Report #1 Example. Semester

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1 Report # Eample Parallel FEM Class n Wnter Semester Kengo aajma Informaton Technology Center Techncal & Scentfc Comptng I (480-07) Semnar on Compter Scence I (480-04)

2 Report-0 D Statc Lnear Elastc Problem F =0 ( mn ) = ma Only deforms n -drecton (dsplacement: ) Sectonal Area A=A + A (>0) Unform Yong s Modls E Bondary Condtons (B.C.) =0 : =0 (fed) = ma : F (aal force) Trss: O bendng deformaton by G-force

3 Report-0 D Statc Lnear Elastc Problem A=A + A (>0) F =0 ( mn ) = ma Only deforms n -drecton (dsplacement: ) Sectonal Area A=A + A (>0) Unform Yong s Modls E Bondary Condtons (B.C.) =0 : =0 (fed) = ma : F (aal force) Trss: O bendng deformaton by G-force

4 Report-0 4 D Statc Lnear Elastc Problem A=A + A (>0) F =0 ( mn ) = ma Eqlbrm Eqaton 0 Stran~ Dsplacement Stress~ Stran E E 0 Governng Eqaton for

5 Report-0 5 Procedres for Comptaton solve eqatons for dsplacement E 0 calc. stran calc. stress E

6 Report-0 6 Analytcal Solton E F A E d d F A A F A log A A C C A 0@ 0 E F A log F EA log A A log A

7 Report-0 7 Report # Implement qadratc nterpolaton on b.c/b.f. ame of the developed code s b.c/b.f Evalate accracy of b and b accordng to effect of nmber of meshes. Confrm the followng eqaton s correct, f sectonal area s constant: V E d T d d d dv De on Agst 9 th (M), 0 at 7:00 Docments Report (Otlne, Reslts, Dscssons) (less than 5 pages) Lst of Sorce Code EA 6L

8 Report-0 8 Report #: Tps () Dervatve of Shape Fnctons =+ =- =0 d d d d d d m A J w E Emat ] [ Vales at Gassan Qad Ponts ( )

9 Report-0 9 Report #: Tps () Jacoban Jacoban at Gassan Qad Ponts ( ) =+ =- =0

10 Report-0 0 Strategy Unform Sect. Area, Lnear Elem. (Analytcal Integraton ) <$fem>/d/d.c Unform Sect. Area, Hgher-Order Elem. <$fem>/d/d.c on-nform Sect. Area, Lnear Elem. (Analytcal ) <$fem>/darea/a.c on-nform Sect. Area, Isoparametrc Lnear Elem. (mercal Integraton ) <$fem>/darea/b.c on-nform Sect. Area, Isoparametrc Hgher- Order Elem. (mercal Integraton ) Startng from d.c s easy.

11 Report-0 Gassan Qadratre On normalzed natral (or local) coordnate system [-,+] Can appromate p to (m-)-th order of fnctons by m qadratre ponts (m= s enogh for qadratc shape fnctons). m f m m m d w f 0.00, , 0.00, w w ,.00 8/ w w / 9 =- =0 = = =+

12 Report-0 Gassan Qadratre can be easly etended to D & D W j W, ), ( j m,n: nmber of qadratre ponts n -drecton : Weghtng Factor : Coordnates of Qad s n j j j m f W W d d f I ), ( ), (

13 Report-0 How to se Gassan Qadratre Coordnate transformaton from [0,L] (or [, ]) to [-,+] s needed. Shape/Interpolaton fnctons mst be handled on natral/local coordnate system. = =0 = =L =- =+ Local node ID s (,, ) are defned as sbscrpts of,, and etc.

14 Report-0 4 Pecewse Lnear Element =- =+,,

15 Report-0 5 nd -order/qadratc Element =+ =- =0,,,,,

16 Report-0 6 Isoparametrc Element Defntons of Isoparametrc Elements n Each element s defned on natral/local coordnate for [-,+] And the components of global coordnate system of each node (e.g.,y,z) for certan nds of elements are defned by shape fnctons [] on natral/local coordnate system, where shape fnctons [] are also sed for nterpolaton of dependent varables. ( ), : n,... n ( )

17 Report-0 7 Integraton over Each Element: [] Ad E dv E T V T V

18 Report-0 8 Partal Dfferentaton on atral/local Coordnate System (/) Accordng to formlae: ( ) can be easly derved accordng to defntons. are reqred for comptatons. s Jacoban (=J) n n J

19 Report-0 9 Partal Dfferentaton on atral/local Coordnate System (/) Target terms are derved as follows: J ) ( ) ( d d J d d J f d f L 0 Integraton on natral/local coordnate system:

20 Report-0 0 Integraton over Each Element: [] (/) m T T T T V T T V A J w E d A J E d J A J J E d J A E d A E dv E

21 Report-0 Integraton over Each Element: [] (/) m m m T A J w E A J w E A J w E, / /

22 Report-0 Eample (/7): Var s/array s #nclde <stdo.h> #nclde <stdlb.h> #nclde <math.h> #nclde <assert.h> nt man(){ nt E,, PLU, IterMa, errno, PLU0; nt R, Z, Q, P, DD, p; doble d, Resd, Eps, Area, F, Yong, Jacob; doble,,, U, U, U, DL, Stran, Sgma, C,, 0, A, A, DISP; doble *U, *Rhs, *; doble *Dag, *AMat; doble **W; nt *Inde, *Item, *Icelnod; doble POI[], WEI[], ddq[], Emat[][]; nt, j, n, n, n,, cel,,,,,,, js; nt ter; FILE *fp; doble Borm, Rho, Rho=0.0, C, Alpha, Dorm; nt err = ;

23 Report-0 Eample (/7) Intalzaton, Array Allocaton /* // // IIT. // */ fp = fopen("npt.dat", "r"); assert(fp!= ULL); fscanf(fp, "%d", &E); fscanf(fp, "%lf %lf %lf %lf %lf", &d, &F, &A, &A, &Yong); fscanf(fp, "%d", &IterMa); fscanf(fp, "%lf", &Eps); fclose(fp); = *E + ; PLU0= * + E* + (E-)*4; U = calloc(, szeof(doble)); = calloc(, szeof(doble)); Dag = calloc(, szeof(doble)); AMat = calloc(plu0, szeof(doble)); Rhs = calloc(, szeof(doble)); Inde= calloc(+, szeof(nt)); Item = calloc(plu0, szeof(nt)); Icelnod= calloc(*e, szeof(nt));

24 Report-0 44 Inpt Data npt.dat E (mber of Elements) e e6,F, A, A, E 00 mber of MA. Iteratons for CG Solver.e-8 Convergence Crtera for CG Solver 0 00 A A A A..5 = 4 5 Element ID ode ID (Global) =0 =5 =50 =75 =00

25 Report-0 5 Eample (/7) Intalzaton, Array Allocaton /* // // IIT. // */ fp = fopen("npt.dat", "r"); assert(fp!= ULL); fscanf(fp, "%d", &E); fscanf(fp, "%lf %lf %lf %lf %lf", &d, &F, &A, &A, &Yong); fscanf(fp, "%d", &IterMa); fscanf(fp, "%lf", &Eps); fclose(fp); = *E + ; PLU0= * + E* + (E-)*4; U = calloc(, szeof(doble)); = calloc(, szeof(doble)); Dag = calloc(, szeof(doble)); AMat = calloc(plu0, szeof(doble)); Rhs = calloc(, szeof(doble)); Inde= calloc(+, szeof(nt)); Item = calloc(plu0, szeof(nt)); Icelnod= calloc(*e, szeof(nt)); Amat: on-zero Off-Dag. Comp. Item: Correspondng Colmn ID

26 D-Part 6 on-zero Off-Dagonal Components odes are connected to node on each element each other ( ) K + ( ) F f 4 +

27 D-Part 7 mber of on-zero Off-Dagonals Dfferent nmber of connectons accordng to locaton of each node 4 for ormal odes for Intermedate odes for Bondary odes

28 D-Part 8 mber of on-zero Off-Dagonals Dfferent nmber of connectons accordng to locaton of each node 4 for ormal odes for Intermedate odes for Bondary odes Bondary ode# Intermedate ode# E ormal ode# E+-= E- ecept Bondary odes Total ode#: = +E+E-= *E + on-zero Off-Dag.#: PLU= * + *E + 4*(E-)

29 Report-0 9 Eample (/7) Intalzaton, Array Allocaton (cont.) W = (doble **)malloc(szeof(doble *)*4); f(w == ULL) { fprntf(stderr, "Error: %s n", strerror(errno)); retrn -; } for(=0; <4; ++) { W[] = (doble *)malloc(szeof(doble)*); f(w[] == ULL) { fprntf(stderr, "Error: %s n", strerror(errno)); retrn -; } } for(=0;<;++) U[] = 0.0; for(=0;<;++) Dag[] = 0.0; for(=0;<;++) Rhs[] = 0.0; for(=0;<plu0;++) AMat[] = 0.0; for(=0;<;++) []= *d*0.5; for(cel=0;cel<e;cel++){ Icelnod[*cel ]= *cel; Icelnod[*cel+]= *cel+; Icelnod[*cel+]= *cel+; } : -coordnate component of each node WEI[0]= +.0; WEI[]= +.0; POI[0]= ; POI[]= ;

30 Report-0 0 Eample (/7) Intalzaton, Array Allocaton (cont.) W = (doble **)malloc(szeof(doble *)*4); f(w == ULL) { fprntf(stderr, "Error: %s n", strerror(errno)); retrn -; } for(=0; <4; ++) { W[] = (doble *)malloc(szeof(doble)*); f(w[] == ULL) { fprntf(stderr, "Error: %s n", strerror(errno)); retrn -; } } for(=0;<;++) U[] = 0.0; for(=0;<;++) Dag[] = 0.0; for(=0;<;++) Rhs[] = 0.0; for(=0;<plu0;++) AMat[] = 0.0; for(=0;<;++) []= *d*0.5; for(cel=0;cel<e;cel++){ Icelnod[*cel ]= *cel; Icelnod[*cel+]= *cel+; Icelnod[*cel+]= *cel+; } WEI[0]= +.0; WEI[]= +.0; POI[0]= ; POI[]= ; Icelnod[*cel] =*cel cel Icelnod[*cel+] =*cel+ Icelnod[*cel+] =*cel+

31 Report-0 Eample (4/7) Global Matr: Colmn ID for on-zero Off-Dag s /* // // COECTIVITY // */ Inde[0]= 0; for(=;<;++) { f (%==) {Inde[]=4; }else {Inde[]=;}} Inde[]= ; Inde[]= ; for(=0;<;++){ Inde[+]= Inde[+] + Inde[];} PLU= Inde[]; for(=0;<;++){ nt js = Inde[]; f( == 0){ Item[jS ] = +; Item[jS+] = +; }else f( == -){ Item[jS ] = -; Item[jS+] = -; }else{ f (%==){ Item[jS ] = -; Item[jS+] = +; } else { Item[jS ] = -; Item[jS+] = -; Item[jS+] = +; Item[jS+] = +;}}} 4 for ormal odes for Intermedate odes for Bondary odes

32 Report-0 Eample (4/7) Global Matr: Colmn ID for on-zero Off-Dag s /* // // COECTIVITY // */ Inde[0]= 0; for(=;<;++) { f (%==) {Inde[]=4; }else {Inde[]=;}} Inde[]= ; Inde[]= ; for(=0;<;++){ Inde[+]= Inde[+] + Inde[];} PLU= Inde[]; for(=0;<;++){ nt js = Inde[]; f( == 0){ Item[jS ] = +; Item[jS+] = +; 両端の }else f( == -){ 節点 : つ Item[jS ] = -; Item[jS+] = -; }else{ f (%==){ Item[jS ] = -; Item[jS+] = +; } else { Item[jS ] = -; Item[jS+] = -; Item[jS+] = +; Item[jS+] = +;}}} Item(jS) Item(jS+) Item(jS) Item(jS+)

33 Report-0 Eample (4/7) Global Matr: Colmn ID for on-zero Off-Dag s /* // // COECTIVITY // */ Inde[0]= 0; for(=;<;++) { f (%==) {Inde[]=4; }else {Inde[]=;}} Inde[]= ; Inde[]= ; for(=0;<;++){ Inde[+]= Inde[+] + Inde[];} PLU= Inde[]; for(=0;<;++){ nt js = Inde[]; f( == 0){ Item[jS ] = +; Item[jS+] = +; }else f( == -){ Item[jS ] = -; Item[jS+] = -; }else{ f (%==){ Item[jS ] = -; 中間節点 Item[jS+] = +; : つ } else { Item[jS ] = -; Item[jS+] = -; Item[jS+] = +; Item[jS+] = +;}}} Item(jS) Item(jS+)

34 Report-0 4 Eample (4/7) Global Matr: Colmn ID for on-zero Off-Dag s /* // // COECTIVITY // */ Inde[0]= 0; for(=;<;++) { f (%==) {Inde[]=4; }else {Inde[]=;}} Inde[]= ; Inde[]= ; for(=0;<;++){ Inde[+]= Inde[+] + Inde[];} PLU= Inde[]; for(=0;<;++){ nt js = Inde[]; f( == 0){ Item[jS ] = +; Item[jS+] = +; }else f( == -){ Item[jS ] = -; Item[jS+] = -; }else{ f (%==){ Item[jS ] = -; Item[jS+] = +; } else { Item[jS ] = -; Item[jS+] = -; 要素両端 Item[jS+] = +; の節点 :4 つ Item[jS+] = +;}}} Item(jS+) Item(jS+) Item(jS) Item(jS+)

35 Report-0 5 Eample (5/7): Matr (/) /* // // MATRI assemble // */ for(cel=0;cel<e;cel++){ n= Icelnod[*cel]; n= Icelnod[*cel+]; n= Icelnod[*cel+]; = [n]; = [n]; = [n]; n cel n n DL = fabs(-); 0 = 0.5 * (+); Emat[0][0]= 0.0; Emat[0][]= 0.0; Emat[0][]= 0.0; Emat[][0]= 0.0; Emat[][]= 0.0; Emat[][]= 0.0; Emat[][0]= 0.0; Emat[][]= 0.0; Emat[][]= 0.0;

36 Report-0 6 Eample (6/7): Matr (/) for(p=0;p<;p++){ ddq[0]= POI[p]; ddq[]= -.0 * POI[p]; ddq[]= POI[p]; = 0 + POI[p]*0.50*DL; Area= A* + A; f(area<= 0.) { fprntf(stderr, "ERROR: Area<0: n"); retrn -; } } Jacob= fabs(ddq[0]* + ddq[]* + ddq[]*); C= Area*Yong/Jacob; Emat[0][0]= Emat[0][0] + C * WEI[p] * ddq[0] * ddq[0]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[];

37 Report-0 7 Dervatves at Gassan Qad. Ponts Dervatve of Shape Fnctons =+ =- =0 d d d d d d m A J w E Emat ] [ Vales at Gassan Qad Ponts ( )

38 Report-0 8 Eample (6/7): Matr (/) for(p=0;p<;p++){ ddq[0]= POI[p]; ddq[]= -.0 * POI[p]; ddq[]= POI[p]; L = 0 + POI[p]*0.50*DL; Area= A* + A; L/ } f(area<= 0.) { fprntf(stderr, "ERROR: Area<0: n"); retrn -; } =- =0 Jacob= fabs(ddq[0]* + ddq[]* + ddq[]*); C= Area*Yong/Jacob; Emat[0][0]= Emat[0][0] + C * WEI[p] * ddq[0] * ddq[0]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; =+ : Global Coordnate for Gassan Qad. Pont Area: Sectonal Area at

39 D-Part 9 -Coord. or Gassan Qad. Ponts nd -Order/Qadratc Element Accordng to defntons of soparametrc elements: =+ =- =0 L L/ = 0 + POI[p]*0.50*DL;

40 Report-0 40 Eample (6/7): Matr (/) for(p=0;p<;p++){ ddq[0]= POI[p]; ddq[]= -.0 * POI[p]; ddq[]= POI[p]; } = 0 + POI[p]*0.50*DL; Area= A* + A; f(area<= 0.) { fprntf(stderr, "ERROR: Area<0: n"); retrn -; } Jacob Jacob= fabs(ddq[0]* + ddq[]* + ddq[]*); C= Area*Yong/Jacob; Emat[0][0]= Emat[0][0] + C * WEI[p] * ddq[0] * ddq[0]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[];

41 Report-0 4 Jacoban at Gassan Qad. Ponts Jacoban Jacoban at Gassan Qad Ponts ( ) =+ =- =0

42 Report-0 4 Eample (6/7): Matr (/) for(p=0;p<;p++){ ddq[0]= POI[p]; ddq[]= -.0 * POI[p]; ddq[]= POI[p]; = 0 + POI[p]*0.50*DL; Area= A* + A; f(area<= 0.) { fprntf(stderr, "ERROR: Area<0: n"); retrn -; } Jacob= fabs(ddq[0]* + ddq[]* + ddq[]*); C= Area*Yong/Jacob; Emat[0][0]= Emat[0][0] + C * WEI[p] * ddq[0] * ddq[0]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[0][]= Emat[0][] + C * WEI[p] * ddq[0] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][0]= Emat[][0] + C * WEI[p] * ddq[] * ddq[0]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; Emat[][]= Emat[][] + C * WEI[p] * ddq[] * ddq[]; } m A J w E Emat ] [

43 Report-0 4 Eample (7/7): Matr (/) Same procedres n d.c Dag[n]= Dag[n] + Emat[0][0]; Dag[n]= Dag[n] + Emat[][]; Dag[n]= Dag[n] + Emat[][]; f (cel==0){=inde[n]; =Inde[n]+; }else {=Inde[n]+; =Inde[n]+;} =Inde[n]; =Inde[n]+; =Inde[n]; =Inde[n]+; n cel n n } AMat[]= AMat[] + Emat[0][]; AMat[]= AMat[] + Emat[0][]; AMat[]= AMat[] + Emat[][0]; AMat[]= AMat[] + Emat[][]; AMat[]= AMat[] + Emat[][0]; AMat[]= AMat[] + Emat[][]; st row: corresponds to n nd row: corresponds to n rd row: corresponds to n

44 Report-0 44 Eample (7/7): Matr (/) Same procedres n d.c Dag[n]= Dag[n] + Emat[0][0]; Dag[n]= Dag[n] + Emat[][]; Dag[n]= Dag[n] + Emat[][]; f (cel==0){=inde[n]; =Inde[n]+; }else {=Inde[n]+; =Inde[n]+;} =Inde[n]; =Inde[n]+; =Inde[n]; =Inde[n]+; n cel n n } AMat[]= AMat[] + Emat[0][]; AMat[]= AMat[] + Emat[0][]; AMat[]= AMat[] + Emat[][0]; AMat[]= AMat[] + Emat[][]; AMat[]= AMat[] + Emat[][0]; AMat[]= AMat[] + Emat[][]; st row: corresponds to n nd row: corresponds to n rd row: corresponds to n

45 Report-0 45 Wnter Semester: Parallel FEM Techncal & Scentfc Comptng II Semnar on Compter Scence II Contents Parallel Programmng sng MPI Data Strctre for Parallel FEM Implementaton of Parallel FEM Eercses sng Fjts PRIMEHPC F0 (Oaleaf-F) Parallelze femd code for D statc lnear-elastc problems n ths semester.

46 Report-0 Techncal & Scentfc Comptng I, II (Fnte Element Method) Instrctor: Kengo aajma Gradate Level Semester & Credts I : Smmer, -Credts II: Wnter, -Credts Overvew (I): Fndamental sses of fnte-element method (FEM) on statc lnear-elastc problems, ncldng lnear eqaton solvers and programmng. (II): Data strctre for parallel FEM, mplementaton of parallel FEM, framewor for development of parallel codes, sch as HPC-MW and ppopen-hpc wth Fjts PRIMEHPC F0 (Oaleaf-F) Strong collaboratons between scence & engneerng, compter scence and nmercal algorthms are reqred towards sccess of largescale scentfc smlatons sng parallel compters. Goal of these classes s that stdents of gradate school of nformaton scence and technologes nderstand reqrement of applcatons and try to develop new nterdscplnary research area. Grade based on reports Smmer (I). Fndamental Theory s for FEM, Statc Lnear-Elastc Problem. FEM by Galern Method. Sparse Lnear Solvers, Precondtoners 4. FEM Programmng D, D 5. ECCS 0 System of ITC Wnter (II). Parallel Programmng sng MPI. Data Strctre for Parallel FEM. Implementaton of Parallel FEM 4. Framewor for Development of Parallel Smlaton Codes sng Large-scale Systems 5. Fjts PRIMEHPC F0 (Oaleaf- F) 46

Numerical Solution of Deformation Equations. in Homotopy Analysis Method

Numerical Solution of Deformation Equations. in Homotopy Analysis Method Appled Mathematcal Scences, Vol. 6, 2012, no. 8, 357 367 Nmercal Solton of Deformaton Eqatons n Homotopy Analyss Method J. Izadan and M. MohammadzadeAttar Department of Mathematcs, Faclty of Scences, Mashhad