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1 ª Œª Œ 6ƒ 4A Á 006 7œ pp. 677 ~ 687 ª p-verso w ³ j ½ w» Stress Recovery Techque by Ordary Krgg Iterpolato p-adaptve Fte Eleet Method Ÿ Á xá Woo, Kwag SugÁJo, Ju HyugÁLee Dog J Abstract Krgg terpolato s oe of the geerally used terpolato techques Geostatstcs feld. Ths techque cludes the experetal ad theoretcal varogras ad the forulato of krgg terpolato. I cotrast to the covetoal least square ethod for stress recovery, krgg terpolato s based o the weghted least square ethod to obta the estated exact soluto fro the stress data at the Gauss pots. The weght factor s detered by varogra odelg for terpolato of stress data apart fro the covetoal terpolato ethods that use a equal weght factor. I addto to ths, the p- level s creased o-uforly or selectvely through a posteror error estato based o SPR (supercoverget patch recovery) techque, proposed by Zekewcz ad Zhu, by auto esh p-refeet. The cut-out plate proble uder teso has bee tested to valdate ths approach. It also provdes valdty of krgg terpolato through coparg to exstg least square ethod. Keywords : krgg terpolato, geostatstcs, a posteror error estato, SPR techque, varogra, stress recovery, adaptve p-refeet j ½ w wù. x j ½ y w swwš. w, ƒ k l w» w ƒ»., w ƒ e w ƒ w w ƒ eƒ. wr, w Zekewcz Zhu w SPR»» z sƒ w p- ³ k ƒ j.» w sq w w. w,» w j ½. w : j ½, w, z sƒ, SPR»,,, p- y 1. w wƒ œw w w w t ew. w ew sww vw. w w w ƒ w ƒ» w w w x w v. w z sƒ(a posteror error estato) w w ƒ. w w w wwš w d wš s ƒw z w w x w ¾ y j w w ƒ ww p- y wù. p- y w 1980» (resdual techque)» Babuska(1980) w. w y w (auto esh)» w w w w ƒ š. w 1980 z Dekowcz Ode(Dekowcz, 1989)» w z sƒ»» w y w ww. wr, 1987 Zekewcz Zhu(wz Z/Z t»)» (recovery techque) w š» w z û w y œw (E-al : kswoo@yu.ac.kr) z w œ 1 (E-al : etaphy@para.co) ( )w» (E-al : o13ra@aver.co) 6ƒ 4A 006 7œ 677

2 h- y ww., 199 Z/Z w SPR» w»» w w w., w w w yw w(supercoverget soluto) w w e w š (algorth) wš w»ƒ rw w z. SPR» h- y w» 1998 z Asworth Seor 003 Ÿ w d x w»» p-verso w w p- y ww ƒ t» w. Z/Z w SPR» w p- y w y yw s ƒwš ³ w w» h- y w r w w ƒ w z w sƒ.» SPR» w (estated exact soluto) syy p- y x w p ƒ w ƒ š ƒ w w ew ƒ ƒ w. z syy w v» š w ƒ v w. j ½ (krgg terpolato) û v eœy D.G. Krgeƒ Ÿ» Ÿ dw» w x š x w (Gudud, 1998). z w ƒ š w š. j ½ ƒ j p w w ƒ ƒ ƒ e w. ƒ e w. w j ½ w š ü w, p w w ƒ š ƒ EFG(eleet free Galerk) j ½ w ƒ t š (Da, 003). w ƒ e w p- y xw p (stress sgularty) y (stress gradet)ƒ j y s y w zw q. w j ½ p- y w w w»(preprocessor)ƒ C ++ - w, w»¾ p- s v ùkü,» w. ƒx (cut-out) ƒ sq ew w w» SPR» w p- y w ³ j ½(ordary Krgg) w p- y w. w w w s ³ j ½ w d» w s w w ƒ¾ ew w.. j ½.1 j ½ w» wÿ dw» w x. Ÿ sƒ x yw ƒ l x y w, y w y w ƒ. j ½ w, Ÿ r Ÿ š e x yw. e ƒ e w w dwš w l d. j ½ w w d w s³ dw ù. Ÿ w e y w w» w. j ½ ³ j ½ w rw w. ³ j ½ rw(ubas) swwš j ½ w yw d w. j ½ w d w w y v w. j ½ 1 w.. p- y w j ½ e, d 1. j ½ w 678 ª Œª Œ

3 4.. h-verso w ƒ e 5.» w syy 3. p-verso w ƒ e ƒ d dw. w» w t œ z d w e w w. SPR» p- verso w» h-verso w SPR» q q ü w ƒ ˆ eƒ w. w ƒ w š ƒ. h-verso w j ½ w ƒ ƒ ƒ š w ƒ e w y ƒ. w h-verso w w q sw ƒ w ˆ w. j ½ w w ü y w d ƒ p-verso w š x w ˆ ƒ j ½ p-verso w yw w.» SPR» w w w σ w. σ sƒ w» SPR» w wš. σ sƒ w. 4 5 w σ w w w w 1 l w, l w w syy. syy w w w j ½ w y ww yw sƒƒ ƒ w w y w w w..3 x (experetal varogra) j ½ w ƒ e w» w ƒƒ w txw w. w w tx ƒ e w ƒ w w. ƒ h r(h) (1) tx w w r(h) () tx (, 00). r( h)=e z( x) z x+ h [( ( )) ] 1 r( h)= z ( x ) z x + h = 1 [ ( )] (1) r(h) w» z(x ) ùkù. sƒ ƒ ƒ¾ (1) () 6ƒ 4A 006 7œ 679

4 Model Lear Sphercal Expoetal Power t 1. w [13,16] Varogra h C --, for h 0 a r( h)= a C, for h a 0 3 h h C r( h)= a a ùkùš ƒ f ù y w w ƒ. t œ w yw 6 e t w - w, x w..4 (theoretcal varogra) x ùkü y(fttg) w w w. x œ txwš œ txw w. œ txw w w w w. () hƒ w ƒ z(x+h) 0 ù y w ƒ š z(x)ƒ ù w ƒ. w w w. w œ w w w ù w» w a (rage) w. w w w tx w t 1. wr, hƒ 0 w 0, > C, for h a 0 > 3h r( h)=c 0 1 exp a r( h)=c 0 1 H 0, for h a ( ( ))+αh β, 0 < β < ; h 0 6. x x w ü 0 0 ùký ½(ugget) w t 1 C o tx. ½ ƒ ùkü y ùkü. w, t 1 h(0) Heavsde w ƒ 0 1 š, 0 p α β. t 1 (Che, 01; Zhag, 1995; Ja, 1996)..5 ³ j ½ (ordary krgg terpolato) t œ w x wš w j ½ yw. w j ½ w j ½ ³ j ½. ³ j ½ w (usapled data) d z 0» w (eghbor data) z w ƒ e w» w ƒ w w. w w ƒ j ½ ƒ j ½ d rw(ubased). j ½ w d (3) š (4) d z 0 w» E txw (Che, 001; Clayto, 1996). z 0 = (3) (4) z 0 š d z 0 ƒ e w w w ƒ e w. (varace) Var(z 0 ) j w ù ƒ ùkü œ (covarace) Cov(z 0, z 0 ) (4) tx w (5) σ. σ =Var(z 0 ) Cov(z 0, )+Var( ) (5) d z 0 ƒ e λ w w ã (6) j ½ ùký (Che, 001; Clayta, 1996). where, λ z = 1 σ =E z 0 z 0 [( ) ] σ =σ z0 λ σ 0 = 1 + σ =Var z 0 z 0 = 1 (6) j ½ ƒ ƒ e λ w j ½ š (6) rw w ³ j ½. ³ j ½ rw s ³ t s³ w ƒ ƒ g rw š w. rw z 0 λ λ j σ j j = 1 z0, z ( ), σ 0 ( ), σ j =Cov z, z j ( ) (3) (4) (6) 680 ª Œª Œ

5 (7) ùký rw w l b w. b =Ez ( 0) E( z z 0 )=E( z 0 ) E λ z = 1 z (7) j ½ w s³ ƒ š (7) rw ƒ 0» w ƒ e w 1 w w. ³ j ½ w (8) sw. 1 λ =0 = 1 (8) wš (6) w ƒ e w ³ j ½ w d. 3. ³ j ½ w p- y 3.1 x SPR» ü r q (patch)» syy ww σ w w q ³ j ½ ww» w t œ. ƒ l š ƒ ƒ ³ j ½ ww. t œ ƒ z ƒ e l x w w. x w š w ƒ š l œ y q w» w w. ƒ ƒ w ƒ. ƒ j ƒ h j hj. š () s ³ w ƒ. ƒ ƒ wš ƒ w yw x 7. w (polyoal odel) w x =0 (7) (8) š w» w ƒ. 3. x» ƒ w w - ùkü tw w. w w v w. w, Power xk š w. w ƒ (weghted least squares) w Ja (1996) Che (001) d de l w w x ƒ ƒà y (fttg) j ½ de ƒ e ƒ yw š. w wš ³ j ½ v g yƒ w 7 w (polyoal odel) w. x w - w txw ww xk w. 5 w w w k 0, 5» w w k1,...k š x(zhag, 1995). rh ( )=k 0 +k 1 h+k 1 h + +k h ³ j ½ y ³ j ½ w rw l j ½ rw w w. (7) rw l (8) wš w ƒ ƒ e w ³ j ½ w d ƒ w. Mze, σ λ σ + 0 = 1 = Subject to, 1 λ =0 = 1 1 λ λ j σ j j = 1, σ j =Cov( z, z j ) (9) (10) (10) ³ j ½ ƒ ùküš. w ƒ w y w (Lagrage ultpler ethod) w w w. w w w w w š w w, w. w w (Lagrage objectve fucto) š w w (Lagrage ultpler) w. ³ j ½ w w (11). L( λ 1, λ,, λ ; ω)= 6ƒ 4A 006 7œ 681

6 σ λ σ 0 + = 1 = 1 L( λ 1, λ,, λ ; ω) : Lagrage Objectve Fuctos ω : Lagrage Multpler λ λ j σ j j = 1 +ω 1 λ = 1 (11) (11) sww xk w, w ƒ e λ ω w ƒƒ r w w w. L = σ λ 0l l + L = 1 λ ω = 1 λ σ l = 1 =0 ω=0, l=1,,, (1) (13) (1) (13) w (14) (15) tx w. e = ( σ σ p ) T [ D] 1 ( σ σ p )dω Ω (17) (17), e -, Ω -, [D] w (costtutve atrx) ùkü. (10) w [D] w L -, y (stress or) š e σ tx. wr, σ-œ σ w w ƒ œ ùkü e σ l w ƒ q w σ-œ d w (11) w. η ƒ w q w t. η Ω w wš w w. Ω e σ η Ω= ( %) e ( Ω + eσ Ω ) 1/ (18) λ σ l = 1 (14) (14) ³ j ½ ùkü. w ³ j ½ w xk tx ƒ w j ½ w rw w w» w w w xkƒ. ³ j ½ l d dw ω=σ 0l ( + 1) +( + 1), l=1,,, j» ƒ w. σ 11 σ 1 σ 1 1 λ1 σ 01 σ 1 σ σ 1 λ σ 0 = λ σ 1 σ σ 1 σ 0 ω (16) rw w w j w j»ƒ f š w. (16) w xk tx ³ j ½ ùküš. 3.4 z sƒ(a posteror error estato) w w w w» w. ƒ w xk» w w w wwš p sww ù y sw x w p ƒ k. p- y p p ƒ j ù ƒ w» w p w v w w w óù z w w z sƒ» v w. SPR p-verso w w ( w σ p ) w ( w σ ) œ l wš l ùkü eσ(=σ σ p ) l ³y w l (eergy or) (17) w. 68 Ω 1/ σ Ω = σ e e = 1 (19) (18) σ w ƒ œ ùkü e σ w ƒ œ ùkü ùküš. (18) w w ³y ùkü. w ƒ x w p ƒ j t w» w ³y w ƒ d wš p ƒ q w. (1) ùküš. SPR» w r q š q x q ƒ. w ƒ q w z j t w w s³ š w. e σ e η e = ( %) ( + eσ ) 1/ e e = e e e σ e = e σ e = e 1 σ e --- = 1 (1) () (3) (3) p ƒ ƒ q w z, η e s³ w w ùkü., (18) (1) w t (4) w. η e ξ e = η Ω ξ e (4) ª Œª Œ

7 11. w p- y (5 w, NDF=168) 8. ep»wx w (4) ξ e > 1 p- ƒ j., η e d η Ω j x w ƒ j p- y ww. wr, SPR w q w r p sww q (patch assebly pot) q w z ƒ» w w w. w σ ³ j ½ w wš w w. q w w w syy w., ƒ yw d yw w w. q ƒ w š w w p ƒ w w j sƒw w. ww w w w. w w š w». w ³ j ½ w š 9. p- y w w» 1. ³ j ½ w p- y (4 w, NDF=13) w š w yw w. w w» w j w e. 4. ew 4.1 Catlever Bea w ew ep w ww. ep 9. ep Ì ƒ s (plae stress) w w w ep w Toseko w. ( )x+( + ν) y D Py u = x 6L 3x 6EI P u = y 3νy L x 6EI (5) (6) p- y w w» 9 š 8 ƒƒ D=10 c, L=10 c, w w P=1 kgf/c. k E= 10 6 kgf/c, ν=0.3 ep» ³ j ½ ƒƒ w ep w w (collecto) 4 ( )+( 4+ 5ν) D x ( 3L x)x w p- w 13. ƒ sq 6ƒ 4A 006 7œ 683

8 그림 로 고정된 체눈설계 14. p=4 그림 그림 전영역 로 유한요소해석 한 후의 가우스적분점에서의 응력분포 15. =0, 즉 그림 p=4 축을 따라 이루어졌다. 그림 10과 같이 정해와 적응적 체눈 세분화의 결과 비 교는 유한요소해석결과와 이론적 정해가 매우 정확한 것을 알 수 있다. 위의 결과에서 보듯이 정해와 같은 수준의 결 과를 도출해내므로 정해를 얻기 위해 지나친 형상함수의 고 차화나 요소재분할이 불필요함을 간접적으로 나타낸다 할 수 있다. 또한 정규 크리깅 보간법을 사용한 적응적 유한요소해 석의 안정성 및 통용성을 입증한다 하겠다. 그림 11과 그림 1에는 SPR기법 사용시 응력복구를 위해 최소제곱법과 정규 크리깅을 사용하였을 때의 최종 적응적 체눈을 나타내며, 그 때의 자유도 비교를 보여주고 있다. 동일한 수준의 정확도를 갖기 위해 정규크리깅 방식(NDF=13)이 최소제곱법 (NDF=168)에 비해 적은 자유도를 필요로 하고, 반복회수도 5회에서 4회로 빠른 시간내에 적응적 -체눈을 결정함을 알 수 있다. 개구부를 갖는 평판 개구부(cut-out)가 존재하는 평판은 개구부의 모서리 부분 에서 응력특이(stress sgularty)가 발생하게 되므로 적응적 유한요소해석을 위한 검증 예제로 많이 사용된다. 다음의 그림 13은 개구부가 있는 평판의 해석 전 영역을 y 16. FEM raw data 17. 최소제곱법 x p- p 4. (Cut-out plate) 그림 18. 크리깅 보간법 보여주고 있다. 그림 13에서 =50 c이며 두께 =1 c이다. 재료상수는 각각 = 10 kgf/c, ν=10으로 설정하고 인장력 σ=1 kgf/c 이 작용하고 있다. 전 해석 영역에서 기하형상과 경계조건, 하중조건 등이 대 칭성을 만족하므로 1/4모델을 통해 해석을 수행하였다. 적응 적 -유한요소해석을 수행하기 전에 형상함수의 차수를 고정 684 a 6 E t p 大韓土木學會論文集

9 3. 1- w w » (p=1) w j ½ w 0. 1-» (p=1) No. of Iterato t. 1- w (η Ω ) ³j ½ NDF η Ω (%) NDF η Ω (%) w w. 3- w j ½ w w z w w wwš ³ j ½ w syy ww. w w w w» w w syy w ƒ z y w. 14 x w w š w. w w ww z ƒ ƒ l w syy» w. 15 w ƒ v q w k ùk, s³ vo-mses w. w ƒƒ, j ½ ww syy š ùkü D esh grd w w l w w w syy. s syy s w j ½ w s wš. p p tx wš. 6ƒ 4A 006 7œ 685

10 5. 1- ƒ, p- y w p- w w ww. w wš» x w p- 1 w sƒ w ƒ w. w z ƒ ƒ x w ùk ü w ƒ ƒ ƒ w. w» w ƒ w w w p- y š (algorth) z hp- y w. ƒƒ w w 1 4 ùkü. ³ w x ƒ w xk ùküš. x ƒ j w, w š x w w w y š. w w w» w w z (terato uber) w. w w w ü w ƒ w ƒ w». ƒ w ƒ sƒ t ùkü. w ƒ, yƒ y 1-10z w, ³ j ½ 8z w w z w. w w z ƒ š y w». j ½ w w ƒ w w w. 5 w w z j ½ j ùkù. j ½ w ˆ w w. z w w w w. ù, 5 3 ƒ ƒ wš ƒ ƒ o x š. t w we ( e ) E ³j ½ No. of Iterato l» w š q. SPR» w Zekewcz/Zhu w w, w w(exact soluto) FEM w w., w w sƒ w ƒ w. w» w t 3 w. t 3 ƒ š.» we(lt value) ƒ w w w(exact soluto) Babuskaƒ w (extrapolato techque) w w w w w sƒ w w SPR» w zw ƒ s ƒw» w. wr, Babuska w we š x(babuska, 1980)»., š SPR» Adaptve Mesh w»¾ w FEMw ww z ƒ ù j ½ w syy(stress soothg) k w(estated exact soluto) ƒ w, w FEMw eergy or w sƒw., (4) t(error dcator)ƒ ξ e >1 p- ƒ j., η e d η Ω j x w ƒ j p- y ww., w p- ƒ k ƒ ƒ w t. w t 3 Babuska (Babuska, 1980) w w (adaptve esh) w»¾ ƒ. ù, ù w» w syy w w FEM w r ƒ» q ù yw w» ƒ. 5. š NDF (%) NDF (%) e E e E y w Zekewcz Zhu w 686 ª Œª Œ

11 SPR p- y w w w SPR w ƒ w š w SPR» w w. w ƒ j ½ œw w.» w w ew ww ƒƒ w ƒ ƒ w. yƒ w we» w w w w ù j ½ w w œw ü š. w w š x w wì w ƒ š yw e w. j ½ w w x w w. w w» w š x w wš w yw ww hp- y zw ƒ. ³ j ½ w w ƒ w y w w w w w. w w w w w w š ƒ. 005 w w w (KRF D00495), ¾ tw. š x Ÿ, x, (003) z sƒ w p- y, w wz, w wz, 3«A y, pp (00) œ -j ½ y»,, pp Asworth, M. ad Seor, B. (1997) A adaptve refeet strategy for hp-fte eleet coputatos, Appl. Nuer. Math., Vol. 101, pp Asworth, M. ad Seor, B. (1997) Aspects of a adaptve hpfte eleet ethod : adaptve strategy, coforg Approxato ad Effcet Solvers, Coput. Meth. Appl. Mech. Egrg., Vol. 101, pp Babuska,G I. ad Rheboldt, W.C. (1980)G Relable error estato ad esh adaptato for the fte eleet ethod, J.T. Ode, Ed., Coput. Meth. Nolear Mech., North-Hollad Publshg Co., pp Clayto V. Deutsch (1996) Correctg for egatve weghts ordary krgg, Coputers & Geosceces, Vol., No. 7, pp Da, K.Y., Lu, G.R., L, K.M., ad Gu, Y.T. (003) Coparso betwee the radal pot terpolato ad the Krgg terpolato used eshfree ethod, Coputatoal Mechacs, Vol. 3, pp Dekowcz, L., Ode, J.T., Rachowcz, W.S ad Hardy, O. (1989) Toward a uversal h-p adaptve fte eleet strategy Part II: A Posteror Error Estato, Coput. Meth. Appl. Mech. Egrg., Vol. 77, pp Gudud Host (1998) Krgg by local polyoal, Coputatoal Statstcs & Data aalyss, Vol. 9, pp Ja, X., Olea, R.A., ad Yu, Y.S. (1996) Sevarogra odelg by weghted least squares, Coputer & Geosceces, Vol., No. 4, pp Jorge Kazuo Yaaoto (000) A alteratve easure of the relablty of ordary krgg estates, Matheatcal Geology, Vol. 3, No. 4, pp Yoglag Che, Xguo Jao (001) Sevarogra Fttg wth Lear Prograg Coputers & Geosceces, Vol. 7, pp Zhag, X.F., Va Ejkere, J.C., ad Heek, A.W. (1995) O the weghted least-square ethod for fttg a sevarogra odel, Coputer & Geosceces, Vol. 1, No. 4, pp Zekewcz, O.C. ad Zhu, J.Z. (1987) A sple error estator ad adaptve procedure for practcal egeerg aalyss, It. J. Nuer. Meth. Egrg., Vol. 4, pp Zekewcz, O.C. ad Zhu, J.Z. (199) The supercoverget patch recovery ad a posteror error estate Part I: The Recovery Techque, It. J. Nuer. Meth. Egrg., Vol. 33, pp Zekewcz, O.C. ad Zhu, J.Z. (199) The supercoverget patch recovery ad a posteror error estate Part II: Error Estate ad Adatvty, It. J. Nuer. Meth. Egrg., Vol. 33, pp ( : / : / : ) 6ƒ 4A 006 7œ 687

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