A DIMENSION-REDUCTION METHOD FOR STOCHASTIC ANALYSIS SECOND-MOMENT ANALYSIS

Transcription

1 A DIMESIO-REDUCTIO METHOD FOR STOCHASTIC AALYSIS SECOD-MOMET AALYSIS S. Rahman Department of Mechanca Engneerng and Center for Computer-Aded Desgn The Unversty of Iowa Iowa Cty, IA June 2003

2 OUTLIE Introducton Dmenson-Reducton Method Appcaton to Stochastc Probems umerca Exampes Concusons Future Works

3 ITRODUCTIO PDF of Input Random Varabe/ Fed Data Anayss Statstcs Dstrbuton Cacuate Reabty CDF Second Moments Confdence Interva Importance Factors Math. Mode Fatgue/Fracture Crashworthness MEMS Mcroeectroncs VH PDF of Output Random Varabe/ Fed Probabstc Methods FORM SORM AMV Monte Caro Imp. Sampng Dr. Smuaton Vadaton Statstcs Dstrbuton If Y = Y( X ) s a response, + + The th moment E Y( X ) y ( x) f ( x) dx X Expectaton operator

4 ITRODUCTIO Anaytca Methods Tayor Expanson (FOSM, SOSM) eumann Expanson Decomposton Method Poynoma Chaos Expanson Statstcay Equvaent Souton Pont Estmate Method Others Smuaton Methods Drect Monte Caro smuaton Quas-Monte Caro smuaton Importance sampng Drectona smuaton Others

5 DIMESIO-REDUCTIO METHOD Mut-dmensona Integraton + [ ( )] a + a (,, ) I y x y x x dx dx a a y y y I[ y( x) ] = I[ y( 0) ] + ( 0) I x + ( ) I x + ( ) I x x + 2! 0 4! 0 2!2! j = x = x < j x xj Tayor expanson at x = 0 Proposed Approxmaton yˆ( x) = yˆ( x,, x ) y(0,,0, x,0,,0) ( ) y(0,,0) = y y I[ yˆ( x) ] = I[ y( 0) ] + ( 0) I x + ( ) I x + 2! 0 4! = x = x xj Tayor expanson at x = 0

6 DIMESIO-REDUCTIO METHOD Resdua Error 4 y I y I y I x x O I x 2!2! x x 2 2 [ ] [ ] ( ) ( 4 ( x) ˆ( x) = 0 ) 2 2 j = < j j Exact Approxmate I[ŷ(x)] represents reduced ntegraton, because ony number of -dmensona ntegratons are requred, as opposed to one -dmensona ntegraton n I[y(x)] If parta dervatves 4 y(0)/ x 2 x j2 and/or I[x 2 x j2 ] are neggby sma, I[ŷ(x)] provdes a convenent approxmaton of I[y(x)]

7 DIMESIO-REDUCTIO METHOD on-symmetrc Doman [ ] b ( ) (,, ) a a b I y x y x x dx dx = = b + a b a x = + ξ, =,, 2 2 b a I[ y( x) ] = ηξ (,, ξ) dξ dξ 2 Arbtrary Expanson Pont (At x = µ) I yˆ( ) I y(,,, x,,, ) ( ) I y(,, ) [ x ] = [ µ µ µ µ ] [ µ µ ] = +

8 STOCHASTIC PROBLEMS Statstca Moments of Response m = E Y( X ) y( x) f ( ) d X x x R Usng the Dmenson-Reducton method: ˆ( ) m E Y Y (,, j, X j, j+,, ) ( ) y (,, ) X = E µ µ µ µ µ µ j= ˆ( ) m E Y X = S ( ) y( µ,, µ ) = 0 j Sj = E Y( µ,, µ j, X j, µ j, +, µ ) ; j =,, ; =,, = Requres ony -D ntegratons

9 STOCHASTIC PROBLEMS Recursve Reaton ( ) S = E Y X, µ 2,, µ ; =,, k k S2 = S Y(, X2, 3,, ) ; k 0 k E µ µ µ = k k S3 = S2 Y(, 2, X3, 4,, ) ; k 0 k E µ µ µ µ = =,, =,, k k Sj = S j Y( µ,, µ j, X j, µ j+, µ ) ; k= 0 k E =,, k k S = S Y( µ,, µ, X) ; k= 0 k E =,,

10 STOCHASTIC PROBLEMS Moment-based Quadrature Rue Dmenson-Reducton Method: ( µ µ µ + µ ) ( µ µ µ + µ ) E Y,,, X,, y,,, x,, f ( x ) dx j j j j j j j X j j n-order Quadrature Rue: n ( µ,, µ j, j, µ j+, µ ) j, ( µ,, µ j, j,, µ j+, µ ) E Y X w y x = w nput moment n n k ( xj xj, k) fx ( x ) ( ) j j dxj j, n k q j, k k =, k µ k= 0 j, = = n n ( x x ) ( x x ) j, jk, j, jk, k=, k k=, k eed to sove near eqn. once

11 STOCHASTIC PROBLEMS Lnear Equbrum Equatons K( X) Y( X) = F( X) where: X R ( m, g) mean of X covarance of X Usng the Dmenson-Reducton method: [ ] = j( X j) j( X j) ( ) ( ) ( ) my =E Y E K F K m F m j= g E YY m m T T = Y Y Y ( X ) ( X ) ( X ) ( X ) ( ) ( ) ( ) ( ) ( ) T T T T T YY = K j j Fj j Fj j K j j K F F K j= E E m m m m

12 EXAMPLES Exampe : Mathematca Functon Y = Y( X, X ) = exp X + 2X2 + X X2 X j j =, 2 2 ( 0, σ ) Standard devaton of response (å Y ) umerca Integraton Dmenson-Reducton 2nd-Order Tayor Expanson Standard devaton of nput varabes (å)

13 EXAMPLES Exampe 2: Stochastc Fnte-Dfference Anayss S x L = 20 n.; = 30 n.; EI = b-n 2 Y(x) ξ(x) L S Lognorma random varabe Mean µ S = 000 b/n; Varance = σ S 2 ξ(x) Homogeneous ognorma fed Mean µ ξ = 2000 b/n 2 ; Varance = σ ξ 2 Exponenta covarance functon 2 3 Dscrete Equbrum Equaton: 7+ζX 4 Y 4 6 X 4 +ζ Y = ζx ζx 3 Y3 X Lognorma random varabes

14 EXAMPLES Exampe 2: Dspacement Statstcs st-order Tayor Expanson Dmenson- Reducton Method Monte Caro Smuaton (0 6 sampes) (a) v ξ =0.3; v S = 0.2 Mean Vector (m Y ) Covarance Matrx (γ Y ) (sym.) (sym.) (sym.) (b) v ξ =0.6; v S = 0.2 Mean Vector (m Y ) Covarance Matrx (γ Y ) (sym.) (sym.) (sym.) st-order Tayor: ( σˆ σ ) ( ˆ ) Y Y Dmenson-Reducton: 0.99 σ σ.04 Y Y

15 EXAMPLES Exampe 3: Stochastc Mesh-Free Anayss x 2 å ì Dð(0,L) Eð(L,L) 20 6 Homogeneous Gaussan RF [ x ] E ( x) = µ +α( ) E 2L Crcuar Hoe 2a Cð(0,a) Að(a,0) 2L = 40, 2a =2 å ì x Bð(L,0) (90 nodes) 2 x Γ α ( x) =σe exp bl µ = ; σ = 0.; = 0.5 E E b (2 Random Varabes) 2L Standard Devaton of Response Locaton Response Varabe 2nd-order eumann Expanson Method 4th-order eumann Expanson Method Dmenson Reducton Method Monte Caro (5000 sampes) A u B u C u D u E u u

16 Exampe 3: CPU Tme EXAMPLES ormazed CPU tme ormazed = CPU.0 CPU CPU by Dmenson Reducton Dmenson Reducton eumann Expanson (2nd Order) eumann Expanson (4th Order) Monte Caro Smuaton (5000 Sampe

17 EXAMPLES Exampe 4: Determnstc onnear FEA P = 40 P k 5 R = 00 mm β = 4.04 h = 2mm Dspacement at center (y), mm Concentrated Load (P),

18 EXAMPLES Exampe 4: Stochastc Fnte-Eement Anayss Statstca Moments (m = E[Y ]) [ ] E ( x) =µ +α( x) Homogeneous Gaussan RF E x Γ =σ µ = = bl 2 α( x) α exp ; E ; b 0.5 st-order Tayor Expanson Method 2nd-order Tayor Expanson Method (a) σ α = 0.2; P = 400 (4 Random Varabes) Dmenson Reducton Method umerca Integraton m m m m m (b) σ α = 0.2; P = 200 m m m m m

19 COCLUSIOS A new dmenson-reducton method has been deveoped for predctng statstca moments Reducton of -dmensona ntegraton to mutpe -dmensona ntegratons Moment-based quadrature rue The method s more accurate than st- and 2 nd -order Tayor seres expanson methods The method s smpe (does not requre cacuaton of parta dervatves and nverson of random matrces) The accuracy of the proposed method s comparabe to the 4 th -order eumann expanson method, but the proposed method s computatonay far more effcent than eumann expanson methods

20 FUTURE WORK Senstvty of statstca moments w.r. t. desgn parameters (robust desgn) Deveopment of dmenson-reducton method for reabty anayss (faure probabty) Potenta appcatons n reabty-based desgn optmzaton