The University of Texas at El Paso http://andrzej.pownuk.com P P - PowerPoint PPT Presentation

  =  ( ) ( ), ( ) f x f x f x        =  '( ) min '( ), '( ) ,max '( ), '( ) f x f x f x f x f x  What is is th the defini initio ion n of the solu lutio ion n of dif iffer ferentia ential l equation? ation?

( ) f x ( ) ( ) ( )    v , x f x f x  ( ) f x     ( ) ( ) ( ) ( ) ( )    v' min ' , ' ,max ' , ' x f x f x f x f x 

Dubois D., Prade H., 1987, On Several Definition of the Differentiation of Fuzzy Mapping, Fuzzy Sets and Systems, Vol.24, pp.117-120 How about integral equations?

Modal interval arithmetic  Affine arithmetic  Constrain interval arithmetic  Ellipsoidal arithmetic  Convex models (equations with the ellipsoidal parameters)  General set valued arithmetic  Fuzzy relational equations  …. Etc. 

  y y ( ) ( ) ( ) ( )  + − + + − ,..., ,..., ... y p p y p p p p p p   1 10 0 1 10 0 m m m m p p 1 m ( ) = ,..., y y p p 0 10 0 m   y y    + +  ... y p p   1 m p p 1 m       −  +  , , y y y y y y y   0 0

 Gradient descent  Interior point method  Sequential quadratic programming  Genetic algorithms  …

 Endpoint combination method  Interval Gauss elimination  Interval Gauss-Seidel method  Linear programming method  Rohn method  Jiri Rohn, "A Handbook of Results on Interval Linear Problems“,2006 http://www.cs.cas.cz/~rohn/publist/handboo k.zip

( )  ( ) ( ) ( ) ( )   −  + ( ) x A,b mid A x rad b rad A x rad b    , A A b b  W. Oettli, W. Prager. Compatibility of approximate solution of linear equations with given error bounds for coefficients and right- hand sides. Numer. Math. 6: 405-409, 1964.

  min max x x i i   ( ) ( ) ( ) ( ) −  −  ( ) ( )   mid A D rad A x b mid A D rad A x b s s   ( ) ( ) ( ) ( ) +  +    ( ) ( ) mid A D rad A x b mid A D rad A x b s s       0 0 x x  H.U. Koyluoglu, A. Çakmak, S.R.K. Nielsen. Interval mapping in structural mechanics. In: Spanos, ed. Computational Stochastic Mechanics. 125-133. Balkema, Rotterdam 1995.

( ) ( )  = −  A mid A D rad A D rc r c  ( ) ( ) = +  b mid b D rad b  r r ( ) = − T  1, 1,1,...,1 r J  ( )  ( ) = A,b A ,b conv conv   rc rc  ( )   = =  : , , conv A ,b conv x A x b r c J  rc rc rc r  = 2 + n n n 2 2 2 2 Rohn’s method Combinatoric solution 2 n n

 For every  r J ( ) ( ) − = 1  Select recommended  c sign mid A b c J r  Solve = A x b rc r ( )  If then register x and go to = sign x c next r   ( )  Otherwise find =  min : k j sign x c j j  Set and go to step 1 . = − c c k k

VERSO SOFT: FT: Veri rifi fica cati tion on software are in MATLA LAB B / I INTLAB LAB http://uivtx.cs.cas.cz/~rohn/matlab/index.html -Real data only: Linear systems (rectangular) -Verified description of all solutions of a system of linear equations -Verified description of all linear squares solutions of a system of linear equations -Verified nonnegative solution of a system of linear inequalities -VERLINPROG for verified nonnegative solution of a system of linear equations -Real data only: Matrix equations (rectangular) -See VERMATREQN for verified solution of the matrix equation A*X*B+C*X*D=F (in particular, of the Sylvester or Lyapunov equation) Etc.

 Find the solution of Ax = b ◦ Transform into fixed point equation g ( x ) = x g ( x ) = x – R ( Ax – b ) = Rb+ ( I – RA ) x ( R nonsingular) ◦ Brouwer’s fixed point theorem If Rb + ( I – R A) X  int (X) then  x  X, Ax = b

 Solve AX AX=b ◦ Brouwer’s fixed point theorem w/ Krawczyk’s operator If R b + ( I – R A) X  int (X) then  (A, b)  X ◦ Iteration  X n +1 = R b + ( I – R A) εX n (for n = 0, 1, 2,…)  Stopping criteria: X n +1  int( X n )  Enclosure:  (A, b)  X n +1

+ −       0 k k k u       = 1 2 2 1       −       k k u p 2 2 2

k 1 = [0.9, 1.1], k 2 = [1.8, 2.2] , p = 1.0 1 1 = k = = [ 0 . 91 , 1 . 11 ] u 1 [ 0 . 9 , 1 . 1 ] 1 + + [ 0 . 9 , 1 . 1 ] [ 1 . 8 , 2 . 2 ] k k = = = 1 2 [ 1 . 12 , 2 . 04 ] u  2 [ 0 . 9 , 1 . 1 ] [ 1 . 8 , 2 . 2 ] k k 1 2 ( overestima tion ) 1 1 1 1 = + = + = ' [ 1 . 36 1 . 67 ] u , 2 [ 0 . 9 , 1 . 1 ] [ 1 . 8 , 2 . 2 ] k k 1 2 ( exact solution)

+ k k = 1 2 u 2 k k 1 2  Two k 1 : the same physical quantity  Interval arithmetic: treat two k 1 as two independent interval quantities having same bounds

 Replace floating point arithmetic by interval arithmetic  Over-pessimistic result due to dependency − −       [ 2 . 7 , 3 . 3 ] [ 2 . 2 , 1 . 8 ] 0 u       = 1       − −  [ 2 . 2 , 1 . 8 ] [ 1 . 8 , 2 . 2 ]     1  u 2 Naïve solution Exact solution −         [ 110 , 112 ] [ 0 . 91 , 1 . 11 ] u u   =     =   1 1         −    [ 134 . 5 , 137 . 5 ]  u    [ 1 . 36 , 1 . 67 ]  u 2 2

 How to reduce overestimation? ◦ Manipulate the expression to reduce multiple occurrence ◦ Trace the sources of dependency + −       0 k k k u     =   1 2 2 1       −       k k u p 2 2 2

 Element-by-Element ◦ K : diagonal matrix, singular p L 2, E 2, A 2 L 1, E 1, A 1 p

 Element-by-element method =  ◦ Element stiffness: + ( ) K d K I i i i       E A E A E A E A   − −   1 1 1 1 1 1 1 1       α   1 0 0   L L L L     = =   +   1 1 1 1 1 K           1 α E A E A     0 1 0 E A E A   − − 1 1 1 1   1 1 1 1   1   L L   L L 1 1 1 1 ◦ System stiffness:  = + ( ) K d K I

 Lagrange Multiplier method ◦ With the constraints: CU – t = 0 ◦ Lagrange multipliers: λ             T T u p u p K C K C             =  =              λ        0   0  t  0  C C

 System equation: Ax Ax = b b       C T u p K       =       λ    0    0 C  rewrite as: + = ( ) A S D x b               T 0 d u p 0 k C k           + =             λ   0 0      0   0   0 0   C 

 x x = [u, λ ] T , u is the displacement vector  Calculate element forces ◦ Conventional FEM: F=k u ( overestimation) Ku = P – C T λ ◦ Present formulation: Ku λ = = L x, p = N b P – C T λ = = p – C T L (x * n+ 1 + x 0 ) P – C T λ = = N b – C T L ( R b – RS M n δ ) P – C T λ = = ( N – C T LR )b + C T LRS M n δ

Click here

http://andrze tp://andrzej.po j.pownuk.com/ wnuk.com/ Click here

 Monotone solution       + u p p 1 1 =  1   1 2    − 1 1  u   p        2 2 p p = + = u 1 p , u 1 1 2 2 2 2 − = 2 4 u p 0  Non-monotone solution = = − 2 2 u p u , p 1 2

 u then = = If  m i n m ax p p p , p 0  p  u then = = If  m i n m ax p p p , p 0  p = = m i n m ax u u p ( ) , u u p ( )

analysis_type linear_static_functional_derivative parameter 1 [210E9,212E9] # E parameter 2 [0.2,0.4] # Poisson number parameter 3 0.1 # thickness parameter 4 [-3,-1] sensitivity # fy point 1 x 0 y 0 point 2 x 1 y 0 point 3 x 1 y 1 point 4 x 0 y 1 point 5 x -1 y 0 point 6 x -1 y 1 rectangle 1 points 1 2 3 4 parameters 1 2 3 rectangle 2 points 5 1 4 6 parameters 1 2 3 load constant_distributed_in_y_local_direction geometrical_object 1 fy 4 load constant_distributed_in_y_local_direction geometrical_object 2 fy 4 boundary_condition fixed point 1 ux boundary_condition fixed point 1 uy boundary_condition fixed point 2 ux boundary_condition fixed point 2 uy boundary_condition fixed point 5 ux boundary_condition fixed point 5 uy #Mesh

( )  =  ( )  =  min u u min u u     or ( )  =  0 C      ,           ,    

   A B A B     =   =       Ω , :  