International Journal on Architectural Science, Volume 3, Number 1, p.1-19, 2002
A REVIEW ON THE APPLICATIONS OF FINITE ELEMENT METHOD TO
HEAT TRANSFER AND FLUID FLOW
T.Y. Chao and W.K. Chow
Department of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, China
(Received 21 November 2001; Accepted 17 January 2002)
ABSTRACT
Practical engineering problems in heat transfer and fluid flow involve one or more governing equations, together
with some boundary conditions over a domain. The domain is often complex and non-uniform. The ability to
use a mesh of finite elements to accurately discretise domain of any size and shape makes the finite element
method a powerful tool to numerically analyse problems in these areas. This paper reviews the applications of
finite element approaches in heat transfer and fluid flow, and highlights some recent advances in this method.
These include improvements in methodology and mesh adaptivity, as well as techniques to improve the
efficiency and estimate the error bounds. Some aspects closely related to the finite volume method have also
been investigated.
1.
This equation has an analytical solution [1]:
INTRODUCTION
In the precomputer era, solving engineering
problems often demanded vast amount of time to
derive analytical or exact solutions. Although
these solutions often provided excellent insight into
the behaviour of some systems, analytical solutions
could be derived for only a limited class of
problems. Since the late 1940s, the widespread
availability of digital computers has led to a
veritable explosion in the use and development of
numerical methods. These techniques can greatly
enhance the capabilities to confront and solve
complex problems, and to handle large systems of
equations, nonlinear behaviour, and complicated
geometries that are often difficult or impossible to
solve analytically.
For example, the governing equation of the
fundamental two-dimensional heat conduction
problem is:
∂ 2u
∂x 2
+
∂ 2u
∂y 2
=0
(1)
where u(x, y) is the temperature distribution in the
Cartesian coordinates x, y, and is defined in a
rectangular region 0 ≤ x ≤ a, 0 ≤ y ≤ b , together
with the boundary conditions:
 u(0, y) = 0 and u(a, y) = 0 for all 0 ≤ y ≤ b


 u(x,0) = 0 and u(x, b) = u for all 0 ≤ x ≤ a
0

(2)
4u
u(x, y) = 0
π
∞
1
k =1 2k + 1
∑
(2k + 1) πy
(2k + 1) πx
a
sin
(2k + 1) πb
a
sinh
a
(3)
sinh
This expression is not simple, and it still demands a
numerical procedure to evaluate. It is desirable to
recast the problem by considering various forms of
discretization. The discretised form of the problem
only requires the solution to be satisfied at a finite
number of points in the region; and in the
remainder of the region, appropriate interpolations
may be used. Thus, the problem is reduced to a
purely algebraic form involving only the basic
arithmetic operations, which could in turn be
solved by numerical methods.
With the arrival and advancement of high-speed
digital computers, the cost-effectiveness of
numerical procedures has been greatly enhanced,
and these methods have become very accurate and
reliable for solving initial and boundary value
problems. One common numerical technique in
engineering analysis is the finite element method
(FEM).
2.
HISTORICAL BACKGROUND
The modern use of finite elements started in the
field of structural engineering. The advent of jet
engine in the 1940s and the resulting changes in
aircraft speeds had led to the change from unswept
to swept wind designs. The first attempt was by
Hrennikoff [2] who developed analogy between
1
International Journal on Architectural Science
actual discrete elements and the corresponding
portions of a continuous solid, and it was adapted
to aircraft structural design.
Based on
displacement assumptions, Turner et al. [3]
introduced the element stiffness matrix for a
triangular element, and together with the direct
stiffness method, described the method for
assembling the elements. Clough [4] introduced
the term ‘finite element’ in a paper describing the
applications in plane elasticity.
Works on the solution of nonlinearity problems had
become more prominent. Incremental technique to
solve geometrical nonlinearity problems was
initiated by Turner et al. [3], and stability problems
were analysed by Martin [5]. Material nonlinearity
problems, such as plasticity and viscoelasticity,
were discussed by Gallagher et al. [6] and
Zienkiewicz et al. [7] respectively.
To find a solution to this system, apply the
weighted residual method [12] and yield:
∫Ω W j (Au − f )dΩ + ∫ Γ W j (Bu − t ) dΓ = 0
where Wj(j = 1,…,n) are weighting functions and
u is an approximation to the unknown u:
u≈u=
n
∑ N ju j
(7)
j=1
in which Nj are some basis functions and uj are the
nodal values of the unknown.
Substituting equation (7) into equation (6), a
system of equations can be obtained:
Ku = f
Melosh [8] utilized the principle of minimum
potential energy and provided the first convergence
proof in the engineering literature. This led to the
use of variational principle that extended the use of
FEM in many new areas. Zienkiewicz and Cheung
[9] examined the solution of Possion’s equation,
and Wilson and Nickell [10] considered the
transient heat conduction problems. The method
also found applications in the field of biomedical
engineering, where geometric and material
nonlinearity would be involved. This problem was
first investigated by Gould et al. [11].
3.
FINITE ELEMENT METHOD
The fundamental idea of the FEM is to discretise
the domain into several subdomains, or finite
elements. These elements can be irregular and
possess different properties so that they form a
basis to discretise complex structures, or structures
with mixed material properties. Further, they can
accurately model the domain boundary regardless
of its shape.
To establish a ‘general purpose’ method for solving
problems in heat transfer and fluid flow, consider
the system of differential equations:
Au = f
in
Ω
in
Γ
where K is a square matrix, and u, f are some
vectors [13].
The Galerkin version of FEM (GFEM) is defined
when the weighting function in equation (6) is:
W j = Nj
(9)
This leads to minimum energy norm errors and
preserves the symmetry of matrix K, and it is the
most frequently used version of FEM. This method
is sometimes called the Bubnov-Galerkin methods
(BGFEM). One popular version of FEM is the
Petrov-Galerkin finite element method (PGFEM)
that uses the sum of the corresponding shape
function plus a perturbation term as each weighting
function [14,15].
Based on similar technique, the finite volume
method (FVM) was also developed. It was first
applied to solve two-dimensional, time-dependent
Euler equations in fluid dynamics by McDonald
[16], and then extended to three-dimensional flows
by Rizzi and Inouye [17]. The idea is to take the
weighting function as:
Wj = I
in
Ωc
(Wj = 0
elsewhere)
(10)
where I is the unity matrix and Ωc is a control
volume which can be discretised in different ways
[12].
(5)
The advantage of FVM is that, for example in fluid
flow, the fluxes are calculated only on twodimensional surfaces of the control volume instead
of on three-dimensional space. Also, this method
allows the shape and location of the finite volumes,
as well as the rules and accuracy for the evaluation
where A is a system of governing equations
defined in the domain Ω, B is a system of some
boundary functions defined in the boundary Γ, and
f, t are some functions. This system governs many
applications in the engineering field.
2
(8)
(4)
with the boundary conditions:
Bu = t
(6)
International Journal on Architectural Science
of the fluxes through the control surface, to vary,
thus giving considerable flexibility to the method.
Naturally, when different types of elements, or
different order approximating functions Nj, are
used in FEM, different numerical results are
obtained. The same is true when different control
volume Ωc is used in FVM. Two main types of
formulations have emerged for FVM to define the
variables: the cell-centred and the cell-vertex
schemes. In the ‘cell-centred’ scheme, the flow
variables are averaged values over the cell and can
be considered as a representation of the central
point of the cell [14], while in the ‘cell-vertex’
version, the variables are attached to the mesh
points on the cell vertex [15], as shown in Fig. 1.
3
C
4
i-1,j
2
i,j+1
9
8
i+1,j E
1
i,j
D
F
B
A
5
6
i,j-1
G
7
K
H
(a) Cell-centred structured finite volume mesh
C
i-1,j+1
D
i-1,j
B
i,j+1
i-1,j-1
i+1,j+1
i+1,j
i,j
G
K
H
The coupling of different versions of FEM and
FVM has also provided extra dimension in solution
methodology. For example, the common features
of GEFM and CVFEM in CFD, such as domain
discretization, interpolation, and the same matrix
form for the resulting systems of discretised
equations, have allowed successful coupling of
these two methods [21]. The hybrid methods have
been proved to be very effective and successful,
and it is obvious that they have great potential for
further investigation.
Comparisons between FEM and FVM have been
regularly featured in many engineering applications,
including CFD and heat transfer, by numerical tests.
It has been found that in some occasions, FVM can
be readily confined to element assemblies and can
be more efficient to approximate coefficients on
interfaces, and, higher order elements can be
implemented without much complication. The
combination of these two methods is capable of
producing a more efficient scheme [12]. For
example, one possibility is to use finite elements
for the diffusive part and finite volumes for the
convective terms, with the help of linear triangular
elements.
F
E
A
its success in fluid flow, FVM for structural
analysis started attracting attention [18]. The
concept of FVM was enhanced by Baliga [19] in
the form of control volume finite element methods
(CVFEM). In the report detailed by Minkowycz et
al. [20], examples of thick plate bending and
welding, compressible flow on a plane nozzle, and
flow in a model gas turbine combustor by FVM
were described. Further, solution procedures by
CVFEM on problems in multidimensional steady,
incompressible fluid flow and heat transfer were
illustrated.
i+1,j-1
i,j-1
(b) Cell-vertex structured finite volume mesh
Fig. 1: Two-dimensional finite volume mesh
systems
Traditionally, computational structural mechanics
was based on FEM, while FVM appeared to be
most widely used and arguably most successful in
computational fluid dynamics (CFD). Because of
4.
TRENDS AND PROBLEMS
FINITE ELEMENT METHODS
IN
The generalized FEM, based on weighted residual
formulations, has become one of the most popular
approaches to solve continuum problems in the
areas of solid mechanics, fluid dynamics, heat and
mass transfer [22]. One of the most important
aspects in finite element computations is the
mapping from the physical coordinate space to the
local coordinates. The mapping depends on the
discretization of the domain and the choice of the
type of elements. Two families of elements are
generally considered: Lagrangian elements require
C0 continuity at the inter-element boundary, and
Hermitian elements impose continuity of higherorder derivatives at the inter-element boundary.
The choice of a suitable subdivision of the region
into finite elements, which sufficiently represents
3
International Journal on Architectural Science
large gradients in the solution and approximates the
bound geometry, is a fundamental consideration in
FEM. New types of elements are regularly
proposed to model different properties and
mechanical responses in finite element modelling.
These include amalgamating two or more standard
types of elements into one, for example, the
coupling of eight-noded isoparametric elements
and three-noded beam elements to model the grout
and the steel bolt respectively of a two-dimensional
rockbolt element [23].
The use of higher order elements is attractive in
terms of computational accuracy. Taking approximation as given by equation (7), a higher order
basis function Nj can be involved. This necessitates
the increase in the number of degrees of freedom
and thus the size of the resulting global system of
equations, which in turn, would impair the
convergence rate in the solution procedure. At
present, some researchers are comparing accuracy
and efficiency between using a finer mesh
discretization (h-version), higher order element or
higher order interpolation polynomials (p-version),
and both (hp-version). The results were first
assessed by Babuška and Dorr [24], yet, no
conclusion can be drawn.
For each element, the weighted residual method
gives rise to element stiffness matrix. When all
elements are dealt with, these element stiffness
matrices are assembled to form a global system Ku
= f. Whence, the next step is to solve this matrix
equation numerically. Two families of methods
can be used: direct and iterative methods. Direct
methods are adequate to solve linear problems; but
for nonlinear problems, iterative methods of
different forms are often used. Linked with
iterative methods, convergence acceleration
techniques, such as preconditioning [25] and
multigrid methods [20], have recently been
developed to dramatically improve the convergence
rate. Different variants of Quasi-Newton method
[26] have also received much attention. Whichever
method one chooses to use, the consistency,
accuracy, stability and convergence of the method
have to be analysed [27,28]. These factors are very
important for the practical use of FEM and are
essential for the reliability of computations.
One further area that is gaining much attention is
the coupling of FEM with other advanced
numerical methods, such as boundary element
method (BEM), discrete element method (DEM)
and structural element method (SEM). This allows
FEM to combine the advantages of one or more
other methods and to minimize disadvantages [23].
For example, the hybrid FEM-BEM uses standard
finite element formulations for the non-overlapping
domain decomposition, and couples these weakly
4
with boundary element formulations over the
coupling boundaries. Computationally, linkage is
achieved by imposing continuity, equilibrium or
other conditions on the common edges. The hybrid
FEM-BEM is perhaps the first hybrid method, and
it was first investigated by Zienkiewicz [29]. The
applications of the hybrid methods have now
extended to a wide range of applications from
geomechanics to magnetic field problems,
biomechanics, vibrations and acoustics.
Treatments on advection-dominated problems in
heat transfer and fluid flows are increasingly
playing an important role in the development of
FEM.
For the simulation of electrophoresis
operation phenomena and the operation of a large
number of chemical reactors, numerical solutions
to diffusion, advection and reaction dominated
problems are often required to represent modelling
of simplified industrial processes. One of the more
popular numerical methods to solve these types of
equations is the Eulerian finite element method.
Giraldo and Neta [30] have shown that the semiLagrangian method could use larger time steps, and
therefore offer greater efficiency and accuracy.
Cubic spline, cubic Hermite and cubic Lagrange
interpolations were used for the trajectory and
departure point calculations, and it has been found
that cubic spline interpolation yielded the best
results.
The Eulerian-Lagrangian Localized Adjoint
Method (ELLAM) has been widely employed to
solve advection-dominated linear transport
problems; and some procedures, based on Picard or
successive approximations, have been used to solve
nonlinear problems. Aldama and Arroyo [31] has
proposed an advanced method that based on the
Taylor-Frechlet expansion of the nonlinear
advection-diffusion-reaction operator.
This
produced an approximate linear problem that could
then be tackled by ELLAM.
5.
APPLICATIONS
OF
FINITE
ELEMENT
METHOD
IN
ENGINEERING
Many of the finite element techniques in use today
were found in structural analysis in the 1960s,
though the original concepts dated back a century
ago. Having found successful applications in linear
and nonlinear structural mechanics, FEM has been
used as a general approximation method for many
physical problems in various engineering fields,
such as computational modelling of forming
processes, geomechanics, solids with evolving
geometries and multiple fracturing, eigenvalue
problems, multi-field problems, contact mechanics
and composite system, adaptive methods for time
International Journal on Architectural Science
dependent problems, time harmonic Maxwell
problems, advection-diffusion problems, heat and
mass transfer, compressible/incompressible flows,
laminar/turbulent boundary layer equation, and
blood flow.
so that the terms involving the weighted integral
∂u
on the boundary vanishes, equation (15)
∂n
becomes:
As an example to show how FEM can be used to
solve a typical engineering problem, consider the
two-dimensional steady-state heat diffusion
problem. The residual RΩ over the region Ω is
defined by:
∫ Ω 
RΩ =
∂
∂u
∂
∂u
(k ) +
(k ) + Q
∂x ∂x
∂y ∂y
(11)
where u is an approximation to the unknown u and
contains trial functions, k is the thermoconductivity, Q is the rate of heat flow, together
with the respective Dirichlet and Neumann
boundary conditions of:
u = u0
on the time boundary
Γu
(12)
and
−k
∂u
=q
∂n
on the boundary
Γq
(13)
(17)
In finite element analysis, the domain Ω is divided
into a number of nonoverlapping subregions of
finite element Ωe. If each element has P degrees of
freedom, the Galerkin representation of the real
solution is:
u e (x, y) =
i =1, P
∑ u ie N ie (x, y)
(18)
Ωe
 ∂u

+ q  dΓ = 0
Wi k
∂
n


 ∂Wi ∂u ∂Wi ∂u 
−
+
k
k  dxdy +

Ω
∂y ∂y 
 ∂x ∂x
∫
∂u
k Wi dΓ +
Γu + Γq
∂n
∫
N ie (x, y) is the basis function, equation (17) can be
modified to:
 ∂N e ∂N ej ∂N e ∂N ej 
i
 i k
 u ej dxdy −
k
+
Ω e  ∂x
∂y
∂y 
∂x


∫Ω
e
N ie Q dxdy +
(14)
∫ Ω Wi Q dxdy +
 ∂u

+ q  dΓ = 0
Wi k
Γq
∂
n


Equation (19) can be written in matrix form as:
K ije u ej = f ie
(i, j = 1, P)
(20)
where the element conductivity matrix K ije and the
K ije =
∫
 ∂N e ∂N ej ∂N e ∂N ej 
i
 i k
 dxdy
k
+
Ω e  ∂x
∂y
∂y 
∂x


(21)
and
Γu
Γq
(i, j = 1, P)
element load vector f ie are respectively defined by:
(16)
on
N ie q dΓ = 0
that portion of the boundary of Ωe which lies on Γq.
Setting the arbitrary weighting functions Wi and
Wi as:
on
qe
with Ωe as the area of an element and Γqe denotes
(15)
 Wi = 0



 Wi = − Wi
∫Γ
(19)
and by Green’s lemma, equation (14) can be
rewritten as:
∫
Wi q dΓ = 0
This is known as the weak form of the steady state
heat conduction equation.
∫
∂

∂u
∂
∂u
Wi  (k ) +
(k ) + Q  dxdy +
Ω
∂
x
∂
x
∂
y
∂
y


q
q
e
The weighted residual method gives:
∫Γ
∫ Ω Wi Q dxdy + ∫ Γ
where u i is the nodal value of the solution u, and
defined in the boundary curve Γ = Γq + Γu .
∫
 ∂Wi ∂u ∂Wi ∂u 
k
k  dxdy −
+
∂y ∂y 
∂x ∂x
f ie =
∫Ω
e
N ie Q dxdy −
∫Γ
qe
N ie q dΓ
(22)
Assembling for all elements for the global system
Ku = f where:
5
International Journal on Architectural Science
K=
M
∑ ℑe K ije ℑe
e =1
T
and
f =
M
∑ ℑe f ie ℑe
T
(23)
APPLICATIONS
TO
HEAT
TRANSFER AND FLUID FLOW
Among the many research groups throughout the
world whose common interest is in the theoretical
development and applications of FEM, the
following is a small selection whose expertise in
these areas may be of great interest to fellow
researchers and practitioners, especially in the field
of Architectural Science. They are listed in
alphabetical order according to their university
name with their research activities outlined:
y
Chung at the University of Alabama at
Huntsville, USA
- CFD by flowfield-dependent variation
(FDV) methods [32].
y
Feistauer at Charles University, Prague, The
Czech Republic
- hybrid schemes for solving nonlinear
convection-diffusion and compressible
viscous flow problems [33].
y
y
Bettess at the University of Durham, UK
- wave envelopes to model progressive
short wave with time independent
potential satisfying the Helmholtz
equation [34];
- two-dimensional wave envelope infinite
element [35].
Minkowycz at the University of Illinois at
Chicago, USA
- spatially periodic flows in irregular
domains [36];
- Sparrow-Galerkin approach to radiation
exchange between surfaces [37].
y
Vanka at the University of Illinois at UrbanaChampaign, USA
- CVFEM and multigrid method for internal
flows and heat transfer [38].
y
Bathe at Massachusetts Institute of
Technology, USA
- acoustic
fluid-structure
interaction
problems [39];
- fluid flows coupled with structural
interactions [40];
- finite element program package ADINA-F
[40].
6
Idelsohn at INTEC, Universidad Nacional del
Litoral, Santa Fe, Argentina
formulation
for
- Petrov-Galerkin
advection-reaction-diffusion
problems
[41].
y
Baines at the University of Reading, UK
- adaptive grid method [42].
y
Hughes at Stanford University, USA
- a priori and a posteriori error estimates for
general linear elliptic operators [43];
- stabilised FEM [44].
y
Baker at the University of Tennessee,
Knoxville, USA
- CFD study of airflow in a mixing box [45];
- Taylor weak statement CFD algorithms
for convection-dominated flows [46,47].
y
J. N. Reddy at Texas A&M University, USA
methods
for
viscous
- multigrid
incompressible flows [48].
y
Babuška at the University of Texas at Austin,
US
- a posteriori error estimators and adaptive
procedures for the h-version [49];
- superconvergence [50];
- p- and hp-versions of FEM for elliptic
equations
(solids)
and
hyperbolic
equations (fluids) [51].
y
Zienkiewicz at the University of Wales,
Swansea, UK
- preconditioning and Galerkin multigrid
method (GMG) [52];
- object-oriented source codes [53].
e =1
in which ℑe is the Boolean matrix representing the
assembly procedure. The global system can then
be solved by any standard numerical method.
Details of this method can be found in Huang et al.
[1].
6.
y
These research activities are briefly described in
the following sections.
7.
RESEARCH BY CHUNG AT THE
UNIVERSITY OF ALABAMA AT
HUNTSVILLE, USA
y
CFD by Flowfield-dependent Variation
(FDV) Methods
In general, solutions are obtained from a single
algorithm dictated by the FDV parameters as
calculated from the current state of flow fields, thus
allowing the governing equations to be modified or
adjusted automatically according to the current
flow field in space and time.
The Navier-Stokes system of equations in
conservation form is expanded by Taylor’s series
up to and include the second order time derivatives
to initially introduce the variation parameters.
They are translated into flow field dependent
physical parameters to characterise fluid flows, and
International Journal on Architectural Science
are based on variables which have the following
properties: the presence of shock waves in
compressible flows is indicated by the sudden
change of Mach number; turbulent microscale
fluctuations for viscous flows are characterised by
rapid changes of Reynolds number; high
temperature gradients are indicated by changes in
Peclet number; finite rate chemistry or stiffness of
species equations are characterised by changes in
Damkohler number; and triple shock wave
turbulent boundary layer interactions within the
secondary separation regions subjected to
separation shock and rear shock are represented by
the simultaneous abrupt changes of Mach number,
Reynolds number and Peclet number. These
changes are recorded between adjacent nodal points.
In addition, adequate numerical controls, such as
artificial viscosity, are automatically activated
according to the current flow field so that both
fluctuating and non-fluctuating parts of the variable
can be addressed. These parameters can also serve
as physical parameters to control numerical
accuracy and stability in the solution process, and
allow the transitions and interactions of different
types of flow to be automatically accommodated.
In the FDV theory, the traditional definitions of
implicit and explicit schemes are significantly
modified. From the current flow field variables,
the variation parameters are calculated which
dictate the numerical accuracy and stability in the
solution procedure, and allow the transitions and
interactions of different types of flows to be
automatically accommodated. Therefore, the FDV
is a powerful scheme that can solve problems
concerning transitions and interactions between
inviscid/viscid, compressible/incompressible and
laminar/turbulent flows. Finite difference method
(FDM) or FEM is employed as a way to discretise
between adjacent nodal points or within an element
and as a solution methodology, but not to dictate
the physics of the problems. Moreover, because of
the FDV’s capability to automatically generate
fluctuation variables, many numerical schemes in
FDM and FEM are shown to be special cases of the
FDV theory. It is shown, in detail, how the
numerical
diffusion
and
shock-capturing
mechanism are built into the FDV equations of
momentum, continuity and energy. Further, the
transitions and interactions between compressible
and incompressible flows, and between laminar and
turbulent flows solved by using the FDV scheme
are fully described.
Several examples are used to validate the FDV
theory. These are contour plots of calculated
variation parameters to test the flow fielddependent properties; shock tube problems to test
the shock capturing ability; driven cavity flow
problems
to
test
the
incompressibility/
compressibility characteristics; and accuracy of
FDV simulation for turbulent flows in supersonic
flows. These sample problems have successfully
verified that the FDV scheme is capable of
demonstrating most of the features available in the
FDV theory. More works are currently being
carried out to include more additional features, and
the various definitions of Damkohler numbers have
been examined to determine how they could be
contributed to aid convergence in stiff equations for
combustion problems [32].
8.
RESEARCH BY FEISTAUER AT
CHARLES UNIVERSITY, PRAGUE,
THE CZECH REPUBLIC
y
Hybrid Schemes for Solving Nonlinear
Convection-Diffusion and Compressible
Viscous Flow Problems
The viscosity and heat conduction coefficients of
gases are small, so that viscous dissipative terms
are often considered as perturbations in the inviscid
Euler system. It implies that an effective numerical
method for solving inviscid flow must be
considered. A hybrid FVM and FEM scheme is
proposed to solve nonlinear convection-diffusion
problems and compressible viscous flow using a
general class of cell-centred flux vector splitting
FVM discretization of inviscid terms together with
FEM discretization of viscous terms over a
triangular grid.
To apply this combined FVM-FEM scheme to a
simplified scalar nonlinear convection-diffusion
conservation law equation, it is important to
analyse the theoretical implication of this scheme.
The convergence analysis and the discretization of
viscous flow are based on FEM. The theoretical
analysis can be generalized to the case of
nonhomogeneous
mixed
Dirichlet-Neumann
boundary conditions on a piecewise-smooth
boundary. Several other mesh systems are being
tried as alternatives, for example, triangular finite
volume-triangular finite elements, and barycentric
finite volumes-nonconforming finite elements; and
the results are being compared. Two ways are
proposed to increase the accuracy – the use of
numerical flux that depends on the values of
second-order recovery of the piecewise constant
FVM solution combined with the use of a suitable
flux limiter; and with the aid of automatic adaptive
mesh refinement in the vicinity of shock waves,
based on a shock indicator using divided
differences of the density and taking into account
the direction of the flow. The inviscid-viscous
operator spitting scheme can be applied to this
hybrid method when the inviscid system and the
7
International Journal on Architectural Science
purely viscous system are split and discretised
separately.
This hybrid method has been applied to several test
problems with an aim to solve the problem of
viscous transonic flow via time stabilization. The
applicability and robustness of this scheme is
justified in the solution of the complete viscous
compressible transonic flow system that consists of
the continuity equation, the Navier-Stokes
equations, the energy equation and the state
equation with experimental data. Particularly good
numerical results are obtained when the OsherSolomon numerical flux is applied to a primary
triangular finite volume mesh and combined with
the FEM discretization on an adjoint triangulation
in examples such as the flow of air through the
GAMM channel and the flow past a cascade of
profiles using the inviscid-viscous operator
splitting method [33].
9.
RESEARCH BY BETTESS AT THE
UNIVERSITY OF DURHAM, UK
y
Wave Envelopes to Model Progressive
Short Wave with Time Independent
Potential
Satisfying
the
Helmholtz
Equation
The goal of this method is to model short wave for
problems like sonar and radar accurately and
economically with a few elements. The complex
potential ϕ in terms of the real wave envelope A
and the real phase p is expressed as ϕ = Aeip so that
in most regions, the functions A and p vary much
more gradually over the domain than the oscillatory
potential ϕ. Nine-noded Lagrange elements are
used, while A and the phase function s are assumed
to have quadratic variation within each element.
Using the standard Galerkin shape and weighting
functions, the wave envelope in integral form is
integrated by parts to give a stiffness matrix which
is Hermitian.
To provide an estimate for the phase, the wave
envelope is determined from the finite element
computation. Let ϕ0 be the potential determined by
the estimate po = ks0 for the phase function, then:
φ 0 = A 0 e iks1
(24)
where A 0 is real (in general the wave envelope
A0 has an imaginary part). Hence, ks1 is used as
the new estimate for the phase function such that:
i A 
s1 = s 0 − ln 0 
k  A 0 
8
(25)
and the error in s0 can be related to the wave
envelope at node n calculated from the element
stiffness matrix equation, whence the element
matrix integral can be evaluated by GaussLegendre approximation.
An iterative process is formed so that in each step,
the error obtained from the resulting finite element
calculation is added to give a better estimate for the
phase. The iteration is found to be always
convergent without exception, even if the initial
convergence is slow due to poor initial estimate.
The convergence is non-uniform, thus further work
is necessary to improve the convergence rate by
investigating the choice of the estimate for the
phase function. This method gives satisfactory
estimation to the phase in several two-dimensional
plane wave diffraction problems, and has great
promise for solving problems where the
wavelength is much smaller than the element size.
Numerical calculations are carried out to first test
the iterative process by using the complete
formulation on the plane wave and Hankel source
in a rectangular domain where the phase value is
perturbed at some nodes by some fixed amount. It
is seen that with different perturbation values or
with a poor initial estimate for the phase, the exact
value of the wave envelope is obtained. For the
diffraction potential for plane waves incident upon
a vertical cylinder, it is found that for the near-field
diffraction problem, the generalized Astley
formulation, which assumes the eikonal equation
holds near the cylinder, does not give convergent
results for the wave envelope and hence it can only
be used far from the body [34].
y
Two-dimensional Wave Envelope Infinite
Element
This is an extension of the FEM for the above so
that they can be used in conjunction with infinite
element analysis to model a two-dimensional wave
diffraction problem with short wavelength with
complex progressive wave potential that satisfies
the Sommerfield condition and the Helmholtz
equation.
The near field is discretised by nine-noded
isoparametric quadrilateral finite elements, while
the far field is analysed by six-noded infinite
elements which are developed in polar coordinates.
For the near field, convectional finite element
interpolation is used for the amplitude A and the
phase s independently, as well as a convectional
iterative method to approximate the phase.
For the far field, the expression for the wave
envelope takes the form:
International Journal on Architectural Science
A(r, θ) =
α(o)
r
+
 1
+ 0 2
r r
r r
β(o)



(26)
and the phase
1
s(r, θ) = r + γ (o) + 0 
r
(27)
where α(o), β(o) and γ(o) are some well-behaved
continuous functions of the polar angle o. The
wave envelope from the differential equation rather
than the potential is modelled, so the usual shape
function variation without the oscillation term eikr is
used. Applying Dirichlet boundary condition to the
governing
finite
element
integral,
three
formulations for infinite elements can be deduced.
To find a better estimate p = ks1 from an estimate
of the phase po = ks0 and from the resulting finite
element calculation for the wave envelope Ao, an
iteration is defined as follows: during the Ith step of
this iterative process, the phase is updated by an
I
amount ε j and the update is determined by:
s Ij+1 = s Ij + ε Ij
for the nodes
j = 1,3,5
(28)
for the nodes
j = 2,4,6
(29)
and
s Ij+1 = s Ij + ε Ij−1
To demonstrate this method numerically, the two
different potentials used are the two-dimensional
Hankel source potential and the wave diffraction
potential past a circular cylinder given by Havelock.
Indifferent results are obtained from the three
formulations, and it can be concluded that the best
results come from Astley’s approach that uses
weighting functions of lower order in the field
variable so that the contour integral contribution in
the governing finite element integral tends to zero
at far field. As for the iteration, it is shown that the
wave number and initial guess for the iteration are
crucial to the convergence of the iteration. Two
choices are recommended – use the ray theory to
estimate the wave direction; or initially solve the
problem for a low wave number, and then use it to
provide the first estimate for a phase at a higher
wave number. It is also found that, for short-wave
diffraction problems, the evaluation of the wave
envelope and phase instead of the potential is more
appropriate. On the whole, the new method would
avoid high computational cost, since each
wavelength requires ten nodes to model the
oscillatory variation of the potential and a fine
mesh for the problem [35].
10. RESEARCH BY MINKOWYCZ AT
THE UNIVERSITY OF ILLINOIS AT
CHICAGO, USA
y
Spatially Periodic Flows in Irregular
Domains
Based on the relative orientation of the modules,
two types of periodicity are considered:
translational and rotational. When the geometry of
the flow problem is complex, periodic boundary
fitted grids are often used over a typical module to
predict such flows. Finite volume nonstaggered
grid methods are often used to discretise the
momentum and continuity equations in fluid flow.
The advantage is that Cartesian velocity
components that are fixed in space are employed as
the primary unknowns. These components are not
periodic in rotationally periodic geometries and
hence it is necessary to consider incorporating the
periodicity conditions over the periodic modules.
In the discretization of the momentum equations,
values of the fluxes are assumed to be available. It
is important that the right face of the discretised
momentum equation lies along a periodic boundary
so that periodicity is preserved for both types of
periodic geometries. To discretise the continuity
equation that involves the balance of mass flow
through the faces of the control volumes, the face
velocities/flow rates in terms of the velocities at the
centre points are calculated, and the velocities on
the two sides of the face are averaged to prevent the
occurrence of the checkerboarding of pressure and
to take care of the sensitivity of the face flow rates
sensitive to the staggered pressure gradient.
Another important consideration for the flow in
periodic geometries is the occurrence of pressure
drop across a periodic module, which arises
through geometries where the throughflow is
driven by the pressure drop across the modules, but
is independent on the type of periodicity in the
geometry. These considerations give rise to a
unified nonstaggered grid for discretising the
governing equations in domains for non-periodic
flows and for both types of spatially periodic flows
in the computation procedure.
The consistency and accuracy of this method is
ascertained by comparing the computed solutions
of the two-dimensional Couette flows in a parallel
plate channel and in a cylindrical annulus. It is
found that the accuracy of the computed results
improves with grid size, and error is almost
negligible when the optimal grid size is used.
Two-dimensional problems in incompressible flow
with translational periodic turbulence and laminar
flow in a cylinder with external longitudinal flow
rotates within a stationary shroud are also
investigated, and again, good results are obtained
[36].
9
International Journal on Architectural Science
y
Sparrow-Galerkin Approach to Radiation
Exchange Between Surfaces
The governing equations for radiation exchange are
Fredholm’s integral equations. Sparrow provided
the formulation of the variational solution and
Galerkin introduced the method of weighted
residuals to solve general differential equations.
The Sparrow-Galerkin method refers to the
extension of Galerkin method to solve radiation
exchange problems. It has been shown that, in
general, this method could provide highly accurate
results, and could converge rapidly to the exact
solutions, except in some cases, for example when
the changes in radiosity cannot be adequately
described even by polynomials of degree 14. This
difficulty could be overcome by subdividing each
surface into smaller surfaces, and each is treated as
a separate surface for inclusion by the SparrowGalerkin method. This becomes the SparrowGalerkin finite element method. This method can
be further enhanced by the use of higher-order
element in the p-version of FEM. Examples of
linear systems have been analysed by this method,
but for nonlinear systems, such as those arise from
thermal radiation in conjunction with conduction or
convection, modification must be made so that
nonlinearity problem can be overcome.
For
nonlinear or integrodifferential equations, a
linearization scheme must first be developed [37].
11. RESEARCH BY VANKA AT THE
UNIVERSITY OF ILLINOIS AT
URBANA-CHAMPAIGN, USA
y
CVFEM and Multigrid Method
Internal Flows and Heat Transfer
for
This procedure combines a CVFEM on general
unstructured grids with multigrid method to speed
up the convergence rate of the numerical solution,
and demonstrates its performance in the
computation of natural convection in square,
triangular and semicircular enclosures with
differently heated walls.
A CVFEM using
triangular elements is employed to discretise the
Navier-Stokes
equations
with
equal-order
interpolations of the flow variables. In the iterative
solution procedure, single and multigrid methods
are employed.
In the single grid method, the iteration starts with
the initial velocity and pressure fields which are
used to obtain a pressure distribution, and in turn,
to solve the momentum equations. The resulting
velocity field is then used to update the solution of
the energy and pressure equations. To ensure
stability of the iteration, part of the change in the
flow field is added to the flow variables, and the
10
remainder comes from the previous solution, so
that the modified momentum equation in the xdirection becomes:
u = χur+1 + (1 - χ) ur
(30)
where ur+1 is the latest iterate, ur is the previous
value, and 0 < χ < 1. This method is convergent,
but the convergence rate is slow if the mesh is
refined, or if more nodes are used to resolve the
flow features accurately.
In the multigrid method, low frequency errors on a
fine mesh are transformed to higher frequency
errors on coarser meshes so that better convergence
rate can be obtained by using a sequence of
increasing coarser grids to which errors are
transferred and subsequently interpolating the
corrections obtained on the coarser grids to finer
grids. Initially, the coarse mesh is read first from
the input file, and mesh refinement is obtained by
successfully subdividing each element.
An
advantage of constructing fine grids embedded
within the coarse grid is that coarse-grid values are
obtained by simply taking the values from fine-grid
nodes that coincide with the coarse-grid. Nonlinear
equations are best tackled by the use of full
approximation scheme to derive the coarse-grid
equations. Further, this method can be easily
extended to solve more complex situations such as
three-dimensional problems and turbulent flows in
complex geometries.
The performance of this method is tested by the
natural-convective flow simulation modelled in
three geometric configurations; namely, square,
triangular and semicircular cavities.
With a
Rayleigh number of 105, it is shown that multigrid
method at several grid densities leads to a
significantly faster convergence rate than single
grid method; but with a bigger Rayleigh number,
the convergence of the solution is much slower for
both methods [38].
12. RESEARCH
BY
BATHE
AT
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY, USA
y
Acoustic Fluid-structure
Problems
Interaction
In this method, the pure displacement-based
formulation is replaced by a displacement/pressure
(u/p) formulation via a variational indicator. The
standard Galerkin finite element discretization
procedure is applied to give the matrix equations of
the u/p formulation. The solvability and stability of
this equation is satisfied under the inf-sup condition
by the use of mixed elements – the u/p elements
International Journal on Architectural Science
correspond to continuous displacements and
discontinuous pressure, whereas the u/p-c elements
yield continuous displacements and pressure across
the element boundaries.
In order to reduce the number of zero frequency
modes, the u-p- Λ formulation is used to consider
the vorticity moment. Again, Galerkin finite
element discretization is used. It is important to
choose appropriate interpolations for the
displacement, pressure and vorticity moment that
satisfy the inf-sup condition in the analysis of solid
and viscous fluids; and for the vorticity moment,
use the same or a lower-order interpolation as for
the pressure. Thus, various types of element are
proposed to use as the basis of the FEM.
If a discontinuous pressure approximation is used,
the degrees of freedom for pressure are statically
condensed out on the element level, so that only the
degrees of freedom for nodal displacement are
present in the assembling process. Slip boundary
conditions of the mass and momentum
conservation around the fluid boundaries and fluidstructure interfaces must be satisfied.
It is
important to allocate appropriate tangential
directions at all boundary nodes so that tangential
boundary conditions can be accommodated,
otherwise, spurious non-zero energy modes are
obtained in the finite element solution. For the
solution of frequencies, the u/p formulation can
predict and obtain the exact number of zero
frequencies, and the number can be reduced by the
use of u-p- Λ formulation. The tall water column
and rigid cavity problems are used as examples to
produce numerical results for comparison. In these
examples, the results agree with the analytical
solutions, and the number of zero frequency modes
is found to be always the same as those from
mathematical prediction [39].
y
Fluid Flows Coupled with Structural
Interactions
In this method, Arbitrary Lagrangian-Eulerian
(ALE) formulation is used to describe the fluid
flow coupled with a Lagrangian formulation for the
structural response. In the evaluation of the total
time derivative of the variables of a fluid particle,
the ALE formulation advocates that the spatial
position is not fixed in space, but is allowed to
move. To solve the governing equations, the
motion is selected, but it does not necessarily
correspond to the particle motion, which would
otherwise be a pure Lagrangian formulation. The
fluids are modelled as compressible or
incompressible media with various material laws.
For low Reynolds and Peclet number flows, the
standard Galerkin procedure is used, and elements
that satisfy the inf-sup conditions are used to
ensure convergent results. For high Reynolds and
Peclet number flows, an upwinding procedure is
embedded in the finite element equations, which
are obtained by the use of Galerkin variational
procedure on the diffusive flux of the equations and
a control volume type procedure on the convective
flux. Upwinding is applied to the flow directed
through the element faces and it can be
implemented very effectively for triangular and
tetrahedral element discretizations.
The element mesh density is prescribed by
assigning either a certain element size to a
complete geometric element or different element
sizes to the points, lines and surfaces defining that
geometry. The meshing is dependent on the
geometry data, and boundary conditions are
automatically transferred to element nodes. If
structural interactions are included, ALE
formulation for the fluid and the kinematical
enforcement of the fluid nodes to lie on the surface
of the structure are used to ensure motions are
compatible between different media.
The solution of the finite element equations is
provided by the biconjugate gradient method or
generalized minimum residual method (GMRES)
together
with
an
incomplete
Cholesky
preconditioner. A specular-diffusive radiation for a
general radiation heat transfer analysis is also
featured. The solution capabilities of this method
represent a powerful tool in the field of CFD, and it
can be implemented to parallel processing machine
for greater efficiency [40].
y
Finite Element
ADINA-F
Program
Package
This software package is designed to analyse
problems in incompressible flows (with or without
heat transfer), compressible flows, and free-surface
flows [40].
13. RESEARCH BY IDELSOHN AT THE
UNIVERSIDAD NACIONAL DEL
LITORAL, ARGENTINA
y
Petrov-Galerkin
Formulation
for
Advection-Reaction-Diffusion Problems
When solving advection-diffusion problems,
difficulties often arise in the spurious oscillations
obtained from Galerkin method when there are
discontinuities in the solution. One of the most
popular methods to overcome these difficulties is
the Streamline Upwind Petrov-Galerkin (SUPG)
method. This scheme is based on the addition of a
perturbation function, which is a function of the
dimensionless Peclet number, to the weight
11
International Journal on Architectural Science
function and hence produce an oscillation-free
solution. Storti et al. [41] introduced a scheme that
included two perturbation functions to the weight
function and the corresponding proportionality
constants. The first one is similar to the one
involved in the SUPG method, while the other one
is symmetric. The combination of these two
functions invariably offers flexibility: for
advection-diffusion problems, it reduces to the
standard SUPG scheme; for reaction-diffusion
problems, the symmetric perturbation function is
used, and this method is called the Centred PetrovGalerkin (CPG) method; and for intermediate
situations, a combination of these two perturbation
functions are used, and it is called the (SU+C)PG
method.
In the latter method, the two
dimensionless numbers, Peclet number and the
reaction number, govern the proportionality
constant for each perturbation. The uniform
convergence of the finite element solution can be
ascertained when the discrete maximum principle
(DMP) is satisfied, and it is found to be dependent
on the region of stability of the Peclet and the
reaction numbers. It is shown that there are some
limits on the region of stability for SUPG and CPG,
but no limit for (SU+C)PG so that it is convergent
in the whole region. It has also been shown that
superconvergence is closely related to the DMP,
and hence (SU+C)PG can be further enhanced to
tackle superconvergence for a broader class of
problems.
the use of moving finite element method, which
can be further enhanced by partially assembling the
matrix to converge faster. In higher dimension, it
may be necessary to introduce regularization such
as small penalty functions to avoid problems with
singularity. An example from the theory of
shallow water flow in a channel is used to illustrate
direct minimization.
14. RESEARCH BY BAINES AT THE
UNIVERSITY OF READING, UK
which equidistributes the arc length s. The discrete
values of the continuous variable defined by:
y
∫ M(x) dx
ξ = ab
~ ~
∫ a M(x) dx
Adaptive Grid Method
Unlike the standard methods, adaptive schemes
have the capability to enhance the preservation of
essential shape properties of the solution, the
capacity for feature capturing, and the provision of
other properties other than those provided by
polynomial accuracy. These schemes are usually
data-dependent and have shape-preserving
properties. They are developed generally for
convection-dominated scalar problems and the
solution of systems of conservation laws
approached through pseudo-time iteration. It is
also possible to adapt the grids by moving or
subdividing the grid so that local features can be
better represented to improve local accuracy.
Two principal ways to represent data are direct
minimization
and
equidistribution.
Direct
minimization arises from the minimization of a
measure of the error directly with respect to nodal
positions as well as the coefficients of the
approximation. The resulting nonlinear equations
require iteration methods to determine the optimal
grids and solutions. One method to achieve this is
12
Another method is equidistribution which is a
standard device to achieve grid relocation. In a
one-dimensional representation, a monitor function
M(x), which is usually derived from a data function,
is first defined. A choice of monitor functions is
available to maximize the efficiency of this method.
For example, if f(x) is a data function, a monitor
function could be defined by:
M(x ) =
df
dx
(31)
which equidistributes f itself, or by:
M( x ) =
ds
dx
(32)
where
s( x ) = ∫
x
x
a
2
 df 
1 +  ~  d~
x
 dx 
~ ~
(33)
relate the gridpoints xj in the physical space a ≤ xj ≤
b to the corresponding gridpoints ξj in the
computational space 0 ≤ ξj ≤ 1. This technique can
be generalized intuitively into higher dimensions.
Grid movement iterations based on equidistribution
can be interleaved with standard schemes. If the
PDE residual is chosen as the monitor function, it
is likely that an equidistribution step will be more
evenly distributed across the grid, and a grid
adaptation allows the scheme to converge more
uniformly as iterations for the PDE converge.
Further, grids generated by equidistribution can be
used as an initial guess in the direct minimization
algorithms. This interleaving technique is
illustrated with good results by three examples:
grids correspond to a Possion equation; a steady
state scalar problem for the advection of a square
profile in a circular trajectory; and a hydraulic
problem concerning the steady flow of water
International Journal on Architectural Science
through a channel with a slight constriction
governed by the shallow water equations. All these
examples employ two-dimensional equidistribution
iterations interleaved with the solver, together with
a two-dimensional equivalent of equation (32).
Continuous efforts are devoted to find a general
method suitable for all grid adaptation purposes. It
is also suggested that a combination of grid
relocation and grid subdivision might offer a better
alternative [42].
15. RESEARCH BY HUGHES AT
STANFORD UNIVERSITY, USA
y
A Priori and A Posteriori Error Estimates
for General Linear Elliptic Operators
This theory is based on the one-dimensional
Galerkin method, but it can be readily converted
into a multidimensional case. The energy norm is
defined by ⋅
E
= B( ⋅ , ⋅ ) , where B( ⋅ , ⋅ ) is the
bilinear form. In the energy norm, symmetric
operators are positive definite, and a priori estimate
for the finite element solution for these operators is
derived by first comparing it with the interpolation
error. Nonsymmetric operators can be decomposed
into a symmetric part and a skew symmetric part,
with the skew norm defined by:
w
skew
= sup
v∈V
1
2
B(w, v) − B(v, w)
v
(34)
E
where V is the space of weighting functions,
whence a priori estimate is found. As an example,
the advection-diffusion equation is analysed, and a
proof follows very closely to the one for the
symmetric positive definite operator.
A posteriori error estimate is based on an explicit
residual-based Galerkin finite element discretization. In the energy norm, the methodology for
deriving a posteriori error estimate for symmetric
and nonsymmetric operators is the same. The
estimate for each interior element is first
investigated, and the boundary terms are treated as
element (interior) quantities. Finally, the global
error is obtained as a summation of the element
contributions. For a priori error estimate in the L2
norm, two cases are studied – positive definite
operators and indefinite operators. For positive
definite operators, a priori error estimate in the
energy norm is recalled, and it is converted to the
L2 norm and the Nitsche trick, in which the extra
power of h (the nondimensional measure of
element length) is extracted to explicitly show the
convergence in the L2 norm. For the indefinite
operators in the Holmholtz equation, the Nitsche
trick is first applied to the error bound; and from
the coercivity of the indefinite operator, the bound
is converted to the L2 norm. A posteriori error
estimate in the L2 norm is independent on the
symmetry and definiteness of the operators, and it
is similar in structure to the one in energy norm
[43].
y
Stabilised FEM
Stabilised methods constitute a systematic
methodology for improving stability behaviour
without compromising accuracy, and provide
fundamental solutions to the problems of discrete
approximations in several practically important
areas, such as convection-diffusion operators. One
approach is the use of residual-free bubbles
whereby Galerkin’s method that involves standard
and simple polynomial finite element spaces, such
as linear triangles, is improved by systematically
enriching the space with residual-free bubbles, or
Galerkin’s method is applied to the enlarged finite
element space consisting of the standard
polynomials and residual-free bubbles. The other
approach is the implementation of the variational
multiscale procedures. The first step in this
approach is to decompose the original problem into
two subproblems, solve for the fine scales in terms
of the coarse scales first, and then substitute the
result into the second subproblem to give a
modified problem that involves only the coarse
scales.
It is shown that these two approaches are
equivalent, so that in practice, they yield the same
results in the enlarged space and hence the same
equation on the original space. Further, they share
the same attributes and shortcomings. For the
shortcomings, both approaches leave an unresolved
part that an analytical solution does not practically
exist except in simple element geometries for
certain linear problems. A number of remedies are
suggested: to approximate the variable coefficients
to simplify the solution at the element level; or to
replace the exact Green’s function by a numerical
approximation. The classical application to the
convection-diffusion operator is used to show how
the above methodology can reproduce the
Streamline-Upwind
Petrov/Galerkin
(SUPG)
stabilization scheme. SUPG method consists of
adding a term which introduces a suitable amount
of artificial viscosity in the direction of streamlines,
but without upsetting the consistency, to the
original bilinear form [44].
16. RESEARCH BY BAKER AT THE
UNIVERSITY
OF
TENNESSEE,
KNOXVILLE, USA
13
International Journal on Architectural Science
y
CFD Study of Airflow in a Mixing Box
In a heating, ventilating and air-conditioning
(HVAC) system, the air-handling unit (AHU) is the
merging point of two air streams. It acts as an
interface between the primary plant and the
secondary system. The mixing box is a component
of this unit. It handles the mixing and the
distribution of temperature across the discharge
plane of the box, and the selection or location of
the mixing air temperature sensors control the
mixing effectiveness.
To study air motion in HVAC systems, the
mathematical modelling of fluid flows considers
the solution of the nonlinear partial differential
conservation equations for mass, momentum and
energy throughout the computation domain, and is
analysed by a commercial program CFX4. The
pre-processor modelled the experimental AHU
developed at the Energy Resource Station of the
Iowa Energy Centre, USA. The inlet ducts are
extended three hydraulic diameters upstream so
that uniform velocity profiles can be imposed at the
duct entrance. To enforce confined flow at the
mixing box outlet, ducts are extended three
hydraulic diameters downstream.
Inside the mixing box, two dampers are placed in
parallel at the end of the inlet ducts, and are
inclined towards each other. They are modelled by
thin flat plates. The ducts and the mixing box are
modelled as adiabatic, and the flow is steady
incompressible turbulent with standard density and
viscosity.
The main aim of this study is to investigate air
distribution at different combinations of damper
positions and air temperatures. It has been found
that the flow in the inlet is fully developed just
upstream of the dampers mostly between the two
damper blades, and eddies from downstream of the
blades at the corners of the mixing box where
separation occurs. At the centre of the mixing box,
the two streams come together and mix slowly as
the flow continuous downstream. A large eddy
forms at the top of the mixing box while the main
flow is in the lower half of the box area. The
computed results agree with expectations and with
field data, and are instrumental in determining the
best location for mixed air temperature sensors and
single-point freezestats, which is found to be about
30% of the distance up from the bottom of the box
in the mixing layer [45]. The next stage of
development is a study on pressure variations and
airflow rates at various damper angles.
•
Taylor Weak Statement CFD Algorithms
for Convection-dominated Flows
14
To improve CFD algorithms for convectiondominated flows, one must consider problems
arising from the Galerkin weak statement (GWS).
The methods of Lax and Wendroff, and Donea
have been generalized as the Taylor weak
statement (TWS). In this generalization, temporal
derivatives are substituted into an explicit Taylor
time series to give the Taylor series-modified semidiscrete form. The TWS family gives rise to a
wide range of independently derived, dissipative
Galerkin weak statement algorithms. They can be
handled by the use of some common methods, such
as SUPG, Taylor Galerkin (TG) and least-squares
(LS) [46].
The modified equation analysis developed for finite
difference method is extended to the GWS-TWS
finite element CFD methods, and it has been shown
that stability for GWS and various TWS
formulations are predictably improved for a range
of traditional and non-traditional Lagrange basis
forms.
The subgrid embedded (SGM) finite element basis
offers another approach to stability. It reduces the
basis to linear basis element rank for any embedded
degree via static condensation, and introduces an
embedded function to augment the diffusion term.
The performance of the SGM element has been
tested for verification and applications, and found
to compare favourably with the off-design de Laval
nozzle shock benchmark problem discussed by
Liou and van Leer [47].
By including the acoustic component, the
contraction of an upstream-bias tensor, and the
convection and pressure decomposition of the
kinetic flux vector, the nonlinear element upstream
weak statement algorithm can be obtained as a
characteristics-based replacement of the TWS
formulation. In this algorithm, the streamwise
dissipation depends on Mach number, while the
cross flow dissipation decreases for increasing
Mach number and becomes supersonic flow.
While the acoustic perturbation is important for
accurate approximation of acoustic wave
propagation, it is also pivotal for global stability
analysis.
17. RESEARCH BY REDDY AT TEXAS
A&M UNIVERSITY, USA
y
Multigrid
Methods
Incompressible Flows
for
Viscous
In order to model complex flows by finite element
method, the use of fine mesh is required to properly
model the details of the flow in local regions, thus
giving rise to a large system of algebraic equations
among nodal variables. To solve such a large
International Journal on Architectural Science
system, iterative solvers are often used because
they do not require matrix inversion or assembly of
global matrix. The accuracy depends on the
convergence parameters used, while the
convergence rate depends on the condition number
of the coefficient matrix. For large values of
penalty parameters, convergence could be slow.
Although this drawback could be overcome by the
use of an iterative penalty function method, a more
efficient way to improve the convergence rate
would be to incorporate the method of successive
refinement, or the multigrid method, to the iterative
solvers.
coarser mesh for a given convergence tolerance.
The accuracy of the multigrid-iterative solver
method is found to be comparable with direct
solvers, but it consumes a fraction of the computer
time [48].
In this method, the analysis starts with a coarse
mesh. A solution of this analysis is taken as an
initial guess for the subsequent series of finer mesh.
This can resolve the high frequency error
components of the coarse mesh to become the low
frequency error components in subsequent finer
meshes, and hence the large wavelength errors that
accompany the initial guesses, which can otherwise
reduce the convergence rate, can be avoided. The
nonlinear equations are linearised and solved by
iterative solvers. Two types of iterative solvers are
considered, and they are variants of the conjugate
gradient algorithms for non-symmetric systems.
One solver minimizes the energy norm of the
system of equations, the other minimizes the
residual norm. They employ element-by-element
data structure of the coefficient matrix and so no
restriction is imposed on the numbering of
elements, and parallel processors can be used to
solve the system efficiently. The mesh is refined
and the above procedure is repeated until the finest
mesh has been used. In each iterative step, the
Newton-Raphson method is used to determine the
location of the fine mesh nodes within a coarse
mesh, and the shape functions at the Gauss point
corresponding to the new node location in the
coarse mesh element.
These interpolation
functions are used to obtain the initial flow field for
subsequent finer meshes. The convergence rate of
the iterative solvers can be further improved by
scaling the coefficient matrix and the force vectors
by the use of diagonal (Jacobi) preconditioning,
and multilevel preconditioning such as LU
decomposition.
Among many estimators for the h-version, residual
estimations and smoothing (recovery) based
estimations are the two commonly used types. For
practical use, it is important to select an estimator
by assessing its quality and robustness. To assess a
particular error indicator for any given translationinvariant mesh (for example, mesh patterns which
are regular, chevron, union jack, or criss-cross
types), it is possible to compute the asymptotic
lower and upper bound CL and CU of the effectivity
index for particular estimators, when the bound is
over a set of solutions and over a class of meshes.
To validate the performance of this method,
examples in incompressible flow through a channel
with a sudden expansion and laminar flow of
incompressible fluid inside a two-dimensional lid
driven square cavity are analysed. To determine its
competitiveness, single grid-iterative solver and
several solvers are used for comparison. It can be
seen that the convergence rate of the single griditerative solver depends on the mesh density and
the solution algorithm. Iterative solvers are found
to be less accurate with a fine mesh than with a
18. RESEARCH BY BABUŠKA AT THE
UNIVERSITY OF TEXAS AT
AUSTIN, USA
y
A Posteriori Error Estimators and
Adaptive Procedures for the h-version
Defining the robustness index R as:

1
1
+ 1−
R = min  1 − C U + 1 − C L , 1 −

CU
CL


 (35)


The index depends on the pattern but only weakly
on the mesh in the neighbourhood of the elements
under consideration. It characterizes the quality of
the estimators well. It has been shown that R = 0
represents the optimal value of the robustness index,
while various estimators that have large robustness
index, especially for elements with large aspect
ratios, anisotropic materials etc. should, in practice,
not be used. In general, the most robust residual
estimator is the equilibrated one. The estimate
should guarantee an upper bound on the entire
domain although the robustness index is not
necessarily small. For elements of higher degree or
when the solution is not smooth (for example, in
the neighbourhood of corners), an adjustment to
this theory has to be made, and that is currently
under investigation. The aims are to produce high
quality estimators in the preasymptotic range that
do not contain any noncomputable terms [49].
y
Superconvergence
Since the finite element solution oscillates about
the exact solution, it is possible to allow some
values to be locally extracted so that higher
accuracy can be obtained from a finite element
solution than direct computation. Therefore, it is
15
International Journal on Architectural Science
necessary to locate the superconvergence points, i.e.
points at which the accuracy, for example in the
first derivative, of the finite element solution are
better by an order than in a general point.
The theory of the computer-based proof of
convergence shows that the superconvergence
point in an element, if exists, can be determined by
a finite number of operations. Like a posteriori
error estimation, the pollution error, which is the
effect of errors originating outside of the element
under consideration of the error within the element,
is required to be negligible and the exact solution is
smooth. As an example, it is found that, for the
Poisson problem −∆u = f for elements of degree
three, there is only one superconvergence point for
a general function f, and six superconvergence
points for a set of harmonic functions. But in the
general case, especially for the elasticity equations,
no superconvergence point exists. It must be noted
that superconvergence in the interior elements has
to be distinguished with elements in the boundary
or mesh with refinement pattern.
Extraction
methods, which are special techniques to obtain the
data of interest, have also been investigated. These
techniques are more accurate than direct
computations, and do not require special meshes,
but are more complicated than the simple pointwise
superconvergence technique [50].
y
p- and hp-versions of FEM for Elliptic
Equations
(Solids)
and
Hyperbolic
Equations (Fluids)
It can be shown that the hp-version converges
exponentially with respect to the number of degrees
of freedom for problems characterized by
piecewise analytic data from the estimate in energy
norm:
u − u FE
E
< Ce − γN
α
(36)
where u and uFE are respectively the exact and the
finite element solutions, and N is the number of
degrees of freedom. The coefficient α depends on
the dimension of the problem (α = 13 for twodimensional and α =
1
5
for three-dimensional
problems), while C and γ > 0 depend on the
strength of the singularity of the solution in the
neighbourhood of the boundary (edges, corners etc.)
and interfaces. It means that the solution, the
domain, the distortion of the elements used, and the
family of meshes used together with the
distribution of the degree of elements all contribute
to the estimate.
From the analysis of the p-version, it can be shown
that its rate of convergence is at least twice that of
the h-version for problems characterised by
16
piecewise analytic data with uniform or quasiuniform meshes. In some specially constructed
meshes, the p- and the hp-versions have been
reported to behave similarly and have almost the
same performance in computation. It is also
remarked that if a higher p is used, the stiffness
matrix will become denser and hence the cost of
construction of the stiffness matrix is higher. From
the numerical results obtained from the elasticity
problem defined on a cracked domain when
homogeneous isotropic behaviour is assumed, it is
shown that the N1/3- log u − u FE E graph is almost
a straight line, which is in agreement with equation
(36) [51].
19. RESEARCH BY ZIENKIEWICZ AT
THE UNIVERSITY OF WALES,
SWANSEA, UK
y
Preconditioning and Galerkin Multigrid
Method (GMG)
The incomplete Cholesky decomposition with no
fill-in scheme (IC(0)) is a commonly used
preconditioning technique to obtain a rapid
convergence rate for an iterative method in the
solution procedure of FEM. The idea is to find an
easily computed close approximation to the
stiffness matrix A and its inverse either explicitly
or implicitly. One way to enhance the performance
of IC(0) is to ensure that a negative diagonal term
is to be avoided in the incomplete decomposition
procedure. To do that, it is suggested to modify the
matrix A by A = A + λD , where D = diag(A) and
λ is the smallest possible positive parameter to
make the diagonal of the incomplete decomposition
of A to be positive. In the process of applying
IC(0) preconditioning to linear matrix equation,
three versions are available: left, right and left-right
preconditionings. Each method leads to a different
pattern, condition number and eigenpair of the
preconditioned matrix, and consequently has
different effects on the convergence of the iterative
methods [52].
The essence of the multigrid method is to consider
two grid forms, one coarse and one fine grid, to
discretise the same geometrical domain. These
meshes may be nested or non-nested, structured or
non-structured. The coarse mesh correction can
substantially reduce the smooth part of the error
induced by the multigrid method (which may not
be swept out effectively by iterative methods). In
GMG, the coarse grid matrix is constructed by
direct projection of the fine grid matrix through a
transfer operator. The transfer operator transfers
displacement correction and residual back and forth
between unrelated unstructured meshes, and
International Journal on Architectural Science
obtains coarse grid approximations. Thus, the
success and efficiency of GMG depend on the
coarse grid, interpolation formulation, and the
quality of the transfer operator. Two types of
transfer operators are looked at: the global based
transfer operator and the more efficient local based
transfer operator.
In geometrically nonlinear
problems, the same transfer operator is used
throughout the whole computational process,
resulting in a further simplification of the
implementation of GMG for these problems.
y
Object-oriented Source Codes
The fundamental aspects of finite elements are
described as objects to improve the underlying
structure of finite element programs so that
research codes are more flexible, more useful, and
can adapt to new ideas promptly [53].
20. CONCLUSIONS
Despite the continuous progress made in other
numerical techniques, FEM offers enormous
flexibility in the treatment of nonlinearities,
inhomogeneities and anisotropy. The objective of
this paper is to identify some trends in FEM and
their relation to research in engineering. Different
versions of FEM have been described. It is
obvious that successful use of FEM depends on the
reliability of the formulations, the parameters used
and proper interpretation of the results. The
coupling of FEM with other advanced numerical
methods has been briefly described, and research in
this area has certainly been picking up momentum.
It is hoped that works from different disciplines,
which common interest is finite element methods,
can promote wider awareness throughout the finite
element community of the latest developments in
engineering and mathematics.
ACKNOWLEDGEMENT
This project is funded by Area of Strategic
Development in Advanced Buildings Technology
in a Dense Urban Environment with account
number 1-A038.
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19