PowerPoint PresentationAlain J. Kassab, Ph.D. Mechanical and Aerospace Engineering
College of Engineering and Computer Science
University of Central Florida
alain.kassab@ucf.edu
1
TOPICS:
1. Introduction: relation of the BEM to Green's function method, Green's free space solutions
and the boundary integral equation.
2. Steady heat conduction problem: theoretical formulation, computational aspects, and
detailed solution of a constant element example.
4. Automating evaluation of the BEM equations, continuous and discontinuous
elements, 2D and 3D examples.
5. Hxfer Boundary conditions, non-linear media, and piecewise non-homogeneous media
6. Large scale problems – conjugate heat transfer coupled BEM/FVM modeling,
domain decomposition, and parallel computing.
7. Transient Heat conduction by examples: hyperbolic heat conduction and ablation model
8. Inverse Problems – inverse geometric problem, heat transfer coefficient reconstruction
9. Anisotropic and othotropic media, axi-symmetric heat conduction and heat conduction
in thin plates
2
(a) Whole Domain Methods
• Domain Grid/Mesh/Point Distribution Generation
- steady problems: direct/iterative techniques
- transient problems: implicit/explicit techniques
Unstructured FVM domain mesh for heat transfer analysis of plenum-cooled blade3
(b) Boundary Methods: Boundary Element Methods (BEM)
• Surface Mesh Generation
analysis of cooled blade
• Green's Third Identity in 2D for Potential Problems:
G.E.:
• The potential Φ at any point ξ interior to the domain Ω
bounded by a curve is given by
where the contour integral in CCW direction and
r = Euclidean distance from source point to field point
=normal derivative
• If there are no internal sources domain integral is zero.
• Exact solution to problem if both variable and normal
derivative known everywhere on
• In a well posed problem either Φ, , or a ratio of the
two is specified at any point on the boundary?
____________________________________________________________
Note: This equation is sometimes taken the point of departure for direct
BEM for potential problems (not a general approach).
x
1. Analytical: Green's function methods
• obtain integral equation
and this function or its derivative vanish appropriately
on boundaries
problems: geometry must lend itself to separable coordinate
system or special types of geometries (method of images)
2. Hybrid Analytical/Numerical: Boundary Element Method
• boundary integral equation (BIE) to form a
boundary value problem
numerically by boundary element method (BEM)
• general method with no geometric limitations.
These two methods are theoretically related
6
Boundary Element Method (BEM) and the Green’s Function Method - theory
• The BEM is a numerical method that finds its roots in the closely related analytical Green’s
function method
• The technique relies on numerical solution of a singular boundary integral equation, this is why
the method was earlier named (60’s and early 70’s) the boundary integral equation method
(BIEM) before C.A Brebbia gave a MWR interpretation and named it the Boundary Element
Method ~ 1980.
Development of Green’s function method solution to Heat conduction:
• consider a simple 2-D heat conduction problem with generation
G.E.: 2T(x,y) +uG(x,y)=0 (x,y) Ω
B.C’s: T|Γ1= 0
q|Γ2= 0
T|Γ3= 100
q|Γ4= 0
T=0 The boundary Γ=Γ1 U Γ2 U Γ3 U Γ4 encloses the domain Ω
where, uG(x,y) is a known generation term, included here for the
sake of general discussion (can always be set to zero).
The notation that q = is used here throughout the presentation.
y
x
Γ1
Γ3
y=0
y=Ly

• The derivation of an integral equation for this problem follows these steps:
1. Introduce a test function T* to be chosen later for convenience of the solution
2. Multiply governing equation by test function and integrate over space (and time in time-
dependent problems)
3. Use Green’s second identity (reciprocal theorem) to move the differential operator
(Laplace in our case) to the test function
4. The next step is to choose T* judiciously to eliminate the area integral by setting
where δ(x-ξ) is the Dirac Delta function shifted at the source point (ξ) and (x) stands for
any coordinate location in 1-D, 2D, or 3D ( in our case it is ξ= (xi , yi )) then using sifting
property of the Dirac delta function
5. Arriving at an integral equation
Green’s 2nd identity: 8
• for our problem, setting ξ=(xi,yi), this equation reads:
problem is that BC’s don’t specify: on Γ1 or Γ3 or T on Γ2 or Γ4
• analytically, we then require that T* also must satisfy additional BC’s specific to the problem
(geometry and specified boundary conditions), namely that T* or vanish where corresponding
BC is not specified
B.C.’s:
• so integrals where we do not know BC’s vanish and then the analytical solution becomes:
T* is the Green’s function
for the problem and T* = T*GF
9
• How to find the Green’s function? solve auxilliary problem with homogeneous BC’s
Auxilliary problem:
B.C’s: T|Γ1= 0
q|Γ2= 0
T|Γ3= 0
q|Γ4= 0
λn , βm eigenvalues of auxilliary problem
N(λn), N(βm) normalizing integral of auxilliary problem
• solution is obtained by variation of parameters as a series solution that can be re-arranged as
•
Γ1
Γ3
The Green’s function for the problem: T * = T *GF
10
X=1
X=0
y=1
y=0
Plot of the Green’s function at (x’,y’)=(0.25,0.5) and (x’,y’)=(0.5,0.75).
GF m m
n n m m
n m n m
T x y x y δ y δ y λ L








,
11
12
Finding the Green’s function is impossible for problems with geometries of any
complexity (even linear problems), that is
~ any problem whose boundaries cannot be framed in a separable
coordinate system Cartesian, cylindrical, spherical, …or whose
geometry does not lend itself to alternative method of images
(semi-infinite media with 1st or 2nd kind BC’s) or similar such
construction
Essentially, Green’s function method is superposition carried out in one integral
relation.
),(4
1
• alternatively, instead of imposing boundary conditions on T*,
1. form a boundary value problem: take source point to the boundary
2. exact boundary integral equation
3. Discretize integral equation and collocate at boundary points
4. simultaneous linear equations solved for T and where not specified by BC’s
• requires T* is available, and that is we can find the particular solution to
• The solution T* to the above is called the Fundamental Solution or Green’s free space solution(*)
• Free space solutions are available for many differential operators and can be found in classical
applied mathematics literature.
3D
_______________________________________________________________________________________________________________________________
(*) T * is the solution in free space of the adjoint differential operator perturbed by a Dirac
Delta function acting at the source point ξ= (xi,yi)
ξ
singular boundary integral equation (BIE) that is
evaluated as a Cauchy principal value (CPV) integral
• the free term Ci is related to the local geometry and often
evaluated numerically by a constant temperature field
argument
• Let us set the generation term to zero (uG=0).
• We can deal with cases where uG is non-zero as special cases using particular solutions when these
are available or the dual-reciprocity method (DRM), which can be thought of a generalized manner
to generate particular solutions for the BEM.
• The BIE is valid everywhere on the boundary and can be used to formulate a boundary
value problem for the unknown temperature or its normal derivative.
x
T(ξ)
(T, )
dΓ
ξ
x
y
over a number of boundary elements, Ne, so that
Γ=Γ1 U Γ2 U Γ3 …ΓNe
linear, quadratic or cubic interpolating polynomials
(shape functions borrowed from FEM) are commonly used, although
other approximations are possible, for instance the B-splines
2. Approximate the the functional dependence of the temperature, T, and its normal
derivative on the boundary in terms of values over a set of nodal points using suitable
interpolation functions: constant, linear or quadratic polynomial interpolation commonly
utilized.
The order of discretization of the temperature and normal derivative need not
be the same as that used for the geometry:
a. sub-parametric: lower order than that used for the geometry
b. iso-parametric: same order than that used for the geometry
c. super-parametric: higher order than that used for the geometry
The temperature and normal derivative are discretized using nodal values whose
location within the element can be chosen to:
a. coincide with the location of the geometric nodes: continuous elements
b. be located offset from the geometric nodes: discontinuous elements
x
y
Ω
Γ
j
Γj-1
(a) (b)
• simplest possible approximation adopts the linear subparametric (constant) boundary elements
that model the geometry as piecewise linear and the dependent variables as constant over the
element. In such a case, the BIE becomes:
x
y
Ω
Γ
j
Γj-1
2-D constant element
• Collocation is used to solve the boundary unknown, by taking
the source point ξ to each boundary node location ξi such that
• defining:
specified• matrices [H] and [G] are fully populated
• elements of [H] and [G] are purely dependent on geometry and are usually evaluated
numerically via adaptive Gauss-Legendre quadratures
• matrix [A] is fully-populated, non-symmetric, and not diagonally dominant
• can be solved by direct methods for small problems and pre-conditioned iterative methods for
large problems (conjugate gradient or bi-conjugate gradient with symmetrized BEM equations
or preferably by the non-symmetric solver GMRES).
• The Gii and Hii coefficients are singular!
Gii is weakly singular, readily evaluated by adaptive quadratures and
the CPV definition of the singular integral.
Hii is strongly singular, however, it is zero and zero for flat surfaces of
constant and linear elements
• However, in general, Hii, can always be evaluated by the argument that when the temperature is
^
B.C’s: T|Γ1= 0
q|Γ2= 0
T|Γ3= 100
q|Γ4= 0
Lx=Ly=1
Γ1
Γ3
y=1
Γ1
Γ3
___________________________________________________________________________________ (*) Note: in practice evaluation of coefficients is automatically carried out numerically
as we will show later.
19
A typical diagonal element:
• assembling the BEM equations:
=
• imposing boundary conditions and moving all known boundary values to the RHS
and unknowns to the LHS (accordingly switching negative of columns between [H] and [G]:
q1
T2
q3
T4
0
0
100
0
0.2695
0.0533
0.0062
0.0533
0.1762
0.5
0.1762
0.1476
0.0062
0.0533
0.2695
0.0533
0.1762
0.1476
0.1762
0.5
0.5
0.1762
0.1476
0.1762
0.0533
0.2695
0.0533
0.0062
0.1476
0.1762
0.5
0.1762
0.0533
0.0062
0.0533
0.2695
0.5
0.1762
0.1476
0.1762
0.1762
0.5
0.1762
0.1476
0.1476
0.1762
0.5
0.1762
0.1762
0.1476
0.1762
0.5
22
j j ij ijif T q then H G i N 12, ...

=
• carrying out vector-matrix multiplication on the RHS lead to standard algebraic form: [A]{x}={b}
-117.4
49.99
117.4
49.99
q1 = q(0,5,0)= -100
q3 = q(0.5,1.0)= 100
T4 = T(0,0.5) =50
• value of temperature (and flux) at any point at the interior is a post-processing operation
y
x
Γ4 Γ2
24
• convergence under point refinement: constant element results for four levels of discretization,
4,8,12, and 16 boundary elements for 16 interior points.
1 2 3
4 5 6
7 8 9
1 2 3
4 5 6
7 8 9
1 2 3
4 5 6
7 8 9
constant element, linear, and quadratic results for four levels of discretization, 4,8,12, and 16
boundary elements utilizing Gauss-Legendre quadratures for the H and G matrices (see
upcoming discussion).
(Tj+2,qj+2)
(xj+1,yj+1)
(Tj,qj)
(Tj+3,qj+3)
• continuous temperature (interpolated over element)
• problem with normal derivative at corner especially
in 3D (need additional equations when temperature
is prescribed)
• more difficult to code and requires much
overhead in 3D
rest) – however can always interpolate later
• no problem with normal derivative at corner
a big advantage especially in 3D ( not star points)
• many more nodes than continuous elements*
• does not require connectivity matrix in 3D
• easier to code in 3D
• more accurate normal derivatives
number of unknowns 26
1. Quadratic sub-parametric element:
• then a typical integral over an element Γj becomes
• where the Jacobian is:
• we evaluate
• model T and q using quadratic shape functions on
each element Γj
and positioned at:
• which is readily evaluated by Gauss-rules as
___________________________________________________________________________________________
Note: better to use adaptive quadratures such as those based on Gauss-Kronrod pairs
(DQAGS in Quadpack) important when dealing with internal points close to boundary,
slender regions, and CPV integrals 28
• NGP= no of Gauss
_____________________________________________________________________________________
31
32
Where the metrics are obtained from the relations for the coordinates (x,y,z) on the ith
boundary element
So that
1. Non-singular elements:
Locations on element Γj
• Dual Reciprocity BEM
• Inverse Problems –heat transfer coefficient reconstruction, inverse geometric problem
• Axi-symmetric heat conduction
34
EXAMPLE:
• re-arranging
Boundary conditions:
• when k = k(T), then classical Kirchhoff transform can be used
• define the Kirchhoff temperature, U(T)
• defined so that so that:
• consequently:
G.E.:
36
o
T
1
r rs s
T U k q k q
n n
function of T can be found analytically
or numerically
o o o
o o o
k k k
β β β
if β use
if β use
T
U(T)
• can write as
• for example for a constant element in case of convective BC
• which is e-arranged as
•or in matrix form if convective conditions are prescribed on the domain boundary
• non-linearity is taken to the boundary. 38
s
o r r r
T U k h T T k h T U U T
n n
n k
k k
i i ij j j ij j ij
j j jo o
h h CU H U U U G T U U G
k k
N N N
j j jo o o
f f
h h h C U H G U U G T U U G
k k k
39
Example:
• Set {fnl (U)}=0 and solve for the first estimate of
the solution {U} (0)
• Iterate until
Piecewise Non-homogeneous Media
• decompose the region into constant conductivity sub-domains (will revisit this
concept later when dealing with large scale problems)
EXAMPLE: two region problem
Transfer Coupled to Conduction Heat Transfer within a Solid
Body, e.g.:
• fuel ejectors
Wq
T
42
• Coupling of a Boundary Element Method (BEM) Internal Heat Conduction
Solver to a Finite Volume Method (FVM) External Navier-Stokes Solver for
the Steady-State Compressible Subsonic CHT Problem over Cooled and
Un-cooled Turbine Blades.
• Temperatures and Heat Fluxes are the Nodal Unknowns in the BEM.
Consequently, Solid/Fluid Interface Continuity is Naturally Provided by the
BEM Conduction Analysis.
44
3-D leads to storage and solution method issues
•Some Solutions:
- iterative solution
2. Preconditioned GMRES(general minimization of
residual) non-symmetric iterative solver for large
systems
Ref: Eduardo Divo, Alain J. Kassab, and Franklin Rodriguez, "Parallel Domain Decomposition Approach for
large-scale 3D Boundary Element Models in Linear and Non-Linear Heat Conduction," Numerical Heat
Transfer, Part B, Fundamentals, Vol. 44, No.5, 2003, pp. 417- 437.
45
• 2D illustration of BEM
4
2Tx,y 0 [H]NXN{T}NX1= [G]NXN{q}NX1
• Application of Boundary Conditions:
[H]NXN{T}NX1= [G]NXN{q}NX1 [A]NXN{x}NX1= {b}NX1
• Algebraic Solution:
N3 flops {x}NX1
11 34
22
42
2
n 2N/(K+1) Boundary Elements
where K: number of sub-domains 11
21
41
1
I31
• Algebraic Analogue to Governing Equation:
2T1x,y 0 [H]nxn{T}nX1= [G]nXn{q}nX1
• Same for all subdomains:
unchanged.
48
• Total reduction in memory allocation:
K[2N/(K+1)]2 vs [N]2 4K/(K+1)2 vs 1K[n]2 vs [N]2
[2N/(K+1)]2 vs [N]2 4/(K+1)2 vs 1[n]2 vs [N]2
(16% vs 100% for K = 4)
(64% vs 100% for K = 4)
• Total FLOPS to solve the linear system: direct
K[2N/(K+1)]3 vs [N]3 8K/(K+1)3 vs 1K[n]3 vs [N]3
(25.6% vs 100% for K = 4)
49
• Application of Boundary Conditions:
[H]nXn{T}nX1= [G]nXn{q}nX1 [A]nXn{x}nX1= {b}nX1
11
21
41
1
50
n3 flops {x}nX1
Same for all subdomains:
iteration step from the “guessed” interface
boundary conditions until convergence
flops/solution (can be carried out if n is small
enough) 51
I
T1 I = (T1
I +T2 I )/2 + R” q1
I /2
I +T2 I )/2 + R” q2
I /2
• Interface Condition “Guessing”:
I
q1 I = q1
I -(q1 I +q2
I )/2
I -(q1 I +q2
I )/2
Aj
Ti
|rij|
Ai
Tj
1
2
Aj
Ti
|rij|
Ai
Tj
1
2
Ti
rij
Ai
Tj
1
2
Aj
Ti
|rij|
Ai
Tj
1
2
Ti
|rij|
Ai
Tj
1
1
2
2
Aj
Ti
|rij|
Ai
Tj
1
1
2
2
Ti
rij
Ai
Tj
1
1
2
2
NT = number of temperature BC nodes
NQ = number of heat flux BC nodes
NH = number of convective BC nodes
55
on a coarse constant element mesh
• once the solution converges it is used as an initial guess
to a finer say bi-linear element mesh until convergence
is again achieved
• this solution can then be used as initial guess to a finer
bi-quadratic mesh etc…
56
terms of temperatures T and not in terms of Kirchhoff
transform U as transformation amplifies the jump at
interface and may lead to the divergence.
- If convective boundary condition is imposed at
exposed surface of subdomain a sub-level iteration is
carried out for that subdomain.
57
mismatched temperatures along all interfaces as:


parallel computation
FORTRAN MPICH implementations
multi-user cluster
problem in all nodes simultaneously and measuring the
computational times to assign an optimal work
proportion FRA(N).
-The fraction of the overall inverse of the time t(N) it
took the Nth processor to compute the test problem is
defined as
-In such a way the fastest processor gets the largest
fraction and slowest gets the lowest fraction. 60
1
1
1
/ (N) ( )
61
• Define the load vector that measures the load of the Nth processor
in solving the problem
- Each k- subdomain has NE elements.
- There are a total of NEtotal element to solve in the problem.
- a=2 if an iterative solver is used and a=3 if a direct solver is used.
• ILOAD is a matrix of assigned regions to each processor
10000100
00010000.
00001001.
010000002
001000101
....321

each node by minimizing the following
objective function using a discrete Genetic
Algorithm:
63
TTk 1)(
)22( 19
conditions on a rectangular slab decomposed into four (K=4)
subdomains each with 150 elements or a total of 600 elements
with 600 dof, 2400 dof, and 4800 dof.
___________________________________________
Z
T: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
• BEM temperature distribution
• Distribution of error w/rt exact
solution. Maximum of 0.0125
2,080 Boundary Elements with 2,080 dof, 8,320 dof, and
16,640 dof in biquadratic element space. Two cases:
7201007.9193.1)( 4
1 TTk
14201053.4133.19)(
14201049.4151.7)(
4
2
4
2
(center 80%)
Cooling ends with h=10 and T=0 and heating through
perimeter with h=1 and T=1000~4000
66
T: 200 400 600 800 1000 1200 1400 1600 1800
Z
X
Y
• BEM solution
for composite
non-linear case:
• Domain decomposition
68
20,056 dof in bilinear element space. Two cases:
7201007.9193.1)( 4
2 TTk
reference temperature varying linearly from 300 to
500 degrees. The outside surface reference
temperature is 1000 and the film coefficient h=10.
The top and bottom end-walls are insulated.
1)(1 Tk
• BEM temperature
distribution for
non-linear case:
T: 600 650 700 750 800 850 900 950 1000
• BEM temperature
distribution for
linear case:
71
elsewhere from CFD solution 1600R ~ 3100R
21,306 Boundary Elements with 21,306 dof and
85,224 dof in bilinear element space. Two cases:
and,
72
73
Typical subdomain
X
Y
Z
T: 1600 1725 1850 1975 2100 2225 2350 2475 2600 2725 2850 2975 3100
X
Y
Z
T: 1600 1725 1850 1975 2100 2225 2350 2475 2600 2725 2850 2975 3100
75
76
q=-100. Cooling through the other end-cap with h=100
and T=0.
178,560 dof when using bilinear element. Two cases:
and,
77
T
400
380
360
340
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
78
79
Addressing the Generation Term
• let us consider the general case where, b(x), is a known generation term so that
1. Point sources: k sources/sinks of strength
no discretization required
• expand the source term in a special series
about N boundary nodes and L internal nodes
• choose the functions in the series to solve
• so that the governing equation is 80
• apply BEM procedure to both sides (dual reciprocity) to yield
• where defined:
• BEM discretization + taking the source point to each at each boundary point
• or in matrix form, this leads to the standard BEM form
• first explain how fk’s and derived uk and pk functions are found before explaining how
β, P and U are found.
81
• most successful functions used are Radial Basis Functions (RBF) that are also used in meshless
methods to be discussed later. Conic RBF’s have been successfully used in DRBEM.
EXAMPLE: conic RBF
r n
k k k
r x y z x x r x y z y y r x y z z and


( , , ) ( ) ( , , ) ( ) ( , , ) (z ) , ,
( , , ) ( , ,z) ( , , )
2
• the expansion coefficient vector β is found by collocation at the dual reciprocity expansion points
where b is the vector of values of b(x) at the N+L DRM expansion points xk
and the interpolant matrix F
has its k-th column comprised of the vectors of values of with j=1,2…N+L.
• as such
• and the final form of the DRBEM equations is
• This is the most useful form as the b can then take on various expressions depending on the
application from transient modeling to non-linear heat transfer to fluid mechanics. 83
2
84
• In the application of the DRBEM to transient heat conduction, the diffusion equation is recast as
T T
α t
2 1
expand the right hand side using the 1+r conic RBF with the expansion coefficients, ,
explicitly as function of time mβ t( )
eN L
mi
T β t f x y x y for i N L
t
Leads to
T F β
β F T
1



11
85
Time integration scheme with denoting the time step and the superscript m denoting
time level
Choice of θ=0,1/2, 1 leads to explicit, Cranck-Nicholson, and fully implicit time
marching by solving
T T T t
1 1 m m m m


_________________________________________________________________________________________________________________________
Divo, E., Kassab, A.J., and Cavalleri, R.J., "Application of the DRBEM to Model Ablation Characteristics of a Thrust Vector Control Vane,"
Engineering Analysis with Boundary Elements, Vol. 23, No. 8, 1999, pp. 693-702.
• Dual reciprocity boundary element method to predict ablation in model TVC vanes
• Moving front algorithm: only surface is re-meshed
• Numerical results:
1. verification vs 1-D analytical model:recession rates
2. results for ablation of quarter-scale and half-scale vanes vs testing
EXAMPLE: BEM Model of Ablation of a Thrust Vector Control Vane
• DRBEM transient equations related time level p+1 values to time level p values:
• where the capacitance matrix is:
Typical heat flux profile
from BL code provided
Quarter scale TVC vane simulations Half scale TVC vane simulations
Limited Data Points:
• DRBEM predicted recession of 0.102 inches after three seconds for the quarter-scale vane
compares well with the experimental result of a mean recession of 0.1 in. This mean value is the
recession of the half-height of the wedge portion of the vane.
• back wall temperature for the half scale vane after seconds is predicted to be 2774 K, while the
measured temperature was 2500K
Error
Estimate
90
Non-linear transient problem
Divo, E.A., and Kassab, A.J., "Transient Non-linear Heat Conduction Solution by a Dual
Reciprocity Boundary Element Method with an Effective Posteriori Error Estimator," CMC:
Computers, Materials, & Continua, Vol. 2, No.4, 2006, pp. 275-288.
Note: properties allowed exact solution in
Kirchhoff transform variable.
2. Time dependent fundamental solutions
3. Dual Reciprocity Method – b taken as a function of time (actually b = )
Example: Hyperbolic heat conduction by Laplace Transform method
91
Nordlund, R.S. and Kassab, A.J., "Non-Fourier Heat Conduction: a boundary element solution of the hyperbolic heat
conduction equation, " Proceedings of the 17th International Conference on Boundary Elements, BEM 17, July 17-19, 1995,
Madison, Wisconsin, pp. 279-286. 92
________________________________________________________________________________________ Kassab, A.J., Divo, E., Kapat, J.S., and Chyu, M.K., "Retrieval of Multi-Dimensional Heat Transfer Coefficient Distributions Using an
Inverse-BEM-Based Regularized Algorithm: Numerical and Experimental Examples," Engineering Analysis,Volume 29, No. 2,2005, pp.150-160.
Silieti, M., Divo, E., and Kassab, A.J.,"An Inverse Boundary Element Method/Genetic Algorithm Based Approach for Retrieval of Multi-dimensional
Heat Transfer Coefficientswithin Film Cooling Holes/Slots," Inverse Problems in Engineering and Science, Vol. 13, No.1, 2005, pp. 79-98.
EXAMPLE: Multidimensional Heat Transfer Coefficient, h, reconstruction using surface temperature
measurements
93
• To find an equation purely in terms of the sought-after fluxes at each time level, F, as
• Sensitivity coefficients:
95
96
2. physical properties
3. boundary conditions
4. initial condition(s)
FIND: the interior geometry which produced the surface temperature pattern
nn 2T(x,y) = 0
Mathematical Formulation:
Introducing the boundary conditions: • internal discretization
not required
processes
boundary data. Parameters (z = 13): in 2-D
x,y: cluster center
rx,ry: axes radii
of approximate adiabatic line that surrounds the singularities.
Parameters (z = 5): 2D
ellipse)
Objective
Function
qN
i
i
q
cluster search. cluster search.
103
First level search: 6 singularities
104 Analogous BC’s to the 2D problem with convection top and bottom
Second level search:
106
• These integrals involve complete elliptic integrals of the 1st kind and second kind
and details can be found in standard BEM references.
Remarks:
- with solid body, no elements on line of symmetry as symmetry is explicitly
accounted for in the axi-symmetric free space solution
- hollow axi-symmetric body requires a closed 2-D contour
- use asymptotic expressions for elliptic integrals with small arguments
- care must be taken with elements close to axis of symmetry
107
108
An interesting Transformation
Shaw, R.P. and Gipson, G.S., A BIE Formulation of Linearly Layered Potential Problems, Engineering
Analysis, Vol. 16, pp. 1–3, 1996.
• For problems with linearly varying thermal conductivity
• The transformation to the new coordinate system
• Transforms
• So that an axisymmetric BEM code can be used to solve such a
non-homogeneous problem.
b ' '
x x y y
x
z
y
δz
109
• where Ko is a modified Bessel function of the second kind of order zero
Note: this fundamental solution is also used in Laplace Transform solution of transient
heat conduction by BEM.
• Body force bi due to Thermal Stresses:
• Boundary Integral Equation (BIE) for Thermoelasticity:
Note: here Gij and Hij are the displacement and traction fundamental
solutions and the kernels Ej and Fj are directly derived from the thermal
fundamental solutions GT and HT.
Thermoelasticity BEM
• Stress (traction) Boundary Integral Equation (BIE) for Thermoelasticity:
Note: here Dijk and Sijk are the stress fundamental solutions and the kernels
Ajk and Bjk are directly derived from the kernels ET and FT.
Note: This BIE can be employed to estimate the stress field and also in
conjunction with the displacement BIE to solve fracture mechanics
problems by the DUAL BEM formulation.
112
Thermo elasticity
Heat conduction
conductivity to axi-symmetic BEM (Shaw and Gipson)
- generalized fundamental solution (Divo and Kassab)
- transformations to Helmholtz type equation and
Dual Reciprocity BEM (A.H.D Cheng et al.,
G. Paulino et al.)
• Transient BEM by time dependent fundamental solutions,
convolution method.
114
• Wrobel, L.C., The Boundary Element Methods - applications to thermofluids and acoustics, McGraw Hill Book Co., New York,
2002.
• Brebbia, C.A. and Dominguez, J., Boundary Elements: an introductory course, McGraw-Hill Book Company, New York, 1992.
• Partridge, P., Brebbia, C.A., and Wrobel, L.C., The Dual Reciprocity Boundary Element Method, Computational Mechanics,
Southampton, UK, 1991.
• Divo, E. and Kassab, A.J., Boundary Element Method for Heat Conduction with Applications in Non-Homogeneous Media,
Wessex Institute of Technology (WIT) Press, Boston, USA, 2003.
• Brebbia, C.A., Telles, J.C.F., and Wrobel, L.C., Boundary Element Techniques in Engineering, Springer-Verlag, New York,
1984.
• Brebbia, C.A. and Walker, S., Boundary Element Techniques in Engineering, Newnes-Butterworths, London, 1980.
• Liggett, J.A. and Liu, P.L.F., Boundary Integral Method for Porous Media Flow, George Allen and Unwin, 1983.
• Banerjee, P.K., Boundary Element Method, McGraw Hill Book, Co, 1992.
• Becker, A.A., The Boundary Element Method in Engineering, McGraw Hill Book Co., New York, 1992.
• Wrobel, L.C. and Brebbia, C.A., Boundary Element Methods in Heat Transfer, Computational Mechanics
Publications, Boston, 1992.
• Bialecki, R.A., Solving Heat Radiation Problems Using the Boundary Element Method, Computational Mechanics
Publications, Boston, 1993.
• Jaswon, M.A. and Symm, G.T., Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, New York,
1977.
• Cruse, T.A., Mathematical Foundation of the Boundary Integral Equation Method in Solid Mechanics, AFSOR-TR-77-1002.
115
B. Some References on Green’s Functions:
• Greenberg, M., Applications of Green’s Functions in Science and Engineering, Prentice Hall, Englewood
Cliffs New Jersey,1971.
• Beck, J.V., Cole, K.D., Haji-Sheik, and Litkouthi, B., Heat Conduction Using Green’s Functions,
Hemisphere Publishing Corporation, Washington, D.C., 1992.
• Zauderer, E., Partial Differential Equations of Applied Mathematics, John Wiley and Sons, New Yok, 1989.
• Kellogg, O.D., Foundations of Potential Theory, Dover Publications, New York, 1953.
• Roach, G.F., Green’s Functions, Cambridge University, New York, 1970.
• Morse, P. and Feshbach, H., Methods of Theoretical Physics, Part I, McGraw Hill Book Co., New York,
1953.
• Courant,R. and Hilbert, D., Methods of Mathematical Physics, Vol. I and II, John Wiley and Sons, New
York, 1962.