1 DERIVATION OF LAGRANGIAN SHAPE FUNCTIONS FOR HEXAHEDRAL ELEMENT HANI AZIZ AMEEN Associate Professor of Engineering and Applied Mechanics Dies and Tools Engineering Department Technical College, Baghdad, Iraq E-mail: [email protected]ABSTRACT This paper introduces a general theory for the derivation of the shape functions for the hexahedral element (20-node) in the interval [0,1] .Two basic procedures are introduced; the first by polynomials equation and the second by superposition .The paper also introduces the formulation of the three-dimensional finite element method for elasticity problems. A complete finite element program is introduced in QBASIC for the three-dimensional problems using either local coordinates [ ] 1 , 1 + − or intrinsic coordinates [ ] 1 , 0 + . INTRODUCTION The Lagrangian quadrilateral family of finite elements appeared very early in the history of the finite element method. Melosh [1] derived the 4-node rectangular element. Pain [2] gave an algorithm for the direct displacement approach with any number of unknown coefficients. The concept of arbitrary-node elements was described by Irons [3]. Argyris [4] derived the 8-node paralleiogramic element. Ergatoudis [5] derived the shape function for some Lagrangian and serendipity elements. Zafrany [6] derived the Lagrangian and Hermitian shape functions for quadrilateral element. The shape functions for 20-node hexahederal element with intrinsic coordinates using Lagrange superposition is presented in this paper. The main concept here is that the geometry of the element is defined using the nodal coordinates and the shape function which are used to interpolate the main unknowns, e.g., displacement or temperature with an iso- parametric formulation which is convenient to express the shape functions in terms of the non- dimensional element coordinates ξ, η and ζ which varies from –1 to +1 over the element for local coordinates and from 0 to +1 over the element for intrinsic coordinates [7]. This coordinates International Electronic Engineering Mathematical Society IEEMS http://www.ieems.org International e-Journal of Abstract and Applied Engineering Mathematics http://www.ieems.org/iejaaem.htm ISSN 2090-5297 Volume (1), January, 2011, pp.1-30
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1
DERIVATION OF LAGRANGIAN SHAPE FUNCTIONS FOR
HEXAHEDRAL ELEMENT
HANI AZIZ AMEEN
Associate Professor of Engineering and Applied Mechanics
ABSTRACT This paper introduces a general theory for the derivation of the shape functions for the hexahedral element (20-node) in the interval [0,1] .Two basic procedures are introduced; the first by polynomials equation and the second by superposition .The paper also introduces the formulation of the three-dimensional finite element method for elasticity problems. A complete finite element program is introduced in QBASIC for the three-dimensional
problems using either local coordinates [ ]1,1 +− or intrinsic coordinates [ ]1,0 + .
INTRODUCTION
The Lagrangian quadrilateral family of finite elements appeared very early in the history of the
finite element method. Melosh [1] derived the 4-node rectangular element. Pain [2] gave an
algorithm for the direct displacement approach with any number of unknown coefficients. The
concept of arbitrary-node elements was described by Irons [3]. Argyris [4] derived the 8-node
paralleiogramic element. Ergatoudis [5] derived the shape function for some Lagrangian and
serendipity elements. Zafrany [6] derived the Lagrangian and Hermitian shape functions for
quadrilateral element. The shape functions for 20-node hexahederal element with intrinsic
coordinates using Lagrange superposition is presented in this paper. The main concept here is
that the geometry of the element is defined using the nodal coordinates and the shape function
which are used to interpolate the main unknowns, e.g., displacement or temperature with an iso-
parametric formulation which is convenient to express the shape functions in terms of the non-
dimensional element coordinates ξ, η and ζ which varies from –1 to +1 over the element for local
coordinates and from 0 to +1 over the element for intrinsic coordinates [7]. This coordinates
International Electronic Engineering Mathematical Society IEEMS
http://www.ieems.org
International e-Journal of Abstract and Applied Engineering Mathematics
http://www.ieems.org/iejaaem.htm
ISSN 2090-5297
Volume (1), January, 2011, pp.1-30
2
system is particularly useful when the numerical integration is adopted to evaluate any element
integrals which are required during the stiffness matrix and load vector calculations.
COMPLETE POLYNOMIALS IN THREE DIMENSIONS
In the three dimensions shown in figure (1), it can be used to provide the terms in a complete
polynomial of nth order [8] which may also be found from the expression
nkjizyxzyxfp
r
kji
r ≤++=∑=
,),,(1
γ
Where the number of terms in the polynomial is given as:
( )( )( ) 6/321 +++= nnnp
Figure 1: Complete polynomials in three dimensions
It can be shown that Ni is identical to what was found previously.
8
Hexahedral Intrinsic Element In this paper the shape functions for the hexahedral element with intrinsic coordinates by superposition method is presented, now similar as before figure (4) shows the superposition of the element.
Figure 4: Superposition for intrinsic coordinates
9
The local number of this element can be shown in the table (2):
Table(2)
Local node ξi ηi ζi
1 0 0 0
2 0 1 0
3 1 1 0
4 1 0 0
5 0 0 1
6 0 1 1
7 1 1 1
8 1 0 1
9 0 0.5 0
10 0.5 1 0
11 1 0.5 0
12 0.5 0 0
13 0 0 0.5
14 0 1 0.5
15 1 1 0.5
16 1 0 0.5
17 0 0.5 1
18 0.5 1 1
19 1 0.5 1
20 0.5 0 1
ΨΨΨΨΨΨ ˆˆ),,( −−++= ζηξζηξ (12)
Applied Lagrange interpolation on the nodes in the ξ direction:
δ is the displacement vector and it takes the following form:
} w vu ........ w vu w v{ 202020222111u=δ
Elasticity Matrix [D] From Hooke's law the stress/ strain relation is
εDσ ][= (18)
Where
`}{ yxyzxyzyx τττσσσσ =
−−
−−
−−
−−
−−
−−
−+−
=
)1(2
)21(00000
0)1(2
)21(0000
00)1(2
)21(000
0001)1()1(
000)1(
1)1(
000)1()1(
1
)21)(1(
)1(][
υ
υ
υ
υυ
υ
υ
υ
υ
υ
υ
υ
υ
υυ
υ
υ
υ
υυ
υED
Where E is the Young modulus of elasticity and υ is the Poisson's ratio.
Stiffness Matrix [K] For many problems of continuum mechanics there are forms of energy balance theorems, which provide variational statement directly. The energy theorem, which is employed in this paper is "the minimum total potential energy theorem". The total potential energy can be expressed as follows [9]:
WUχ −=
Where U is the strain energy stored in the system
14
W is the work done by the external loads. Of all possible displacement states (u,v,w) a body can assume which satisfy compatibility and given kinematic or displacement boundary equilibrium equation makes the total potential energy
assuming a minimum value, i.e., =χ minimum and the variation of 0=δχ
Hence:
σε2
1=U per volume
For increment ∫∫∫= dxdydzεσUt
)(2
1
And
δFWt.=
Where F is the force vector which takes the following form:
}Fz Fy Fx ..................... Fz Fy { 202020111FxF =
Thus:
∫∫∫ −= Fδdxdydzεσχtt
)(2
1 (19)
Substitute equations (17) and (18) into equation (19) it can be deduced that:
( ) FδδBDBδχttt −∫∫∫= dzdy dx ]][[][
2
1
Minimization 0 , =∂∂δ
χχ , it can be shown that FδK =][
Where
∫∫∫= dzdy dx ]][[][][ BDBK t (20)
When modified Gaussian quadratic product rule is adopted, the integration in equation (20) is carried out using the expression [10]:
∫ ∫ ∫=1
0
1
0
1
0
d d d det[J] ),,(][ ζηξζηξfK (21)
Where
]][[][),,( BDBζηξf t=
ζηξdx d d d det[J] dzdy =
Thus equation (21) can be written:
∫ ∫ ∫=1
0
1
0
1
0
d d d ),,(][ ζηξζηξgK
∑ ∑ ∑== = =
NQ
k
NQ
j
NQ
iiiikji ζηξgwwwK
1 1 1
),,( ][
Where NQ is the number of modified Gaussian points, table(3). Table 3: Modified Gaussian points
NQ iii ζηξ ,, wi,wj,wk
2 0.2113249 0.7886751
0.5 0.5
3 0.11270167 0.5
0.88729833
0.27777778 0.44444444 0.27777778
15
COMPUTER PROGRAM The FE- program (FEEL3D) is built in QUIKBASIC language as shown in appendix. The flow chart of the program is illustrated in figure (5).
Figure 5: Flow chart for the program
COMPUTER SOLUTION EXAMPLE The cantilever beam with concentrated load is employed. The physical properties are P=20 kN L= 4 m I = 104 * 106 mm4 E=100 GN/m2
3.0=υ
Figure 6 Finite element mesh for this problem can be shown in figure (6).
Figure 7: FE-mesh for cantilever beam
16
Table (3) shows the deflection of the beam for the two element coordinates (local, intrinsic) and it can be deduced that the results for the intrinsic coordinates element is more efficient than that for local coordinates element and the error for intrinsic coordinates at maximum deflection is much better than that for local intrinsic coordinates.
Length (mm)
Theoretical deflection*
(mm)
Deflection* for intrinsic
coordinates element (mm)
Deflection* for local
coordinates element (mm)
1000 2.08 2.41 2.32
2000 8.97 9.4 9.19
3000 21.63 20.3 19.27
4000 41.02 33.72 31.13
* In the downward direction
The error for local coordinate at maximum deflection = 24.07% The error for intrinsic coordinates at maximum deflection =17.75%
CONCLUSIONS The authors have introduced a general theory for Lagrangian superposition shape functions of hexahedral element (20-node). This theory is simpler than that given in reference [8] and also the authors used their method (superposition) to derive the shape functions for hexahedral element in intrinsic coordinates, i.e., in the interval [0,1] such shape functions have been programmed by the authors in their finite element system the FEEL3D system. The results have been obtained from the intrinsic element were more accurate than for local element. Thus the hexahedral element for intrinsic coordinates is more efficient than that for local coordinates.
REFERENCES
(1) R. J. Melosh, 'Basis for Derivation of Matrices for the Direct Stiffness Method', AIAA
Journal, vol. (1),1631-1637 ,1963.
(2) T. H. Pain, 'Derivation of Element Stiffness Matrices'. AIAA, Journal, vol. (2), pp. 576-
577, 1964.
(3) B. M. Irons, 'Engineering Applications of Numerical Integration in Stiffness Methods',
AIAA, Journal, vol. (4), pp. 2035-2037, 1966.
(4) J. H. Argyris, 'Continua and Discontinua', Proc. Conf. Matrix Method in Struct. Mech.
Airforce Inst. of Tech.,Wrigh Patterson, A.F. Base ,1965.
(5) I. Erqatoudis, B. M. Irons and O. C. Zienkiewicz, 'Curved isoparametric quadrilateral
elements for finite element analysis', Int. J. Solid and Struct., vol. 4, pp. 31-42,1968.
(6) A. EL-Zafrany and R. A. Cookson, 'Derivation of Lagrangian and Hermitian Shape
functions for quadrilateral elements,' Int. J. Num. Meth. Eng., vol.(23), pp. 1939-1958,1986.
(7) E. Hinton and D. R. J. Owen, 'An Introduction to Finite Element Computations',
Pineridge Press, 1981.
(8) O. C. Zienkiewicz, 'The Finite Element Method', Mc Graw-Hill book company limited,
17
1977
(9) S. S. Rao, 'The Finite Element Method in Engineering', 2nd Edition Pergamon Press,
1989.
(10) A. H. Stroud and Don Secrest, 'Gaussian Quadrature Formulas', Prentice- Hall,
Englewood Cliffs, N. J. , 1966.
(11) P. A. F. Martins and M. J. M. Barata Marques, 'Model 3-A three Dimensional mesh
generator', Computers and Structures, vol. (42), No. (4), pp. 511-529, 1992.
Appendix
In this appendix a list of the FEEL3D program is presented and the mesh for the FEEL3D can be
made by using Reference [11], if the complex problems existed.
DECLARE SUB XJA() DECLARE SUB JINVERS() DECLARE SUB MATV (M!, N!, A!(), B!(), C(()! DECLARE SUB REACTION() DECLARE SUB CARTD() DECLARE SUB JACOB() DECLARE SUB INTRD() DECLARE SUB MATI (N!, A!(), ND(! DECLARE SUB DMATRIX() DECLARE SUB BMATRIX() DECLARE SUB MATM (M!, N!, L!, A!(), B!(), C(()! DECLARE SUB MATT (M!, N!, A!(), AT(()! DECLARE SUB ESMG() DECLARE SUB REDUCER() DECLARE SUB SOLVER() DECLARE SUB MATS (N!, A!(), B!(), DV!, ND(! DECLARE SUB LOCAD() DECLARE SUB DATIn() DECLARE SUB ASSEMBLY() DECLARE SUB DISPLACEMENT() DECLARE SUB STRESS() DECLARE SUB LOCACO() DECLARE SUB INTRCO()
'+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 'PROGRAM FINITE ELEMENT METHOD FOR 'THREE DIMENSIONAL ELASTICITY 'PROBLEMS (FEEL3D.BAS)
ASST. PROF. DR. HANI AZIZ AMEEN TECHNICAL COOLEGE – BAGHDAD – IRAQ. NOV. 2010
'========================================================================= 'MAIN PROGRAM
'========================================================================= DIM SHARED E, P, IDOF, ITY, NN, NEL, NT, NR, DETJ DIM SHARED XI, ETA, ZITA CLS INPUT "NAME OF DATA FILE"; A$ INPUT "NAME OF DISPLACEMENT FILE"; DISP$
18
INPUT "NAME OF REACTION FILE"; REAC$ INPUT "NAME OF STRESS FILE"; STRS$ OPEN A$ FOR INPUT AS #1 OPEN DISP$ FOR OUTPUT AS #2 ' DISPLACEMENT OPEN REAC$ FOR OUTPUT AS #3 ' REACTION OPEN STRS$ FOR OUTPUT AS #4 ' STRESS INPUT #1, NN, NEL, IDOF NT = NN * IDOF $'DYNAMIC DIM SHARED SKE(60, 60), SKG(NT, NT), FG(NT), RG(NT), DG(NT) DIM SHARED XG(NN), YG(NN), ZG(NN) DIM SHARED ITA(NEL, 20) DIM SHARED ISW(NT), PDIS(NT( DIM SHARED XE(20), YE(20), ZE(20), DE(60) DIM SHARED DSFX(20), DSFY(20), DSFZ(20( DIM SHARED DSFXI(20), DSFETA(20), DSFZITA(20) DIM SHARED XJINV(3, 3), XJ(3, 6) DIM SHARED B(6, 60), D(6, 6), DB(6, 60), BT(60, 6), BTDB(60, 60) DIM SHARED WQ(3), XQ(3) DIM SHARED XXI(20), EETA(20), ZZITA(20) DIM SHARED S(6), EE(6), SP(6), EP(3), NC(NN), EN(NN, 6), SN(NN, 6) DATIn DIM SHARED SKR(NR, NR), FR(NR), DR(NR) ASSEMBLY REDUCER SOLVER DISPLACEMENT REACTION STRESS SCREEN 1 PRINT "FINSHED WITH HELP OF GOD" END REM $STATIC SUB ASSEMBLY ========================================================================' 'ASSEMBLY OF EQUATION FOR THE WHOLE DOMAIN
'========================================================================= FOR IE = 1 TO NEL FOR I = 1 TO 20 XE(I) = XG(ITA(IE, I)) YE(I) = YG(ITA(IE, I)) ZE(I) = ZG(ITA(IE, I)) NEXT I CALL ESMG FOR I = 1 TO 20 FOR IR = 1 TO IDOF IL = (I - 1) * IDOF + IR IG = (ITA(IE, I) - 1) * IDOF + IR FOR J = 1 TO 20 FOR S = 1 TO IDOF JL = (J - 1) * IDOF + S JG = (ITA(IE, J) - 1) * IDOF + S SKG(IG, JG) = SKG(IG, JG) + SKE(IL, JL) NEXT S, J, IR, I NEXT IE END SUB SUB BMATRIX
'======================================================================== CALL JACOB FOR I = 1 TO 20 DSFX(I) = XJINV(1, 1) * DSFXI(I) + XJINV(1, 2) * DSFETA(I) + XJINV(1, 3) * DSFZITA(I) DSFY(I) = XJINV(2, 1) * DSFXI(I) + XJINV(2, 2) * DSFETA(I) + XJINV(2, 3) * DSFZITA(I) DSFZ(I) = XJINV(3, 1) * DSFXI(I) + XJINV(3, 2) * DSFETA(I) + XJINV(3, 3) * DSFZITA(I) NEXT I END SUB SUB DATIn
'-------------------------------------------------------------- 'READ MATERIAL DATA
'-------------------------------------------------------------- INPUT #1, E, P, ITY
'-------------------------------------------------------------- 'READ MESH DATA
'-------------------------------------------------------------- FOR II = 1 TO NN INPUT #1, I, XG(I), YG(I), ZG(I) NEXT II FOR II = 1 TO NEL INPUT #1, IE FOR I = 1 TO 20 INPUT #1, ITA(IE, I) NEXT I NEXT II
'---------------------------------------------------------------- 'READ BOUNDARY CONDITION DATA
'---------------------------------------------------------------- INPUT #1, NRN FOR I = 1 TO NRN INPUT #1, IG
20
FOR L = 1 TO IDOF INPUT #1, ISW(IDOF * (IG - 1) + L), PDIS(IDOF * (IG - 1) + L) NEXT L NEXT I NT = NN * IDOF NR = NT FOR I = 1 TO NT IF ISW(I) <> 0 THEN NR = NR - 1 NEXT I
'-------------------------------------------------------------- 'READ LOADING DATA
'--------------------------------------------------------------- INPUT #1, NLN FOR I = 1 TO NLN INPUT #1, IG FOR IR = 1 TO IDOF INPUT #1, FG(IDOF * (IG - 1) + IR) NEXT IR NEXT I
'---------------------------------------------------------------- CLOSE #1 END SUB SUB DISPLACEMENT
'========================================================================= PRINT #2, "DISPLACEMENT" IG = 0 IR = IG FOR I = 1 TO NN FOR LR = 1 TO IDOF IG = IG + 1 IF ISW(IG) = 1 THEN GOTO 10 IR = IR + 1 DG(IG) = DR(IR) GOTO 50 10 DG(IG) = PDIS(IG) 50 NEXT LR PRINT #2, I, XG(I), YG(I), ZG(I) FOR LR = 1 TO IDOF PRINT #2, DG(IDOF * (I - 1) + LR) NEXT LR NEXT I CLOSE #2 END SUB SUB DMATRIX
'======================================================================= ' D MATRIX SUBROUTINE
'======================================================================= FOR I = 1 TO 6 FOR J = 1 TO 6 D(I, J) = 0! NEXT J, I Q = (E * (1 - P)) / ((1 + P) * (1 - 2 * P)) Q1 = P / (1 – P) Q2 = (1 - 2 * P) / (2 * (1 – P)) D(1, 1) = Q D(1, 2) = Q * Q1 D(1, 3) = Q * Q1 D(2, 1) = Q * Q1
'--------------------------------------------- N = 3 FOR K = 1 TO N - 1 FOR I = K + 1 TO N A = XJ(I, K) / XJ(K, K) FOR J = K TO N XJ(I, J) = XJ(I, J) - XJ(K, J) * A NEXT J NEXT I NEXT K C = 1! FOR I = 1 TO N C = C * XJ(I, I) NEXT I DETJ = ABS(C) END SUB SUB JINVERS
'-------------------------------------------------- CALL XJA N = 3 N1 = N + 1 N2 = N * 2 FOR I = 1 TO N FOR J = N1 TO N2 XJ(I, J) = 0! IF I <> (J - N) THEN GOTO 20 XJ(I, J) = 1! 20 NEXT J NEXT I FOR K = 1 TO N A = XJ(K, K( FOR J = 1 TO N2 XJ(K, J) = XJ(K, J) / A NEXT J FOR I = 1 TO N IF I = K THEN GOTO 150 A = XJ(I, K( FOR J = 1 TO N2 XJ(I, J) = XJ(I, J) - A * XJ(K, J) NEXT J 150 NEXT I NEXT K KK = 0 FOR I = 1 TO N FOR J = N1 TO N2 KK = KK + 1 XJINV(I, KK) = XJ(I, J) NEXT J KK = 0 NEXT I END SUB SUB LOCACO
'======================================================================== 'LOCAL COORDINATE OF ELEMENT NODES
'======================================================================== FOR I = 1 TO N FOR J = 1 TO N A(I, J) = 0! NEXT J, I END SUB SUB MATM (M, N, L, A(), B(), C(()
'========================================================================= FOR I = 1 TO M FOR K = 1 TO L C(I, K) = 0 FOR J = 1 TO N C(I, K) = C(I, K) + A(I, J) * B(J, K) NEXT J, K, I END SUB SUB MATS (N, A(), B(), DV, ND)
'========================================================================= FOR I = 1 TO N FOR J = 1 TO N B(I, J) = B(I, J) + DV * A(I, J( NEXT J, I END SUB SUB MATT (M, N, A(), AT(()
'========================================================================== FOR I = 1 TO M FOR J = 1 TO N AT(J, I) = A(I, J) NEXT J, I END SUB SUB MATV (M, N, A(), B(), C(()
'========================================================================== 'MULTIPLIER MATRIX BY VECTOR SUBROUTINE
'========================================================================== FOR I = 1 TO M C(I) = 0! FOR J = 1 TO N C(I) = C(I) + A(I, J) * B(J) NEXT J, I END SUB SUB REACTION
IG = 0 FOR I = 1 TO NN FOR NR = 1 TO IDOF IG = IG + 1 RG(IG) = -FG(IG( FOR JG = 1 TO NT RG(IG) = RG(IG) + SKG(IG, JG) * DG(JG) NEXT JG, NR PRINT #3, I, XG(I), YG(I), ZG(I) FOR L = 1 TO IDOF PRINT #3, FG(IDOF * (I - 1) + L), RG(IDOF * (I - 1) + L) NEXT L NEXT I CLOSE #3 END SUB SUB REDUCER
'======================================================================== 'APPLICATION OF BOUNDARY CONDITION REDUCES OF EQUA.
'======================================================================== IR = 0 FOR I = 1 TO NT IF ISW(I) = 1 THEN 100 IR = IR + 1 FR(IR) = FG(I) JR = 0 FOR J = 1 TO NT IF ISW(J) = 1 THEN 110 JR = JR + 1 SKR(IR, JR) = SKG(I, J) GOTO 90 110 FR(IR) = FR(IR) - SKG(I, J) * PDIS(J) 90 NEXT J 100 NEXT I END SUB SUB SOLVER
'========================================================================= FOR K = 1 TO NR - 1 IF ABS(SKR(K, K)) = 0 THEN GOTO 1 FOR I = K + 1 TO NR Q = SKR(I, K) / SKR(K, K) FR(I) = FR(I) - Q * FR(K) FOR J = K + 1 TO NR SKR(I, J) = SKR(I, J) - Q * SKR(K, J) NEXT J, I, K FOR I = NR TO 1 STEP -1 SUM = FR(I) FOR J = I + 1 TO NR SUM = SUM - SKR(I, J) * DR(J) NEXT J DR(I) = SUM / SKR(I, I) NEXT I GOTO 60 1 PRINT "GAUSS FAILES" STOP 60 END SUB SUB STRESS