partial differential equation and its application in computational - PowerPoint PPT Presentation

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 29/138 http://zeus.polsl.gliwice.pl/~pownuk

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 30/138 http://zeus.polsl.gliwice.pl/~pownuk

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 31/138 http://zeus.polsl.gliwice.pl/~pownuk

Random sets interpretation of fuzzy sets 1   =  +  = ( P ) ( ) { } { } P P P   1 1 2 F F 2  = ( ) P 1 2 F ˆ    ( ) { } P H   4 4 ˆ    ( ) H { } P   3 3  = ( ) P 0.5 1 F  ˆ   { } P ( ) H   2 2  ˆ   { 1 } P ( ) H   1  P  = { } 1 P P P i  i   1 2 Andrzej Pownuk 32/138 http://zeus.polsl.gliwice.pl/~pownuk

This is not a probability density function or a conditional probability and cannot be converted to them. Andrzej Pownuk 33/138 http://zeus.polsl.gliwice.pl/~pownuk

ˆ ˆ    →   : ( ) ( ) X X I R      →   : ( ) X X R   ˆ        , ( ) ( ) X X   =    P X ([ , ]) { : ( ) [ , ]}  a b P X a b    ˆ =      ([ , ]) { : ( ) [ , ] } Pl a b P X a b    P X ([ , ]) ([ , ])  a b Pl a b  Andrzej Pownuk 34/138 http://zeus.polsl.gliwice.pl/~pownuk

ˆ ˆ    →   : ( ) ( ) H H I R Random sets   Probabilistic + −  =  =  ( ) ( ) ( ) H H H    methods ˆ ˆ ˆ Fuzzy       ( ) ( ) ... ( ) H H H    1 2 n methods ˆ ˆ ˆ  =  = =  ( ) ( ) ... ( ) H H H Semi-probabilistic    1 2 n methods or (interval methods) another procedures. Andrzej Pownuk 35/138 http://zeus.polsl.gliwice.pl/~pownuk

Design of structures with fuzzy parameters =   0 { ( ) 0 } P Pl g h P f f =  ( ) P sup h f F  : ( ) 0 h g h Andrzej Pownuk 36/138 http://zeus.polsl.gliwice.pl/~pownuk

Equation with fuzzy and random parameters    →   : ( ) , X X R   ˆ ˆ ˆ  =       →   : ( ) ( ), ( ) { : ( )}. h P h H H H I R     F + ˆ =      {( , ) : ( ( ), ( )) 0 } P f P g X H      Andrzej Pownuk 37/138 http://zeus.polsl.gliwice.pl/~pownuk

+ ˆ =      {( , ) : ( ( ), ( )) 0 } P f P g X H       =  ( ) ( ) x sup h ( ) x F F  : ( , ) 0 h g x h  + =   { } ( ) P P x x  ( ) f x F x +   =  =  ( ) ( ) ( ( )) P x dP x E x   ( ) f x F F −  Andrzej Pownuk 38/138 http://zeus.polsl.gliwice.pl/~pownuk

General algorithm ˆ ˆ    →   : ( ) ( ), H H I R   ˆ  =    ( ) { : ( )}. h P h H  F =  ( ) , L u, h f(h) u V = y = ( ) ( ), ( h ) K h u Q h g  =  ( ) ( ) y sup h ( ) g F F = : ( ) h y g h + =  =  ( ) ( ) h P sup sup y ( ) f F g F   : ( ) 0 : 0 h g h y y Andrzej Pownuk 39/138 http://zeus.polsl.gliwice.pl/~pownuk

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

Numerical methods of solution of partial differential equations Andrzej Pownuk 41/138 http://zeus.polsl.gliwice.pl/~pownuk

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 42/138 http://zeus.polsl.gliwice.pl/~pownuk

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 43/138 http://zeus.polsl.gliwice.pl/~pownuk

Algorithm     2 2 u u −  +  =   , f x     2 2   x y =    0 , u x     2 2 u u     − +  =  vd fvd     2 2   x y         u v u v    +   =  d fvd         x x y x   Andrzej Pownuk 44/138 http://zeus.polsl.gliwice.pl/~pownuk

      u v u v  = +    ( , ) a u v d         x x y x    =  ( ) l fvd    = , ( ) ( ) v V a u,v l v Andrzej Pownuk 45/138 http://zeus.polsl.gliwice.pl/~pownuk

 =  i V h  i  V n   2 1   =   =   ( ) ( ) , ( ) ( ) u x u x v x v x h i i h i i i i  =  ( ) - shape functions i x ij j Andrzej Pownuk 46/138 http://zeus.polsl.gliwice.pl/~pownuk

  = , ( ) ( ) v V a u ,v l v h h h h h System of linear algebraic equations Ku = Q =   =  ( , ), ( ) K a Q l ij i j i i Andrzej Pownuk 47/138 http://zeus.polsl.gliwice.pl/~pownuk

Approximate solution − = 1 u K Q  =   ( ) ( ) , u x u x h i i i  ( ) ( ) u h x u x Andrzej Pownuk 48/138 http://zeus.polsl.gliwice.pl/~pownuk

Numerical methods of solution of fuzzy partial differential equations Andrzej Pownuk 49/138 http://zeus.polsl.gliwice.pl/~pownuk

Application of finite element method to solution of fuzzy partial differential equations. Parameter dependent boundary value problem. =   ( ) ( ), , L x, u, h f x, h u V h F =  ( ) ( ), K h u Q h h F =  ( ), u u h h F  n ( ) u F R F Andrzej Pownuk 50/138 http://zeus.polsl.gliwice.pl/~pownuk

 -level cut method ˆ =    { : ( ) } h h h  F ˆ = =  ˆ { : ( ) ( ), } u u K h u Q h h h    =   ˆ ( ) { : } u sup u u  ( ) u F The same algorithm can be apply with BEM or FDM. Andrzej Pownuk 51/138 http://zeus.polsl.gliwice.pl/~pownuk

ˆ = =  ˆ { : ( ) ( ), } u u K h u Q h h h   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 52/138 http://zeus.polsl.gliwice.pl/~pownuk

Solution set of system of linear interval equations is very complicated.       [1,2] [2,4] [-1,1] x 1  =       [2,4] [1,2] [1,2]       x 2 2 ( ) A , B  3 ( ) hull A , B  3 3 1 3 Andrzej Pownuk 53/138 http://zeus.polsl.gliwice.pl/~pownuk

Monotone functions + = + ( ) u u h u = ( h ) u − = − ( ) u u h h − + h h − = + = − + ( ), ( ). u u h u u h Andrzej Pownuk 54/138 http://zeus.polsl.gliwice.pl/~pownuk

ˆ = =  ˆ { : ( ) ( ), } u u K h u Q h h h   ˆ  m h R  m system equations have to be solved. 2 Sensitivity analysis  u − − + + = =  If , then ( ), ( ) 0 u u h u u h  h  u − + + − = =  If , then ( ), ( ) 0 u u h u u h  h 1+2 n system of equation (in the worst case) have to be solved. Andrzej Pownuk 55/138 http://zeus.polsl.gliwice.pl/~pownuk

Multidimensional algorithm ˆ =  ( ) ( ), K h u Q h h h        ( ) ( ) ( ) u h Q h K h   = − = 0 0 0 ( ) ( ), 1,..., K h u h i m    0 0 h h h i i i ˆ  = ( ) h h mid  0         u u      = =  i i  ,..., , 1 ,..., S sign sign i n       i      h h  1 m Andrzej Pownuk 56/138 http://zeus.polsl.gliwice.pl/~pownuk

Calculate unique sign vectors  = * , 1 ,..., . S q k q      = −  . S S If ( 1 ) , then S S i j i j Calculate unique interval solutions ˆ ˆ    = −  ˆ * * * [ ( , ), ( , ( 1 ) )] u u h S u h S   i i i Calculate all interval solutions  =      ˆ ˆ * { 1 ,..., }, { 1 ,..., }, i n j k u u i j Andrzej Pownuk 57/138 http://zeus.polsl.gliwice.pl/~pownuk

Computational complexity 1 +2 n system of equation (in the worst case) have to be solved.  s i Unique All sign sign vectors vectors     u u 1 1 ...           * S S h h   1 1 1     m ... ... ...   ... ...        u u u         =  =  =   * i ... i * S S S S       i   q  h h h 1 m   ... ...     ... ... ...           u u *     S S  n n  ... n k     h h 1 m Andrzej Pownuk 58/138 http://zeus.polsl.gliwice.pl/~pownuk

This method can be applied only when the relation between the solution and uncertain parameters u = ( h ) u is monotone. Andrzej Pownuk 59/138 http://zeus.polsl.gliwice.pl/~pownuk

According to my experience (and many numerical results which was published) in problems of computational mechanics ˆ the intervals are usually narrow h  and the relation u = u ( h ) is monotone . Andrzej Pownuk 60/138 http://zeus.polsl.gliwice.pl/~pownuk

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) 93-111 McWilliam S., Anti-optimization of uncertain structures using interval analysis, Computers and Structures, 79 (2000) 421-430 Noor A.K., Starnes J.H., Peters J.M., Uncertainty analysis of composite structures, Computer methods in applied mechanics and engineering, 79 (2000) 413-232 Andrzej Pownuk 61/138 http://zeus.polsl.gliwice.pl/~pownuk

Valliappan S., Pham T.D., Elasto-Plastic Finite Element Analysis with Fuzzy Parameters, International Journal for Numerical Methods in Engineering, 38 (1995) 531-548 Valliappan S., Pham T.D., Fuzzy Finite Analysis of a Foundation on Elastic Soil Medium. International Journal for Numerical Methods and Engineering, 17 (1993) 771-789 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 62/138 http://zeus.polsl.gliwice.pl/~pownuk

Particular case - system of linear interval equations     F F   F ... K K X Q 11 1 1 1 n       = ... ... ... ... ...             F F F ...   K K X Q     1 n nn n n ˆ ˆ ˆ          ... K K X Q 11 1 1 1 n       = ... ... ... ... ...           ˆ ˆ ˆ      ...   K K X Q     1 n nn n n Andrzej Pownuk 63/138 http://zeus.polsl.gliwice.pl/~pownuk

   = K u Q 0 0 0       u Q K   = − 0 0 0 K u 0 0    h h h i i i         u u     = i  i i  ,..., S sign sign              h h   1 m = j = * i j , where S S C   i ˆ ˆ ˆ = −  * * * i i i [ ( , ), ( , ( 1 ) )] X X h S X h S      ˆ ˆ = = * j , where X X j C   i i i Andrzej Pownuk 64/138 http://zeus.polsl.gliwice.pl/~pownuk

Computational complexity of this algorithm 1+2 p - system of equations. * i S p - number of independent sign vectors .  p  [ n 1 , ] + + [ 1 2 , 1 2 ] n - system of equations n - number of degree of freedom. Andrzej Pownuk 65/138 http://zeus.polsl.gliwice.pl/~pownuk

Calculation of the solution between the nodal points e u 6 e u 5 3 x 3 x 0 = e e e ( ) ( ) u x N x u  e e u 2 e e u 4 u 1 1 e u 3 2 x 1 x 2 Andrzej Pownuk 66/138 http://zeus.polsl.gliwice.pl/~pownuk

= e e e ( , ) ( , ) ( ) u x h N x h u h 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         e e ( , ) ( , ) x h x h u u     = e  0 0 0 0  , ... , S sign sign         h h       1 m Andrzej Pownuk 67/138 http://zeus.polsl.gliwice.pl/~pownuk

Calculation of extreme solutions between the nodal points. 1) Calculate sensitivity of the solution. (this procedure use existing results of the calculations)         e e ( , ) ( , ) x h x h u u     = e  0 0 0 0  , ... , S sign sign         h h       1 m 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 68/138 http://zeus.polsl.gliwice.pl/~pownuk

L Numerical q example 4 , E L 3 2 Plane stress problem L in theory of elasticity 1 Andrzej Pownuk 69/138 http://zeus.polsl.gliwice.pl/~pownuk

Plane stress problem in theory of elasticity E E + +  =   = 0 , , 1,2 u u f      , , +  −  2 ( 1 ) 2 ( 1 ) =    * , u u x   u  =    * , n t x      - mass density, E ,  - material constant, f - mass force.   u  = u ,    x  Andrzej Pownuk 70/138 http://zeus.polsl.gliwice.pl/~pownuk

Finite element method Ku = Q =    =   + T T  T , Q N f d N t dS , K B D B d     e  x = = ( ) ( ) , u N u x ( ) ( ) . u x N x u i ij j Andrzej Pownuk 71/138 http://zeus.polsl.gliwice.pl/~pownuk

e u 6     e x u u 5 1  = x = 1 , 3  x x u   3     u 2 2  e = ( ) ( ) u N x u x e u 2 e e u 4 u 1 1 e u 3 2 x 1 x 2   u 1   u   2       ( ) ( ) 0 ( ) 0 ( ) 0 N x N x N x u u x = 1 = 1 2 3  3   ( ) u x      ( )   0 ( ) 0 ( ) 0 ( )  u x N x N x N x u   2 1 2 3 4   u 5       u 6 Andrzej Pownuk 72/138 http://zeus.polsl.gliwice.pl/~pownuk

  3 e x u 6 = 1   x e 3 u 5 3   x 3 2 x 3    e 2 x = 1   x 2   2 1 e   x x u 2 e = e u 4 2 1   u 1 x 1 1 1 e   x u 3 2 2 x 1 x 2 Andrzej Pownuk 73/138 http://zeus.polsl.gliwice.pl/~pownuk

= + + ( ) N x a b x c x 1 2 i i i i =  ( x ) N i j ij − + − + − 2 3 3 2 2 3 3 2 ( ) ( ) x x x x x x x x x x = 1 2 1 2 2 2 1 1 1 2 ( ) N x 1  1 1 1 x x 1 2 Etc.  = 2 2 1 x x 1 2 3 3 1 x x 1 2 Andrzej Pownuk 74/138 http://zeus.polsl.gliwice.pl/~pownuk

=   T , K B D B d  e      N N N 1 2 3  0 0 0     x x x   1 1 1      N N N = 1 2 3 0 0 0 B      x x x 2 2 2         N N N N N N   1 1 2 2 3 3           x x x x x x 1 2 2 1 2 1    1 0   E   =  1 0 D −  2   −  1 1 0 0     2 Andrzej Pownuk 75/138 http://zeus.polsl.gliwice.pl/~pownuk

Geometry of the problem L Fuzzy parameters: q , , , E E E E 1 2 3 4 Real parameters: 4 , E L ,  3 , q L 2 L 1 Andrzej Pownuk 76/138 http://zeus.polsl.gliwice.pl/~pownuk

Numerical data   kN =  = 1 , q 0 . 3 .   L=1 [ m ],   m  =0  =1  ˆ  [189, 231] [ GPa ] 210 [ GPa ] 1 E ˆ  2 [189, 231] [ GPa ] 210 [ GPa ] E ˆ  3 [189, 231] [ GPa ] 210 [ GPa ] E ˆ  4 [189, 231] [ GPa ] 210 [ GPa ] E Andrzej Pownuk 77/138 http://zeus.polsl.gliwice.pl/~pownuk

Numerical results Fuzzy stress  =0  =1   y [0.96749, 0.974493] [ kPa ] 0.971063 [ kPa ] ˆ  1  y 2 [1.02833, 1.02955] [ kPa ] 1.02894 [ kPa ] ˆ   y ˆ  3 [0.98086, 1.01719] [ kPa ] 0.999086 [ kPa ]  y 4 [0.982807, 1.01914] [ kPa ] 1.00091 [ kPa ] ˆ  Fuzzy displacement  =  =  = ˆ ˆ ˆ Nr Nr Nr , 0 [ m ] , 0 [ m ] , 0 [ m ] u u u    i i i 1 [0, 0] 5 [3.2517e-14,7.49058e-13] 9 [-1.5134e-12,1.0498e-12] 2 [0, 0] 6 [3.81132e-12, 4.692e-12] 10 [8.1381e-12,9.9465e-12] 3 [0, 0] 7 [-1.5243e-12,-4.9879e-13] 11 [-3.1758e-12,-1.7949e-13] 4 [0, 0] 8 [ 4.4199e-12, 5.4275e-12 ] 12 [8.7620e-12,1.0709e-11] Andrzej Pownuk 78/138 http://zeus.polsl.gliwice.pl/~pownuk

Numerical example Truss structure Andrzej Pownuk 79/138 http://zeus.polsl.gliwice.pl/~pownuk

Numerical example (truss structure)    d du + =    0 EA n    dx dx   Boundary conditions L L  du dv =  = + ( , ) , ( ) ..., a u v EA dx l v nvdx dx dx 0 0   = , ( ) ( ) v V a u,v l v Andrzej Pownuk 80/138 http://zeus.polsl.gliwice.pl/~pownuk

P 3 P P 1 2 P =10 [ kN ] Young’s modules the same like in previous example.  = 0 . 3 L =1 [ m ] Andrzej Pownuk 81/138 http://zeus.polsl.gliwice.pl/~pownuk

Interval solution: axial force [ N ] 1 [ 3145.34, 4393.45 ] 21 [ -751.05, -742.133 ] 41 [ -194.644, -208.406 ] 61 [ 1686.62, 1641.68 ] 2 [ 1482.48, 1914.16 ] 22 [ 453.902, 470.55 ] 42 [ -2188.83, -2205.43 ] 62 [ 1528.04, 1545.77 ] 3 [ -172.138, -221.845 ] 23 [ -1417.47, -1433.55 ] 43 [ 275.268, 294.73 ] 63 [ -343.334, -358.339 ] 4 [ 164.454, 279.737 ] 24 [ 6437.89, 6417.04 ] 44 [ -7448.38, -7428.59 ] 64 [ 2470.18, 2524.72 ] 5 [ -958.619, -936.417 ] 25 [ -7444.75, -7432.58 ] 45 [ -194.644, -208.406 ] 65 [ -947.416, -949.597 ] 6 [ 2459.35, 2536.53 ] 26 [ -200.408, -202.065 ] 46 [ 6417.52, 6439.45 ] 66 [ 253.654, 185.319 ] 7 [ 1527.83, 1546.14 ] 27 [ -2196.2, -2197.33 ] 47 [ 451.658, 473.02 ] 67 [ 1683.18, 1701.27 ] 8 [ -343.544, -357.966 ] 28 [ 283.42, 285.763 ] 48 [ -1419.72, -1431.08 ] 68 [ -188.192, -202.832 ] 9 [ 1708.72, 1617.27 ] 29 [ 4020.01, 4013.59 ] 49 [ -738.486, -755.954 ] 69 [ 3683.74, 3761.16 ] 10 [ -840.883, -841.035 ] 30 [ -200.408, -202.065 ] 50 [ -166.773, -171.028 ] 11 [ 1132.62, 1189.25 ] 31 [ -9461.8, -9431.91 ] 51 [ 4242.96, 4244.56 ] 12 [ 1532.73, 1547.37 ] 32 [ 3589.87, 3583.79 ] 52 [ 1655.57, 1672.95 ] 13 [ -338.641, -356.736 ] 33 [ -3488.96, -3478.74 ] 53 [ -215.805, -231.149 ] 14 [ 3028.51, 2962.81 ] 34 [ 713.715, 704.035 ] 54 [ -266.518, -258.031 ] 15 [ -932.071, -929.76 ] 35 [ 4929.89, 4924.37 ] 55 [ -930.146, -931.887 ] 16 [ -278.358, -245.009 ] 36 [ 720.439, 696.638 ] 56 [ 3007.62, 2985.78 ] 17 [ 1656.79, 1671.62 ] 37 [ 3580.36, 3594.25 ] 57 [ 1531.23, 1549.04 ] 18 [ -214.586, -232.489 ] 38 [ -3482.95, -3485.36 ] 58 [ -340.144, -355.068 ] 19 [ 4264.06, 4221.36 ] 39 [ -9466.06, -9427.23 ] 59 [ 1144.66, 1176 ] 20 [ -169.222, -168.335 ] 40 [ 4010.55, 4024 ] 60 [ -839.969, -841.95 ] Andrzej Pownuk 82/138 http://zeus.polsl.gliwice.pl/~pownuk

Truss structure (Second example) Andrzej Pownuk 83/138 http://zeus.polsl.gliwice.pl/~pownuk

n L L L L L P P E i , ˆ  A Andrzej Pownuk 84/138 http://zeus.polsl.gliwice.pl/~pownuk

Data ˆ =  = [ 189 , 231 ] [ ], 0, E GPa  ˆ =  = [ 210 , 210 ] [ ], 1, E GPa  L = 1 m [ ], A = 2 0 . 0001 [ ], m ˆ =  = [ 9 , 11 ] [kN], 0, P  ˆ =  = [ 10 , 10 ] [kN], 1. P  Andrzej Pownuk 85/138 http://zeus.polsl.gliwice.pl/~pownuk

Time of calculation n DOF Elements Time 200 804 1000 00:02:38 300 1204 1500 00:08:56 400 1604 2000 00:20:46 500 2004 2500 00:39:45 Processor: AMD Duron 750 MHz RAM: 256 MB Andrzej Pownuk 86/138 http://zeus.polsl.gliwice.pl/~pownuk

Monotonicity tests (point tests) Andrzej Pownuk 87/138 http://zeus.polsl.gliwice.pl/~pownuk

Monotone solutions. (Special case)  = = α ( ) Ku Q h j h j j =     ( h ) , Q h R i ij j ij j    1  j   Q = = = α ... const    j h    j   nj Andrzej Pownuk 88/138 http://zeus.polsl.gliwice.pl/~pownuk

 K = 0  h j      u Q K   − − = − = = 1 1 α K q K const      j h h h   j j j u = - linear function. ( h ) u Andrzej Pownuk 89/138 http://zeus.polsl.gliwice.pl/~pownuk

Natural interval extension = − 2 ( ) , f x x x 2 − ˆ = ˆ ˆ ˆ ( ) f x x x ˆ − = −  − − − = ([ 1 , 2 ]) [ 1 , 2 ] [ 1 , 2 ] [ 1 , 2 ] f = − + − = − [ 2 , 4 ] [ 2 , 1 ] [ 4 , 5 ] −  1 − = ([ 1 , 2 ]) , 2 f     4 ˆ  ˆ ˆ ( ) ( ) f x f x Andrzej Pownuk 90/138 http://zeus.polsl.gliwice.pl/~pownuk

Monotonicity tests   m  2  ( ) ( ) ( ) h h h u u u = + − 0 0 0 ( ) h h     j j h h h h = i i i j 1 j If ˆ   m  ˆ 2  ( ) ( ) ( ) h h h u u u ˆ   = + − 0 0 0 0 ( ) h h      j j h h h h = i i i j 1 j then function u = ( h ) u is monotone. Andrzej Pownuk 91/138 http://zeus.polsl.gliwice.pl/~pownuk

High order monotonicity tests   m  m m  2 2   ( ) ( ) ( ) 1 ( ) u h u h u h u h = + − 0 + − 0 − 0 + 0 0 0 ( ) ( )( ) ... h h h h h h     j j   j j k k 2 h h h h h h = i i i j i j 1 j j k If ˆ   m  ˆ 2  ( ) ( ) ( ) h h h u u u ˆ   = + − + 0 0 0 0 ( ) ... h h      j j h h h h = i i i j 1 j then function u = ( h ) u is monotone. Andrzej Pownuk 92/138 http://zeus.polsl.gliwice.pl/~pownuk

Numerical example (Reinforced Concrete Beam) Data Concrete Steel Geometry     4 5 a =     0 . 127 m E 1.3,1.5 10 MPa E 2 . 0 , 2 . 2 10 MPa    ct =   b = 0 MPa 0 . 2 , 0 . 3 0 . 152 m  = 2 A = 0 0.019 m Numerical result     −  =0:   4 0 . 182 , 0 . 200 10 u x m 2     −  =1:   4 0 . 190 , 0 . 190 10 u x m 2 Andrzej Pownuk 93/138 http://zeus.polsl.gliwice.pl/~pownuk

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 94/138 http://zeus.polsl.gliwice.pl/~pownuk

Taylor model ( ) m  0  ( ) u h ˆ  = 0 + − 0 0 = ( ) ( ) , ( ) u h u h h h h mid h      i i h = i 1 i 0 ( ) du h  = + − 0 0 ( ) ( ) ( ) u h u h h h   dh ( h ) u u h h 0 Andrzej Pownuk 95/138 http://zeus.polsl.gliwice.pl/~pownuk

Approximate interval solution 0  ( ) m  0 ( ) u h ˆ ˆ  = = + − 0 ˆ ˆ ( ) ( ) , u u h u h h h       i i h = i 1 i ˆ   ˆ ( ). u u h  Andrzej Pownuk 96/138 http://zeus.polsl.gliwice.pl/~pownuk

Computational complexity − 1 0 - 1 solution of K ( ) u h   0 (  ) h u − 1 K - the same matrix  h i 1 - point solution Andrzej Pownuk 97/138 http://zeus.polsl.gliwice.pl/~pownuk

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. 93-111     ( ) 1 ( ) u h u h = + − + − − 0 0 0 0 0 ( ) ( ) ( ) ( )( ) h h u u h h h h h h 0 L i i i i j j    2 h h h i i j i i j  ( ) ( ) h h u L u = ( h ) u u L + ( ) u L h  u = ( h ) u + ( ) u h  + h 0 h h   Andrzej Pownuk 98/138 http://zeus.polsl.gliwice.pl/~pownuk

Finite difference method +  − −  0 0 0 ( ) ( ) ( ) du h u h h u h h      2 dx h +  −  + −  2 0 0 0 0 ( ) ( ) 2 ( ) ( ) d u h u h h u h u h h      ( ) 2 2  dx h 0 2 0 ( ) ( ) ( ) du h du h d u h    + − 0 ( ) h h  2 dx dx dx Andrzej Pownuk 99/138 http://zeus.polsl.gliwice.pl/~pownuk

Monotonicity test based on finite difference method (1D) 0 2 0 ( ) ( ) ( ) du h du h d u h    + − = 0 ( ) 0 h h  2 dx dx dx 0 ( )   du h  +  − −   0 0 ( ) ( ) u h h u h h h dx   = − = − 0 0 h h h   +  −  + −  2 0 0 0 0 ( ) ( ) 2 ( ) ( ) d u h u h h u h u h h     2 dx ˆ  h If h function is monotone.  Andrzej Pownuk 100/138 http://zeus.polsl.gliwice.pl/~pownuk