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FEM-BEM Coupling
Dr. Hatem R. Wasmi
A. Prof. in applied Mechanics
The advantages of FEM and BEM
coupling has been investigated
extensively in several engineering fields,
such as geomechanics , andelectromagnetics and there are several
different methods of coupling BEM and
FEM
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FINITE ELEMENT method
General characteristics
Continuous (but not smooth) base as wellas weighting functions
Suitable for complicated geometries and
structural problems
Combination of fluid and structures (solid-
fluid interaction)
http://en.wikipedia.org/wiki/Finite_element_methodhttp://en.wikipedia.org/wiki/Finite_element_method
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FINITE ELEMENT method
12
3
x
y
4
5
6
Base functions Ni(x), Ni(x,y) or Ni(x,y,z) and corresponding weight functions
are defined in each finite element (section, triangle, cube) separately as apolynomial (linear, quadratic,…). Continuity of base functions is assured by
connectivity at nodes. Nodes x j are usually at perimeter of elements and are
shared by neighbours.
Base function Ni (identical with weight function wi) is associated with node
xi and must fulfill the requirement: (base function is 1 inassociated node, and 0 at all other nodes)
ij ji x N )(
In CFD (2D flow) velocities are approximated by quadratic polynomial (6
coefficients, therefore 6 nodes ) and pressures by linear polynomial (3
coefficients and nodes ). Blue nodes are prescribed at boundary.
Verify number
of coeffs.!
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FINITE ELEMENT example
),(2
2
2
2
y x f xT
yT
0)),(()),((
2
2
2
2
d y xwf y
T
y
w
x
T
x
wd y x f
x
T
y
T w
)()( x N T xT j j
fd N T d
y
N
y
N
x
N
x
N i
n
j
j
ji ji
1
)(
fd N T A i
n
j
jij
1
Poisson’s equation
MWR and application of Green’s theorem
Base functions are identical with weight function (Galerkin’s method)
w i( x )=N i( x )
Resulting system of linear algebraic equations for T i
Derive Green’s
theorem!
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methodBOUNDARY element
General characteristics
Analytical (therefore continuous) weighting
functions. Method evolved from method of
singular integrals (BEM makes use analytical
weight functions with singularities, so calledfundamental solutions).
Suitable for complicated geometries (potential
flow around cars, airplanes… )
Meshing must be done only at boundary. No
problems with boundaries at infinity.
Not so advantageous for nonlinear problem.
http://en.wikipedia.org/wiki/Boundary_element_methodhttp://en.wikipedia.org/wiki/Boundary_element_method
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BOUNDARY element example
),(2
2
2
2
y x f x
T
y
T
Poisson’s equation
WR and application of Green’s theorem twice (second
derivatives transferred to w)
Weight functions are solved as a fundamental solution of adjoined equation
Green’s
theorem!
fwdxdywd
y
T n
x
T ndxdy
y
w
y
T
x
w
x
T y x )()(
fwdxdyd
y
wT
y
T wn
x
wT
x
T wndxdy
y
w
x
wT y x ))()(()( 2
2
2
2
),(2
2
2
2
iiii y y x x
yw
xw
i
i
r
y xw 1
ln
2
1),(
22 )()( iii y y x xr
Singularity: Delta function at a pointxi,yi
Delta function!Solution (called Green’s function) is
Verify!
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BOUNDARY element example
Substituting w=wi (Green’s function at point i)
fwdxdyd
ywT
yT wn
xwT
xT wndxdy
yw
xwT y x ))()(()( 2
2
2
2
dxdy fwd
y
wn
x
wnT
y
T n
x
T nw y xT i
i
y
i
x y xiii )]()([),(
N
j
j j N T T 1
)()(
N
j
jnj N T n
T
1
)()(
Solution T at arbitrary point xi,yi is expressed in terms of boundary values
dΓ
Γ2 (normal
derivative)
Γ1 (fixed T)
At any boundary point must
be specified either T or
normal derivative of T , notboth simultaneousl .
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BOUNDARY element example
dΓ
Γ2 (normal
derivative)Γ
1 (fixed T)
dxdy fwd N
y
wn
x
wnT wT i j
i
y
i
x jinj )]([0
dxdy fwd N
y
wn
x
wnT d N wT i j
i y
i x j jinj )(
Values at boundary nodes not specified asboundary conditions must be evaluated from the
following system of algebraic equations:
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FEM-BEM coupling
This problem is closely related to the multi-regionproblem of the BE method such as presented in
Figure 1. The multi-region analysis has to fulfil
continuity and equilibrium conditions along the
interface line Γ betweenΩ
andΩ
regions.
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