A Six-node Triangle Finite Volume Method for Solids with a Rotational Degree of
Freedom
Wenke Pan*, Marcus Wheel
Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK.
Abstract:
A novel vertex centred finite-volume (FV) method is presented for determining the
displacement field in loaded two-dimensional structures. The solid is discretized into
six-node finite elements (FE) and the standard six-node FE shape function is employed
to describe the displacement variation in each element. However, the displacements of
the midside nodes are replaced with the displacements and rotations of the two
neighbouring edge nodes. Thus, only the control volumes centred around each vertex
node of the triangular element are required. These control volumes are constructed by
joining lines between the centre of the element and the midside nodes. The force
equilibrium in two perpendicular directions and one in-plane moment equilibrium
equation are derived for each control volume. By solving the system equations, the
displacements and rotations of the vertex nodes can be obtained and then the
displacements of the midside nodes can be calculated. Following this, the strain and
stress can be determined through the relationship between strain and displacement, and
stress and strain, respectively. The results of simulation show that this novel FV method
is not only better than the previously derived three-node triangle FV method with
rotational degrees of freedom, but also better than the three-node FE method with
rotations, and can also compete with the six-node triangle FE method. In addition, no
locking problem was encountered as the value of Poisson’s ratio of the elastic material
approached 0.5.
1
Keywords: Vertex centred finite volume method; Rotational degree of freedom; Sixnode triangle element; Moment equilibrium equations.
1. Introduction
The finite element method has dominated the simulation of solids and structures for
many years, while the FV method has been used successfully for fluid dynamic
analysis. Over the last fifteen years, there has been some progress is using the FV
method for solids and structures. Initially, the FV method was used for small strain
elastic compressible and incompressible material for 2D plane strain, plane stress, axialsymmetry and 3D analysis [1-6], and also dynamic analysis [7]. Following this, it was
used for plate analysis [8-9] and then for large deformation hyperelastic [10-11] and
elastic-plastic materials [12]. The FV method has also been used for coupled analysis of
interacting fluids and structures in which the same discretization method is used for
both media [13-14]. This unified method may show some potential advantage for this
type of analysis.
Two common methods to construct the FV control volume are the cell vertex method
and the cell centre method. For the cell-centre method, the displacement field of the
shape function cannot be higher than second order, whilst for the cell-vertex FV
method, the standard FE shape function can be used directly.
Some special FE methods incorporating a rotational degree of freedom(DOF) for 2D
analysis have been presented and their shape functions can also be used for FV analysis.
2
Allman [15-16] presented a compatible triangular finite-element including vertex
rotations for plane elasticity analysis. When replacing the middle-node displacement
with the displacements and rotations of the neighbouring vertex node, Cook et al [17]
compared the results between 3-node, 6-node triangular and 4-node and 8-node
quadrilateral finite elements and pointed out that there are some advantages for the
inclusion of rotations in some cases. Onate et al [18] presented an FV method for beam
analysis which includes a rotation, the discritization procedure involving the four types
of equations of bending moment, shear force, rotation and deflection. Based on [15],
Wenke et al. [19] included the Allman rotation for 3-node FV analysis for 2D elasticity
problems, the results showing that there are some advantages over the 3-node FE
method with or without rotations and the FV method without rotations for some cases.
In this paper, a 6-node triangular finite-element shape function was used for 2D elastic
analysis. To incorporate rotations in the present FV analysis, control volumes were
constructed around each vertex node of the six-node triangle only; there are no volumes
around the midside node unlike other reported methods [20]. Two force and a moment
equilibrium equations were written for each control volume. By using the assumption in
[17], the total system of equilibrium equations can be transformed to including only the
vertex node displacements and rotations and thus the DOF of each triangular element is
then reduced from 12(6*2, 6 nodes with displacement components u and v) to 9 (3*3, 3
vertex nodes with u, v and rotation ). Because of the special structure of the CV, the
total number of unknown variables is equal to the total number of equilibrium
equations. By solving these equations, all unknown edge displacements and rotations
can be obtained, and then the midside node displacement can be determined. The
3
second section of this paper gives an overview of the triangular element and the third
section discusses the construction of the CV, gives an overview of the formulation and
details the displacement and stress boundary conditions. The fourth section provides
some examples for testing the method and also a comparison between the different
methods. Finally, conclusions are given.
2. Displacement, strain and stress fields for the 6 node triangular finite element
2.1 Displacement fields
Fig. 1 shows a six-node 2-D triangular element. A conventional six-node triangular
polynomial shape function is used and the displacement components, u and v, within the
triangles are assumed to be
u  a1  a 2 x  a3 y  a 4 x 2  a5 xy  a6 y 2

2
2
 v  b1  b2 x  b3 y  b4 x  b5 xy  b6 y
(1)
where x and y are the Cartesian coordinates of any point in the interior of the element.
Applying equations (1) at every node of the triangle, gives:
a  A 1u

1
b  A v
where
a  a1
(2)
a2 ... a6  ,
T
b  b1 b2 ... b6 
T
and
u  u1 u 2 ... u6 
T
v  v1 v2 ... v6  , while matrix A is given in the Appendix.
T
2.2 Strain fields
The expressions of the strain within the triangle can be deduced from the strain and
displacement relationship to provide the following:
4

u
 xx 
 B xa

x

v

 yy 
 B yb

y

  v  u  B b  B a
x
y
 xy x y

(3)
where  xx ,  yy and  xy are the usual direct and shear strain components and matrices
B x and B y are given in the Appendix.
2.3 Stress fields
Elastic isotropic or anisotropic material can be considered, using Hook’s law, for the
stress and strain relationship of the material:
σ  Dε
(4)
T
where the stress and strain vectors are σ   xx  yy  xy  and ε   xx
 yy  xy T
and
 D11
D   D21
 D31
D12
D22
D32
D13   D1T 


D23   D 2T 
D33  D3T 


(5)
The vector D1, D2 and D3are the first, second and third row of the matrix D
respectively.
Thus:
 D1T ε
 xx 


σ   yy   Dε  D2T ε
 
D3T ε
 xy 


(6)
3. Control volume and the equilibrium equations
3.1 Control volume
5
The conventional cell vertex based FV method is incompatible with the six-node
triangle element unless the construction of the CV’s is modified. An obvious
modification is shown in Fig. 2(a). Another approach is described in [20]. There is a
control volume surrounding every node including the midside ones. The control
volumes are multi-faced polygons and are constructed by connecting the centre of the
triangles to points midway between the midside and vertex nodes. For each vertex node,
there are two CV edges located within each of the elements connected to it. The multifaced polygons are the control volumes associated with a vertex nodes (edge node A
and midside node B) located inside the solid and a vertex node (Edge node C) on the
boundary. For a 2D problem, each control volume has two force equilibrium equations
along perpendicular directions and each node has two unknown displacement variables,
so that the total number of equations is equal to the total number of unknown variables.
Solving these equations, the unknown nodal displacements can be obtained and the
strains and stresses determined subsequently.
In this paper, rotational degrees of freedom are incorporated in the analysis (see Fig.3),
by replacing the midside node displacements with the displacements and rotations of the
vertex nodes. In so doing, the final unknown variables will consist of only the
displacements and rotations of the vertex nodes. The modified method for the
construction of the control volumes in a mesh of 6 node triangles cannot be used here,
however conventional vertex based CVs can be constructed. There are several ways to
build these. The easiest way to construct them for this purpose is shown in Fig. 2(b).
The control volume is constructed by the connection of the centre point of the
neighbouring triangular element to the neighbouring midside node. The control volume
for an internal node, A, and a boundary edge node, C, are shown in Fig. 2(b). Fig. 4 (a)
6
details the control volume around a vertex node. Each control volume has three
equations that include two forces and a moment equilibrium equation. Thus, the total
number of discrete equilibrium equations will be the same as the total number of
unknown variables that comprise two displacements and a rotation for each node.
3.2 Equilibrium equations and stiffness matrix
From Fig. 4(b), for any point P(x,y) along the line AB:
 x  x A  S sin 

 y  y A  S cos 
(7)
where  is the angle between the normal of line AB and the x-axis and S is the distance
from point A to P.
The resultant force components Fx and Fy acting in x and y directions are:
 FX   S AB ( xx cos   xy sin  )ds
0

S AB
 FY  0 ( xy cos   yy sin  )ds
(8)
Also, the resultant moment about the origin of the coordinate system is:
M o  
S AB
0
( xx cos   xy sin  ) yds  
S AB
0
( xy cos    yy sin  ) xds
(9)
Also, the resultant forces and moments can be integrated analytically, to provide the
following:
  B x ds
0  1

 A
FX  D1 cos   D3 sin   0
 B y ds   0
  B y ds  B x ds  


0  u 
 
A 1   v 
(10.1)
  B x ds
0  1

 A
FY  D 2 sin   D3 cos   0
 B y ds   0
  B y ds  B x ds  


0  u 
 
A 1   v 
(10.2)


T
T
T
T


7
and:
Mo 

  B x yds


T
T
  D1 cos   D3 sin   0

  B y yds



  B x xds

T
T
 D 2 sin   D3 cos   0

  B y xds






0  
 
yds
B
 y  

 B x yds    A 1
 0
0 


 B y xds 

 B x xds 
0  u 
 
A 1   v 
(10.3)
where the derivation of the matrices  B x xds ,  B x yds ,  B y xds and  B y yds are given
in the Appendix. For each control volume, the forces acting in the x and y directions and
the moment about the origin of the coordinate system must be equivalent to those acting
externally, where these can be expressed as follows:
 j n j
FX  FXE  0

j 1
 j  n j
E
  FY  FY  0
 j j n1
  M Oj  M OE  0
 j 1
(11)
where FXE , FYE and M OE are the sum of any external forces and moments applied to the
control volume, and n is the total number of CV internal edges around each boundary
edge node. The moment equilibrium equation for each volume is necessary to enable the
rotation to be introduced and this is different from the standard finite-volume method
for solids. For the conventional 2D FV method for solids, there is no need to consider
the moment equilibrium equation, but for the FV method presented in this paper, the
impending introduction of the extra unknown variable (the rotation), requires that the
moment equilibrium equation must be considered.
8
By assembling all equations of every control volume the total equilibrium equations of
the problem considered can be written as follows:
uv
K Xv

  FXE 

 u  
K Yvuv       FYE 
v
uv  
 M E 

M mv
O 

uv
 K Xu
 uv
 K Yu
uv
M mu
(12)
The unknown variables in the above equations are the displacements of all element
nodes and the stiffness matrix is therefore not square. The displacements of the midside
nodes of each element can be replaced by the displacements and rotations of the two
neighbouring vertex nodes. Assuming that the transformation matrix is T [17], the
following equation shows the relationship between the unknown variables of the
midside nodes and that of the vertex nodes:
u
1
v1 ... v6   T * u1 u2
T
... u6
u3
v1 v2
v3 1 2 3  (13)
T
For each finite element the linear transformation matrix T is a 12*9 matrix and can be
found in [17]. Substituting (13) into (12) finally gives the system equations as follows:
 K Xu
K
 Yu
K Mu
K Xv
K Yv
K Mv
K X   u    FXE 
  

K Y   v     FYE 
K M  ω  M oE 
(14)
where:
v ω  u1 ... u np
u
T
F
E
X
FYE
M OE   FxE1
T
v1 ... vnp
... FnpE
FyE1
1 ... np 
T
... FnpE
M 1E
... M npE 
T
and ‘np’ is the total number of vertex nodes. By solving Eqn. (14), the vertex
displacements and rotations can be obtained, after which, using Eqn. (13), the midside
node displacements can be calculated. From Eqns. (3) and (6), the strain and stress field
can be determined subsequently.
9
4. Boundary conditions
Displacement boundary conditions can be one or all of the three types:
u p  u 0p , v p  v 0p and  p  w0p
They can be applied directly to the system equations (14) as in the conventional FE
method.
The force boundary can be a concentrated point force or a distributed stress. For a
concentrated force at the edge node, FXp  FXp0 , FYp  FYp0 and M p  M p0 . They can be
put into Eqn. (14) directly in the usual way.
For a distributed stress, for example the uniformly distributed stress shown in Fig. 5:
FXA   0 S AC cos  , FYA   0 S AC sin  and
M OA   0 S AC cos  * ( xC  x A ) / 2   0 S AC sin  * ( yC  y A ) / 2
which in effect enable the distributed loads to be replaced by equivalent nodal forces
and moments.
5. Numerical results
A number of benchmark problems, including simple tension and simple shear are
considered. Then convergence and performance near incompressibility are tested for a
square plate under a pure bending loading. Then the case of a rectangular plate with
different ratios of length-to-width subjected to pure bending is investigated. Finally, a
comparison of the present FV method and previous FV methods and FE methods for a
cantilever beam subjected to varied load is performed. The values of Young’s modulus,
E, and Poisson’s ratio, ν, were chosen as 1.0GPa and 0.3 respectively for test one to test
four (but not for the near incompressibility tests). For the last test, the material
properties are the same as that of the reference [15] for simplicity of the comparison.
10
5.1 Test 1: simple tension of a square plate
The first test problem is the simple tension of a square thin plate with the length and
width both being 2.0m. A quarter model is simulated, shown in Fig. 6, for which the
boundary conditions are UA=0, VA=0, UB=0,VC=0 and also A=B=C=0. A constant
horizontal tensile stress of 1MPa is applied on the right side of the plate by imposing
equivalent forces to the right edge nodes and moments to the corner nodes as required.
The whole model is discritized as four 6-node elements and hence there are five
corresponding control volumes. Table 1 shows the comparison of the calculated and
theoretical results of displacement, strain and stress at point D(0.75m, 0.2m). As
expected the results are exact.
5.2 Test 2: simple shearing of a square plate
The simple shear model is shown in Fig. 7. Constant shear stresses are applied along all
the plate(size 1m*1m) sides, the value of the shear stress being 1.0MPa. The boundary
conditions are as follows: UA=0, VA=0, UB=0, A=0. The same discritization is used as
for Test 1, and the investigation point is point D(0.75m, 0.2m). Table 2 shows the
comparison of the simulated results and the exact results. From Table 1 and Table 2, the
simulated results are noted to be the same as the exact results within the limits of
computational accuracy. Actually, by investigating the entire set of results for both
models, a linear displacement field and a constant strain and stress field are observed.
Thus, both of the patch tests have been passed. Although the midside node
displacements are replaced by the neighbouring edge node displacements and rotations,
due to the assumption of quadratic displacement field (see equation (1)), the
displacement field within the 6-node triangle element is still in quadratic form, the
simulation results can predict the exact results for constant stress cases.
11
5.3 Test 3: pure bending of a square plate
5.3.1 Convergence
The material and geometry of the third benchmark problem are the same as those of
Test 2, but this test is not that of a constant-stress state. For the convergence test, a
series of mesh refinements were performed. The whole model is discritized into 6-node
triangular elements with different mesh sizes. A typical mesh is shown in Fig 8. The
meshes were all constrained by suppressing the edge normal displacements and
rotations of all of the left edge nodes as well as the tangential displacement of the node
midway along the edge. A linear normal stress distribution, varying from 1.0MPa at the
top corner to –1.0MPa at the bottom corner, was then applied to the right edge of the
plate by applying suitable edge normal forces and moments to the right edge nodes.
Table 3. shows the displacement component values predicted at the lower right corner
of the plate with mesh refinement by the present method and that with previous twofield and three-field FV formulations. The results of U and V displacements with the
present method and also the exact results are shown in the fourth and the seventh
columns respectively. From these results, it can be seen that the simulated values
converge to the exact values.
For the coarsest mesh 4*4, by considering the U displacement, the relative error of the
current method is less than 0.7%, while for the two-field formulation, it is 11.5% and
for the three-field formulation, it is 1.4%. For the V displacement, the relative error of
the present method and two-field and three-field are 0.6%, 10.9% and 1.3%
respectively. Comparing all the cases in this table, the results show that at any given
level of mesh refinement, the present method is more accurate than its three-field
12
counterpart when the number of degrees of freedom used in each method is the same.
The number of degrees of freedom of the present method is about 50% greater than that
of the two-field formulation with same mesh, but the results for the U and V
displacements of the present method with the 8*8 mesh (135 degrees of freedom) are
better than that of two-field 16*16 mesh(degrees of freedom 578). From Table.3, it can
be seen that present method is better than both the two-field and previous three-field
finite-volume methods.
5.3.2 Poisson’s ratio
The same model as 5.3.1 is used for this test except Poisson’s ratio is now varied. When
the value of Poisson’s ratio for the material approaches 0.5, the 3-node triangular finite
element will exhibit locking. To investigate if this behaviour occurs for the present
method, the value of Poisson’s ratio varying from 0.49 to 0.49….9 (15 digits of ‘9’) is
performed. Fig. 9 shows the top right corner node displacements versus the logarithm
function of Poisson’s ratio. The theoretical solution for completely incompressible
material (Poisson’s ratio equals 0.5) is also shown in this figure. From both Fig. 9 (a)
and (b), it can be seen that as the mesh is refined the computed results tend towards the
exact results at all values of ν investigated. Clearly, the results do not appear to diverge
from the exact solution as Poisson’s ratio approached 0.5.
5.4 Test 4: Pure bending of a slim cantilever with a high aspect ratio
The fourth verification test involved predicting the deformation of a rectangular plate
with an aspect ratio of 8 (L/H=8.0) under a state of pure bending. Two meshes with
nx*ny of 8*2 and 16*4 were used for the comparison of the three FV methods. In this
13
case the number of elements in the horizontal direction was four times the number in the
vertical direction. Thus, the aspect ratio or length of the horizontally aligned side
relative to the vertically aligned side of each element was 2.0. The same boundary
conditions as those of the previous case were used. The right edge had suitable forces
and moments applied so as to impose a linear normal stress distribution, varying from
1.0MPa at the top right corner to –1.0MPa at the bottom right corner. Table 4 shows the
comparison of the results from the three FV methods. From this table, for 81 degrees of
freedom, the U and V displacements of the present method are much better than that of
two-field FV method with 170 degrees of freedom. For the 8*2 mesh, the values for the
present FV method are the same as those for three-field FV method, for only one layer
of element above and under the neutral axis of the plate, while for the 16*4 mesh, the
values for the present method are worse than those for the previous three-field method.
This may be caused by the replacement of the middle node displacements by the
displacements and rotations of the two neighbouring edge nodes and the greater aspect
ratio of the horizontally aligned side being twice that of the vertically aligned side of the
element.
Although the quadratic displacement fields have been used for this study, the linear
stress fields should be predicted exactly for pure bending cases, the simulation results
only converge with mesh refinement. The reason is that the rotation boundary
conditions used are not exactly the same as the true ones.
5.5 Test 5: bending of a cantilever subjected to an end load
The last test problem is a cantilevered rectangular plate subjected to a transverse end
load. This problem was previously used to assess the performance of FE procedures
14
incorporating rotations [15, 17]. The geometry of the plate and the associated material
properties are shown in Fig. 10. For direct comparison with previous FE and FV results,
the imperial units were not converted into metric units. The two meshes employed
previously [15, 19] were again used to represent the geometry of the plate. The
transverse end load, W = 40×103 lbs, was administered by applying a parabolic shear
stress distribution (  xy
x L

3W
y
[1  4( ) 2 ] ) across the right end of the cantilever.
2 Ht
H
The equivalent forces and moments were applied to the nodes on the right edge of the
plate. The left edge of the plate was constrained in the manner employed in the
preceding tests.
Table 5 shows a comparison the results of the present FV method and those of the other
FE and FV methods, which include the constant strain triangle (CST), the Allman
triangle, the linear strain triangle(LST), a two-field FV and the three-field FV [15, 19].
For the present method with 81 degrees of freedom, the prediction of the vertical
deflection of the lower right corner of the plate is 0.3483mm, which is better than that
of the CST with 170 DOF and of the Allman Triangle and the two-field and three-field
FV with the same degrees of freedom. With 255 DOF, the present method is also better
than the aforementioned methods. For the comparison between the LST and the present
method, it is seen that they are both converging quickly to exact results, with the LST
performing slightly better than the present method.
6. Conclusions
A new FV procedure for predicting deformations in two-dimensional plane problems
has been presented. Control volumes centred on the vertices of a triangular FE mesh
have been used for the construction of the force and moment equilibrium equations. In
15
contrast to previous work the six-node standard polynomial shape function is used for
the displacement field. The displacements of the midside nodes are replaced by the
displacements and rotations of the neighbouring vertex nodes. By using the side CVs
and the replacement of the unknown displacements of the midside nodes, the total
number of system unknown variables is the same as the total number of equilibrium
equations. The simulation results show that the present method passes the patch tests
with constant strain states and converges to the exact results for linear strain states,
under mesh refinement. No locking phenomenon is observed when the value of
Poisson’s ratio approaches 0.5. The method is better than two-field finite volume
methods and also the constant strain triangle and the Allman triangle based finite
element methods. It is also better than three-field FV method for regular mesh and can
also compete with the linear strain triangle finite element method.
Appendix
The formulation of matrix A in Eqn. (2) is as follows:
1

1
A
...

1
x1
x2
...
y1
y2
...
x12
x 22
...
x1 y1
x2 y 2
...
x6
y6
x62
x6 y 6
y12 

y 22 
... 

y 62 
(A-1)
B x and B y in Eqn. (2) are given as follows:
T

B x  0 1 0 2 x y 0

T

B y  0 0 1 0 x 2 y 
(A-2)
The closedform integration of B x and B y in Eqns. (10.1 to 10.3) are given as follows:
 B x ds  0 s 0 2 x y 0
(A-3)
16
 B y ds  0
0 s 0 x
2 y
(A-4)
with:
1
2
x  x P S AB  sin S AB
2
(A-5)
1
2
cos S AB
2
(A-6)
y  y P S AB 
Where S AB is the distance between point A and B.
References
[1] I. Demirdzic, S. Muzaferija. Finite volume method for stress analysis in complex
domains. International Journal for Numerical Methods in Engineering 1994; 37: 37513766.
[2] C. Bailey, M. Cross. A finite volume procedure to solve elastic solid mechanics
problems in three dimensions on an unstructured mesh. International Journal for
Numerical Methods in Engineering 1995; 38: 1757-1776.
[3] M.A. Wheel. A mixed finite volume formulation for determining the small strain
deformation of incompressible materials. International Journal for Numerical Methods
in Engineering 1999; 44: 1843-1861.
[4] M.A. Wheel. A finite volume approach to the stress analysis of pressurized
axisymmetric structures. International Journal of Pressure Vessels and Piping 1996;
68: 311-317.
[5] K. Zarrabi, A. Basu. A finite volume element formulation for solution of elastic
axisymmetric pressurized components. International Journal of Pressure Vessels and
Piping, 2000; 77: 479-484.
17
[6] J. Fainberg, H.-J. Leister. Finite volume miltigrid solver for thermo-elastic stress
analysis in anisotropic materials. Computer Methods in Applied Mechanics and
Engineering 1996; 137: 167-174.
[7] A.K. Slone, C. Bailey, M. Cross. Dynamic solid mechanics using finite volume
methods. Applied Mathematical Modelling 2003; 27: 69-87.
[8] M.A. Wheel. A finite volume method for analysing the bending deformation of thick
and thin plates. Computer Methods in Applied Mechanics and Engineering 1997; Issues
1-2, 147: 199-208.
[9] N. Fallah. A cell vertex and cell centred finite volume method for plate bending
analysis. Computer Methods in Applied Mechanics and Engineering 2004; 193: 34573470.
[10] K. Maneeratana, A. Ivankovic. Finite volume method for geometrically nonlinear
stress analysis applications. Proceeding of the 7th Annual ACME Conference, School of
Engineering, University of Durham, 29-30 March, 1999.
[11] W. Pan, M. A. Wheel. A finite volume method for predicting finite strain
deformations in incompressible materials. European Conference on Computational
Mechanics Solids, Structures and Coupled Problems in Engineering, Munich, Germany.
August 31– September 3, 1999.
[12] G Taylor, C. Bailey, M. Cross. A vertex based finite volume method applied to
non-linear material problems in computational solid mechanics. International Journal
for Numerical Methods in Engineering 2003; 56: 507-529.
[13] A.B. Oldroyd, M.A. Wheel, T.J. Scanlon. An integrated finite volume based
approach for analysing flow induced vibrations. Proceeding of European Conference on
Computational Mechanics, Munich, Germany, August 31– September 3, 1999.
18
[14] Slone, A.K., Bailey, C., Pericleous, K.A., Cross, M. Dynamic fluid structure
interaction using finite volume unstructured mesh procedures. Computers and
Structures, 2002; 80: 371-390.
[15] D. J. Allman. A compatible triangular element include vertex rotations for plane
elasticity analysis. Computer & Structures 1984; 19: 1-8.
[16] D. J. Allman. Evaluation of the constant strain triangle with drilling rotations. .
International Journal for Numerical Methods in Engineering 1988; 26: 2645-2655.
[17] R.D. Cook. On the Allman triangular and a related quadrilateral element.
Computers & Structures 1986; 22: 1065-1067.
[18] E. Onate, M. Cervera, O.C. Zienkiewicz. A finite volume format for structural
mechanics. International Journal for Numerical Methods in Engineering 1994; 37:
181-201.
[19] W. Pan, M. A. Wheel. A finite volume method for solid mechanics incorporating
rotational degrees of freedom. Computers & Structures, 2003; Issue 5, 81: 321-329.
[20] G.I. Tsamasphyros, C.D. Vrettos. Higher order finite volume formulation for the
solution of classical 1D and 2D elasticity. Submitted to Computational Mechanics,
2008.
19
Table 1 Comparison of the displacement, strain and stress between the calculated and
exact results under the simple tension condition.
UD
(*10-4 m)
VD
(*10-4 m)
xxD
(*10-4 )
yyD
(*10-4 )
xyD
(*10-4 )
xxD
(MPa)
yyD
(Mpa)
xyD
(MPa)
FV
0.0750
-0.0060
0.1000
-0.0300
0.0000
1.000
0.000
0.000
Exact
0.075
-0.006
0.1
-0.3
0.0
1.0
0.0
0.0
Table 2 Comparison of the displacement, strain and stress between the calculated and
the exact results under shearing conditions.
UD
(*10-4 m)
VD
(*10-4 m)
xxD
(*10-4 )
yyD
(*10-4 )
xyD
(*10-4 )
xxD
(MPa)
yyD
(Mpa)
xyD
(MPa)
FV
0.0000
0.1950
0.0000
0.0000
0.2600
0.000
0.000
1.000
Exact
0.0
0.195
0.0
0.0
0.26
0.0
0.0
1.0
20
Table 3 The convergence test of a square plate subject to pure bending loading by the
present method and also a comparison between the present method and two-field and
three-field methods. The displacement shown in this table is at point C.
Mesh
size
(nx*ny)
Two-field
(u, v)
UC(*103
m)
from[19]
Threefield
(u, v, )
UC(*103
m)
from[19]
Current
method
UC(*103
m)
Two-field
(u, v)
VC(*103
m)
from[19]
Threefield
(u, v, )
VC(*103
m)
from[19]
Current
method
VC(*103
m)
4*4
-0.88556
-0.986084
0.993694
-0.95784
-1.06055
-1.06811
8*8
-0.96310
-0.995865
0.998107
-1.03924
-1.07088
-1.07306
12*12
-0.98160
-0.997991
0.999076
-1.05772
-1.07305
-1.07408
16*16
-0.98886
-0.998802
0.999447
-1.06475
-1.07385
-1.07446
24*24
---
---
0.999733
---
---
-1.07475
Exact
1.0
1.0
1.0
-1.075
-1.075
-1.075
21
Table 4 Comparison of the results between the three FV methods: the present method,
and the two-field and three-field methods [19].
Mesh
size
UC
(u, v)
-0.373781
(54)
UC
(u, v, )
3-node
triangle
-0.765342
(81)
UC
(u, v, )
6-node
triangle
-0.765342
(81)
8×2
16×4
-0.608709
(170)
-0.790950
(255)
Exact
-0.8000
-0.8000
VC
(u, v)
-2.986773
(54)
VC
(u, v, )
3-node
triangle
-6.12700
(81)
VC
(u, v, )
6-node
triangle
-6.12700
(81)
-0.816499
(255)
-4.886667
(255)
-6.33638
(255)
-6.54391
(255)
-0.8000
-6.4075
-6.4075
-6.4075
22
Table 5 Comparison of exact solution for and end loaded rectangular cantilever with
predictions provided by various FE and FV procedures using three different mesh
configurations (number of degrees of freedom in parenthesis) [15,19]
Mesh
(Ref. 15,19)
CST
Allman
Triangle
I
0.0909
(20)
0.2696
(30)
II
0.1988
(54)
III
Exact
LST
FV
(u, v)
FV
(u, v, )
Current
method
(u, v, )
-------
0.0903
(20)
0.3226
(30)
-------
0.3261
(81)
0.3550
(170)
0.1985
(54)
0.3431
(81)
0.3483
(81)
0.3115
(170)
0.3471
(255)
0.3556
(594)
0.3056
(170)
0.3520
(255)
0.3561
(255)
0.3558
0.3558
0.3558
0.3558
0.3558
0.3558
23
v3
y
3
u3
v5
v6
5
6
v1
u6
6
u5
v4
v2
2
u2
4
u4
1
O
u1
x
Fig. 1. A two-dimensional 6-node triangular element with a degree of translation only.
24
Edge node of the
triangle element
Midside node
Centre of the
triangle element
B
A
C
C
A
(a)
(b)
Fig. 2. Discritization of a sample model with six-node triangle elements and CVs
associated with typical nodes. (a) Conventional FV discritization, ‘A’ edge node(inside
the model), ‘B’ midside node (inside the model), ‘C’ edge node(on the boundary of the
model). (b) New FV discritization, ‘A’ edge node (inside the model), ‘C’ edge node (on
the boundary of the model).
25
y
v3
3
3
u3
v2
2
v1
1 1
2
u2
u1
x
Fig. 3. A two-dimensional 3-node triangular element with degrees of translation and
rotation.
26
Edge node
Centre of the element
B
yy
P
yx

xy xx
S
A
O
O
N
A
(b)
(a)
Fig. 4. A typical FV control volume. (a) control volume of an edge node, (b) the
stresses on the one side of the profile.
27
y
B
0
N
S AC

( xC C
, yC )
A( x A , y A )
x
O
Fig. 5. Uniformly distributed stresses on the boundary.
28
  1.0MPa
B
(4)
(1)
(3)
y
(2)
A
x
D
C
Fig. 6. Test 1: a quarter model of square plate subject to uniform tensile stress.
29
  1.0MPa
B
(4)
(1)
(3)
y
(2)
A
D
x
Fig. 7. Test 2: The simple shear of a square plate.
30
A
C
  1.0MPa
.
B
Fig. 8. Test 3: A square plate under pure bending.
31
1.01
0
5
10
15
20
1.00
V ( *10-3m)
-3
U ( *10 m)
-1.07
0.99
0.98
Exact
Mesh 16*16
Mesh 8*8
Mesh 4*4
0.97
0.96
Exact
Mesh 16*16
Mesh 8*8
Mesh 4*4
-1.09
-1.11
0.95
-1.13
0
5
10
-log10(0.5-v)
(a)
15
20
-log10(0.5-v)
(b)
Fig. 9. Top right corner node displacements against logarithmic (  log 10 (0.5   ) )
variation in Poisson’s ratio: (a) horizontal displacement; (b) vertical displacement.
32
E = 30 10 6psi

thickness = 1.0 inch
W = 40 10 3 lbs
y
x
H = 12 inch
y
E = 30×106psi

L = 48 inch
x
Fig. 10. End loaded cantilever test problem.
33