07 finite elements weq
TRANSCRIPT
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1Finite element method
Finite Elements: 1D acoustic wave equation
Helmholtz (wave) equation (time-dependent)
Regular gridIrregular grid
Explicit time integration
Implicit time integratonNumerical Examples
Scope : Understand the basic concept of the finite elementmethod applied to the 1D acoustic wave equation.
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2Finite element method
Acoustic wave equation in 1D
How do we solve a time-dependent problem suchas the acoustic wave equation?
where v is the wave speed.using the same ideas as before we multiply this equation withan arbitrary function and integrate over the whole domain, e.g. [0,1], and
after partial integration
f uvut = 22
dx f dxuvdxu j j jt =1
0
1
0
21
0
2
.. we now introduce an approximation for u using our previous
basis functions...
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3Finite element method
Weak form of wave equation
)()(~1
xt cuu i N
ii
=
=
together we obtain
)()(~1
222 xt cuu i N
iit t t
=
=
note that now our coefficients are time-dependent!... and ...
=+1
0
1
0
21
0
2 j j
iii j
iiit f dxcvdxc
which we can write as ...
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4Finite element method
Time extrapolation
=+1
0
1
0
21
0
2 j j
iii j
iiit f dxcvdxc
... in Matrix form ...
gc Avc M T T =+ 2&&
M A b
... remember the coefficients c correspond to theactual values of u at the grid points for the right choiceof basis functions ...
How can we solve this time-dependent problem?
stiffness matrixmass matrix
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6Finite element method
Mass matrix
=+ 10
1
0
21
0
2 j j
iii j
iiit f dxcvdxc
... lets recall the definition of our basis functions ...
=1
0
dx M jiij ii
i
ii
i x x x
elsewhere
h x for h x
xh for h
x
x =
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Matrix assembly
% ass embl e mat r i x Mi j
M=zer os( nx) ;
f o r i =2: nx- 1,
f o r j =2: nx- 1,
i f i ==j ,
M( i , j ) =h( i - 1) / 3+h( i ) / 3;
el sei f j ==i +1
M( i , j ) =h( i ) / 6;
el sei f j ==i - 1
M( i , j ) =h( i ) / 6;el s e
M( i , j ) =0;
end
end
end
% assembl e mat r i x Ai j
A=zer os( nx) ;
f o r i =2: nx- 1,
f o r j =2: nx- 1,
i f i ==j ,
A( i , j ) =1/ h( i - 1) +1/ h( i ) ;
el sei f i ==j +1
A( i , j ) =- 1/ h( i - 1) ;
el sei f i +1==j
A( i , j ) =- 1/ h( i ) ;el s e
A( i , j ) =0;
end
end
end
Mij A ij
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Numerical example
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Implicit time integration
... let us use an implicit finite-difference approximation for the time derivative ...
gc Avc M T T =+ 2&&
... leading to the solution at time t k+1 :
gc Avdt
ccc M k
T k k T =+
+
++
12
211 2
[ ] ( )( )12122
1 2
+ ++=
k
T T T
k cc M gdt Adt v M c
How do the numerical solutions compare?
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Summary
The time-dependent problem (wave equation) leads to theintroduction of the mass matrix.
The numerical solution requires the inversion of a systemmatrix (it may be sparse).
Both explicit or implicit formulations of the time-dependent partare possible.
The time-dependent problem (wave equation) leads to theintroduction of the mass matrix.
The numerical solution requires the inversion of a systemmatrix (it may be sparse).
Both explicit or implicit formulations of the time-dependent partare possible.