بيانات الباحث الإ سم : عبدالله حسين زكي الدرجات العلمية : 1- بكالوريوس هندسة الطيران – جامعة القاهرة - بتقدير جيد جداً مع مرتبة الشرف 2- ماجستير في الرياضيات الهندسية كلية الهندسة-جامعة القاهرة كلية الهندسة-جامعة القاهرة 3- دكتوراه الفلسفة في الرياضيات الهندسية كلية الهندسة-جامعة القاهرة كلية الهندسة-جامعة القاهرة

Stochastic Finite Element Method (SFEM) Transformation and Spectral Approaches

Outline 1. Objectives 2. The Stochastic Finite Element Methods SFEM (Literature Review) 3. Transformation SFEM Approach (FEM-RVT Technique) 4. Spectral SFEM Approach 5. Conclusions 6. Recommendations for Future work

1 - Objectives Proposing a new SFE technique ( FEM-RVT) to find the approximate complete solution of a SDE Proposing a new SFE technique ( FEM-RVT) to find the approximate complete solution of a SDE  Explaining the theoretical basis of the spectral SFEM approach Verification of this technique through some numerical applications Verification of this technique through some numerical applications Using this technique to solve a problem that does not have an exact solution Using this technique to solve a problem that does not have an exact solution  Verification of the results of some applications found in the literature  Introducing a new application of that approach Making a literature review Making a literature review

2-The Stochastic Finite Element Methods SFEM (Literature Review) Definition   The stochastic finite element method (SFEM) is a new method to solve stochastic differential equation (SDE) using the known deterministic finite element Method (FEM) adapted to stochastic techniques. Examples of SFEM approaches  Perturbation SFEM  Spectral SFEM

3-Transformation SFEM Approach (FEM-RVT Technique) Random D.E Approximate p.d.f

System of Transformation 3-1 RVT Theory

3.2 Finite Element Formulation of a Differential Equation are the unknown d.o.f. per element.

3-2 Brief Mathematical Description of FEM-RVT are the random coefficients of the operator All processes take the form: Using our proposed technique the following steps are performed: 1-Apply the FE formulation to get: The problem is:

Fictitious outputs Fictitious outputs System of Transformation Main output 3- We introduce (n-1) fictitious random outputs 4- Using RVT theory 5- Finally the marginal p.d.f of the solution process is:

Numerical Applications 1- Cantilever Beam With Random Load With B.C. s L x z

Case 1: Uniform distribution Three cases for the distribution of g 0 :  FEM-RVT Solution ( two finite elements):

Exact p.d.f and p.d.f using FEM-RVT technique computed at x=3L/4 in case of Uniform dist. of load intensity at mid span of the beam (+Exact; F.E) x=3L/4 Let us take x=3L/4

Case 2 : Atx=3L/4 At x=3L/4 Exact p.d.f and p.d.f using FEM-RVT technique computed at x=3L/4 in case of Exponential dist. of load intensity at mid span of the beam (+Exact; -----F.E)

Case 3: Atx=3L/4 At x=3L/4 Fig. 2.8 Exact p.d.f and p.d.f using FEM-RVT Technique computed at x=3L/4 in case of Normal dist. of load intensity at mid span of the beam (+Exact; -----F.E)

2- Cantilever Beam with Random Load and Random Bending Rigidity  Mathematical Model with SDE Fictitious output

Case 1:Uniform variation: Three cases for the distributions FEM-RVT Solution :

At x=3L/4 and for: Exact P.d.f and P.d.f using FEM-RVT technique computed at x=3L/4 in case of uniform load amplitude and uniform rigidity

Case 2. Exponential-Uniform: The exact p.d.f is: FEM-RVT Solution:

At x=3L/4 and for Exact P.d.f and P.d.f using FEM-RVT technique computed at x=3L/4 in case of Exponential dist. of load amplitude and uniform rigidity

Case 3. Normal variations: FEM-RVT solution: At x=3L/4 and for

Exact P.d.f and P.d.f using FEM-RVT technique computed at x=3L/4 in case of Normal dist. of load amplitude and Normal rigidity

3- Cantilever Beam With Random Bending Rigidity Represented by Random process Deterministic excitation and random operator Consider two cases for the distribution of 

p.d.f of the tip deflection using FEM-RVT technique in case of Exponential dist. of  ( =1) in case of Exponential dist. of  ( =1)

p.d.f of the tip deflection using FEM-RVT technique in case of Normal dist. of  (  = 0,  2 =1)

4- Spectral SFEM Approach 4.1 Introduction  Second order stochastic processes (random fields) defined only by their means and covariance functions. 4-2 Karhunen-Loeve (K-L) Expansion  K-L expansion of a stochastic process  Any second order stochastic process can be represented as:

Properties of The K-L Expansion 1] Convergence of the expansion The K-L expansion is mean square convergent irrespective of the probabilistic structure of the random field being expanded, provided that it has a finite variance. 2] Uniqueness of The Expansion The random variables appearing in K-L expansion of a random field are orthonormal if and only if the orthonormal functions and the constants are representing the eigenfunctions and eigenvalues of the covariance kernel of that field. 3] Error Minimizing Property The generalized coordinate system defined by the eigenfunctions of the covariance kernel is optimal in the sense that, the mean-square error resulting from a finite representation of the process is minimized.

4.2.2 Exact Solution of The Integral Equation 1- Exponential covariance model The solution of the following integral equation is the key of spectral SFEM approach

2]Triangular Covariance Model 3] Wiener process (nonstationary Gaussian)

4.3 K-L Expansion in The Framework of SFE Formulation : (Response Representation and Statistics) Stochastic Finite Element Formulation of The Problem

Imposing B.C. s where

4.3.2 Statistical Moments of The Response Vector

4-4 Numerical Applications Random Field For The Excitation Function (Simply supported beam under Random Load) (Simply supported beam under Random Load) with In this problem The response vector is

 Statistical Moments of The Response Vector Results for Exponential covariance of the excitation function X M=2  2 *10^ 5 M=4  2 *10^ 5 M=6  2 *10^ 5 Exact  2 *10^

Variance of the deflection along the beam, Exponential Covariance model for the load (Ten finite element mesh size ).

4.4.2 Random Field For The Operator Coefficient (Cantilever Beam With Random Bending) In this problem The response vector Statistical moments for the third order Neumann expansion (P=3)

Results : Results : » Results for Exponential covariance model Standard deviation of the deflection along the beam, Exponential covariance model of the bending rigidity, l cor =1.0,  EI =0.3 Exponential covariance model of the bending rigidity, l cor =1.0,  EI =0.3

» Results for Triangular covariance model Standard deviation of the deflection along the beam, Triangular covariance model of the bending rigidity, ( l cor =2.0,  EI =0.3)

» Results for Wiener process Standard deviation of the deflection along the beam, Wiener model for the bending rigidity,  2 =1.0 Wiener model for the bending rigidity,  2 =1.0

4.4.3 Random Fields For Operator and Excitation (Simply Supported Beam With Random Bending Rigidity and Random Excitation) In this case the mathematical model is: The response vector is With B.C. s

Statistical moments for the second order Neumann expansion (P=2)

» Results for Exponential model for the excitation and Triangular model for the operator coefficient Deflection variance along the beam, Exponential covariance model for the load and Triangular covariance model for the bending rigidity

» Results for Exponential model for both the excitation and the operator coefficient Deflection variance along the beam, Exponential covariance model for both load and bending rigidity

5-Conclusions 1- The proposed FEM-RVT technique gives almost exact p.d.f. for the solution process 3- FEM-RVT technique can handle the problems that do not have exact solution and introduce a good approximate p.d.f of the solution process. 2- FEM-RVT technique is possible when we have randomness in both the operator and the excitation of the SDE.. 1- The spectral SFEM is suitable for problems in which the stochastic processes are defined only by their means and covariance functions 2- The solutions of a SDE with random excitation or random operator showed that this method is reliable. 3- A new application( SDE with random excitation and random operator) of this method illustrated the power of this method.  Spectral SFEM  Transformation SFEM

6-Recommendations 1-Using FEM-RVT technique for solving stochastic PDE. 2- Using the K-L expansion in the FEM-RVT technique to get the p.d.f of the solution process when the input processes are of the type dealt with the spectral SFEM.

Thank you Any questions?????