computer program fem1d(ch7).ppt - tamu...

1

The Finite Element MethodComputer Program FEM1D

Read: Chapter 7

Review of FE Models of Chs. 3 – 6 Numerical integration in 1-D Logical units of a FEA program Flow chart of a typical processor unit Element calculations Computer program FEM1D Input data to FEM1D Example problems for FEM1D Summary

CONTENTS

Numerical Integration - 2

REVIEW OF THE 1-D FE MODELS-1

2 2 2

02 2 2 ( ),u u u uc m a b c u f x tt t x x x x

∂ ∂ ∂ ∂ ∂ ∂ + − + + = ∂ ∂ ∂ ∂ ∂ ∂

Weak formulation

Governing equation (generalized to include all 1-D problems discussed in Chapters 3-6)

2 2

02 2

2 2

02 2

1

2

3 2 4

2

2

2

0 ,

( ) ( ) ( ) ( )

a

b

a

b

x

x

x

x

a b a b

u uc mt t

u uc mt

u uv a b c u f x t dxx x x x

v u v ua b c vu dxx x x x

v x Q v x

v f

Q x Qt

Q x

( )

θ θ

∂ ∂ ∂ ∂ = − + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + ∂ ∂ ∂ ∂ −

∂ ∂+∂ ∂

∂ ∂+

− − −

∂

− + ∂


REVIEW OF THE 1-D FE MODELS-2

22

02 2

b b

a a

b

ae

x xe eij i j ij i jx x

xj je ei iij i j i i ix

C c dx, M m dx

d dd dK a b c dx, F f dx Qdx dx dx dx

W

= =

æ ö÷ç ÷ç= + + = +÷ç ÷ç ÷çè ø

ò ò

ò ò

jj jj

j jj jjj j

Full discretization

Semidiscretization

( ) ( )

K u FK K C

F C K u F F

,ˆ ˆˆ

ˆ ( ) ( )

s s s s

s ss

s s s s ss s

t

t t

a

a a a

+ + +

+ +

+ + +

=

= D +

= - - D +D + -

1 1 1

1 1

1 1 11 1

( )( )

K u FK K M C

F F M u u u

C u u u

ˆ ˆˆ

ˆ

s s s

s s s s

s s s s s s

s s s s

a a

a a a

a a a

+ + +

+ + + +

+ + +

+

=

= + +

= + + +

+ + +

1 1 1

1 1 1 13 5

1 1 13 4 5

15 6 7

e e e e e e e+ + = C u M u K u F

( ) ( ) General

( ) ( ) Gauss rule

b

a

NPTx

ij ij ij I Ix

I

NGP

ij ij ij I II

G F x dx F x W

K F d F Wx x x

=

+

- =

= » -

= » -

åò

åò

1

1

11

NUMERICAL INTEGRATION


,

NGPor NGPor NGP

pNGP

p

p

p

é ù+ê ú=ê úë û

= == == =

12

1 1

2 3 2

4 5 3

Nearest larger integer equal to (p+1)/2

( ) [ ]

( ) ( )

iF Fh

I F d F F

F F W

x x

x

+

-

+= = + =

= ´ =

ò1

1 21 2

1

1 1

22 2

2 0

•

x=-1

•

( )F 0

x=+1x

( )F x

5

NUMERICAL INTEGRATION-6

6

NUMERICAL INTEGRATION-7

● ●

x x ξ

01

eax x

xξ

===-

1 2

eh

1

be

e

x xx hξ

===+

12

12

1 12 2

(1 )(1 )

(1 ) (1 )

ea

eea ee ea b

x x xx hx x h

x x

ξξ

ξ ξ

= += += + += - + +

121 1 1

122 2 2

( ) ; ( ) 1 ; ( ) (1 )

( ) ; ( ) ; ( ) (1 )

ee e eb

e e

ee e ea

e e

x x xx xh h

x x xx xh h

-= = - = -

-= = = +

y y y x x

y y y x x

21 1 12 2 21 2 2( ) (1 ), ( ) (1 ), ( ) (1 )e e e=- - =- + = -y x x x y x x y x x x

● ●

ξ

1●

eh

2 3

Computer Implementation: 7

LOGIAL UNITS OF A FEA PROGRAM

POSTPROCESSORCompute the solution

and its derivatives atdesired points of thedomain

Print/plot the results

PREPROCESSOR

Read geometry and material data finite element mesh data boundary (and initial) conditions

PROCESSOR Generate finite element mesh Calculate element matricesAssemble element equationsApply boundary conditions Solve the equations Check error tolerance and maximum

number of allowable iterations

JN Reddy

PREPROCESSOR

PROCESSOR

POSTPROCESSOR

ECHODATA and

MESH1D

ASSEMBLE

BOUNDARY

EQNSOLVRor

EIGNSLVR

POSTPROC or

REACTION

COEFFCNTor

TRANSFRM

SHAPE1D

FLOW CHART OF PROGRAM FEM1D

Computer Implementation: 8JN Reddy

Initialize global K and F

DO N = 1 to NEM

Call ELMATRCS to calculate K(N)

and F(N), and assemble to form global K and F

Transfer global information(material properties, geometry,

and solution) to element

Print solution STOP

Flow Chart of a Typical PROCESSOR Unit

Call BOUNDARY to impose boundary conditions and call

SOLVER to solve the equations

Computer Implementation: 9JN Reddy

10

Variables used in the program

NPE - nodes per element, ELX( ) - Global coordinate of the th node of element , ELK( , ) - Element coefficient, ELF( ) - Element coefficient,

0, 1 Coefficients in the definition o

ei

eij

ei

ni i e xi j Ki f

AX AX − f ( ) : ( ) 0 1*SFL( ) Element hape (or approximation) funtion, DSFL( ) Derivative of the th shape function with respect to

the local (normalized) coordinate :

GDSFL( ) Derivative of the

ei

i

a x a x AX AX xi s

i idd

i

= +−

−

−

y

yx

x

1

th shape function with respect

to the global coordinate : i i i

id d ddx Jdx d dx d

− = ≡

y y yxx x

FINITE ELEMENT PROGRAM FEM1D-1

11

1 11

1

1 1

( ) ( ) * ( ),

1 1( ) ( ) ( ) ( )

ˆ ˆ( ) (GAUSPT( , )) * ( , )

1 1ˆ ( ) ( ) ( )

b

a

n ne ej j

j jx

j ji ii j i j e

e exNGP NGP

ij I I ijI I

iij i j

e

x x ELX j SFL j

d dd dx x dx J ddx dx J d J d

F w F I NGP GAUSWT J NGP

dFJ d J

= =

−

= =

= =

+ = +

≈ =

= +

y x

y yy yy y y x y x x

x x

x

yx y x y x

x,

1 , or 0.5

( , ) thGauss point, ,for the -point Gauss rule( , ) thGauss weight, ,for the -point Gauss rule

je

e

i i ie e e

e

I

I

dJ

dd d dd dxdx J d J hdx dx d J d d

GAUSPT I j I jGAUSWT I j I w j

= = = = =

−−

y

xy y yx

xx x x

x

Numerical Integration

FINITE ELEMENT PROGRAM FEM1D-2

12

NSPV – Number of specified primary variables of the problem.

ISPV(I,J) – Array containing the information about the global node number and the local

degree of freedom that is specified.

ISPV(I,1) – For the Ith boundary condition, the global node number at which the BC is specified.

ISPV(I,2) – For the Ith boundary condition, the local degree of freedom that is specified.

VSPV(I) – The specified value of the deg. of freedom.

Similar meaning for NSSV, ISSV, and VSSV for specified secondary variables

IMPOSOITION OF BOUNDARY CONDITIONS

13

••

••

•

•

•

•• •

•

•••

•

• •• •

•

••

••

• • •••

• •••

••

••

•

•NDF = 36

I

( )16 17 18, ,U U U

( )1 2 3

1

, ,( ) *

K K KU U UK I NDF

+ + +

= -

•

NDF= number of primary degrees of freedom at a node

IMPOSOITION OF BOUNDARY CONDITIONS

14

_______________________________________________________________* Data Card 1: TITLE* Data Card 2: MODEL, NTYPE, ITEM

MODEL=1, NTYPE=0: A problem of MODEL EQUATION 1MODEL=1, NTYPE=1: A circular DISK (PLANE STRESS)MODEL=1, NTYPE>1: A circular DISK (PLANE STRAIN)MODEL=2, NTYPE=0: A Timoshenko BEAM (RIE#) problemMODEL=2, NTYPE=1: A Timoshenko PLATE (RIE) problemMODEL=2, NTYPE=2: A Timoshenko BEAM (CIE##) problemMODEL=2, NTYPE>2: A Timoshenko PLATE (CIE) problemMODEL=3, NTYPE=0: A Euler-Bernoulli BEAM problemMODEL=3, NTYPE>0: A Euler-Bernoulli Circular plateMODEL=4, NTYPE=0: A plane TRUSS problemMODEL=4, NTYPE=1: A Euler-Bernoulli FRAME problemMODEL=4, NTYPE=2: A Timoshenko (CIE) FRAME problemITEM=0, Steady-state solutionITEM=1, Transient analysis of PARABOLIC equationsITEM=2, Transient analysis of HYPERBOLIC equationsITEM=3, Eigenvalue analysis

* Data Card 3: IELEM, NEMIELEM=0, Hermite cubic finite elementIELEM=1, Linear Lagrange finite elementIELEM=2, Quadratic Lagrange finite element

Table 7.3.2: Description of the input variablesto FEM1D (revised from the book)-1

15

* Data Card 4: ICONT, NPRNTICONT=1, Data (AX,BX,CX,FX and mesh) is continuousICONT=0, Data is element dependentNPRNT=0, Not print element or global matrices

but postprocess the solution and printNPRNT=1, Print Element 1 coefficient matrices only

but postprocess the solution and printNPRNT=2, Print Element 1 and global matrices but

NOT postprocess the solutionNPRNT>2, Not print element or global matrices and

NOT postprocess the solutionSKIP Cards 5 through 15 for TRUSS/FRAME problems (MODEL = 4), andread Cards 5 through 15 only if MODEL.NE.4._____________________

* Data Card 5: DX(I)Array of element lengths. DX(1) denotes the global coordinate ofNode 1 of the mesh; DX(I) (I=2,NEM+1) denotes the length of the (I-1)st element. Here NEM denotes the number of elements in themesh.

Table 7.3.2: Description of the input variablesto the program FEM1D -2

16

Cards 6 through 9 define the coefficients in the model equations. All coefficients are expressed in terms of GLOBAL coordinate x. See Table 7.3 for the meaning of the coefficients, especially for deformation of circular plates and Timoshenko elements.

* Data Card 6: AX0, AX1, AX2

* Data Card 7: BX0, BX1

* Data Card 8: CX0, CX1

SKIP Card 9 for eigenvalue problems (i.e. ITEM=3)

* Data Card 9: FX0, FX1, FX2

* Data Card 10: NNM


17


* Data Card 11: DCAX(n,i), i=1,2,3

* Data Card 12: DCBX(n,i), i=1,2

* Data Card 13: DCCX(n,i), i=1,2

* Data Card 14: DCFX(n,i), i=1,2,3

READ Cards 15 through 20 for TRUSS/FRAME problems (MODEL = 4).SKIP Cards 15 through 20 if MODEL.NE.4 ___________________

* Data Card 15: NNM

SKIP Cards 16 through 18 for TRUSS problems (NTYPE = 0). Cards

16 through 18 are read for each element, i.e., NEM times _________

SKIP Cards 11 through 16 if data is continuous (ICONT.NE.0). Cards 11-14 are read for each element (i.e., NEM times). All coefficients are expressed in terms of the GLOBAL coordinates.

18


* Data Card 16: PR, SE, SL, SA, SI, CS, SNPR - Poisson's ratio of the material#SE - Young's modulus of the materialSL - Length of the elementSA - Cross-sectional area of the elementSI - Moment of inertia of the elementCS - Cosine of the angle of orientation of the elementSN - Sine of the angle of orientation of the element; the

angle is measured counter-clock-wise from x axis

* Data Card 17: HF, VF, PF, XB, CNT, SNTHF - Intensity of the horizontal distributed forceVF - Intensity of the transversely distributed forcePF - Point load on the elementXB - Distance from node 1, along the length of the

element, to the point of load application, PFCNT - Cosine of the angle of orientation of the load PFSNT - Sine of the angle of orientation of the load PF; the

angle is measured counter-clock-wise from x axis

* Data Card 18: NOD

READ Cards 19 and 20 only for TRUSS problems (NTYPE = 0). Cards19 and 20 are read for each element; i.e. NEM times ______________

* Data Card 19: SE, SL, SA, CS, SNSE - Young's modulus of the materialSL - Length of the elementSA - Cross-ectional area of the elementCS - Cosine of the angle of orientation of the elementSN - Sine of the angle of orientation of the element

Angle is measured counter-clock-wise from x axisHF - Intensity of the horizontal distributed force

* Data Card 20: NOD

* Data Card 21: NCON

19


20

SKIP Card 22 if no constraint conditions are specified (NCON = 0). RepeatCard 22 NCON times:__________________________________________

* Data Card 22: ICON (i), VCON(i) * Data Card 23: NSPV

SKIP Card 24 if no primary variables is specified (NSPV=0). RepeatCard 24 NSPV times ______________________________________

* Data Card 24: ISPV(I,1), ISPV(I,2), VSPV(I) (I = 1 to NSPV)

ISPV(I,1) - Node number at which the PV is specifiedISPV(I,2) - Specified local primary DOF at the nodeVSPV(I) - Specified value of the primary variable (PV)

(will not read for eigenvalue problems)

SKIP Card 25 for eigenvalue problems (i.e. when ITEM=3) __________

* Data Card 25: NSSV


21


SKIP Card 26 if no secondary variable is specified (NSSV=0). RepeatCard 26 NSSV times __________________________________________

* Data Card 26: ISSV(I,1), ISSV(I,2), VSSV(I) (I = 1 to NSSV)ISSV(I,1) - Node number at which the SV is specifiedISSV(I,2) - Specified local secondary DOF at the nodeVSSV(I) - Specified value of the secondary variable (PV)

* Data Card 27: NNBC

SKIP Card 28 if no mixed boundary condition is specified (NNBC=0).The mixed boundary condition is assumed to be of the form: SV+VNBC*(PV-UREF) = 0. Repeat Card 28 NNBC times ____________________

* Data Card 28: INBC(I,1), INBC(I,2), VNBC(I), UREF(I) (I = 1 to NNBC)INBC(I,1) - Node number at which the mixed B.C. is specifiedINBC(I,2) - Local DOF of the PV and SV at the nodeVNBC(I) - Value of the coefficient of the PV in the B.C.UREF(I) - Reference value of the PV

22


* Data Card 29: NMPCSKIP Card 30 if no multipoint constraints are specified (NMPC=0).Repeat Card 30 NMPC times ________________________

* Data Card 30: I=1, NMPC(IMC1(I,J),J=1,2),(IMC2(I,J),J=1,2),(VMPC(I,J),J=1,4)

SKIP Card 31 if ITEM=0 (read for time-dependent and eigenvalue problems)* Data Card 31: CT0, CT1

CT0 - Constant part of CT = CT0 + CT1*XCT1 - Linear part of CT = CT0 + CT1*X

SKIP remaining cards if steady-state or eigenvalue analysis is to beperformed (ITEM=0 or ITEM=3) __________________

* Data Card 32: DT, ALFA, GAMADT - Time increment (uniform)

ALFA - Parameter in the time approximation schemeBETA - Parameter in the time approximation scheme

23


* Data Card 33: INCOND, NTIME, INTVL

INCOND- Indicator for initial conditions:INCOND=0, Homogeneous (zero) initial conditionsINCOND>0, Nonhomogeneous initial conditionsNTIME - Number of time steps for which solution is soughtINTVL - Time step intervals at which solution is to printed

SKIP Cards 34 and 35 if initial conditions are zero (INCOND=0) ___

* Data Card 34: GU0(I){GU0} - Array of initial values of the primary variables

SKIP Card 34 for parabolic equations (ITEM=1) ________________

* Data Card 35: GU1(I){GU1} - Array of initial values of the first time-derivatives of the

primary variables

24

TO RUN THE EXECUTABLE PROGRAM FEM1DF15.EXE

Executable Computer Programs from the book, AnIntroduction to the Finite Element Method by J. N. Reddy,3rd ed., McGraw--Hill, 2006.

Notes:Programs FEM1D and FEM2D are a revised versions of theprograms from the second edition of the book. The revisions areminor. Both FEM1D and FEM2D were compiled using theMicrosoft Fortran compiler, with a fixed array dimensions.Hence, only a limited size problem can be analyzed using thecompiled versions of the programs. The programs were compiledwith a maximum number of degrees of freedom of 2,500. If yourcomputer has the storage and you have the source programs, youmay recompile the program after changing the DIMENSIONstatements in the programs.

25

TO RUN THE EXECUTABLE PROGRAM FEM1D.EXE

To run a program on a PC, the files should be downloaded to your PCinto a folder (say, FEM_Reddy). The user is required to prepare a datafile for each problem he or she wants to solve, using the instructions inthe book (use Table 7.3.2 for FEM1D and Table 13.4.1 for FEM2D). Mosterrors are mistakes made in the preparation of the data files. Therefore,you must check the data files when you see 'run-time error' message orthe program is not executed (by returning just the echo of the input datafile). All files should be in the same folder where FEM1D.EXE andFEM2D.EXE are placed.

To run the program: Double click on the executable file (FEM1D.EXE or FEM2D.EXE).

A window (called Command Prompt window) will pop open (withwhite letter and black background). It will read

File name missing or blank – please enter file nameUNIT 5?

26

TO RUN THE EXECUTABLE PROGRAM FEM1D.EXE

Type the input file name (with its extension) and press Enter. Forexample, if the file name you prepared is labeled as Prob1.inp, youmay type it as prob1.inp (not case sensitive). The Command Promptwindow will now display

File name missing or blank – please enter file nameUNIT 6?

Type the output file name (with its extension) and press Enter. Forexample, if you want the computer to return the output to fileProb1.out, you may type it as prob1.out. Note that you do not preparethis file; it will be created by the computer. The Command Promptwindow will disappear, indicating that it has taken the data file andexecuted the program. You will find the file prob1.out in the samefolder where you are running the program. Depending on the resultsyou see, you may have to correct the data file.

27

InsulatedL

Convection heat transferthrough surface

L = 0.05, c = 400, β = 100, k = 50

(all with proper units)(a)

3000 =θ

x

(b) 1 2 3 4 5

300)0(1 ==θU 042

===Lxdx

dQ θ

)3DX()2DX(000 .xx == )4DX( )5DX(5,4,3,2I

0125.0DX(I)0.0)1DX( 0

=====

hx

Example 1: One-Dimensional Heat transfer in a Rod2

2 0 0,d c x Ldx

qq- + = < < 2 400Pc m

Akb

= = =

EXAMPLE PROBLEMS FOR PROGRAM FEM1D

28

Example 1: Heat transfer in a rod (4 linear elements) 1 0 0 MODEL, NTYPE, ITEM 1 4 IELEM, NEM 1 1 ICONT, NPRNT

0.0 0.0125 0.0125 0.0125 0.0125 DX(1)=X0; DX(2), etc. 1.0 0.0 0.0 AX0, AX1, AX2 0.0 0.0 BX0, BX1

400.0 0.0 CX0, CX1 0.0 0.0 0.0 FX0, FX1, FX2

1 NSPV 1 1 300.0 ISPV(1,1), ISPV(1,2), VSPV(1) 0 NSSV 0 NNBC0 NMPC

One-Dimensional Heat transfer in a Rod

29

Example 2: 1-D Heat transfer in a composite wall

Material 1, k1 Material 2, k2 Material 3, k3

Surface area,A (=1)

T∞= 50ºCT0= 200ºC

h3h1 h2

•1 2 3 4

x• • • )CW/(m10C50

cm 4 cm 2.5

cm 2 )CW/(m 20 ,)CW/(m 40 )CW/(m 70

2

3

2

1

3

2

1

⋅=

=

===

⋅=

⋅=

⋅=

∞

βThhhkkk

1 200U =(3)

42 ( ) 0Q U Tb ∞+ − =

30

1-D Heat transfer in a composite wallExample 2: Heat transfer in a composite wall

1 0 0 MODEL, NTYPE, ITEM 1 3 IELEM, NEM 0 1 ICONT, NPRNT 0.0 0.02 0.025 0.04 DX(I)

70.0 0.0 0.0 AX0, AX1, AX2 Data for 0.0 0.0 BX0, BX1 Element 1 0.0 0.0 CX0, CX1 0.0 0.0 0.0 FX0,FX1,FX2


31


1 NSPV 1 1 200.0 ISPV(1,1), ISPV(1,2), VSPV(1) 0 NSSV 1 NNBC 4 1 10.0 50.0 INBC(1,1),INBC(1,2),VNBC(1),UREF(1) 0 NMPC

1-D Heat transfer in a composite wall

VNBC(1) = , UREF(1) = b T∞

32

Example 3: Axial deformation of a composite bar

Al Al

d = 4 in.

12 in 4 in8 in

d = 2 in.d = 2 in.

Rigid plateSteel (St) , Es = 30 × 106 psi

Aluminum (Al), Ea = 10 × 106 psi

P = 100 kips = 105 lb

StP

P

1010 lb/ink =Linear elastic spring

● ● ● ●1 2 3 4

1 0U =(1) (2)2 1 2Q Q P+ = − (2) (3)

2 1 0Q Q+ =

(3)42 0Q kU+ =

33

Example 3: Axial deformation of a composite bar1 0 0 MODEL, NTYPE, ITEM1 3 IELEM, NEM0 2 ICONT, NPRNT

0.0 12.0 8.0 4.0 DX(I)

12.56637E07 0.0 0.0 AX0, AX1, AX20.0 0.0 BX0, BX10.0 0.0 CX0, CX10.0 0.0 0.0 FX0, FX1, FX2



Element 1

Element 2

34



1 NSPV1 1 0.0 ISPV(1,1),ISPV(1,2),VSPV(1)1 NSSV2 1 −2.0E05 ISSV(1,1),ISSV(1,2),VSSV(1) 1 NNBC4 1 1.0E10 0.0 INBC(1,1),INBC(1,2),VNBC(1),UREF(1) 0 NMPC

Element 3

35

Example 4: Bending of a Beam

Linear spring, k = 10-4 EI (N/m)

1,000 N/m

5 m

2,500 N

x

•c

2.5 m5 m

•

1,250 N-mEI = 2×106 N-m2z

● ● ●1 2 3

1

2

00

UU

==

3

(1) (2)4 2

01,250

UQ Q

=+ =

(2)53

(2)4

2,5000

Q kUQ

+ = −=

36

Example 4: Bending of a beam

Example 4: Clamped and Spring-supported Beam (EBT)3 0 0 MODEL, NTYPE, ITEM0 2 IELEM, NEM0 1 ICONT, NPRNT0.0 5.0 5.0 DX(I)

0.0 0.0 0.0 AX0, AX1, AX2 Data for2.0E6 0.0 BX0, BX1 Element 10.0 0.0 CX0, CX1

−1.0E3 0.0 0.0 FX0, FX1, FX2 0.0 0.0 0.0 AX0, AX1, AX2 Data for2.0E6 0.0 BX0, BX1 Element 20.0 0.0 CX0, CX1 0.0 0.0 0.0 FX0, FX1, FX2

37

3 NSPV1 1 0.0 ISPV(1,1), ISPV(1,2), VSPV(1)1 2 0.0 ISPV(2,1), ISPV(2,2), VSPV(2)2 1 0.0 ISPV(3,1), ISPV(3,2), VSPV(3)

2 NSSV2 2 1250.0 ISSV(1,1), ISSV(1,2), VSSV(1)3 1 -2500.0 ISSV(2,1), ISSV(2,2), VSSV(2)

1 NNBC (with transverse spring)3 1 2.0E02 0.0 INBC(1,1),INBC(1,2),VNBC(1),UREF(1)0 NMPC

Example 4: Bending of a beam

SUMMARY

Computer Implementation: 38

Review of FE Models of Chapters 3 – 6 Numerical integration in 1-D Logical units of a FEA program Flow chart of a typical processor unit Element calculations Computer program FEM1D Input data to FEM1D Example problems for FEM1D Summary

We have discussed the following topics in this presentation:

computer program fem1d(ch7).ppt - tamu...

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