Mathematical modeling of uncertainty in computational mechanics

1. Mathematical modeling of uncertainty in computational mechanics Andrzej Pownuk Silesian University of Technology Poland andrzej@pownuk.com http://andrzej.pownuk.com

2. Schedule • Different kind of uncertainty • Design of structures with uncertain parameters • Equations with uncertain parameters • Overview of FEM method • Optimization methods • Sensitivity analysis method • Equations with different kind of uncertainty in parameters • Future plans • Conclusions

5. Rod under tension • Differential form of equilibrium equation E – Young modulus. A – area of cress-section. n – distributed load parallel to the rod, u – displacement

6. Different kind of uncertainty

7. Floating-point and real numbers - parameter e.g. Floating-point numbers

8. Uncertain parameters • Taking into account uncertainty using deterministic corrections. • Control problems • Gregorian and Julian calendar vs astronomical year(commonyears and leap years) steering wheel is necessary

9. Uncertain parameters • Semi-probabilistic methods This method is currently used in practical civil engineering applications (worst case analysis) - safety factor Some people believe that probability doesn't exist. - partial safety factor Law constraints

10. Uncertain parameters • Random parameters Using probability theory one can say that buildings are usually safe ...

11. Uncertain parameters • Bayesian probability Cox's theorem - "logical" interpretation of probability

12. Uncertain parameters • Interval parameters Interval parameter is not equivalent to uniformly distributed random variable

13. Uncertain parameters Set valued random variable Upper and lower probability

14. Uncertain parameters • Nested family of random sets

15. Uncertain parameters • Fuzzy sets Extension principle

16. Uncertain parameters • Fuzzy random variables • Random variables with fuzzy parameters Etc.

17. Design of structures with uncertain parameters

18. Design of structures • Safety condition P – load, A – area of cross-section σ – stress

19. Safe area

20. Design of structures with interval parameters

21. Design of structures with interval parameters

22. More complicated cases - design constraints

23. Design constraints

24. Geometrical safety conditions

25. Applications of united solution set • In general solution set of the design process is very complicated. • In applications usually only extreme values are needed.

26. Different solution sets • United Solution Set • Controllable Solution Set • Tolerable Solution Set

27. Example United Solution Set Tolerable Solution Set Controllable solution set

28. Example • United solution set • Tolerable solution set • Controllable solution set

29. Safety of the structures - true but not safe - unacceptable solution

30. Safety of the structures • Definition • of safe cross-section or • Definition • of safe cross-section

31. More complicated safety conditions

32. It is possible to check safety of the structure using united solution sets

33. Equations with uncertain parameters

34. Equations with uncertain parameters • Let’s assume that u(x,h) is a solution of some equation. How to transform the vector of uncertain parameters through the function u in the point x?

35. Transformation of uncertain parameters through the functionux

36. Transformation of interval parameters

37. Transformation of random parameters Transformation of probability density functions. - the PDF of the uncertain parameter h is known. PDF of the results

38. Transformation of random parameters

39. Main problem The solution ux(h) is known implicitly and sometimes it is very difficult to calculate the explicit description of the function u=ux(h).

40. Analytical solution • In a very few cases it is possible to calculate solution analytically. After that it is possible to predict behavior of the uncertain solution ux(h) explicitly. • Numerical solutions have greater practical significance than analytical one.

41. Newton method or Etc.

42. Continuation method • Continuation methods are used to compute solution manifolds of nonlinear systems. (For example predictor-corrector continuation method).

43. Many methods need the solution of the system of equations with interval parameters

44. Interval solution of the equations with interval parameters - smallest interval which contain the exact solution set.

45. Methods based on interval arithmetic • Muhanna’s method • Neumaier’s method • Skalna’s method • Popova’s method • Interval Gauss elimination method • Interval Gauss-Seidel method • etc.

46. Methods based on interval arithmetic • These methods generate the results with guaranteed accuracy • Except some very special cases it is very difficult to apply them to some real engineering problems

47. Overview of FEM method

48. Finite Element Method (FEM)

49. Real world truss structures

50. Truss structure