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7.19  Legacy information

• Legacy 2D elements
• Rules for elements in of_mk_subs
• hexa8, hexa20
• penta6, penta15
• tetra4, tetra10
• q4p (plane stress/strain)
• q4p axisymmetric
• q5p (plane stress/strain)
• q8p (plane stress/strain)
• q8p axisymmetric
• t3p (plane stress/strain)
• t3p axisymmetric
• t6p (plane stress/strain)
• t6p axisymmetric

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

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

• q4p (plane stress/strain) uses the et*2q1d routines for plane stress and plane strain.
• q4p (axisymmetric) uses the et*aq1d routines for axisymmetry. The radial u_r and axial u_z displacement are bilinear functions over the element.
• q5p (plane stress/strain) uses the et*5noe routines for axisymmetry.

  There are five nodes for this incompressible quadrilateral element, four nodes at the vertices and one at the intersection of the two diagonals.

• q8p uses the et*2q2c routines for plane stress and plane strain and et*aq2c for axisymmetry.
• q9a is a plane axisymmetric element with Fourier support. It uses the e*aq2c routines to generate matrices.
• t3p uses the et*2p1d routines for plane stress and plane strain and et*ap1d routines for axisymmetry.

  The displacement (u,v) are assumed to be linear functions of (x,y) (Linear Triangular Element), thus the strain are constant (Constant Strain Triangle).

• t6p uses the et*2p2c routines for plane stress and plane strain and et*ap2c routines for axisymmetry.

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

ω_i = 1 /8 for i=1,4

b_i for i=1,4 as below, with z=α₁

b_i for i=4,8 as below, with z=α₂

hexa8 surfaces are integrated using a 4 point rule

ω_i = 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 ω_l = w_i w_j w_k for i,j,k=1,3

with

w₁ = w₃ = 5/18 and w₂ = 8/18 b_l = (α_i, α_j, α_k) 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 ω_k = w_i w_j for i,j=1,3 with

w_i as above and b_k = (α_i, α_j) 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

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

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 ω_i = 1 /4 for i=1,4 and b_i=S_i the i-th element vertex.

tetra4 surfaces are integrated at the 3 vertices with ω_i = 1 /3 for i=1,3 and b_i=S_i 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

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

λ_j for j=1,4 are barycentric coefficients for each vertex S_j :

b_k=∑_j=1,4λ_j S_j for k=1,15

tetra10 surfaces are integrated using a 7 point rule

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

λ_j for j=1,3 are barycentric coefficients for each surface vertex S_j :

b_k=∑_j=1,3λ_j S_j 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 ω_i=1/4 and b_i=S_i for i=1,4.

For edges, q4p uses numerical integration at each corner node with ω_i=1/2 and b_i=S_i for i=1,2.

q4p axisymmetric

For surfaces, q4p uses a 4 point rule with

• ω_i = 1 /4 for i=1,4
• b₁ = (α₁,α₁) , b₂ = (α₂,α₁) , b₃ = (α₂,α₂) , b₄ = (α₁,α₂)
  with α₁ = 1 /2 − 1 /2 √3 = 0.2113249 and α₂ = 1 /2 + 1 /2 √3 = 0.7886751

For edges, q4p uses a 2 point rule with

• ω_i = 1 /2 for i=1,2
• b₁= α₁ and b₂ = α₂ the 2 gauss points of the edge.

q5p (plane stress/strain)

For surfaces, q5p uses a 5 point rule with b_i=S_i 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

• ω_k = w_i w_j for i,j=1,3 with w₁ = w₃ = 5/18 et w₂ = 8/18
• b_k = (α_i, α_j) for i,j=1,3 with α₁ = 1 − √3/5 /2 , α₂ = 0.5 and α₃ = 1 + √3/5/2

For edges, q8p uses a 3 point rule with

• ω₁ = ω₂ = 1/6 and ω₃ = 4/6
• b_i=S_i for i=1,2 corner nodes of the edge et b₃ the midside.

q8p axisymmetric

For surfaces, q8p uses a 9 point rule with

• ω_k = w_i w_j for i,j=1,3
  with w₁ = w₃ = 5/18 and w₂ = 8/18
• b_k = (α_i, α_j) for i,j=1,3
  with α₁ = 1 − √3/5/2 , α₂ = 0.5 and α₃ = 1 + √3/5/2

For edges, q8p 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

t3p (plane stress/strain)

For surfaces, t3p uses a 3 point rule at the vertices with ω_i = 1 /3 and b_i=S_i.

For edges, t3p uses a 2 point rule at the vertices with ω_i = 1 /2 and b_i=S_i.

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 ω_i = 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

• ω_i = 1 /3 for i=1,6
• b_i = S_i+3,i+4 the three midside nodes.

For edges, t6p uses a 3 point rule

• ω₁ = ω₂ = 1 /6 and ω₃ = 4 /6
• b_i=S_i, i=1,2 the i-th vertex of the actual edge and b₃ = S_i,i+1 the midside.

t6p axisymmetric

For surfaces, t6p uses a 7 point rule

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

λ_j for j=1,3 are barycentric coefficients for each vertex S_j :

b_k=∑_j=1,3λ_j S_j 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