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A finite volume method for large strain analysis of incompressible hyperelastic materials

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This paper describes development of a displacement–pressure based finite volume formulation for modelling of large strain problems involving incompressible hyperelastic materials. The method is based on the solution of the integral conservation equations governing momentum balance in total Lagrangian description. The incompressibility constraint is enforced by employing the integral form of the mass conservation equation in deformed configurations of the body. A Mooney–Rivlin incompressible material model is used for material description. A collocated variable arrangement is used and the spatial domain is discretized using finite volumes of an arbitrary polyhedral shape. A segregated approach is employed to solve resulting set of coupled non-linear algebraic equations, utilizing a SIMPLE based algorithm for displacement–pressure coupling. Comparisons of numerical and analytical results show a very good agreement. For the limited range of cell topologies tested the developed method appears to be locking free. Copyright © 2005 John Wiley & Sons, Ltd.
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A finite volume method for large strain analysis of
incompressible hyperelastic materials
I. Bijelonja
Maˇsinski fakultet Sarajevo, Vilsonovo ˇsetaliˇste 9, 71000 Sarajevo,
Bosnia and Herzegovina,
I. Demirdˇzi´c and S. Muzaferija
CD-adapco, D¨urrenhofstrasse 4, D-90402 N¨urnberg, Germany
This paper describes development of a displacement-pressure based finite volume for-
mulation for modelling of large strain problems involving incompressible hyperelastic ma-
terials. The method is based on the solution of the integral conservation equations govern-
ing momentum balance in total Lagrangian description. The incompressibility constraint
is enforced by employing the integral form of the mass conservation equation in deformed
configurations of the body. A Mooney-Rivlin incompressible material model is used for
material description. A collocated variable arrangement is used and the spatial domain is
discretised using finite volumes of an arbitrary polyhedral shape. A segregated approach
is employed to solve resulting set of coupled non-linear algebraic equations, utilising a
SIMPLE based algorithm for displacement-pressure coupling. Comparisons of numerical
and analytical results show a very good agreement. For the limited range of cell topologies
tested the developed method appears to be locking free.
Keywords: finite volume method; hyperelastic incompressible body
1 INTRODUCTION
There are several materials such as elastomers, polymers, foams and biological tissues
which can undergo large deformations and exhibit large nonlinear elastic behaviour. Some
of them exhibit a practically incompressible response which is associated with a highly
nonlinear constraint on the deformation field.
The interest in finite volume (FV) application to the nonlinear structural analysis
problems involving large strains has grown recently [1, 2, 3]. Of particular interest here
1
are the applications of FV to incompressible materials. Henry and Collins [4] introduced
the linear elastic incompressible material model undergoing small axisymmetric deforma-
tion using finite volume modelling. To account for volumetric invariance, a condition
between the radial and axial displacements and the hydrostatic pressure as the addi-
tional dependent variable are used. The rectangular cells, simple differencing scheme and
SIMPLEC [6] algorithm were employed in discretisation. Later, the model was incorpo-
rated into a commercially available FLOW3D code, in order to model the blood flow in
artery [7].
A small strain incompressible material model was introduced to a geometrically un-
restricted finite volume model in the form of the mixed formulation [8, 9]. The auxiliary
pressure equation was used to impose incompressibility constraints. From the results of
several benchmark solutions, the finite volume method appeared to offer a number of
advantages over equivalent finite element models as the finite volume solution was con-
servative and incompressibility was satisfied exactly for each discretisation element of the
solution domain, rather than in an average sense. Although certain restrictions on mesh
configuration had to be imposed to avoid locking, these restrictions were less severe than
those of the equivalent finite element meshes.
The incompressible small strain model was also extended to large deformation using
Mooney-Rivlin hyperelastic material model which employed the second Piola-Kirchhoff
stress and the Green strain tensor under the total Lagrangian description [10]. Governing
equations were solved in their incremental form. The incompressibility condition was
enforced using differential form of the mass conservation law in the total Lagrangian
viewpoint. Effects of quadrilateral and triangular control volumes on the accuracy of
results were studied. Numerical calculation with meshes consisting of triangular cells
showed excellent agreement with analytical results. Meshes consisting of quadrilateral
FV cells displayed too stiff behaviour, indicating a locking phenomenon.
This paper outlines the development of a computational method based on the finite vol-
ume discretisation of the governing equations describing hyperelastic material behaviour.
The method is based on the solution of an integral form of incremental conservation
2
equation governing momentum balance in the total Lagrangian viewpoint. For a ma-
terial description, the two term Mooney-Rivlin incompressible material model is used.
Incompressibility constraint is enforced using an integral form of the incremental mass
conservation law in the Eulerian description and the hydrostatic pressure as an additional
dependent variable.
The method employs numerical meshes consisting of contiguous control volumes of
arbitrary topology. A collocated variable arrangement is used, and a segregated approach
is employed to solve resulting set of coupled non-linear algebraic equations, making use
of a SIMPLE [11] based algorithm for calculation of hydrostatic pressure featuring in the
constitutive equation. The method is applied to several test cases for which analytical or
numerical solutions exist.
In the next section the governing equations together with appropriate constitutive
relations and initial and boundary conditions are given. This is followed by a brief de-
scription of the FV discretisation procedure and the solution algorithm. Finally, the
method’s capabilities are demonstrated by applying it to a number of test cases.
2 MATHEMATICAL FORMULATION
In this section the mathematical model of deformation of hyperelastic bodies is presented.
It includes an integral form of conservation equations governing mass and momentum bal-
ance, constitutive relations, and boundary conditions. The two term Mooney-Rivlin con-
stitutive equations are used, although the method is applicable to any type of constitutive
equation with a relationship between stress and strain tensor.
2.1 Governing equations
The behaviour of a continuum is governed by the following momentum and mass transport
equations:
d
dtZ
V
ρvdV+Z
A
ρv(vvA)·da=Z
A
σ·da+Z
V
ρfbdV(1)
3
d
dtZ
V
ρdV+Z
A
ρ(vvA)·da= 0 (2)
which are valid for an arbitrary part of continuum with volume Vbounded by the surface
A, with the surface vector apointing outwards. In Equations (1) and (2), ρis the mass
density, vis the velocity, vAis the velocity of the control volume boundary, fbis the
resulting body force and σis the Cauchy stress tensor. It is assumed here that angular
momentum balance equation is satisfied identically due the shear stresses conjugate prin-
ciple. The change of the volume Vand the velocity of the control volume boundary vA
satisfy the space conservation low [5]:
d
dtZ
V
dVZ
A
vA·da= 0 (3)
In the case of an incompressible material and Lagrangian approach, ρ= const. and
vA=v, Equation (2) is satisfied identically, and Equations (1) and (3) become:
D
Dt Z
V
ρvdV=Z
A
σ·da+Z
V
ρfbdV(4)
Z
A
v·da= 0 (5)
The balance law of linear momentum (4) can be written with respect to the undeformed
configuration as:
∂t Z
V0
ρ0
u
∂t dV0=Z
A0
(S·FT)·da0+Z
V0
ρ0fbdV0(6)
where uis the displacement vector, Sis the second Piola-Kirchhoff stress tensor and
Fis the deformation gradient tensor defined as F=I+ grad u, where Iis the identity
tensor. The displacement vector u=xx0relates the position of a material particle
in the reference configuration x0to the position of the material point in the current
configuration x. In Equation (6), ρ0is the material density, and V0,A0, and a0denote the
volume, the surface, and the surface vector of the element in the reference configuration,
respectively, and are related to their counterparts in the current configuration by the
following equations:
dV=JdV0,da=J(F1)T·da0(7)
4
where for an incompressible material the Jacobian J= det F= 1, and the second Piola-
Kirchhoff stress tensor Srelates to Cauchy stress tensor σby:
σ=F·S·FT(8)
Equation (6) describes equations of motion in the total Lagrangian description, and its
incremental form can be written as:
∂t Z
V0
ρ0
∂δu
∂t dV0=Z
A0
(δS·FT+S·δFT+δS·δFT)·da0+Z
V0
ρ0δfbdV0(9)
where δF= grad δuis the deformation gradient tensor increment and δSis the second
Piola-Kirchhoff stress tensor increment which is defined by a constitutive model. In
Equation (9) Sand Frefer to the beginning of the current time increment.
2.2 Constitutive laws
The constitutive equation for a hyperelastic material is defined in terms of a strain energy
density function. The strain energy density function for a hyperelastic material which
is isotropic with respect to the initial, unstressed configuration, depends upon the strain
tensor only through its principal invariants. Here a two-term Mooney-Rivlin type material
is considered with the modified strain energy density function:
W=C1(I13) + C2(I23) 1
2p(I31) (10)
where C1and C2are material constants, I1,I2and I3are the three invariants of the right
Cauchy-Green strain tensor C=FT·F, and the hydrostatic pressure pis an unknown
variable that can be determined from the condition of incompressibility. For incompress-
ible materials I3= 1 and Equation (10) reduces to the conventional Mooney-Rivlin strain
energy density function.
The strain energy density function (10) relates the right Cauchy-Green strain tensor
to the second Piola-Kirchhoff stress tensor through constitutive relation:
S= 2W
C= 2C1I+ 2C2(I1IC)pC1(11)
5
If material is stress free in the reference configuration, the hydrostatic pressure for unde-
formed body follows from Equation (11):
p0= 2(C1+ 2C2) (12)
Taking the material time derivative of Equation (11) and then multiplying it by the time
increment δt the incremental constitutive equation is obtained:
δS= 2C2(tr δC I δC) + pC1·δC·C1δpC1(13)
where δp is the hydrostatic pressure increment and δCis the increment of the right
Cauchy-Green strain tensor:
δC=δFT·F+FT·δF+δFT·δF(14)
or
δC= grad δu+ (grad δu)T+ grad u·(grad δu)T+
(grad u)T·grad δu+ (grad δu)T·grad δu(15)
2.3 Resulting set of equations
Introducing constitutive relations (13) into equations governing momentum balance (9)
the following equation for the Cartesian displacement increment components δuican be
obtained:
∂t Z
V0
ρ0
∂δui
∂t dV0+Z
A0
µ0grad δui·da0=Z
A0
qA0·da0+Z
V0
ρ0δfbidV0(16)
where:
qA0={[2C2(trδC I δC) + pC1·δC·C1δpC1]·(FT+δFT) +
S·δFT+µ0grad δu} · ii(17)
and iiare the Cartesian unit base vectors. Note that the diffusion term R
A0
µ0grad δui·
da0in Equation (16) is constructed adding the term R
A0
µ0grad δu·da0to both sides
of Equation (9). For the diffusivity coefficient µ0the small strain shear modulus µ0=
6
2(C1+C2) is chosen. The construction of the diffusion term in this way simplifies the
derivation of the discrete counterpart of Equation (16).
The space conservation law equation (5) becomes
Z
A
u
∂t ·da= 0 (18)
Equations (16) and (18) make a close set of four equations with four unknown func-
tions, δuiand δp, of spatial coordinates and time.
2.4 Initial and boundary condition
To complete the mathematical model, initial and boundary conditions have to be specified.
As initial conditions, the displacements, and in transient cases velocities, have to be
specified at all points of the solution domain. Boundary conditions have to be specified
at all times at all solution domain boundaries.
3 NUMERICAL METHOD
In this section Equations (16) and (18) are discretised by employing the finite volume
method described in detail in [12], and an algorithm for the solution of discretised equa-
tions is outlined.
3.1 Space and time discretisation
In order to obtain the discrete counterpart of Equation (16), the time interval of interest
is divided into an arbitrary number of time steps δt, and the space is discretised by a
number of contiguous, non-overlapping control volumes (CV), with computational points
at their centres, (Figure 1). The boundary nodes, needed for the specification of boundary
conditions, reside at the centre of boundary cell faces.
7
P0
P1
P
d
a
rPo
a1
i1i2
i3JJ
J
.
.
.
Figure 1: A control volume of an arbitrary shape.
3.2 Discretisation of momentum equation
After the space discretisation, the momentum equation (16) is written for each control
volume as follows:
∂t Z
V0
ρ0
∂δui
∂t dV0+
nf
X
j=1 Z
A0j
µ0grad δui·da0=
nf
X
j=1 Z
A0j
qA0·da0+Z
V0
ρ0δfbidV0(19)
Rate of change Diffusion Source terms
where nfis the number of faces enclosing the cell P0. In order to evaluate integrals in the
above equation, distributions of dependent variables and physical properties of material
in space and time have to be assumed.
Spatial distribution The following linear spatial distribution is employed:
ψ(r) = ψP0+ (grad ψ)P0·(rrP0) (20)
where ψstands for dependent variables δuior δp, or a physical property of the material,
rP0is the position vector of the control volume centre P0(Figure 1). The unknown vector
(grad ψ)P0is calculated by ensuring a fit to a set of sampling points consisting of the
nearest neighbours of point P0, i.e. by solving the following set of equations:
dj·(grad ψ)P0=ψPjψP0(j= 1, ..., nf) (21)
8
where dj=rPjrP0is a distance vector joining point P0with its neighbour Pj(Figure
1). To solve this over-determined set of equations, the least square method is used.
Equation (20) is used to calculate values of ψat the cell-faces jnecessary for evaluation
of the surfaces integrals in (19), leading to a second-order symmetric formula:
ψj=1
2(ψP0+ψPj) + 1
2[(grad ψ)P0·(rjrP0) + (grad ψ)Pj·(rjrPj)] (22)
where rjis the position vector of the cell-face centre.
Temporal distribution In evaluating the transient term a linear variation of dependent
variables in time is assumed. According to the adopted fully implicit time discretisation
scheme, diffusion and source terms are evaluated at the current instant of time tm, where
mis the time step counter. Henceforth, the time step counter will normally be omitted,
and all values of δuiand δp will refer to the current time tm, unless indicated otherwise.
Rate of change The transient term in Equation (19) is approximated as:
∂t Z
V0
ρ0
∂δui
∂t dV0(ρ0V0)P0
δt2δui2δum1
i+δum2
iP0
(23)
where the volume integral is approximated using the midpoint rule, and δt is the time
increment.
Diffusion Using the midpoint rule approximation of the surface integral the diffusive
flux of δuithrough an internal cell-face jcan be approximated by:
Z
A0j
µ0grad δui·da0µ0j(grad δui)
j·(a0)j(24)
where µ0jstands for the cell-face mean value of the diffusivity, obtained by using Equation
(22), and the value of (gradδui)
jis constructed in the following manner:
(grad δui)
j= (grad δui)j+"(δui)Pj(δui)P0
|(d0)j|grad δui·(d0)j
|(d0)j|#|(d0)j|(a0)j
(d0)j·(a0)j
(25)
where the overbar denotes the arithmetic average of the values calculated at nodes P0
and Pj. The second term on the right hand side (term in [] brackets) represents the
9
difference between the second-order central difference approximation of the derivative in
the direction of vector (d0)jand the value obtained by interpolating cell-center gradients.
It vanishes if the spatial variation of δuiis linear or quadratic. Otherwise, its magnitude is
proportional to the second-order truncation error of the scheme and reduces accordingly
with grid refinement. This correction term detects and smoothes out any unphysical
oscillations that might occur in the iteration process when solving resulting set of algebraic
equations. The first part of this term is treated implicitly, and the rest of the diffusion
flux explicitly.
Source terms The surface and volume integrals in Equation (19) are calculated using
again the midpoint rule:
Z
A0j
qA0·da0qA0j·(a0)j(26)
and
Z
V0
ρ0δfbidV0(ρ0δfbi)P0(V0)P0(27)
Initial and boundary conditions To start the calculation, all dependent variables
featuring in Equations (16) and (17) have to be set to their initial values. For a stress
free body S=0,F=I,C1=Iand the initial value for total pressure pis given
by Equation (12). The displacement vector components u1
iat time t1=t0δt are
required. They are calculated from the known initial velocity field, applying backward
differencing scheme.
Boundary conditions have to be applied on the faces coinciding with the boundary of
the solution domain. In case of Dirichlet boundary condition, the expressions for diffusion
fluxes and sources remain valid, except for replacing (δui)Pjby the boundary value (δui)B.
On the boundary regions where Neumann boundary conditions are specified, the boundary
fluxes are added to the source term, while the variable values at the boundary are obtained
using a second-order extrapolation.
10
Resulting algebraic equations After assembling all terms featuring in Equation (19),
for each CV a following non-linear algebraic equation, which links the value of dependent
variable δuiat the control volume centre with the values at the points in the neighbour-
hood, is obtained:
aδui0(δui)P0
ni
X
j=1
aδuij(δui)Pj=bδ ui(28)
where niis the number of internal faces of the cell P0,
aδuij=µ0j
(a0)j·(a0)j
dj·(a0)j
aδui0=
nf
X
j=1
aδuij+(ρ0V0)P0
δt2
bδui=
nf
X
j=1
µ0j"(grad δui)j·(a0)jgrad δui·(d0)j
(a0)j·(a0)j
(d0)j·(a0)j#+ (29)
nf
X
j=1 qA0j·(a0)j+ (ρ0δfbi)P0(V0)P0+
(ρ0V0)P0
δt22δum1
iδum2
iP0
+
nB
X
B=1
aδuiB(δui)B,
and nB=nfniis the number of boundary faces surrounding cell P0.
3.3 Discretisation of space conservation equation and calcula-
tion of pressure
It can be noted that the pressure increment which appears in the source term of the
discretised momentum equation is unknown, while at the same time space conservation
equation (18) has not been used yet. However, the pressure increment does not feature in
the space conservation equation which comes as an additional constraint on the displace-
ment field. In order to calculate the pressure field and to couple it with the displacement
field, a pressure-correction method of the SIMPLE-type [11] is used.
Writing the space conservation Equation (18) for the control volume configuration
occupied by the body at the instant of time t=tm1+δt/2 and using the central
11
differencing scheme for the velocity approximation, the space conservation equation can
be written as follows:
Z
A
u
∂t ·da
nf
X
j=1 Z
Am1/2
umum1
δt ·da=
nf
X
j=1 Z
Am1/2
δu
δt ·da= 0 (30)
Multiplying last equation by δt and using the midpoint rule for surface integral approxi-
mation, one gets:
nf
X
j=1
δu
j·(am1/2)j=
nf
X
j=1
δVj= 0 (31)
where δV jdenotes the volume swept by the face j. According to the SIMPLE algorithm,
the displacement at the cell face δu
jis constructed in the following manner:
δu
j=δuj V0
aδu0!"δpPjδpP0
|(dm1/2)j|grad δp ·(dm1/2)j
|(dm1/2)j|#|(dm1/2)j|(am1/2)j
(dm1/2)j·(am1/2)j
(32)
where δujis the spatially interpolated displacement, the overbar denotes the arithmetic
average, and aδu0is the corresponding momentum equation central coefficient. The second
term in Equation (32) is a third-order pressure diffusion term, analogous to the term
introduced by Equation (25), when the diffusive transport of variable δuiwas discussed.
Its role is to smooth out the oscillatory pressure profile, and at the same time introduce the
pressure into the space conservation equation in such a manner that a pressure-correction
equation can easily be constructed. This is achieved by employing the predictor-corrector
procedure which will be briefly outlined here.
The so-called predictor stage values of δuand δp (featuring in expression (32) for δu
j),
which satisfy the (linearised) momentum equation, do not necessarily satisfy the space
conservation equation (31). By correcting the cell-face displacements, e.g.:
δu∗∗
j=δu
j+ V0
aδu0!δp0
Pjδp0
P0
|(dm1/2)j|
|(dm1/2)j|(am1/2)j
(dm1/2)j·(am1/2)j
(33)
and requiring that the double starred cell-face displacement satisfy the space conservation
Equation (31), an equation for the pressure correction δp0is obtained:
aδp0
0δp0
P0
nf
X
j=1
aδp0
jδp0
Pj=bδp0(34)
12
with coefficients:
aδp0
j= V0
aδu0!(am1/2)j·(am1/2)j
(dm1/2)j·(am1/2)j
, aδp0
0=
nf
X
j=1
aδp0
j, bδp0=
nf
X
j=1
δV j(35)
where all variables have their predictor stage values. After the field of pressure correction
δp0is obtained, the displacement and pressure increments are corrected via:
δuP0=δuP0,pred +δu0
P0=δuP0,pred 1
aδu0
nf
X
j=1
δp0
j(am1/2)j
δpP0=δpP0,pred +βpδp0
P0(36)
where βpis an under-relaxation factor (typically βp= 0.2 to 0.3).
The boundary conditions for the pressure-correction equation (34) depend on the
boundary conditions for the momentum equations. On those boundaries where displace-
ment is prescribed, pressure correction is zero, which implies a zero-gradient Neumann
boundary condition on the pressure correction. If the traction is prescribed at the bound-
ary, the pressure is calculated via constitutive relation and its correction is zero, leading
to a Dirichlet boundary condition for the pressure correction.
3.4 Solution algorithm
After assembling equations (28) and (34) for all CVs, four sets of Ncoupled non-linear
algebraic equations is obtained, where Nis the number of CVs. Due to the nonlinearity
of the underlying equations, the solution of this system of algebraic equations is obtained
by employing a segregated iterative algorithm. It consists of linearisation and temporary
decoupling of equations (28) and (34) by assuming that coefficients and source terms are
known (calculated by using dependent variable values from the previous iteration or the
previous time step). As a result, a set of linear algebraic equations for each dependent
variable is obtained:
Aφφ=bφ(37)
where φstands for δuior δp0,Aφis an N×Nmatrix, vector φcontains values of
dependent variable φat Nnodal points and bφis the source vector. The matrices Aφ
13
resulting from the discretisation method described above are sparse, with the number of
nonzero elements in each row equal to the number of nearest neighbours plus one, ni+ 1.
They are also diagonally dominant, which makes the equation system (37) easily solvable
by an iterative method which retains the sparsity of the matrix A.
The solution strategy used here can be outlined by the following sequence:
1. Provide initial values of all dependent variables.
2. Assemble and solve Equation (37) for displacement increment components, employ-
ing the currently available dependent variable.
3. Assemble and solve Equation (37) for the pressure correction and use calculated
values to correct displacement vector components and pressure.
4. Repeat the sequence of steps 2–3 by updating the coefficient matrix and source term
until a converged solution is obtained, i.e. until the residual norm,
||rφ|| =||Aφφbφ|| (38)
is reduced by a prescribed number of orders of magnitude.
5. Update total displacement and total pressure
um=um1+δum
pm=pm1+δpm
as well as the total second Piola-Kirchhoff stress tensor, Sm=Sm1+δSm, and
inverse of the right Cauchy-Green strain tensor, C1= (FT·F)1.
6. Advance the time by δt and return to step 2; repeat until the prescribed number
of time steps is completed (In a steady case, the load is divided into mincrements
and δt is a pseudo time increment).
Equations (37) are solved using the conjugate gradient method with an incomplete Cholesky
preconditioning. There is no need to solve them to a tight tolerance, since the coefficients
14
and sources are only an approximation (based on the values of dependent variables from
the previous iteration or time step) and reduction of the sum of absolute residuals by an
order of magnitude normally suffices.
Note that in order to promote stability of the solution method, an under-relaxation is
often necessary.
4 TEST CASES
In this section a set of test cases chosen to demonstrate capabilities of the method is
presented. The results of calculations are compared with analytical results and results
obtained by the finite element method.
4.1 Long wall
As a first test problem, an infinitely long wall of constant thickness loaded uniformly in
two directions, as shown in Figure 2, is considered. The material of the wall is assumed
incompressible with Mooney-Rivlin constants C1 = 80 MPa and C2 = 20 MPa.
x
y
z
50 MPa
100 MPa
1 m
2 m
Figure 2: Biaxially loaded long wall.
15
Considering the geometry and the loads used, the problem can be reduced to a plain-
strain two dimensional simulation. The finite volume grid consists of 3x3 cells.
The numerical results for different values of the load increment and values obtained
from the analytical solution of various field variables are shown in Table 1, where uxand
uyare displacements in xand ydirection, σxx and σyy are the corresponding components
of Cauchy stress tensor, pis the hydrostatic pressure, and Exx and Eyy are the Green-
Lagrangian strain tensor components. It is obvious that the computed results agree very
well with those obtained from the analytical solution. A reason for getting such good
numerical results with very coarse mesh is that the analytical solution corresponds to a
homogeneous (constant strain) deformation of the wall and this deformation mode is also
included in the above described numerical procedure.
Table 1: Numerical and analytical results for the long wall problem
No. of load inc. Exx Eyy σxx(MPa) σyy(MPa) ux(m) uy(m) p(MPa)
10 0.1530 0.2200 49.0 102 0.1683 0.4009 228.5
50 0.1532 0.2210 49.8 100 0.1677 0.4019 228.6
100 0.1533 0.2213 49.9 100 0.1676 0.4022 228.6
Analytical 0.1535 0.2215 50.0 100 0.1675 0.4025 228.6
4.2 Cylindrical pressure vessel
As a second test a homogeneous thick-wall cylindrical pressure vessel with inner radius
Ri= 7 m, outer radius Ro= 18.625 m, and loaded internally with pressure pi= 100 MPa
is analysed. The Mooney-Rivlin constants are C1 = 80 MPa and C2 = 20 MPa. The
problem is considered as a plane strain, with the space domain made of a ten degrees
16
segment of the cylinder with 10 cells in the radial direction. The total inner pressure load
is applied in 100 equal increments.
0
1
2
3
4
0 20 40 60 80 100
Internal pressure (MPa)
Radial displacement (m)
analytical
numerical
Figure 3: Internal pressure vs. radial displacement of the inner surface of the pressure
vessel.
Results presented in Figure 3 show numerical and analytical [13] calculations of radial
displacement of the inner surface of the vessel versus internal pressure. The maximum
relative error is less then 1% for internal pressure of 100 MPa. The variation of the
circumferential Cauchy stress through the thickness of the cylinder is plotted in Figure 4
and the variation of the radial Cauchy stress across the thickness of the cylinder is plotted
against analytical results in Figure 5. The numerical results agree very well with those
obtained from the analytical solution.
17
0
50
100
150
200
250
300
350
400
7 9 11 13 15 17 19
Undeformed radial distance (m)
Hoop stress (MPa)
analytical
numerical
Figure 4: Hoop Cauchy stress vs. undeformed radius.
-120
-100
-80
-60
-40
-20
0
7 9 11 13 15 17 19
Undeformed radial distance (m)
Radial stress (MPa)
analytical
numerical
Figure 5: Radial Cauchy stress vs. the undeformed radius.
18
4.3 Pressing a long rubber cylinder
An infinitely long homogeneous cylinder with the diameter D= 0.4 m is pressed between
two frictionless rigid plates. The material of the cylinder is a rubber with Mooney-Rivlin
constants C1 = 0.293 MPa and C2 = 0.177 MPa.
D
solutiondomain
Figure 6: Rubber cylinder problem sketch (left) and finite volume mesh (right).
Figure 7: Deformed rubber cylinder finite volume mesh for total displacement of 0.2 m.
The loading is displacement-controlled with the total vertical displacement of the top
rigid plate of 0.2 m (Figure 6). The problem is considered as the plane strain. Due to the
geometric and loading symmetry only a quarter of the cylinder is analysed. The numerical
analysis is performed for two different finite volume meshes consisting of 82 and 190 cells
respectively. The coarser FV mesh is shown in Figure 6. The total displacement load is
19
imposed in 30 equal increments. During compression, initially free surface of the cylinder
comes into contact with rigid plates. At this point, the stress free boundary condition
is changed into the mixed boundary condition – prescribed displacement in the vertical
direction and the zero shear stress in the plane normal to that direction.
In Figure 7 the deformed finite volume mesh for the total displacement load of 0.2 m
is shown. The force-displacement curves for the two finite volume meshes are shown in
Figure 8. It can be seen that two FV solutions are almost coincident. Figure 8 also shows
the finite element results reported in [14], where an almost incompressible material is
assumed (Poisson’s ratio ν= 0.49967) and the same Mooney-Rivlin constants as in the
present calculation are employed. The difference between the finite volume and the finite
element simulations is not larger then 2.5 percent.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.00 0.05 0.10 0.15 0.20 0.25
Plate displacement (m)
Force x 106(N/m)
FVM (82 CVs)
FVM (190 CVs)
FEM
Figure 8: The force vs. displacement simulations of a rubber cylinder. Comparison of
FVM and FEM [14] results.
20
5 CONCLUSIONS
In this paper a finite volume based numerical method for predicting the large strain
behaviour of hyperelastic bodies is presented. The method solves the momentum and
mass balance equations in an integral form. The incompressibility constraint is enforced by
employing the hydrostatic pressure as an additional dependent variable. For the material
description a Mooney-Rivlin incompressible material model is used.
The presented test cases, chosen for the availability of the analytical solution or the
finite element simulation, demonstrate very good accuracy of the method. The locking
phenomenon, common in numerical descriptions of incompressible material behaviour, is
not registered. It could be that conservative nature of the method, which assures that the
kinematic constraint equation of incompressibility (resulting from the space conservation
law) is satisfied for each control volume, leads to a volumetric locking-free scheme.
References
[1] Maneeratana K, Ivankovic A. Finite volume method for large deformation with liner
hypoelastic materials. In Finite Volumes for Complex Applications II, Vilsmeier R,
Benkhaldoun F, Hanel D (eds), HERMES Science Publications, Paris, 1999; 459–466.
[2] Maneeratana K, Ivankovic A. Finite volume method for structural applications in-
volving material and geometrical non-linearities. Proceedings of European Conference
on Computational Mechanics, ECCM’99, Munich, Germany, 1999.
[3] Fallah N, Bailey C, Cross M, Taylor GA, Comparison of finite element and finite
volume methods application in geometrically nonlinear stress analysis, Appl. Math.
Modelling 2000; 24:439–455.
[4] Henry FS, Collins MW. Prediction of transient wall movement of an incompressible
elastic tube using a finite volume procedure. Proceedings of the Second International
Conference on Computers in Biomedicine, BIOMED 93, Bath, UK, 1993.
21
[5] Demirdˇzi´c I, Peri´c M. Space conservation law in finite volume calculations of fluid
flow, International Journal for Numererical Method in Fluids 1988;8:1037–1050.
[6] Patankar SV. Numercal Heat Transfer and Fluid Flow. McGraw-Hill, 1980.
[7] Henry FS, Collins MW. A novel predictive model with compliance for arterial flows.
Proceedings of the 1993 ASME Winter Annual Meeting, New Orleans, 1993.
[8] Wheel MA. Modelling incompressible materials using a mixed structural finite volume
approach. Computational Mechanics in UK 1997, ACME 5, London, 1997.
[9] Wheel MA. A mixed finite volume formulation for determining the small strain de-
formation of incompressible materials. International Journal for Numerical Methods
in Engineering 1999; 44:1843–1861.
[10] Wenke P, Wheel MA. A finite volume method for predicting finite strain deforma-
tiona in incompressible materials. Proceedings of the European Conference on Com-
putational Mechanics, ECCM’99, M¨unchen. 1999.
[11] Patankar SV, Spalding DB. A calculation procedure for heat, mass and momentum
transfer in three-dimensional parabolic flows, International Journal for Heat and
Mass Transfer 1972; 15:1787–1806.
[12] Demirdˇzi´c I, Muzaferija S. Numerical method for coupled fluid flow, heat transfer and
stress analysis using unstructured moving meshes with cells of arbitrary topology,
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[13] Green AE, Zerna W. Theoretical Elasticity. Oxford University Press, 1968.
[14] Ansys-6.1, Verification Manuel.
http://www.oulu.fi/atkk/tkpalv/unix/ansys-6.1/content/Hlp V VM201.html [10
February 2004]
22
... This benchmark consists in a wall with 1 m × 2 m (width × height) and assumed infinitely long in length, i.e. along the z axis (Fig. G.1a). Hence, due to symmetry, it can be assumed to be (230), and the analytical solution as also presented by Bijelonja et al. (230). All numerical solutions were found after solving for 100 incremental steps along time. ...
... This benchmark consists in a wall with 1 m × 2 m (width × height) and assumed infinitely long in length, i.e. along the z axis (Fig. G.1a). Hence, due to symmetry, it can be assumed to be (230), and the analytical solution as also presented by Bijelonja et al. (230). All numerical solutions were found after solving for 100 incremental steps along time. ...
... This problem was investigated by Sussman et al. (229) and also by Bijelonja et al. (230). The latter modeled the material of the cylinder as purely incompressible and the former assumed a nearly-incompressible behavior, which is characterized by the authors as when "the ratio of the bulk modulus to the shear modulus is typically several thousand". ...
Thesis
Full-text available
Intracranial aneurysms (IAs) are abnormalities formed in the cerebral arteries characterized by outpouching regions of their walls. The danger with these lesions occurs if they rupture, which causes intracranial hemorrhage and possibly leads to the death of the patient, presenting a mortality rate as high as 50 %. The rupture event is hard to predict, though, and, currently, surgical treatments also pose risks to the patient. Numerical simulations of the blood flow inside IAs have been extensively used to study them because of the well-known connections between hemodynamics and their inception, growth, and rupture. Physically, although it should be modeled as a Fluid-Solid Interaction (FSI) problem, the majority of those works have solely focused on the hemodynamics while either ignoring the wall tissue motion entirely, through rigid-wall modeling, or using limited assumptions for it. One possible explanation is the scarcity of measurements of their wall mechanical properties and also its thickness, which limits the use of better modeling options. Consequently, few works have investigated the impact of tissue modeling on their mechanical response, an important endeavor to try to predict the likelihood of rupture, because it is a wall-exclusive event that theoretically depends on the level of stress. In this context, this work investigated the influence of different hyperelastic laws and the material properties and thickness heterogeneity on the wall mechanics of IAs, given their rupture status. Pulsatile numerical simulations with patient-specific vascular geometries harboring Ias were carried out using the one-way fluid-solid interaction solution strategy implemented in solids4foam, an extension of OpenFOAM ® , in which the blood flow is solved and applied as the driving force of the wall motion. First, it was found that different wall morphology models yielded smaller absolute differences in the mechanical response than different hyperelastic laws. Second, the stretch levels of IAs walls were more sensitive to the hyperelastic laws and material constants than the stress, especially for ruptured IAs, allowing the identification of these by the higher stretch levels instead of stress levels. Additionally, the morphology variable that best correlated with regions of high stress and stretch was the wall curvature. Finally, these findings could be used to guide modeling decisions on IA simulations and also suggest new metrics based on the wall curvature to predict the likelihood of rupture.
... Among the numerical algorithms, finite volume 2 and finite element 3 methods are the most widespread discretization methods; either of which possesses its own distinct features. While the latter is known as an unchallenged discretization method in the area of stress analysis, the former is preferred in the analysis of incompressible solids [4][5][6] as it strongly enforces the conservation and incompressibility condition both locally and globally and not just in an average sense. Some studies [7][8][9] have compared the performances of finite element and finite volume-based solvers in terms of accuracy and convergence rate. ...
... Alternatively, other researchers have introduced hydrostatic pressure as an unknown variable which must be sought in a way that ensures satisfaction of the incompressibility condition. The latter approach has led to displacementpressure formulation which has been adopted in various studies with the aim of calculating deformation for small strain problems in linear elasticity [26][27][28][29] as well as large deformation for hyperelastic models [5,30,31] in such a way that spurious hydrostatic pressure fluctuations are eliminated when the material becomes nearly or truly incompressible. Alternatively, it has been shown by Giannopapa [32] that the inclusion of velocity, instead of displacement, results in a more accurate solution for a given discretization method. ...
... The distinguished feature of the proposed solver comes from the fact that the computational domain for the solid is generally unknown in each time step and must be sought during the solution process. This is in contrast to the conventional methods, such as linear elastic approach, in which it is assumed that the computational domain is fixed due to small deformation-and Updated Lagrangian [40] or Total Lagrangian [5] approaches, which use the last calculated or initial configuration as the computational domain, respectively. From this point of view, the formulation can be considered as a more general form of the deformed Lagrangian method [41], since ALE allows moving the mesh nodes independent of the material points. ...
Article
Unification of the numerical models and methods in computational solid and fluid dynamics has been a research objective with at least two major advantages in mind. The first benefit of such unification is the more efficient data transfer between fluid and solid media, and the second advantage is the possibility of developing a better solver as compared to the separate existing solid and fluid solvers. In this paper, a conservative fluid-like pressure-velocity-based formulation is proposed that simulates large deformation of a weakly compressible hyper-elastic solid on an Arbitrary Lagrangian-Eulerian (ALE) framework. The proposed solver, which is implemented in OpenFOAM software, allows for flexible grid movement, i.e. mesh points are not forced to follow material points, as well as for the employment of various material models such as Mooney-Rivlin and Neo-Hookean constitutive laws. Three challenging 2-D and 3-D test cases including torsion, bending and pressing of solid objects are presented to examine and discuss the accuracy and flexibility of the proposed solver. Furthermore, more light is shed on the concept of the pressure in compressible solids and appropriate boundary conditions at traction boundaries.
... In the past decades, significant advancements have been made in the field of nonlinear elasticity enabling the study of the complex nature of hyperelastic materials and their composites. Computational methods, specifically the finite element method, have been widely employed to study the mechanical response of these materials under large deformations [5][6][7][8][9][10], providing valuable insights for engineers and researchers. Research on finite element modelling of nonlinear elasticity tends to focus on the three-dimensional problem or the plane strain approximation thereof [11][12][13][14][15]. ...
... The strain energy density function may be additively decomposed based on J and C as 8) or, for the case of isotropy, in terms of J and b as ...
Preprint
Modelling the large deformation of hyperelastic solids under plane stress conditions for arbitrary compressible and nearly incompressible material models is challenging. This is in contrast to the case of full incompressibility where the out-of-plane deformation can be entirely characterised by the in-plane components. A rigorous general procedure for the incorporation of the plane stress condition for the compressible case (including the nearly incompressible case) is provided here, accompanied by a robust and open source finite element code. An isochoric/volumetric decomposition is adopted for nearly incompressible materials yielding a robust single-field finite element formulation. The nonlinear equation for the out-of-plane component of the deformation gradient is solved using a Newton-Raphson procedure nested at the quadrature point level. The model's performance and accuracy are made clear via a series of simulations of benchmark problems. Additional challenging numerical examples of composites reinforced with particles and fibres further demonstrate the capability of this general computational framework.
... Incompressibility is addressed in the literature based on various strategies. For a specific class of finite volume methods [7], Bijelonja et al. [8] discretise the integral balance of momentum in its second-order form, as well as the integral version of the incompressibility constraint J ≡ 0 obtained through time differentiation of the initial restriction. Several computational strategies have also been proposed in the Finite Element literature, including mixed methods and stabilised methods [9][10][11][12], as well as penalisation methods [13], including computational models that account for viscoelastic material behaviour [14,15]. ...
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The formation of shear shock waves in the brain has been proposed as one of the plausible explanations for deep intracranial injuries. In fact, such singular solutions emerge naturally in soft viscoelastic tissues under dynamic loading conditions. To improve our understanding of the mechanical processes at hand, the development of dedicated computational models is needed. The present study concerns three‐dimensional numerical models of incompressible viscoelastic solids whose motion is analysed by means of shock‐capturing finite volume methods. More specifically, we focus on the use of the artificial compressibility method, a technique that has been frequently employed in computational fluid dynamics. The material behaviour is deduced from the Fung–Simo quasi‐linear viscoelasiticity (QLV) theory where the elastic response is of Yeoh type. We analyse the accuracy of the method and demonstrate its applicability for the study of nonlinear wave propagation in soft solids. The numerical results cover accuracy tests, shock formation and wave focusing.
... The second terms on the left and right-hand sides come from the Rhie-Chow stabilisation. The described approach is similar to the formulation proposed by Bijelonja et al. [47][48][49][50] for incompressibility, quasi-incompressibility and elastoplasticity. The scalar coefficient ensures the stabilisation is scaled appropriately. ...
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This article presents the implementation of the canonical Lemaitre and Gurson–Tvergaard–Needleman (GTN) damage models and a more recent phase‐field type model within a Lagrangian, geometrically nonlinear, cell‐centred finite volume framework. The proposed segregated solution procedure uses Picard‐type defect (deferred) outer corrections, where the primary unknowns are cell‐centre displacements and pressures. Spurious zero‐energy modes (numerical oscillations in displacement and pressure) are avoided by introducing stabilisation (smoothing) diffusion terms to the pressure and momentum equations. Appropriate scaling of the momentum “Rhie–Chow” stabilisation term is shown to be important in regions of plasticity and damage. To accurately predict damage and fracture in wire drawing where hydrostatic pressure is high, novel variants of the Lemaitre model with crack‐closure and triaxiality effects are proposed. The developed methods are validated against the notched round bar and flat notched bar experimental cases and subsequently applied to the analysis of axisymmetric wire drawing. It is shown that the proposed finite volume approach provides a robust basis for predicting damage in wire drawing, where the proposed novel Lemaitre model with crack‐closure effects was shown to be the most suitable for predicting experimentally observed fracture.
... Incompressibility is addressed in the literature based on various strategies. For a specific class of finite volume methods 7 , Bijelonja et al. 8 discretise the integral balance of momentum in its second-order form, as well as the integral version of the incompressibility constraintJ ≡ 0 obtained through time-differentiation of the initial restriction. Several computational strategies have also been proposed in the Finite Element literature, including mixed methods and stabilised methods 9,10,11,12 , as well as penalisation methods 13 , including computational models that account for viscoelastic material behaviour 14,15 . ...
Preprint
Full-text available
The formation of shear shock waves in the brain has been proposed as one of the plausible explanations for deep intracranial injuries. In fact, such singular solutions emerge naturally in soft viscoelastic tissues under dynamic loading conditions. To improve our understanding of the mechanical processes at hand, the development of dedicated computational models is needed. The present study concerns three-dimensional numerical models of incompressible viscoelastic solids whose motion is analysed by means of shock-capturing finite volume methods. More specifically, we focus on the use of the artificial compressibility method, a technique that has been frequently employed in computational fluid dynamics. The material behaviour is deduced from the Fung-Simo quasi-linear viscoelasiticity theory (QLV) where the elastic response is of Yeoh type. We analyse the accuracy of the method and demonstrate its applicability for the study of nonlinear wave propagation in soft solids. The numerical results cover accuracy tests, shock formation and wave focusing.
... If mentioned, the resulting gap (penetration) is calculated relatively as the ratio between the gap distance and contacting cell height. Incompressible and nearly incompressible materials are not considered because the employed displacement-based formulation is inappropriate for such problems; instead, a mixed should be employed, for example, as proposed by Bjelonja et al. 3,4 ...
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... As mentioned before, the SIMPLE algorithm with Rhie and Chow interpolation, which is usually used for flow solutions, can be applied in solids to avoid non-physical oscillation of the pressure. In the case of large deformation of incompressible solids, Bijelonja et al. 30 developed displacement-pressure-based finite volume formulation for the simulation of hyperelastic materials undergoing large deformations. They used Mooney-Rivlin incompressible material model to simulate material deformation. ...
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In the single domain method for solving fluid‐solid interaction (FSI) problems, a unified formulation is used for the entire computational domain. In such monolithic FSI solvers, all of the governing equations are solved simultaneously. In the present study, the single domain method is further extended to an interface‐tracking FSI solver which accounts for mesh movement via an Arbitrary Lagrangian‐Eulerian (ALE) description of the governing equations. The focus is on internal flow problems with large deformation. Pressure and velocity are selected as the dependent variables for both solid and fluid parts of the computational domain. A distinguishing feature of the proposed method is that the governing equations at the interface are discretized in a conservative manner. Interfacial boundary conditions are enforced via a pressure‐velocity splitting method to convert the kinematic and dynamic conditions at the interface into pressure‐velocity relations. A PISO‐like procedure is used to solve the discretized equations. In order to evaluate the proposed solver, strongly‐coupled FSI benchmark test cases are employed. The performance of the proposed method and computational results are also compared with those obtained by a conventional partitioned solver. The results show that the proposed solver provides more accurate results on a coarser mesh compared to the benchmark solutions. The proposed method is also capable of solving strongly coupled problems for which the partitioned solver fails to converge.
... With a similar philosophy, an alternative version of cell centred FVM (via a nodebased discretisation of the numerical fluxes) originally proposed by Mazeran and Després [102], and later explored in [103][104][105], where gas dynamics applications were adapted to the case of hyperelastic solids [106,107]. In application of fully incompressible materials (biomedical and structural engineering), a mixed-based displacement-rotation VCFVM framework [108][109][110] and a displacement-pressure cell centred finite volume algorithm [53,111] are presented in order to account for incompressibility constraint. In parallel, Scovazzi and co-authors [25,[112][113][114][115] also used a mixed-based approach for a linear tetrahedral element by utilising a Variational Multi-Scale method. ...
Thesis
Full-text available
It is evidently not trivial to analytically solve practical engineering problems due to their inherent nonlinearities. Moreover, experimental testing can be extremely costly and time-consuming. In the past few decades, therefore, numerical techniques have been progressively developed and utilised in order to investigate complex engineering applications through computer simulations. In the context of fast thermo-elastodynamics, modern commercial packages are typically developed on the basis of second order displacement-based finite element formulations and, unfortunately, that introduces a series of numerical shortcomings (e.g. detrimental locking, hour-glass modes, spurious pressure oscillations). To rectify these drawbacks, a mixed-based set of first order hyperbolic conservation laws for thermo- elastodynamics is presented in terms of the linear momentum per unit undeformed volume, the deformation gradient, its co-factor, its Jacobian and the balance of total Energy. Interestingly, the conservation formulation framework allows exploiting available CFD techniques in the context of solid dynamics. From a computational standpoint, two distinct spatial discretisations are employed, namely, Vertex-Centred Finite Volume Method (VCFVM) and Smooth Particle Hydrodynamics (SPH). A linear reconstruction procedure together with a slope limiter is employed in order to ensure second order accuracy in space whilst avoiding numerical oscillations in the vicinity of sharp gradients. The semi-discrete system of equations is then temporally discretised using a second-order Total Variation Diminishing (TVD) Runge-Kutta time integrator. Finally, a wide spectrum of challenging examples is presented in order to assess both the performance and applicability of the proposed schemes. The new formulation is proven to be very efficient in nearly incompressible thermoelasticity in comparison with classical finite element displacement-based approaches.
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We present a novel convex formulation that models rigid and deformable bodies coupled through frictional contact. The formulation incorporates a new corotational material model with positive semi-definite Hessian, which allows us to extend our previous work on the convex formulation of compliant contact to model large body deformations. We rigorously characterize our approximations and present implementation details. With proven global convergence, effective warm-start, the ability to take large time steps, and specialized sparse algebra, our method runs robustly at interactive rates. We provide validation results and performance metrics on challenging simulations relevant to robotics applications. Our method is made available in the open-source robotics toolkit Drake.
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A novel three-dimensional finite volume (FV) procedure is described in detail for the analysis of geometrically nonlinear problems. The FV procedure is compared with the conventional finite element (FE) Galerkin approach. FV can be considered to be a particular case of the weighted residual method with a unit weighting function, where in the FE Galerkin method we use the shape function as weighting function. A Fortran code has been developed based on the finite volume cell vertex formulation. The formulation is tested on a number of geometrically nonlinear problems. In comparison with FE, the results reveal that FV can reach the FE results in a higher mesh density.
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A finite volume formulation for determining small strain deformations in incompressible materials is presented in detail. The formulation includes displacement and hydrostatic pressure variables. The displacement field varies linearly along and across each cell face. The hydrostatic pressure field associated with each face is uniform. The cells that discretize the structure are geometrically unrestricted, each cell can have an arbitrary number of faces. The formulation is tested on a number of linear elastic plane strain benchmark problems. This testing reveals that when meshes of multifaceted cells are employed to represent the structure then locking behaviour is exhibited, but when triangular cells are used then accurate predictions of the displacement and stress fields are produced. Copyright © 1999 John Wiley & Sons, Ltd.
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In the numerical solutions of fluid flow problems in moving co-ordinates, an additional conservation equation, namely the space conservation law, has to be solved simultaneously with the mass, momentum and energy conservation equations. In this paper a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases. The results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices. It is also demonstrated, by analysis and test calculations, that not satisfying the space conservation law in a numerical solution procedure introduces errors in the form of artificial mass sources. These errors can be made negligible only by choosing a sufficiently small time step, which sometimes may be smaller than required by the temporal discretization accuracy.
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A numerical method is presented that can be used for both solid body stress analysis and fluid flow predictions, independently as well as in a coupled manner. The method uses an integral form of equations governing mass, energy and momentum balance for an arbitrary control volume. A detailed description is provided of a novel second-order accurate spatial discretisation technique which can accommodate unstructured moving meshes with cells of arbitrary topology. This is accompanied by a fully implicit temporal discretisation, which makes the method stable for any time step size. The resulting set of coupled non-linear algebraic equations is solved by employing a segregated approach, leading to a decoupled set of linear algebraic equations for each dependent variable, with a sparse diagonally dominant coefficient matrix. These equations are solved by an iterative conjugate gradient solver which retains the sparsity of the coefficient matrix, thus achieving a very efficient use of computer resources.The method has been tested on a number of fluid flow and stress analysis problems and numerical calculations show favourable agreement with analytical and/or experimental results. In order to demonstrate the full capabilities of the present method, an example of numerically complex coupled fluid flow, stress analysis and heat transfer calculations is presented.
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A general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction. Such flows give rise to parabolic differential equations and so can be called three-dimensional parabolic flows. The procedure can be regarded as a boundary-layer method, provided it is recognised that, unlike earlier published methods with this name, it takes full account of the cross-stream diffusion of momentum, etc., and of the pressure variation in the cross-stream plane. The pressure field is determined by: first calculating an intermediate velocity field based on an estimated pressure field; and then obtaining appropriate correction so as to satisfy the continuity equation. To illustrate the procedure, calculations are presented for the developing laminar flow and heat transfer in a square duct with a laterally-moving wall.
A novel predictive model with compliance for arterial flows
  • F S Henry
  • M W Collins
Henry FS, Collins MW. A novel predictive model with compliance for arterial flows. Proceedings of the 1993 ASME Winter Annual Meeting, New Orleans, 1993.