بيانات الباحث

1. 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 3- دكتوراه الفلسفة في الرياضيات الهندسية كلية الهندسة-جامعة القاهرة

2. بسم الله الرحمن الرحيم Stochastic Finite Element Method (SFEM) Transformation and Spectral Approaches

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

4. 1- Objectives • Making a literature review • Proposing a new SFE technique ( FEM-RVT) to find the approximate complete solution of a SDE • Verification of this technique through some numerical applications • Using this technique to solve a problem that does not have an exact solution • Explaining the theoretical basis of the spectral SFEM approach • Verification of the results of some applications found in the literature • Introducing a new application of that approach

5. 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

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

7. System of Transformation 3-1 RVT Theory

8. 3-2 Brief Mathematical Description of FEM-RVT The problem is: 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:

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

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

11. Case 1:Uniform distribution Three cases for the distribution ofg0 : • FEM-RVT Solution ( two finite elements):

12. Let us take x=3L/4 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)

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

14. Case 3: Atx=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)

15. with SDE Fictitious output 2-Cantilever Beam with Random Load and RandomBending Rigidity • Mathematical Model

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

17. 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

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

19. 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

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

21. 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

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

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

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

25. K-L expansion of a stochastic process 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 • Any second order stochastic process can be represented as:

26. 4.3 K-L Expansion in The Framework of SFEFormulation:(Response Representation and Statistics) 4.3.1 Stochastic Finite Element Formulation of The Problem

27. Imposing B.C.s where

28. 4.3.2 Statistical Moments of The Response Vector

29. 4-4 Numerical Applications 4.4.1 Random Field For The Excitation Function (Simply supported beam under Random Load) with In this problem The response vector is

30. Statistical Moments of The Response Vector 4.4.3 Results for Exponential covariance of the excitation function

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

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

33. Results : » Results for Exponential covariance model Standard deviation of the deflection along the beam, Exponential covariance model of the bending rigidity, lcor =1.0, sEI =0.3

34. » Results for Triangular covariance model Standard deviation of the deflection along the beam, Triangular covariance model of the bending rigidity, ( lcor =2.0, sEI =0.3)

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

36. 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: With B.C.s The response vector is

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

38. »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

39. » 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

40. 5-Conclusions • Transformation SFEM 1- The proposed FEM-RVT technique gives almost exact p.d.f. for the solution process 2- FEM-RVT technique is possible when we have randomness in both the operator and the excitation of the SDE.. 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. • Spectral SFEM 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.

41. 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.

42. Thank you Any questions?????