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7.19  Legacy information

This section gives data that is no longer used but is important enough not to be deleted.

7.19.1  Legacy 2D elements

These elements support isotropic and 2-D anisotropic materials declared with a material entry described in m_elastic. Element property declarations are p_solid subtype 2 entries

 [ProId  fe_mat('p_solid','SI',2)  f  N  0]

Where

fFormulation : 0 plane stress, 1 plane strain, 2 axisymmetric.
NFourier coefficient for axisymmetric formulations
Integset to zero to select this family of elements.

The xy plane is used with displacement DOFs .01 and .02 given at each node. Element matrix calls are implemented using .c files called by of_mk_subs.c and handled by the element function itself, while load computations are handled by fe_load. For integration rules, see section 7.19.2. The following elements are supported

7.19.2  Rules for elements in of_mk_subs

hexa8, hexa20

The hexa8 and hexa20 elements are the standard 8 node 24 DOF and 20 node 60 DOF brick elements.

The hexa8 element uses the et*3q1d routines.

hexa8 volumes are integrated at 8 Gauss points

ωi = 1 /8 for i=1,4

bi for i=1,4 as below, with z1

bi for i=4,8 as below, with z2

hexa8 surfaces are integrated using a 4 point rule

ωi = 1 /4 for i=1,4

b1= ( α1 , α1 ) , b2= ( α2 , α1 ) , b3= ( α2 , α2 ) and b4= ( α1 , α2 )

with α1= 1 /2−1 /2 √3=0.2113249 and α2= 1 /2+1 /2 √3=0.7886751.

The hexa20 element uses the et*3q2c routines.

hexa20 volumes are integrated at 27 Gauss points ωl = wi wj wk for i,j,k=1,3

with

w1 = w3 = 5/18 and w2 = 8/18 bl = (αi, αj, αk) for i,j,k=1,3

with

α1 = 1 − √3/5 /2 , α2 = 0.5 and α3 = 1 + √3/5 /2

α1 = 1 − √3/5 /2 , α2 = 0.5 and

hexa20 surfaces are integrated at 9 Gauss points ωk = wi wj for i,j=1,3 with

wi as above and bk = (αi, αj) for i,j=1,3

with α1 = 1 − √3/5 /2 , α2 = 0.5 and α3 = 1 + √3/5 /2 .

penta6, penta15

The penta6 and penta15 elements are the standard 6 node 18 DOF and 15 node 45 DOF pentahedral elements. A derivation of these elements can be found in [43].

The penta6 element uses the et*3r1d routines.

penta6 volumes are integrated at 6 Gauss points

Points bkxyz
1aac
2bac
3abc
4aad
5bad
6abd

with a=1 /6=.16667, b=4/6=.66667, c=1 /2−1 /2 √3=.21132, d= 1 /2+1 /2 √3=.78868

penta6 surfaces are integrated at 3 Gauss points for a triangular face (see tetra4) and 4 Gauss points for a quadrangular face (see hexa8).

penta15 volumes are integrated at 21 Gauss points with the 21 points formula

a= 9 − 2 √15 /21 , b= 9 + 2 √15 /21 ,

c= 6 + √15 /21 , d= 6 − √15 /21 ,

e= 0.5 ( 1 − √ 3 /5 ),

f= 0.5 and g= 0.5 ( 1 + √ 3 /5 )

α = 155 − √15 /2400 , β = 5 /18 ,

γ = 155 + √15 /2400 , δ = 9 /80 and є = 8 /18 .

Positions and weights of the 21 Gauss point are

Points bkxyzweight ωk
1dde α.β
2bdeα.β
3dbeα.β
4caeγ.β
5cceγ.β
6aceγ.β
7 1 /3 1 /3 eδ.β
8ddfα.є
9bdfα.є
10dbfα.є
11cafγ.є
12ccfγ.є
13acfγ.є
141 /3 1 /3 fδ.є
15ddgα.β
16bdgα.β
17dbgα.β
18cagγ.β
19ccgγ.β
20acgγ.β
211 /3 1 /3 gδ.β

penta15 surfaces are integrated at 7 Gauss points for a triangular face (see tetra10) and 9 Gauss points for a quadrangular face (see hexa20).

tetra4, tetra10

The tetra4 element is the standard 4 node 12 DOF trilinear isoparametric solid element. tetra10 is the corresponding second order element.

You should be aware that this element can perform very badly (for poor aspect ratio, particular loading conditions, etc.) and that higher order elements should be used instead.

The tetra4 element uses the et*3p1d routines.

tetra4 volumes are integrated at the 4 vertices ωi = 1 /4 for i=1,4 and bi=Si the i-th element vertex.

tetra4 surfaces are integrated at the 3 vertices with ωi = 1 /3 for i=1,3 and bi=Si the i-th vertex of the actual face

The tetra10 element is second order and uses the et*3p2c routines.

tetra10 volumes are integrated at 15 Gauss points

Points bkλ1λ2λ3λ4weight ωk
1 1 /4 1 /4 1 /4 1 /4 8 /405
2baaaα
3abaaα
4aabaα
5aaabα
6dcccβ
7cdccβ
8ccdcβ
9cccdβ
10eeffγ
11feefγ
12ffeeγ
13effeγ
14efefγ
15fefeγ

with a = 7 − √15 /34 = 0.0919711 , b = 13 + 3 √15 /34 = 0.7240868 , c = 7 + √15 /34 = 0.3197936 ,
d = 13 − 3 √15 /34 = 0.0406191 , e = 10 − 2 √15 /40 = 0.0563508 , f = 10 + 2 √15 /40 = 0.4436492

and α = 2665 + 14 √15 /226800 , β = 2665 − 14 √15 /226800 et γ = 5 /567

λj for j=1,4 are barycentric coefficients for each vertex Sj :

bk=∑j=1,4λj Sj for k=1,15

tetra10 surfaces are integrated using a 7 point rule

Points bkλ1λ2λ3weight ωk
1cdcα
2dccα
3ccdα
4bbaβ
5abbβ
6babβ
7 1 /3 1 /3 1 /3 γ

with γ = 9 /80 = 0.11250 , α = 155 − √15 /2400 = 0.06296959 , β = 155 + √15 /2400 = 0.066197075 and a = 9 − 2 √15 /21 = 0.05961587 , b = 6 + √15 /21 = 0.47014206 , c = 6 − √15 /21 = 0.10128651 , d = 9 + 2 √15 /21 = 0.797427

λj for j=1,3 are barycentric coefficients for each surface vertex Sj :

bk=∑j=1,3λj Sj for k=1,7

q4p (plane stress/strain)

The displacement (u,v) are bilinear functions over the element.

For surfaces, q4p uses numerical integration at the corner nodes with ωi=1/4 and bi=Si for i=1,4.

For edges, q4p uses numerical integration at each corner node with ωi=1/2 and bi=Si for i=1,2.

q4p (axisymmetric)

For surfaces, q4p uses a 4 point rule with

For edges, q4p uses a 2 point rule with

q5p (plane stress/strain)

For surfaces, q5p uses a 5 point rule with bi=Si for i=1,4 the corner nodes and b5 the node 5.

For edges, q5p uses a 1 point rule with ω = 1 /2 and b the midside node.

q8p (plane stress/strain)

For surfaces, q8p uses a 9 point rule with

For edges, q8p uses a 3 point rule with

q8p (axisymmetric)

For surfaces, q8p uses a 9 point rule with

For edges, q8p uses a 3 point rule with

t3p (plane stress/strain)

For surfaces, t3p uses a 3 point rule at the vertices with ωi = 1 /3 and bi=Si.

For edges, t3p uses a 2 point rule at the vertices with ωi = 1 /2 and bi=Si.

t3p (axisymmetric)

For surfaces, t3p uses a 1 point rule at the barycenter (b1=G) with ω1 = 1 /2 .

For edges, t3p uses a 2 point rule at the vertices with ωi = 1 /2 and b1 = 1 /2 − 2 /2 √3 and b2 =1 /2 + 2 /2 √3 .

t6p (plane stress/strain)

For surfaces, t6p uses a 3 point rule with

For edges, t6p uses a 3 point rule

t6p (axisymmetric)

For surfaces, t6p uses a 7 point rule

Points bkλ1λ2λ3weight ωk
1 1 /3 1 /3 1 /3 a
2αββb
3ββαb
4βαβb
5γγδc
6δγγc
7γδγc

with :

a = 9 / 80 = 0.11250 , b = 155 + √15/2400 = 0.066197075 and
c = 155 − √15/2400 = 0.06296959

α = 9 − 2 √15/ 21 = 0.05961587 , β = 6 + √15/ 21 = 0.47014206
γ = 6 − √15/ 21 = 0.10128651 , δ = 9 + 2 √15/ 21 = 0.797427

λj for j=1,3 are barycentric coefficients for each vertex Sj :

bk=∑j=1,3λj Sj for k=1,7

For edges, t6p uses a 3 point rule with ω1 = ω3 = 5/18 , ω2 = 8/18

b1 = 1− √3/5/2 = 0.1127015 , b2 = 0.5 and b3 = 1 + √ 3/5 / 2 = 0.8872985


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