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Journal ArticleDOI

An improved boundary element method for analysis of profile polymer extrusion

01 Jul 1997-Engineering Analysis With Boundary Elements (Elsevier)-Vol. 20, Iss: 1, pp 81-89

Abstract: An improved boundary element method for non-linear viscoelastic flow analysis is reported. In this method, the domain integral representing the non-linear effects is calculated using a more efficient approximating technique. This is achieved by first transforming the domain integral into a form that can be approximated by particular solutions to the original problem. These particular solutions are in fact expressed as a linear combination of radial basis functions, whose coefficients are found by data fitting technique. As a result, the numerical computation of the volume integral is eliminated and a significant reduction of CPU time is achieved. The routines are tested with simple flows and then applied to solve complex three-dimensional direct and inverse extrusion problems of polymeric fluids, such as thermoplastic melts. Inverse extrusion process, where an extrusion die profile needs to be computed for a given profile extrudate, is a very important practical engineering application which is successfully analysed by the present method. Relative to a previous BEM implementation where the volume integral is computed directly using numerical quadratures, a CPU time reduction ranging from 40 to 70% is achieved.

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ELSEVIER
PII:SO955-7997(97)OOO60-X
Engineering Analysis with Bounda~ Elements
20 (1997) 81-89
© 1997 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0955-7997/97/$17.00
An improved boundary element method for
analysis of profile polymer extrusion
T. Nguyen-Thien a, T. Tran-Cong a'* & N. Phan-Thien b
aFaculty of Engineering and Surveying, University of Southern Queensland, Toowoomba QLD 4350, Australia
bDepartment of Mechanical and Mechatronic Engineering, The University of Sydney, Sydney NSW 2006, Australia
(Received 19 February 1997; accepted 29 May 1997)
An improved boundary element method for non-linear viscoelastic flow analysis is
reported. In this method, the domain integral representing the non-linear effects is
calculated using a more efficient approximating technique. This is achieved by first
transforming the domain integral into a form that can be approximated by particular
solutions to the original problem. These particular solutions are in fact expressed as a
linear combination of radial basis functions, whose coefficients are found by data
fitting technique. As a result, the numerical computation of the volume integral is
eliminated and a significant reduction of CPU time is achieved. The routines are tested
with simple flows and then applied to solve complex three-dimensional direct and
inverse extrusion problems of polymeric fluids, such as thermoplastic melts. Inverse
extrusion process, where an extrusion die profile needs to be computed for a given
profile extrudate, is a very important practical engineering application which is
successfully analysed by the present method. Relative to a previous BEM
implementation where the volume integral is computed directly using numerical
quadratures, a CPU time reduction ranging from 40 to 70% is achieved. © 1997
Elsevier Science Ltd.
Keywords:
Boundary element method, domain integral transformation, particular
solution, profile polymer extrusion.
1 INTRODUCTION
Boundary element methods (BEM) have become popular
techniques for solving boundary value problems in solid
and fluid mechanics. Many linear problems involving partial
differential equation (PDE) such as potential flow (the
Laplace equation), linear elasticity (Navier's equation)
and viscous creeping flow (the Stokes equation) have been
solved successfully using BEM. Obviously, for the prob-
lems involving homogeneous PDE this technique has cer-
tain advantages over the finite element and the finite
difference methods because it requires only discretization
of the boundary of the domain, thus reducing the dimension-
ality of the problems by one. Unfortunately, this advantage
is greatly offset in non-linear problems such as viscoelastic
flows. In these problems, the non-linearities can be formu-
lated into a body force term and the problem can be solved
*To whom correspondence should be addressed.
81
iteratively. At each stage of the iterative process, a linear
problem is solved, with the non-linear terms estimated from
the results of the previous iteration. 1-3 The resulting integral
equations include domain integrals. These integrals are
usually computed by numerical quadrature techniques
which require domain discretization. 4"5 However, a draw-
back of this procedure is the high demand of CPU time in
the computation of the domain integral, which is about 60-
65% of the total CPU time required for a three-dimensional
extrusion problem using a coarse mesh such as MSHI.
When a fine mesh (such as MSH2) is used, this percentage
is even higher. When the number of volume nodes is large
(for a fine mesh) this procedure is apparently inefficient.
Moreover, it is inconvenient from the numerical point of
view, especially near corner points where the stresses are
infinitely large. 6
Alternative methods of calculating the domain integral
have been used recently in order to make the BEM more
effective. 6-1° Most of the proposed techniques that have

82
T. Nguyen-Thien
et al.
been used so far belong to either methods of particular
solutions or methods of transforming domain integrals to
boundary ones. The concept in both methods is similar in
principle; the difference is in the way the total solution for
the problem is obtained. In the former method, particular solu-
tions satisfying the inhomogeneous PDE are first found, and
the remainder of the solutions satisfying the homogeneous
PDE with appropriate boundary conditions, which must be
adjusted to ensure the correct boundary conditions for the
total solution, is obtained. Then the total solution is found by
adding the particular solutions to homogeneous ones. 6'1°' I I In
the latter method, the divergence theorem or the reciprocity
theorem is applied instead to convert domain integrals (also
called pseudo-body forces) into the boundary ones. 7-932
The functions used to represent the non-homogeneous
part of the problem could be chosen among a great number
of approximants. A good choice, however, is important for
numerical efficiency. Radial basis functions in recent years
have become popular for multidimensional interpolation. 13.
Zheng
et
al. 6"14
and Zheng and Phan-Thien 12 showed some
advantages and successful results of the radial basis func-
tions in the particular solutions methods, as applied to some
inhomogeneous potential problems. We also prefer the use
of radial basis functions in this study.
The main purpose of the present work is to extend the
study of three-dimensional extrusion of viscoelastic
fluids 535 using the methods of particular solutions. In
the next section we briefly recall the basic equations govern-
ing viscoelastic flow problems and their integral equation
representation, followed by a derivation of the approximate
pseudo-body force using radial basis functions. Next the
details of the numerical treatment and solution scheme are
described, which are validated with some test results involving
simple flows. The method is then applied to study complex
three-dimensional direct extrusion and inverse extrusion flows
of viscoelastic fluids. The efficiency of the method is demon-
strated by CPU time-saving ranging from 40 to 70%.
is the upper-convected derivative of the extra stress tensor,
with L the velocity gradient tensor and L x its transpose.
With the introduction of the relaxation time, a new dimen-
sionless group, the Wiessenberg number, arises. It is
defined as
Wi= k'~,
where ~, is a typical shear rate.
In BEM applications, eqn (1) in conjunction with eqn (2)
is written as
o = -
pl +
2~pD + e (4)
where 2~lpD represents the total (arbitrary) linear part of the
stress tensor and e the non-linear part. From this equation
and balance of mass and momentum, we obtain the follow-
ing integral equation:
Cij(x)uj(x) = ~nUij* (x,
y)tj(y) dI'(y)
- f~Dt~(x,y)uj(y)
dr(y)
_ JfDeJk(Y )
Ou~(x,y)
dfl(y) (5)
cgx k
where u is the velocity field, t the traction field, u*(x,y) and
t*(x,y) are known kernels (Stokeslet and its associated trac-
tion field, see, for example, Tran-Cong and Phan-Thien 5'15
and Tran-Cong16). The last integral on the RHS of eqn (5),
considered as a pseudo-body force, does not introduce any
unknown. The use of gaussian quadrature formulas to cal-
culate this integral, however, as mentioned above, could be
time consuming, and sometimes it is practically impossible
for studying complex problems such as extrusion process at
high Wiessenberg numbers, especially at refined mesh.
Alternatively this domain integral can be calculated
approximately using particular soulutions, as described in
the next section.
2 BOUNDARY INTEGRAL FORMULATION
We consider a steady, isothermal flow of viscoelastic fluid
with a single relaxation time, where the stress tensor can be
arbitrarily decomposed as: 5
a= -pl +
2~TsD + T (1)
Here p is the hydrostatic pressure which arises due to the
incompressibility constraint, 1 is the unit tensor, Os is 'sol-
vent' viscosity, D is rate of strain tensor, r is the extra-
stress tensor which is governed by a differential constitu-
tive equation of the type
X AT
~-+R=0 (2)
in which 3, is the relaxation time, R is model dependent, and
AT Jr
-- q- u.Vr - LT - TL + (3)
At at
3 APPROXIMATION OF THE VOLUME
INTEGRAL
The volume integral in eqn (5) can be converted onto the
boundary, using the divergence theorem after integrating by
parts as follows:
f ou~(x,
b i = JneJk(y)
aXk Y) dfl(y)
= [aoejk(y)uij(x,y)nk
dF(y)
fDu~.(x,
Y) 0ej~(x, y)
- Oxk
df/(y)
= J;~iejk(y)uij(x,
y)n~ dF(y) + u p (6)

Analysis of profile polymer extrusion
83
A
i
'- - n '~, x
I Ty=0. ! Vx= I
I Tz-"O.,I "~Vy=0.
~ '- - ~, %v~.
vy=o. i . ~Tx=O. ~
~,._-o. i IIS,1
, ,trio.,
i
I ' ""~" I ~
k.Tz---O.l h
I, I / \
-.:.2.2.].:.:.:.2.2.:.:.:.:.[.I.:.2.:.2.:[:.:.:::::::~:::.2::::."
". ~./
:i!iiii!i!iiiiiii !!i!i!iii! !ii!ivx o !:!
x -_ "':::::~:2:2:2:2:::I::2::I:::2:2:I:2Vy=0.:22222~22:2:2::2:1 -.
Fig. 1. Boundary conditions for Couette flow. V and T are velocity and traction vectors.
in which u p is defined as
u p = - ~ u*'x
0ej,(x,
Y)
D ij( ,y )
" ~
dfl(y)
= - J V~(x,y)~j(y) dfl(y) (7)
where
1 0ej,(x, y) (8)
~PJ(Y)- 2
Oxk
and
I'ij(x,
y)
=
2uij(x ,
Y)
Parton and Perlin 17 have proved that the solution for u p in
eqn (7) can be found by solving the Navier equations
(
1 p ~2uP )
k, 1 --S-~ vv'u + = - 2~,(x) (9)
#
Here ~ is the Poisson's ratio. We will consider the limiting
case where 1, ---* 1/2 in the final results, since this limit is
regular.
To transform eqn (9) into the biharmonic equation we use
the 'Galerkin' form of solution given by
u p=VV.G --1 V2 G (10)
2(1 - x,)
where G is the Galerkin vector (see, for example, Phan-
Thien and Kim18). Substitution of eqn (10) into eqn (9)
yields the following equation:
V2V2G = f (11 )
where
f= -2 -~
02)
#
If a function G satisfying eqn (11) can be found then the
corresponding u p is determined by eqn (10). Note that
in this final form of the particular solution, the limit of
---, 112 is not singular. One way to find G is to approximate f
X
" ::::::: :::::::::::r::::
============================================
~-iiiiiiiV~-i:i:i!!i!:i:i:!:!:!i~-!:~
~:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:T-~.:.-:.--:-...-y ~
ITx=O. I )TxffiO. Ii ~ L
= I,,.._~ I ivy-~., I I ^'1 F
I'ql
,v~,t
IVy=°'I /
i*~:1 '- - -,i ~Vz---0.1 I~
B .............................~..:.:.......:.:........~_~j [ h
.....-...:.:.:.:. ==========================================
Fig. 2. Boundary conditions for Poiseuille flow. V and T are velocity and traction vectors.

84
T. Nguyen-Thien
et al.
Table 1. Maximum differences of velocity and stresses between solutions obtained from the two methods and analytical solutions for
Couette flow using a coarse mesh MSI-I1 (note that P1 = NTDBEM and P2 = NTDBEM96)
Wi
% Av .u max
I %Ar~,I ma~,
%At ~:(
m~ I % S
AV
P 1 P2 P 1 P2 P 1 P2
0.1 0.51 0.50 6.65 6.51 2.48 2.62 40
0.2 0.55 0.61 5.46 5.56 2.67 2.86 42
0.3 0.67 0.71 5.26 5.35 3.12 3.27 40
0.4 0.74 0.81 5.37 5.45 3.55 3.85 43
0.5 0.93 0.92 6.05 6.25 4.07 4.26 42
0.6 1.05 1.15 6.35 6.41 5.02 5.2 40
in terms of radial basis functions if:
N
f/(r) = y.
Olin~/(~'n)
(13)
n=]
in which i = 1, 2, 3 for three-dimensional problems, N is
the number of distributed points in the domain and
Ir-r~l
rn = (14)
where r~, ~n are suitably chosen constants and ~b is a func-
tion of a single variable] 2. Now eqn (1 l) reduces to, if its
right-hand side is replaced by a radial basis function (and
therefore a particular solution is also a radial basis func-
tion), the following:
1 O(~2O( 1 0(~20~)))
~2~\ ~\g~ =~(~) (15)
When the radial basis function ~b is chosen to be exp( - ~2)
for three-dimensional problems, a particular solution of eqn
(15) is given by
1(( ' ) v/~erf(~) + exp( _ ~2) _ 2)
(16)
The Galerkin vector in (10) corresponding to this particular
solution will be given by
N
Gi(r) -= E 4ffr.)ai (17)
n=l
(see Coleman
et al.
for more details).
Finally, with the substitution of the derivatives of
G i
from
eqns (15), 0 and (17) into eqn (10), the displacement u p can
be written as
1
u p ~- aiq~l - 1 - p(o~i4~2 +
~,i~,k~k4~3)
where
4,~ - q~"+ 2_4¢
r
(18)
(19)
1
~b 2 ~- -~b' (20)
F
1
~3~ ''- 7~ (21)
r
in which ? i =
O~'lOx,.
4 NUMERICAL IMPLEMENTATION
A decoupled technique, similar to that reported by Bush 3 is
implemented here. The procedure for finding the extra-
stress (the non-linear part) is similar to that of Tran-Cong
and Phan-Thien 5 and the details will not be repeated here.
The new feature of the present method is the way in which
the domain integral is computed. The data fitting techniques
will be used instead of numerical quadrature techniques.
Ifain, fin
and rn in eqns (13), (14), 0 and (17) are known
from eqns (19)-(21) and (18), the pseudo-body force b in
eqns (6) and (7) is also determined. In principle, the three
parameters can be found by using available non-linear data-
fitting techniques. However the calculation is performed
only by iteration and therefore such a scheme could be
Table 2. Maximum differences of velocity and stresses between solutions obtained from the two methods and analytical solutions for
Couette flow using a fine mesh MSH2 (note that P1 = NTDBEM and P2 = NTDBEM96)
Wi
%AY ~(maxl
%Ar~(,,a~l %Arx:,
,,ax~ %SAV
PI P2 PI P2 PI P2
0.1 0.13 0.14 2.25 2.41 1.48 1.51 55
0.2 0.19 0.21 1.55 1.70 1.66 1.81 55
0.3 0.28 0.30 1.86 2.15 2.08 2.12 56
0.4 0.37 0.39 2.92 3.02 2.25 2.32 58
0.5 0.41 0.42 3.42 3.61 2.55 2.67 58
0.6 0.52 0.58 3.55 3.82 2.80 2.93 58

Analysis of profile polymer extrusion
85
Table 3. Maximum differences of velocity and stresses between solutions obtained from the two methods and analytical solutions for
Poiseuille flow using a coarse mesh MSH1 (note that PI = NTDBEM and P2 = NTDBEM96)
Wi
%Avx(max) %ATxx(max) %AT"xz(max ) %SAV
P1 P2 P1 P2 P1 P2
0.1 1.43 1.54 1.10 1.05 1.81 2.01 41
0.2 1.42 1.50 1.05 1.06 1.85 2.2 42
0.3 1.30 1.43 1.02 1.06 1.80 2.3 42
0.4 1.10 1.05 1.04 1.05 1.90 2.3 43
0.5 1.00 0.92 1.02 1.04 1.85 2.1 43
0.6 0.85 0.71 1.00 1.02 1.85 2.1 43
time consuming. Alternatively we can use linear data-fitting
techniques by choosing/3, and rn
apriori
and leave o~in to be
determined. Zheng
et
al. 6"14 and Coleman
et al.~°
obtained
good results following this approach. Recommendations for
/3~ and r, are also given. In this work rn are chosen to be
suitably distributed points in the domain (here we use the
points resulted from a typical FE-type discretization). The
/3n for each point are chosen to be the average distance from
its immediate neighbours multiplied by a weighting factor,
and the weighting factor can vary in a range from 0.5 to 2.0
by numerical experiments. In order to obtain accurate
solutions, i3~ (or the weighting factor) must be carefully
chosen. This matter will be discussed in the next section.
To find oti, one can use either a collocation method or a
least-square fitting method. Both of those methods result in
a set of linear algebraic equations in which c~i, are unknown.
The use of the gaussian distribution as the basis functions
makes the system matrix effectively sparse, for which either
the Gauss elimination (GE) or an iterative solution method
can be used, such as the conjugate gradient method
(CGM) 19 which has been known to be more efficient for
large systems. The results shown in the next section were
obtained with both GE and CGM methods. The differences
in the results by the two methods are in the range of 0.01-
0.05%, which must be due to the iterative nature of the
CGM method.
5 NUMERICAL RESULTS
5.1 Test problems
In this section we show some test results obtained from the
procedure described above. The code was tested in some
simple flows (Couette and Poiseuille flows) for which
analytical solutions are known. The pseudo-body force
was calculated using gaussian quadrature formulas; 5 the
same procedure is adopted here. For each problem a
coarse mesh, MSH1 (with 73 boundary nodes and 79
domain nodes) and a fine mesh, MSH2 (with 709 boundary
nodes and 1549 domain nodes) were used.
Couette and Poiseulle flows of an upper-convected Max-
well fluid are used to test the program. The boundary con-
ditions for the two flows are illustrated in Figs 1 and 2. The
velocity and stress fields are obtained analytically by sol-
ving the field equations with appropriate boundary con-
ditions. 21 Field variables are non-dimensionalized
according to
t~ t X'---- X_, u'----U, 7"' 7"
-----~' a -------U
7--
a
where t is time, x is the position vector, u is the velocity
vector, r is the extra stress tensor and ~ is the viscosity. X is
the fluid relaxation time, a is a typical length and U is a
typical flow speed. Then, for Couette flow, the velocity
field v and stress field 7 are given as
V~
[!
t/p
, 7 ~
L2wi%~
10
For Poiseuille flow, we have
V=
~h 2
~-~(1 - z 2)
0
0
[
2~pWiz 2 0 -z%
0 0
L - Z~p 0 0
In these formulas 5' denotes shear rate,
Wi
= ~,'i' is the
Table 4. Maximum differences of velocity and stresses between solutions obtained from the two methods and analytical solutions for
Poiseuille flow using a fine mesh MSH2 (note that P1 = NTDBEM and P2 = NTDBEM96)
Wi
%Avx(max) °~AT"xx(max ) °~mTxz(max ) %SAV
P1 P2 PI P2 PI P2
0.1 0.83 0.92 0.23 0.23 0.65 0.59 55
0.2 0.80 0.83 0.26 0.25 0.69 0.63 55
0.3 0.78 0.75 0.25 0.26 0.71 0.65 56
0.4 0.74 0.67 0.27 0.28 0.71 0.66 56
0.5 0.60 0.58 0.30 0.31 0.73 0.66 56
0.6 0.58 0.41 0.31 0.30 0.81 0.67 56

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References
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16 Feb 2013-
TL;DR: This well written book is enlarged by the following topics: B-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for theLR and QR algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations and preconditioning techniques.
Abstract: This well written book is enlarged by the following topics: $B$-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for the $LR$ and $QR$ algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations, preconditioning techniques and convergence rate of the conjugate gradient algorithm and multigrid methods for boundary value problems. Cf. also the reviews of the German original editions.

6,228 citations


Book
01 Jan 1991-
Abstract: 1 Introduction.- 2 The Boundary Element Method for Equations ?2u = 0 and ?2u = b.- 2.1 Introduction.- 2.2 The Case of the Laplace Equation.- 2.2.1 Fundamental Relationships.- 2.2.2 Boundary Integral Equations.- 2.2.3 The Boundary Element Method for Laplace's Equation.- 2.2.4 Evaluation of Integrals.- 2.2.5 Linear Elements.- 2.2.6 Treatment of Corners.- 2.2.7 Quadratic and Higher-Order Elements.- 2.3 Formulation for the Poisson Equation.- 2.3.1 Basic Relationships.- 2.3.2 Cell Integration Approach.- 2.3.3 The Monte Carlo Method.- 2.3.4 The Use of Particular Solutions.- 2.3.5 The Galerkin Vector Approach.- 2.3.6 The Multiple Reciprocity Method.- 2.4 Computer Program 1.- 2.4.1 MAINP1.- 2.4.2 Subroutine INPUT1.- 2.4.3 Subroutine ASSEM2.- 2.4.4 Subroutine NECMOD.- 2.4.5 Subroutine SOLVER.- 2.4.6 Subroutine INTERM.- 2.4.7 Subroutine OUTPUT.- 2.4.8 Results of a Test Problem.- 2.5 References.- 3 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y).- 3.1 Equation Development.- 3.1.1 Preliminary Considerations.- 3.1.2 Mathematical Development of the DRM for the Poisson Equation.- 3.2 Different f Expansions.- 3.2.1 Case f = r.- 3.2.2 Case f = 1+ r.- 3.2.3 Case f = 1 at One Node and f = r at Remaining Nodes.- 3.3 Computer Implementation.- 3.3.1 Schematized Matrix Equations.- 3.3.2 Sign of the Components of r and its Derivatives.- 3.4 Computer Program 2.- 3.4.1 MAINP2.- 3.4.2 Subroutine INPUT2.- 3.4.3 Subroutine ALFAF2.- 3.4.4 Subroutine RHSVEC.- 3.4.5 Comparison of Results for a Torsion Problem using Different Approximating Functions.- 3.4.6 Data and Output for Program 2.- 3.5 Results for Different Functions b = b(x,y).- 3.5.1 The Case ?2u = ?x.- 3.5.2 The Case ?2u = ?x2.- 3.5.3 The Case ?2u = a2 ? x2.- 3.5.4 Results using Quadratic Elements.- 3.6 Problems with Different Domain Integrals on Different Regions.- 3.6.1 The Subregion Technique.- 3.6.2 Integration over Internal Region.- 3.7 References.- 4 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u).- 4.1 Introduction.- 4.2 The Convective Case.- 4.2.1 Results for the Case ?2u = ??u/?x.- 4.2.2 Results for the Case ?2u = ?(?u/?x+ ?u/?y).- 4.2.3 Internal Derivatives of the Problem Variables.- 4.3 The Helmholtz Equation.- 4.3.1 DRM Formulations.- 4.3.2 DRM Results for Vibrating Beam.- 4.3.3 Results for Non-Inversion DRM.- 4.4 Non-Linear Cases.- 4.4.1 Burger's Equation.- 4.4.2 Spontaneous Ignition: The Steady-State Case.- 4.4.3 Non-Linear Material Problems.- 4.5 Computer Program 3.- 4.5.1 MAINP3.- 4.5.2 Subroutine ALFAF3.- 4.5.3 Subroutine RHSMAT.- 4.5.4 Subroutine DERIVXY.- 4.5.5 Results of Test Problems.- 4.6 Three-Dimensional Analysis.- 4.6.1 Equations of the Type ?2u = b(x, y, z).- 4.6.2 Equations of the Type ?2u = b(x, y, z, u).- 4.7 References.- 5 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u, t).- 5.1 Introduction.- 5.2 The Diffusion Equation.- 5.3 Computer Program 4.- 5.3.1 MAINP4.- 5.3.2 Subroutine ASSEMB.- 5.3.3 Subroutine VECTIN.- 5.3.4 Subroutine BOUNDC.- 5.3.5 Results of a Test Problem.- 5.3.6 Data Input.- 5.3.7 Computer Output.- 5.3.8 Further Applications.- 5.3.9 Other Time-Stepping Schemes.- 5.4 Special f Expansions.- 5.4.1 Axisymmetric Diffusion.- 5.4.2 Infinite Regions.- 5.5 The Wave Equation.- 5.5.1 Infinite and Semi-Infinite Regions.- 5.6 The Transient Convection-Diffusion Equation.- 5.7 Non-Linear Problems.- 5.7.1 Non-Linear Materials.- 5.7.2 Non-Linear Boundary Conditions.- 5.7.3 Spontaneous Ignition: Transient Case.- 5.8 References.- 6 Other Fundamental Solutions.- 6.1 Introduction.- 6.2 Two-Dimensional Elasticity.- 6.2.1 Static Analysis.- 6.2.2 Treatment of Body Forces.- 6.2.3 Dynamic Analysis.- 6.3 Plate Bending.- 6.4 Three-Dimensional Elasticity.- 6.4.1 Computational Formulation.- 6.4.2 Gravitational Load.- 6.4.3 Centrifugal Load.- 6.4.4 Thermal Load.- 6.5 Transient Convection-Diffusion.- 6.6 References.- 7 Conclusions.- Appendix 1.- Appendix 2.- The Authors.

999 citations


Journal ArticleDOI
R. R. Martin, J. Tinsley Oden1Institutions (1)

192 citations



Book
01 Oct 1985-
Abstract: Mathematical methods of the theory of elasticity , Mathematical methods of the theory of elasticity , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

99 citations