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Numerical solutions of fuzzy partial differential equation and its application in computational mechanics

Numerical solutions of fuzzy partial differential equation and its application in computational mechanics. Andrzej Pownuk Char of Theoretical Mechanics Silesian University of Technology. Numerical example. Plane stress problem in theory of elasticity. Plane stress problem

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Numerical solutions of fuzzy partial differential equation and its application in computational mechanics

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  1. Numerical solutions of fuzzy partial differential equation and its application in computational mechanics Andrzej Pownuk Char of Theoretical Mechanics Silesian University of Technology Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  2. Numerical example Plane stress problem in theory of elasticity Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  3. Plane stress problem in theory of elasticity  - mass density, E, - material constant, - mass force. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  4. Triangular fuzzy number Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  5. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  6. Data Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  7. Time of calculation Processor: AMD Duron 750 MHz RAM: 256 MB Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  8. Numerical example Shell structure with fuzzy material properties Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  9. Equilibrium equations of shell structures where Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  10. Numerical data (=0) L=0.263 [m], r=0.126 [m], F=444.8 [N], t= Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  11. Numerical results (fuzzy displacement) =0: =1: u = -0.04102 [m]. Using this method we can obtain the fuzzy solution in one point. The solution was calculated by using the ANSYS FEM program. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  12. The main goal of this presentation is to describe methods of solution of partial differential equations with fuzzy parameters. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  13. Basic properties of fuzzy sets Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  14. Fuzzy sets Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  15. Extension principle Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  16. Fuzzy equations Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  17. Fuzzy algebraic equations Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  18. Fuzzy differential equation (example) Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  19. Definition of the solution of fuzzy differential equation Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  20. Fuzzy partial differential equations Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  21. Algebraic solution set United solution set Controllable solution set Tolerable solution set Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  22. Remarks Buckley J.J., Feuring T., Fuzzy differential equations. Fuzzy Sets and System, Vol.110, 2000, 43-54 Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  23. This derivative leads to another definition of the solution of the fuzzy differential equation. - Goetschel-Voxman derivative, - Seikkala derivative, - Dubois-Prade derivative, - Puri-Ralescu derivative, - Kandel-Friedman-Ming derivative, - etc. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  24. Applications of fuzzy equations in computational mechanics Physical interpretations of fuzzy sets Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  25. Equilibrium equations of isotropic linear elastic materials Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  26. Uncertain parameters - Fuzzy loads, - Fuzzy geometry, - Fuzzy material properties, - Fuzzy boundary conditions e.t.c. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  27. Semi-probabilistic methods Modeling of uncertainty Probabilistic methods Usually we don’t have enough information to calculate probabilistic characteristics of the structure. We need another methods of modeling of uncertainty. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  28. Random sets interpretation of fuzzy sets Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  29. Dubois D., Prade H., Random sets and fuzzy interval analysis. Fuzzy Sets and System, Vol. 38, pp.309-312, 1991 Goodman I.R., Fuzzy sets as a equivalence class of random sets. Fuzzy Sets and Possibility Theory. R. Yager ed., pp.327-343, 1982 Kawamura H., Kuwamato Y., A combined probability-possibility evaluation theory for structural reliability. In Shuller G.I., Shinusuka G.I., Yao M. e.d., Structural Safety and Reliability, Rotterdam, pp.1519-1523, 1994 Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  30. Bilgic T., Turksen I.B., Measurement of membership function theoretical and empirical work. Chapter 3 in Dubois D., Prade H., ed., Handbook of fuzzy sets and systems, vol.1 Fundamentals of fuzzy sets, Kluwer, pp.195-232, 1999 Philippe SMETS, Gert DE COOMAN, Imprecise Probability Project, etc. Nguyen H.T., On random sets and belief function, J. Math. Anal. Applic., 65, pp.531-542, 1978 Clif Joslyn, Possibilistic measurement and sets statistics. 1992 Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  31. Ferrari P., Savoia M., Fuzzy number theory to obtain conservative results with respect to probability, Computer methods in applied mechanics and engineering, Vol. 160, pp. 205-222, 1998 Tonon F., Bernardini A., A random set approach to the optimization of uncertain structures, Computers and Structures, Vol. 68, pp.583-600, 1998 Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  32. 1 0.5 Random sets interpretationof fuzzy sets P Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  33. This is not a probability density function or a conditional probability and cannot be converted to them. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  34. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  35. Random sets Probabilistic methods Fuzzy methods Semi-probabilistic methods (interval methods) or another procedures. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  36. Design of structures with fuzzy parameters Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  37. Equation with fuzzy and random parameters Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  38. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  39. General algorithm Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  40. Other methods of modeling of uncertainty: - TBM model (Philip Smith). - imprecise probability (Imprecise Probability Project, Buckley, Thomas etc.). - etc. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  41. Numerical methods of solution of partial differential equations Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  42. Numerical methods of solution of partial differential equations - finite element method (FEM) - boundary element method (BEM) - finite difference method (FDM) 1) Boundary value problem. 2) Discretization. 3) System of algebraic equations. 4) Approximate solution. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  43. Finite element method Using FEM we can solve very complicated problems. These problems have thousands degree of freedom. Curtusy to ADINA R & D, Inc. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  44. Algorithm Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  45. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  46. - shape functions Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  47. System of linear algebraic equations Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  48. Approximate solution Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  49. Numerical methods of solution of fuzzy partial differential equations Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

  50. Application of finite element method to solution of fuzzy partial differential equations. Parameter dependent boundary value problem. Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk

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