SLIDE 1
GENERAL PROCEDURE OF FE SOFTWARE FE SOFTWARE PROCEDURE PRE - - PowerPoint PPT Presentation
GENERAL PROCEDURE OF FE SOFTWARE FE SOFTWARE PROCEDURE PRE - - PowerPoint PPT Presentation
GENERAL PROCEDURE OF FE SOFTWARE FE SOFTWARE PROCEDURE PRE PROCESSING SOLVING POST PROCESSING SOLVING GENERAL PROCEDURE ENGINEERING MATHEMATICS (PDE) FEM DERIVATION COMPUTER PROGRAMMING CODE VALIDATION FREE VIBRATION
SLIDE 2
SLIDE 3
SOLVING
ENGINEERING MATHEMATICS (PDE) COMPUTER PROGRAMMING CODE VALIDATION
GENERAL PROCEDURE
FEM DERIVATION
SLIDE 4
COLOR GRAPHICS COMPUTER PROGRAMMING
FREE VIBRATION
FINITE ELEMENT METHOD ENGINEERING MATHEMATICS
C C ELEMENT CONDUCTION MATRIX: C DO 100 I=1,3 DO 100 J=1,3 AKC(I,J) = 0. DO 110 K=1,2 AKC(I,J) = AKC(I,J) + BT(I,K)*B(K,J) 110 CONTINUE AKC(I,J) = TK*AREA*THICK*AKC(I,J) 100 CONTINUE DO 120 I=1,3 DO 120 J=1,3 AKE(I,J) = AKE(I,J) + AKC(I,J) 120 CONTINUE C C ELEMENT CONVECTION MATRICES: C IF(LTYPE(IE,3).NE.1) GO TO 300 FAC = H*AREA/12. DO 230 I=1,3 DO 230 J=1,3 AKH(I,J) = FAC 230 CONTINUE DO 240 I=1,3 AKH(I,I) = 2.*FAC 240 CONTINUE FAC = H*AREA*TI/3. DO 250 I=1,3 QH(I) = FAC 250 CONTINUE DO 260 I=1,3 QE(I) = QE(I) + QH(I) DO 260 J=1,3 AKE(I,J) = AKE(I,J) + AKH(I,J) 260 CONTINUE 300 CONTINUE
2 2 2 2 2 2
d x d y d z m k x y z dt dt dt
2
K M
mx kx
SLIDE 5
- Free vibration system
k
x(t)
m
Newton’s 2nd law of motion
x
F ma mx
mx kx
ENGINEERING MATHEMATICS (PDE)
FREE VIBRATION ANALYSIS
SLIDE 6
mx kx
Mass matrix Stiffness matrix Displacement ( 3 dimensions )
FEM DERIVATION
FREE VIBRATION ANALYSIS
K M
- Finite element equation
SLIDE 7
2
M K
FEM DERIVATION
FREE VIBRATION ANALYSIS
Circular frequency Mode shapes
- Harmonics motion
time, t
Applitude,
Period, T = 1/ f
x
( ) sin( ) t t 2 f
Finite element equation for free vibration
SLIDE 8
FREE VIBRATION ANALYSIS
SLIDE 9
FREE VIBRATION ANALYSIS
SLIDE 10
COMPUTER PROGRAMMING
FREE VIBRATION ANALYSIS
Example in FORTRAN
SLIDE 11
T e
K V B C B
1 1 1 1 2 2 (1 )(1 2 ) 1 2 2 1 2 2 E C
1 2 3 4 1 2 3 4 1 2 3 4 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4
1 6 b b b b c c c c d d d d B c b c b c b c b V d c d c d c d c d b d b d b d b
SLIDE 12
CODE VALIDATION
FREE VIBRATION ANALYSIS
Exact Solution Numerical Solution (FDM , FEM, FVM, etc.) Commercial Software (ANSYS, NASTRAN, etc.) Experimental Result
0.0 0.8 0.6 0.4 0.2 1.0 0.0 0.8 0.6 0.4 0.2 1.0
Present Exact solution
u
2 90 4 180 6 8 10 270 360 Yoon, et al. Present
Nu
- 0.4
- 0.6
20 10
- 0.2
30 0.0 0.2 0.4 40 50 60 0.6 70 Present Malan et al. Sampaio
v t
SLIDE 13
HEAT TRANSFER ANALYSIS SOLVING
FE ANALYSIS MUST SATISFY ENERGY EQUATION FOURIER’S LAW
SLIDE 14
COLOR GRAPHICS COMPUTER PROGRAMMING
HEAT TRANSFER
FINITE ELEMENT METHOD ENGINEERING MATHEMATICS
C C ELEMENT CONDUCTION MATRIX: C DO 100 I=1,3 DO 100 J=1,3 AKC(I,J) = 0. DO 110 K=1,2 AKC(I,J) = AKC(I,J) + BT(I,K)*B(K,J) 110 CONTINUE AKC(I,J) = TK*AREA*THICK*AKC(I,J) 100 CONTINUE DO 120 I=1,3 DO 120 J=1,3 AKE(I,J) = AKE(I,J) + AKC(I,J) 120 CONTINUE C C ELEMENT CONVECTION MATRICES: C IF(LTYPE(IE,3).NE.1) GO TO 300 FAC = H*AREA/12. DO 230 I=1,3 DO 230 J=1,3 AKH(I,J) = FAC 230 CONTINUE DO 240 I=1,3 AKH(I,I) = 2.*FAC 240 CONTINUE FAC = H*AREA*TI/3. DO 250 I=1,3 QH(I) = FAC 250 CONTINUE DO 260 I=1,3 QE(I) = QE(I) + QH(I) DO 260 J=1,3 AKE(I,J) = AKE(I,J) + AKH(I,J) 260 CONTINUE 300 CONTINUE
t T c Q z q y q x q
z y x
T K K K T C
r h c
r h q Q c
Q Q Q Q Q
T K K K T C
r h c
r h q Q c
Q Q Q Q Q
SLIDE 15
STRUCTURAL ANALYSIS SOLVING
FE ANALYSIS MUST SATISFY EQUILIBRIUM EQUATION STRESS – STRAIN REL. STRAIN – DISPLACEMENT REL.
SLIDE 16
COLOR GRAPHICS COMPUTER PROGRAMMING
SOLID MECHANICS
FINITE ELEMENT METHOD ENGINEERING MATHEMATICS
DO 20 I=1,3 DO 30 J=1,6 B(I,J) = B(I,J)/(2.*AREA) BT(J,I) = B(I,J) 30 CONTINUE 20 CONTINUE C C ELASTICITY MATRIX: C FAC = ELAS/(1.-PR*PR) C(1,1) = FAC C(1,2) = FAC*PR C(1,3) = 0. C(2,1) = C(1,2) C(2,2) = C(1,1) C(2,3) = 0. C(3,1) = 0. C(3,2) = 0. C(3,3) = FAC*(1.-PR)/2. C C ELEMENT STIFFNESS MATRIX: C DO 100 I=1,3 DO 100 J=1,6 DUMA(I,J) = 0. DO 200 K=1,3 DUMA(I,J) = DUMA(I,J) + C(I,K)*B(K,J) 200 CONTINUE 100 CONTINUE
SLIDE 17
FLUID DYNAMICS ANALYSIS SOLVING
FE ANALYSIS MUST SATISFY CONSERVATION OF MASS CONSERVATION OF MOMENTUM CONSERVATION OF ENERGY
SLIDE 18
LOW-SPEED INCOMPRESSIBLE FLOW
COLOR GRAPHICS COMPUTER PROGRAMMING ENGINEERING MATHEMATICS FINITE ELEMENT METHOD
DO 110 I=1,6 DO 110 J=1,3 DO 110 K=1,3 DO 110 L=1,6 CXX = CXX + A(IA,I)*B(I,J)*A(IB,L)*B(L,K)*G(J,K) CYY = CYY + A(IA,I)*C(I,J)*A(IB,L)*C(L,K)*G(J,K) CXY = CXY + A(IA,I)*C(I,J)*A(IB,L)*B(L,K)*G(J,K) CYX = CYX + A(IA,I)*B(I,J)*A(IB,L)*C(L,K)*G(J,K) 110 CONTINUE SXX(IA,IB) = 2.*ANEW*CXX + ANEW*CYY SXY(IA,IB) = ANEW*CXY SYX(IA,IB) = ANEW*CYX SYY(IA,IB) = ANEW*CXX + 2.*ANEW*CYY 100 CONTINUE C C COMPUTE [HX] AND [HY] MATRICES: C DO 150 IA=1,3 DO 150 IB=1,6 CX = 0. CY = 0. DO 160 I=1,6 DO 160 J=1,3 CX = CX + A(IB,I)*B(I,J)*G(J,IA) CY = CY + A(IB,I)*C(I,J)*G(J,IA) 160 CONTINUE HX(IA,IB) = CX/DEN HY(IA,IB) = CY/DEN 150 CONTINUE
SLIDE 19
HIGH-SPEED COMPRESSIBLE FLOW
COLOR GRAPHICS COMPUTER PROGRAMMING ENGINEERING MATHEMATICS FINITE ELEMENT METHOD
DO 110 I=1,6 DO 110 J=1,3 DO 110 K=1,3 DO 110 L=1,6 CXX = CXX + A(IA,I)*B(I,J)*A(IB,L)*B(L,K)*G(J,K) CYY = CYY + A(IA,I)*C(I,J)*A(IB,L)*C(L,K)*G(J,K) CXY = CXY + A(IA,I)*C(I,J)*A(IB,L)*B(L,K)*G(J,K) CYX = CYX + A(IA,I)*B(I,J)*A(IB,L)*C(L,K)*G(J,K) 110 CONTINUE SXX(IA,IB) = 2.*ANEW*CXX + ANEW*CYY SXY(IA,IB) = ANEW*CXY SYX(IA,IB) = ANEW*CYX SYY(IA,IB) = ANEW*CXX + 2.*ANEW*CYY 100 CONTINUE C C COMPUTE [HX] AND [HY] MATRICES: C DO 150 IA=1,3 DO 150 IB=1,6 CX = 0. CY = 0. DO 160 I=1,6 DO 160 J=1,3 CX = CX + A(IB,I)*B(I,J)*G(J,IA) CY = CY + A(IB,I)*C(I,J)*G(J,IA) 160 CONTINUE HX(IA,IB) = CX/DEN HY(IA,IB) = CY/DEN 150 CONTINUE
{U} {E} {F} t x y
2 2
u v u u p uv {U} ;{E} ; {F} v uv v p u pu v pv
n * i i u
M U t C uU R
2 n k us j i us
t u K u U R 2
2 n k ps ps
t u K p R 2