fem-bem coupling.pdf

8/18/2019 FEM-BEM Coupling.pdf

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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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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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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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fem-bem coupling.pdf

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