Xp + ifh and [(/ — \jn + jfh elements are 1 and whose remaining n^-\ (if i=j) or n^-2 (if i>j) elements are 0. //„ is the inverse matrix of G„. The Jacobean associated with the above transformation is simply the determinant of the matrix , again by using theorem 16.4.2 of [6] one obtains:
This is the generalization of the simple case of the PTM-FEM. This expression of the Jacobean of the inverse matrix transformation plays the central role in the determination of the response of a linear stochastic system. In the series of papers of Soize [7-8], the p.d.f of stiffness matrix has a Gamma distribution and the mean of the stiffness matrix can be evaluated from the stochastic finite element method or from Monte-Carlo simulation. Once, the p.d.f of the inverse of stiffness matrix is known, then using (3) the calculation of the p.d.f of the joint probability density function of the response is straightforward.
f^{u) = \j\-'"''\Mk) CONCLUSION In this paper, the generalization of the Probabilistic Transformation Method with finite element analysis is developed. The main disadvantage of the PTM-FEM technique is applicable on a special case, especially when the analyzed structure is homogenize, i.e. when the form of the stiffness matrix is simple. So the idea of generalization is to work with the whole matrix and not with its element. The main difficulty to work with the whole stiffness matrix is in finding the p.d.f of the inverse of stiffness matrix which becomes solved by using our developed formulae.
REFERENCES [1]. M. El-Tawil, W. El-Tahhan, A. Hussein, A proposed technique of SFEM on solving ordinary random differential equation, J. Appl. Math. Comput., 35-47. 2005. [2]. S. Kadry, R. Younes. Etude Probabiliste d'un Systeme Mecanique a Parametres Incertains par une Technique Basee sur la Methode de transformation. Proceeding of CANCAM 2004. [3]. S. Kadry, A. Chateauneuf, K. El-Tawil. Random eigenvalue problem of Stochastic Systems. Proceedings of the 8th Int. Conf. on Computational Struct. Technology, Civil-Comp Press, Spam, 2006a. [4]. S. Kadry, A. Chateauneuf, K. El-Tawil. One-Dimensional Transformation Method in Reliability Analysis. Proceedings of the 8th Int. Conf. on Computational Struct. Technology, Civil-Comp Press, Spain, 2006b. [5]. A. Papoulis, Probability, Random Variables and Stochastic Processes, 4th Edition McGrawHill, Boston, USA, 2002. 1508 Downloaded 09 Mar 2010 to 194.146.153.202. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/proceedings/cpcr.jsp
[6]. A. Gupta, D. Nagar, Matrix Variate Distributions, Monographs & Surveys in Pure & Applied Mathematics, Chapman & Hall/CRC, London, 2000. [7]. C. Soize, Random matrix theory and non-parametric model of random uncertainties in vibration analysis. Journal of Sound and Vibration 263 (4) (2003) 893-916. [8]. C. Soize, Non-gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Computer Methods in Applied Mechanics and Engineering 195 (1-3) (2006) 26-64.
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