1
A
Finite Volume Method for Solids with a Rotational Degree of Freedom
Based on the
6

Node Triangle
W. Pan
*
¹, M.A. Wheel¹ and Y. Qin
2
1
. Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK.
2. Department of Design, Manufacture Engin
eering and Management, University of Strathclyde,
Glasgow, UK.
* corresponding author, email: wenke.pan@strath.ac.uk
Abstract:
A finite

volume (FV) cell vertex method is presented for determining the
displacement field for solids exhibiting with
incompre
ssibility
. The solid is discr
e
ti
z
ed
into six

node finite elements and the standard six

node finite

element shape function is
employed for each element. Only control volume
s
around vertex node of the triangular
element are considered. For considering the ma
terial incompressibility, a constant
hydrostatic pressure as an extra unknown variable within each element is assumed. The
force equilibrium in two perpendicular directions and one in

plane moment equilibrium
equation are derived for each control volume. T
he volume conservation is satisfied by
setting the integration of volumetric strain as zero within each element. By solving the
system control equations, the displacements and rotations of the vertex nodes and the
hydrostatic pressure for each element can
be obtained and then the displacements of the
midside nodes can be calculated. The
simulation
results show that this FV method
passes the patch tests and converges to theoretical results under mesh refinement
for
material behaviour incompressibility
.
Keyw
ords
:
Finite Volume Method,
Control volume, Vertex centred method,
Rotational
degree, Incompressibility.
2
1. Introduction
Finite element (FE) method has dominated solid and structure analysis for many years,
while
FV
method has been successfully
used
for
fluid dynamic analysis
.
Over the last
fifteen years, some progress
es
have been made for
employing
FV method for solids and
structures.
Apart from the application of FV method to elastic
[1

6
], elastic

plastic
[
7
],
and plate analysis [
8
,
9
] for compressible
materials
. FV method has also been applied to
large deformation compressible and near

incompressible hyperelastic material
analysis[10,11].
For
incompressible material deformation analysis
, a t
hree

node and
four

node element has been used for analysis 2D e
lastic media by using the cell central
FV method [
12
],
a
bi

linear displacement field was assumed within the quadrilateral
constructed by the connection of cell central and adjoining edge nodes, and also the
constant hydrostatic pressure within each eleme
nt.
Mixed
form
FV
method
was also
presented for 2D and 3D
and also fracture problems with
incompressible linear elastic
materials
[
1
3
]
, the non

physical
stresses
oscillations
near crack tip
was eliminated.
FV
method that
included the Allman degree of rotat
ion for 3

node[
1
4
] and 6

node
FV [15]
analysis for 2D elastic problems
were presented, some advantages have been
demonstrated
for some cases. In this paper,
a
mixed form
FV cell vertex method is
presented for determining the displacement field for solids e
xhibiting incompressibility.
The similar discretization method with linear strain distribution six

node triangle
element
as [15] is used
. To include the rotation degree of freedom, the midside
displacements are replaced by its neighbouring vertex nodes[
16]
,
and t
he constant
hydro

static pressure within each element is assumed although there is no difficulty for
the assumption of a
linear distribution of hydrostatic pressure within each element.
The
3
force equilibrium in two perpendicular directions and one
in

plane moment equilibrium
equation are derived for each control volume.
T
he volume conservation is satisfied by
setting the integration of volumetric strain as zero within each element
for small strain
elastic problem
. The total number of the force and m
oment equilibrium equations plus
the
number of
volume conservation equation
s
is
the same as the number of the total
unknown variables due to the special construction of the
CV.
By solving these
equations, the displacements and rotations of the vertex nodes
and the hydrostatic
pressure for each element can be obtained and then the displacements of the middle

side
nodes can be calculated.
In section 2,
formulation for
displacement,
strain and stress
filed will be given
, and also the
compensation equation
rela
ted with material
incomp
ressibility
.
The
construction of
special
CV and the
system equations are
given in
section 3,
and the
test problems
will be followed in section 4, and the
final section
is the
conclusions
.
2.
Formulation
for d
isplacement, strain and
stress fields
The six

node triangle element is considered for the discretization of the 2D
incompressible material problem.
The same formulation
for
displacement
and
strain
field within each element
as [
15
] are used in this paper
and are
summarized
as
fol
lowing:
2.1 Displacement fields
The displacement
components,
u
and
v
, within a
six

node triangular
element
are
:
(1)
4
Where
and
are coefficients. By a
pplying
equations (1) at every no
de of the triangle,
gives:
(2)
Where
, and
u
i
,
v
i
, are displacement components of
vertex nodes (for i=1,2,3) and midside nodes (for i=4,5,6) along x and y direction
respectively
.
For the introduction o
f rotation degree of freedom of each vertex nodes,
t
he midside
node displacement
will be replaced by the neighbouring vertex node displacements and
rotations. The relationship can be repressed by the transformation matrix
T
,
(3)
W
here
i
,
(i=1,
2,
3)
are rotations of the vertex nodes of 6

node triangle,
and
matrix
T
can be found in [
16
].
With the use of relationship between strain and displacement,
the
direct strain
components
,
and
shear strain
are
(
4
)
Where
A
is a constant
matrix
,
the components of v
ector
and
are constant or
linear function of coordinators x or y, and
A
,
and
can be found in
[
15
].
By using
the
constitutive relationship between stresses and strain and the hydrostatic
pressure
, the stress components are
5
(
5
)
where
G
is the shear modulus
,
p
is hydro
static pressure
.
Put (
4
) into (
5
),
gives
(
6
)
3. Control volume and the equilibrium equations
3.1 Control volume for cell vertex FV procedure
One difference between FE and
FV method is that equilibrium equations must be built
on e
ach individual
CV
for FV, under the cell centre FV procedure, the CV is the same
as FE element, while for cell vertex FV procedure, the CV is completely different from
FE element
.
There are some ways to build CV for cell vertex FV procedure.
One method t
o construct
CV
for
cell vertex FV
procedure
is
to
connect the middle point
between neighbouring nodes and the centre of the element
to form a multi

side complex
shape CV
.
Other method was presented in [
17
],
some position of the points needed to be
connecte
d to construct CV was obtained by
trial and error
method.
In this paper, due to
the introduction of rotational degree of freedom, special CV
as used in [
15
] was
adopted. Figure
1
(a)
shows the
typical
CV
s
(constructed by dotted lines) around
a
vertex node
A
and B inside and on the boundary of the model respectively
.
Assuming the total number of vertex node and element are
‘
nd
’
and ‘
ne
’ respectively,
thus the total numer of the system unknown variables will be ‘3*
nd
+
ne
’. There are two
force equilibrium equat
ions along perpendicular directions and one moment equilibrium
equation for each CV, so the total number of equation related with forces and moment
are ‘3*
nd
’, and it is less than the total number of unknown variables. Therefore, the
6
compensation equation
has to be built. The incompressibility condition for each element
can be used for this purpose.
Figure
1
Typical
CV
s
and inner stresses
.
Solid circle, open circle and small triangle
represent vertex nodes, midside nodes and geometrical centre o
f FE element. (a
) CVs
around the inside
vertex
node A and boundary
vertex
node B (b) in
n
er stresses on one
of the segments of the profile of CV.
3.2
Force and moment e
quilibrium equations
To
obtain
the force and
moment
equilibrium equations for each CV,
the
resultant forces
and moment have to be integrated along the profile of each CV. From
formula
(
5
), it can
be seen that stress components within each CV is linearly distributed, so the resultant
forces and moment can be integrated analytically along
the
profile of each CV.
For one
typical segment of the CV shown in Figure
1
(b), the coordinators
of any point
along the
segment can be expressed as
x
C
y
P
S
xx
xx
yx
xy
(a)
(b)
A
B
O
D
C
D
7
(7)
where
the angle between the normal of line CD and the x

axis is
and the distance
from point
C
to P
is
S
.
By
integrating
explicitly
t
he
stresses
components
along the profile of each CV, the
resultant force components
and
acting in
x
and
y
directions
are
.
(8.1)
(8.2)
T
he resultant moment about the origin of the coordinat
e
system
is
:
(
8.3
)
By summarizing all the resultants force and moment along the profile of each CV
,
the
equilibrium equation for
eac
h
CV
can be written as following:
(9)
Where
,
and
are external forces and moment acting on the CV.
3
.3
Compensation equation for incompressibility
For linear e
lastic material, the incompressibility of each element can be expressed as
8
(10)
For simplicity, the strain
components
at the centre of the triangular element
were used
for above formula (
10
), therefore,
(11)
Each element has one compensation equation and one unknown hydrostatic pressure
variable.
3.4 system equations
Combining formulation (
9
) and (
11
), the system equation can be
assembled
as
following:
(12)
where
3.5
. Boundary conditions
For concentrated point force or moment
acting on each CV
, they can be direct
ly
put into
the right

hand side of equation
s
(
12
). For distributed stresses
on
boundar
ies
, result
ant
forces alone x and y dir
ection and moment to the origin of coordinator can be easily
integrated along the profile of CV, then
the
same procedure as concentrated forces and
moments can be used. The stiffness matrix of the whole structure shown in equation
s
9
(
12
) is
a
singular matri
x, displacement and rotation boundary
conditions
have to be
applied
to the equations
, the
standard
procedure as the treatment of displacement
boundary conditions
for
FE can be used.
That is, l
et the diagonal element of the
stiffness
matrix
corresponding to
the
i
th
constraint
as one, while the
other
s
in the same row as
zero
, also let the
i
th
element of the
right

hand side
vector
as the exact value of the
constraint.
By solving
the modified
Eqn. (
12
), the vertex displacements
,
rotations
and
hydrostatic pressu
re
can be obtained, after which, using Eqn. (
3
), the midside node
displacements can be calculated. From Eqns. (
4
) and (6), the strain and stress field can
be determined subsequently.
4
. Numerical
test
results
Four 2D
plane strain
benchmark problems
with
incompressible material behaviour
were
considered. The first and the second examples are isotropic plate un
der
simple tension
and
simple
shear
, the third one is case of a
square plate subject to pure bending. The
fourth case is an
infinite long tube subje
cte to internal pressure.
The value of
material’s
shear
modulus,
G
,
is
chosen as 1.0GPa
.
For
each case, the calculated results were
compared with that of theoretical results.
4
.1 Test 1
and 2
: simple tension
and shear
of a square plate
An infinitely long s
quare plate with the length and width of 2m*2m subjected to simple
tension
stress 1.0MPa
is shown
by the solid lines
in Figure 2
. A
quarter model is
analysed
and the model is discretized into four different size 6

node triangle elements.
The dotted lines i
n Figure 2 show the expected deformed mesh. The calculated results
show that the constant strain field is exactly predicted.
Figure 3 shows the initial and the
deformed mesh of an infinitely long square plate with length and width of 1m*1m
10
subject to simpl
e shear. As the simple tension case, the solid and dotted lines represent
initial and deformed mesh. From this figure, the exact deformation shape is obtained.
4.2 Test problem 3: Square plate subjected to pure bending
The third test example, sh
own in Figure
4
(a half model)
, is an isotropic square plate
with length 2.0m
subject to pure bending loading
and the linearly varying edge normal
stress is given by
(13)
Theoretical results
of the displacement components are as f
ollowing
(14)
where
Young’s modulus
E
=2
G
(1+
)
is and
=0.5
.
x
igure
2
deformed and undeformed
mesh of a plate subjected to simple
tension, magnify factor 1e5
Figure
3
deformed and undeformed
mesh of a plate subjected
to simple
shear, magnify factor 2e4
B
y
11
Figure 4
A
thick
square
plate subject to pure bending
Figure 5 Top right corner vertical displacement changes with the number of division
along x dir
ection
Figure 6 Top right corner horizontal displacement changes with the number of division
along x direction
The half plate model is tested using a consistent refinement of a regular mesh with the
same number of division along x and y direction
s. The boundary conditions are: The
horizontal displacements and in

plane rotations along left edge are zeros; the vertical
displacement at the middle point of the left edge is zero.
Figure 5 show
s
the comparison
between the calculated and theoretical resu
lts of vertical displacement of top right
corner under mesh refinement. From this figure, it can be seen that the calculated results
12
converge to the theoretical results. Figure 6 shows the results of the horizontal
displacement at the top right corner, sim
ilar trend as Figure 5 can be observed although
the horizontal displacement converges slower than that of vertical displacement.
4.3 Test problem 4: Inflation of an infinitely long tube
under internal pressure
Figure
7(a)
is a schematic drawing of an infi
nitely long tube
(a quarter model)
subjected
to internal pressure, the tube material is linear incomprensible material with shear
modulus 1
00
.0GPa. The analytical solution of radial displacement is
(15)
Figure
7
Infin
itely long tube subjected to internal pressure. (a) a quarter model with
loading and boundary shown, (b) the u
ndeformed
(solid line) and deformed
m
esh(dashed line)
,
solid and empty circle represent nodes
attached to undeformed and
deformed mesh respectivel
y, the number of
division along the radial and hoop
directions are all 4
, d
isplacement ampli
fication
factor
is
6
e4
.
(a)
(b)
A
B
C
D
13
Figure 8
The relati
ve
error
varies with the number
of
division along radial direction
Figure 9
The comparison between numerical
and analytical results
The tube inner and outside radius are 0.1m and 0.4m respectively, the internal pressure
is 1.0MPa. The symmetry boundary conditions were applied for the
line AB and CD
,
and the rotations along these two lines are all constrained.
F
igure 7(b) shows
a
the
undeformed and deformed mesh
of a special case
with the number of division along
radial and hoop direction are all 4.
From this figure, the axi

symmetrical deformation is
observed.
Figure 8 shows the relative error of the inner born
radial displacement
against the number of divisions along the radial direction. With the mesh refinement,
the relative error becomes smaller and smaller.
In
Figure 9
, with the division along
radial and hoop direction of 32,
the comparison between the nume
rical and analytical
results of radial displacement profile along boundary AB is presented.
It can be clearly
seen that the presented method can give good results.
5. Conclusions
14
Six

node triangle element with h
igh order displacement field
was used for c
ell vertex
FV procedure to tackle the 2D problem with material behaviour incompressibility.
Constant hydrostatic pressure within each element was assumed. W
ith special
construction of CV and the replacement of midside node displacement by neighbouring
edge
node displacement and rotations, the total number of unk
n
own variables equals to
the total
number of
force and moment equilibrium equations. The benchmark test
examples shows that the presented FV method can predict the constant strain field
exactly, and
can
also predict the linear
and more complex strain field
with the mesh
refinement
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