Andrzej Pownuk 1/138 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 2/138 Numerical example Plane stress problem in theory of elasticity

Andrzej Pownuk 3/138 Plane stress problem in theory of elasticity  - mass density, E, - material constant, - mass force.

Andrzej Pownuk 4/138 Triangular fuzzy number

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Andrzej Pownuk 6/138 Data

Andrzej Pownuk 7/138 Time of calculation Processor: AMD Duron 750 MHz RAM: 256 MB

Andrzej Pownuk 8/138 Numerical example Shell structure with fuzzy material properties

Andrzej Pownuk 9/138 Equilibrium equations of shell structures where

Andrzej Pownuk 10/138 Numerical data (  =0) L=0.263 [m], r=0.126 [m], F=444.8 [N], t=

Andrzej Pownuk 11/138 Numerical results (fuzzy displacement)  =0:  =1: u = [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 12/138 The main goal of this presentation is to describe methods of solution of partial differential equations with fuzzy parameters.

Andrzej Pownuk 13/138 Basic properties of fuzzy sets

Andrzej Pownuk 14/138 Fuzzy sets

Andrzej Pownuk 15/138 Extension principle

Andrzej Pownuk 16/138 Fuzzy equations

Andrzej Pownuk 17/138 Fuzzy algebraic equations

Andrzej Pownuk 18/138 Fuzzy differential equation (example)

Andrzej Pownuk 19/138 Definition of the solution of fuzzy differential equation

Andrzej Pownuk 20/138 Fuzzy partial differential equations

Andrzej Pownuk 21/138 Algebraic solution set United solution set Controllable solution set Tolerable solution set

Andrzej Pownuk 22/138 Remarks Buckley J.J., Feuring T., Fuzzy differential equations. Fuzzy Sets and System, Vol.110, 2000, 43-54

Andrzej Pownuk 23/138 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 24/138 Applications of fuzzy equations in computational mechanics Physical interpretations of fuzzy sets

Andrzej Pownuk 25/138 Equilibrium equations of isotropic linear elastic materials

Andrzej Pownuk 26/138 Uncertain parameters - Fuzzy loads, - Fuzzy geometry, - Fuzzy material properties, - Fuzzy boundary conditions e.t.c.

Andrzej Pownuk 27/138 Modeling of uncertainty Probabilistic methods Semi-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 28/138 Random sets interpretation of fuzzy sets

Andrzej Pownuk 29/138 Dubois D., Prade H., Random sets and fuzzy interval analysis. Fuzzy Sets and System, Vol. 38, pp , 1991 Goodman I.R., Fuzzy sets as a equivalence class of random sets. Fuzzy Sets and Possibility Theory. R. Yager ed., pp , 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 , 1994

Andrzej Pownuk 30/138 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 , 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 , 1978 Clif Joslyn, Possibilistic measurement and sets statistics. 1992

Andrzej Pownuk 31/138 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 , 1998 Tonon F., Bernardini A., A random set approach to the optimization of uncertain structures, Computers and Structures, Vol. 68, pp , 1998

Andrzej Pownuk 32/138 Random sets interpretation of fuzzy sets P 1 0.5

Andrzej Pownuk 33/138 This is not a probability density function or a conditional probability and cannot be converted to them.

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Andrzej Pownuk 35/138 Random sets Probabilistic methods Fuzzy methods Semi-probabilistic methods (interval methods) or another procedures.

Andrzej Pownuk 36/138 Design of structures with fuzzy parameters

Andrzej Pownuk 37/138 Equation with fuzzy and random parameters

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Andrzej Pownuk 39/138 General algorithm

Andrzej Pownuk 40/138 Other methods of modeling of uncertainty: - TBM model (Philip Smith). - imprecise probability (Imprecise Probability Project, Buckley, Thomas etc.). - etc.

Andrzej Pownuk 41/138 Numerical methods of solution of partial differential equations

Andrzej Pownuk 42/138 Numerical methods of solution of partial differential equations - finite element method (FEM) - boundary element method (BEM) - finite difference method (FDM) 1) Boundary value problem. 3) System of algebraic equations. 4) Approximate solution. 2) Discretization.

Andrzej Pownuk 43/138 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 44/138 Algorithm

Andrzej Pownuk 45/138

Andrzej Pownuk 46/138 - shape functions

Andrzej Pownuk 47/138 System of linear algebraic equations

Andrzej Pownuk 48/138 Approximate solution

Andrzej Pownuk 49/138 Numerical methods of solution of fuzzy partial differential equations

Andrzej Pownuk 50/138 Application of finite element method to solution of fuzzy partial differential equations. Parameter dependent boundary value problem.

Andrzej Pownuk 51/138  -level cut method The same algorithm can be apply with BEM or FDM.

Andrzej Pownuk 52/138 Computing accurate solution is NP-Hard. Kreinovich V., Lakeyev A., Rohn J., Kahl P., 1998, Computational Complexity Feasibility of Data Processing and Interval Computations. Kluwer Academic Publishers, Dordrecht We can solve these equation only in special cases.

Andrzej Pownuk 53/138 Solution set of system of linear interval equations is very complicated.

Andrzej Pownuk 54/138 Monotone functions

Andrzej Pownuk 55/138 system equations have to be solved. Sensitivity analysis If, then If, then 1+2n system of equation (in the worst case) have to be solved.

Andrzej Pownuk 56/138 Multidimensional algorithm

Andrzej Pownuk 57/138 Calculate unique sign vectors If, then Calculate unique interval solutions Calculate all interval solutions

Andrzej Pownuk 58/138 Computational complexity 1+2n system of equation (in the worst case) have to be solved. All sign vectors Unique sign vectors

Andrzej Pownuk 59/138 This method can be applied only when the relation between the solution and uncertain parameters is monotone.

Andrzej Pownuk 60/138 According to my experience (and many numerical results which was published) in problems of computational mechanics the intervals are usually narrow and the relation u=u(h) is monotone.

Andrzej Pownuk 61/138 Akpan U.O., Koko T.S., Orisamolu I.R., Gallant B.K., Practical fuzzy finite element analysis of structures, Finite Elements in Analysis and Design, 38 (2000) McWilliam S., Anti-optimization of uncertain structures using interval analysis, Computers and Structures, 79 (2000) Noor A.K., Starnes J.H., Peters J.M., Uncertainty analysis of composite structures, Computer methods in applied mechanics and engineering, 79 (2000)

Andrzej Pownuk 62/138 Valliappan S., Pham T.D., Elasto-Plastic Finite Element Analysis with Fuzzy Parameters, International Journal for Numerical Methods in Engineering, 38 (1995) Valliappan S., Pham T.D., Fuzzy Finite Analysis of a Foundation on Elastic Soil Medium. International Journal for Numerical Methods and Engineering, 17 (1993) Maglaras G., Nikolaidids E., Haftka R.T., Cudney H.H., Analytical-experimental comparison of probabilistic methods and fuzzy set based methods for designing under uncertainty. Structural Optimization, 13 (1997) 69-80

Andrzej Pownuk 63/138 Particular case - system of linear interval equations

Andrzej Pownuk 64/138

Andrzej Pownuk 65/138 Computational complexity of this algorithm p - number of independent sign vectors. 1+2p - system of equations. n - number of degree of freedom. - system of equations

Andrzej Pownuk 66/138 Calculation of the solution between the nodal points

Andrzej Pownuk 67/138 Extreme solution inside the element cannot be calculated using only the nodal solutions u. (because of the unknown dependency of the parameters) Extreme solution can be calculated using sensitivity analysis

Andrzej Pownuk 68/138 Calculation of extreme solutions between the nodal points. 1) Calculate sensitivity of the solution. (this procedure use existing results of the calculations) 2) If this sensitivity vector is new then calculate the new interval solution. The extreme solution can be calculated using this solution. 3) If sensitivity vector isn’t new then calculate the extreme solution using existing data.

Andrzej Pownuk 69/138 Numerical example Plane stress problem in theory of elasticity

Andrzej Pownuk 70/138 Plane stress problem in theory of elasticity  - mass density, E, - material constant, - mass force.

Andrzej Pownuk 71/138 Finite element method Ku=Q

Andrzej Pownuk 72/138

Andrzej Pownuk 73/138

Andrzej Pownuk 74/138 Etc.

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Andrzej Pownuk 76/138 Geometry of the problem Fuzzy parameters: Real parameters:

Andrzej Pownuk 77/138 Numerical data L=1 [m],

Andrzej Pownuk 78/138 Numerical results Fuzzy displacement Fuzzy stress

Andrzej Pownuk 79/138 Numerical example Truss structure

Andrzej Pownuk 80/138 Numerical example (truss structure)

Andrzej Pownuk 81/138 P=10 [kN] Young’s modules the same like in previous example. L=1 [m]

Andrzej Pownuk 82/138 Interval solution: axial force [N]

Andrzej Pownuk 83/138 Truss structure (Second example)

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Andrzej Pownuk 85/138 Data

Andrzej Pownuk 86/138 Time of calculation Processor: AMD Duron 750 MHz RAM: 256 MB

Andrzej Pownuk 87/138 Monotonicity tests (point tests)

Andrzej Pownuk 88/138 Monotone solutions. (Special case)

Andrzej Pownuk 89/138 - linear function.

Andrzej Pownuk 90/138 Natural interval extension

Andrzej Pownuk 91/138 Monotonicity tests If then function is monotone.

Andrzej Pownuk 92/138 High order monotonicity tests If then function is monotone.

Andrzej Pownuk 93/138 Numerical example (Reinforced Concrete Beam) Numerical result  =0:  =1:

Andrzej Pownuk 94/138 In this example commercial FEM program ANSYS was applied. Point monotonicity test can be applied to results which were generated by the existing engineering software.

Andrzej Pownuk 95/138 Taylor model

Andrzej Pownuk 96/138 Approximate interval solution

Andrzej Pownuk 97/138 Computational complexity - 1 solution of - the same matrix 1 - point solution

Andrzej Pownuk 98/138 Akapan U.O., Koko T.S., Orisamolu I.R., Gallant B.K., Practical fuzzy finite element analysis of structures. Finite Element in Analysis and Design, Vol. 38, 2001, pp

Andrzej Pownuk 99/138 Finite difference method

Andrzej Pownuk 100/138 function is monotone.If Monotonicity test based on finite difference method (1D)

Andrzej Pownuk 101/138 Monotonicity test based on finite differences and interval extension (1D) then function is monotone. If

Andrzej Pownuk 102/138 Monotonicity test based on finite difference method (multidimensional case)

Andrzej Pownuk 103/138 We can check how reliable this method is.

Andrzej Pownuk 104/138 In this procedure we don’t have to solve any equation. Monotonicity test based on finite differences and interval extension (multidimensional case)

Andrzej Pownuk 105/138 More reliable monotonicity test

Andrzej Pownuk 106/138 Subdivision

Andrzej Pownuk 107/138 If width of the interval i.e. is sufficiently small, then extreme values of the function u can be approximated by using the endpoints of given interval.

Andrzej Pownuk 108/138 Exact monotonicity tests based on the interval arithmetic

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Andrzej Pownuk 110/138 Numerical example

Andrzej Pownuk 111/138

Andrzej Pownuk 112/138 Sometimes system of algebraic equations is nonlinear. In this case we can apply interval Jacobean matrices.

Andrzej Pownuk 113/138

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Andrzej Pownuk 115/138 Regular interval matrix

Andrzej Pownuk 116/138 It can be shown that if the following interval Jacobean matrices are regular, then solutions of parameter dependent system of equations are monotone.

Andrzej Pownuk 117/138 Numerical example Uncertain parameters: E,A,J.

Andrzej Pownuk 118/138 Equilibrium equations of rod structures

Andrzej Pownuk 119/138 L=H=1 [m],P=1 [kN].

Andrzej Pownuk 120/138 Optimization methods

Andrzej Pownuk 121/138

Andrzej Pownuk 122/138 These methods can be applied to the very wide intervals Function doesn't have to be monotone.

Andrzej Pownuk 123/138 Numerical example

Andrzej Pownuk 124/138 Numerical data Analytical solution

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Andrzej Pownuk 126/138 Other methods and applications

Andrzej Pownuk 127/138 Popova, E. D., On the Solution of Parametrised Linear Systems. In: W. Kraemer, J. Wolff von Gudenberg (Eds.): Scientific Computing, Validated Numerics, Interval Methods. Kluwer Acad. Publishers, 2001, pp Muhanna L.R., Mullen L.R., Uncertainty in Mechanics. Problems - Interval Based - Approach. Journal of Engineering Mechanics, Vol. 127, No.6, 2002, pp Iterative methods

Andrzej Pownuk 128/138 Inner solutionOuter solution

Andrzej Pownuk 129/138 Valliappan S., Pham T.D., 1993, Fuzzy Finite Element Analysis of a Foundation on Elastic Soil Medium. International Journal for Numerical and Analytical Methods in Geomechanics, Vol.17, s The authors were solved some special fuzzy partial differential equations using only endpoints of given intervals. In some cases we can prove, that the solution can be calculated using only endpoints of given intervals.

Andrzej Pownuk 130/138 Load combinations in civil engineering Many existing civil engineering programs can calculate extreme solutions of partial differential equations with interval parameters (only loads) e.g: - ROBOT ( - CivilFEM ( These programs calculate all possible combinations and then calculate the extreme solutions (some forces exclude each other).

Andrzej Pownuk 131/138 Fuzzy eigenvalue problem

Andrzej Pownuk 132/138 Upper probability of the stability

Andrzej Pownuk 133/138 Random set Monte Carlo simulations In some cases we cannot apply fuzzy sets theory to solution of this problem.

Andrzej Pownuk 134/138 Conclusions

Andrzej Pownuk 135/138 Conclusions 1) Calculation of the solutions of fuzzy partial differential equations is in general very difficult (NP-hard). 2) In engineering applications the relation between the solution and uncertain parameters is usually monotone. 3) Using methods which are based on sensitivity analysis we can solve very complicated problems of computational mechanics. (thousands degree of freedom)

Andrzej Pownuk 136/138 4) If we apply the point monotonicity tests we can use results which was generated by the existing engineering software. 5) Reliable methods of solution of fuzzy partial differential equations are based on the interval arithmetic. These methods have high computational complexity. 6) In some cases (e.g. if we know analytical solution) optimization method can be applied.

Andrzej Pownuk 137/138 7) In some special cases we can predict the solution of fuzzy partial differential equations. 8) Fuzzy partial differential equation can be applied to modeling of mechanical systems (structures) with uncertain parameters.