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 52245 June 2003
OUTLIE Introducton Dmenson-Reducton Method Appcaton to Stochastc Probems umerca Exampes Concusons Future Works
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
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
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! 2 4 4 2 4 2 2 2 4 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! 2 4 2 4 2 2 2 = x = x xj Tayor expanson at x = 0
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)]
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 ] = +
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
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 =,,
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
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
EXAMPLES Exampe : Mathematca Functon Y = Y( X, X ) = exp 2 2 2 2 2 + 00X + 2X2 + X X2 X j j =, 2 2 ( 0, σ ) Standard devaton of response (å Y ) 0.3 0.2 0.2 0. 0. umerca Integraton Dmenson-Reducton 2nd-Order Tayor Expanson 0.0 0.02 0.04 0.06 0.08 0.0 0.2 0.4 0.6 0.8 Standard devaton of nput varabes (å)
EXAMPLES Exampe 2: Stochastc Fnte-Dfference Anayss S x L = 20 n.; = 30 n.; EI = 6.45 0 8 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 4 5 2 2 4 +ζx 3 Y3 X Lognorma random varabes
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 ) 0.285 0.457 0.373 0.0065 0.0079 0.00 0.006 0.026 (sym.) 0.0206 0.297 0.480 0.39 0.0066 0.03 0.009 0.098 0.059 (sym.) 0.030 0.297 0.480 0.392 0.0070 0.02 0.0096 0.022 0.070 (sym.) 0.039 (b) v ξ =0.6; v S = 0.2 Mean Vector (m Y ) Covarance Matrx (γ Y ) 0.285 0.457 0.373 0.063 0.057 0.0320 0.07 0.0296 (sym.) 0.0653 0.328 0.540 0.439 0.063 0.0292 0.023 0.0535 0.0428 (sym.) 0.0348 0.328 0.539 0.439 0.073 0.032 0.0247 0.0573 0.0459 (sym.) 0.0373 st-order Tayor: 0.69 2.25 ( σˆ σ ) ( ˆ ) Y Y Dmenson-Reducton: 0.99 σ σ.04 Y Y
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) 2 8 4 0 0 4 8 2 6 20 (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.06 0 -. 0 -.09 0 -.2 0 - B u 4.36 0-4.60 0-4.52 0-4.6 0 - C u 2 2.36 0-2.48 0-2.43 0-2.46 0 - D u 2.24.33.30.33 E u 5.74 0-5.94 0-5.83 0-5.99 0 - u 2.30.37.35.38
Exampe 3: CPU Tme EXAMPLES ormazed CPU tme 00.0 0.0.0 ormazed = CPU.0 CPU CPU by Dmenson Reducton 2.6 8. 29.2 0. Dmenson Reducton eumann Expanson (2nd Order) eumann Expanson (4th Order) Monte Caro Smuaton (5000 Sampe
EXAMPLES Exampe 4: Determnstc onnear FEA P = 40 P k 5 R = 00 mm β = 4.04 h = 2mm Dspacement at center (y), mm 4 3 2 0 0 50 00 50 200 250 300 350 400 Concentrated Load (P),
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 4.7535 4.7625 4.7626 4.7629 m 2 22.624 22.70 22.7 22.74 m 3 07.8 08.42 08.43 08.46 m 4 54.36 58.30 58.36 58.53 m 5 2457. 2480.7 248. 2482.2 (b) σ α = 0.2; P = 200 m.604.943.9674.9742 m 2 2.8458 4.4572 4.343 4.479 m 3 5.3958 2.6826 0.766.493 m 4 0.844 46.08 29.937 32.339 m 5 22.9 22.36 92.555 96.850
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
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