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CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
HW 5
• Repeat the HW associated with the FD LBI except that you will now use 1-D FEM with linear elements. Also, you will only consider the isothermal case with the non-dimensionalized temperature set to 1. A sample program has been provided for you. Among the tasks are:– Check the sample program for bugs (especially the matrix element
equations).
– Place the boundary conditions into the matrix.
– Modify the matrix storage for use of PETSC.
– Look at different values of dissipation (both varying in time and space) and see how they affect the solution.
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
HW 5 (cont’d 2)
The sample code stores the matrix in a banded format:
11 12
21 22 23
32 33 34
4543 44
54 55
0 0 0
0 0
00
0 0
0 0 0
k k
k k k
k k kk
k k
k k
11 12
21 22 23
32 33 34
43 44 45
54 55
k k
k k k
k k k
k k k
k k
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Higher Order Elements in 1-D
The Lagrange family provides basis functions to approximatethe solution to any degree polynomial in 1-D. To approximatea kth degree polynomial, we need k+1 nodes:
Element m
1mx 2
mxmkx 1
mkx
1( )
0 at other nodes
mm jj
x xx
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Higher Order Elements in 1-D (cont’d 2)
One possibility:
1 2 1 1 1( )( ) ( )( ) ( )m m m m m mj j j kx x x x x x x x x x
The above expression does not satisfy the condition that
1 2 1 1 1
1, at . In fact,
( ) ( )( ) ( )( ) ( )
m mj j
m m m m m m m m m m m mj j j j j j j j j k
x x
x x x x x x x x x x x x
Therefore,
1 2 1 1 1
1 2 1 1 1
( )( ) ( )( ) ( )
( )( ) ( )( ) ( )
m m m m mj j km
j m m m m m m m m m mj j j j j j j k
x x x x x x x x x x
x x x x x x x x x x
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Higher Order Elements in 1-D (cont’d 3)
Example for quadratic (k=2)
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
2-D Finite Element Methodfor Isothermal Flow of Ionized Gas
Through a Nozzle
Robert Lee
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Differences from 1-D Case
• Method of weighted residuals applied over a surface rather than a line.
• Unknowns increase because velocity along two directions.
• Finite element basis functions are 2-D.
• The resulting matrix equation produces general sparse matrices.
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Differences from 1-D Case (cont’d 2)
• Gridding of geometry now becomes complicated.• There are numerous choices for element shapes• Errors present in both discretization of unknowns
and the geometry.
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Equation of Interest
After application of the method of weighted residuals,the equation of interest is
, , , , ,m
m mm mj jm m i i
j i
d dd df dxdy
dx dx dy dy
Thus, the only thing that we must find is
the basis function ( , ) and its derivative
in and .
mj x y
x y
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Review of FEM
• The basis functions j(x)are generated by simple functions defined piecewise (element by element) over the FEM grid.
• The basis must be smooth enough such that their derivatives in the weight residual equation exists (assume nth order derivatives), i.e.,
FEM provides a systematic and very general way ofgenerating the basis functions (usually polynomialapproximations). The criteria are:
2njnxdx
is element of interest 2n
jnydx
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Review of FEM (cont’d 2)
• The basis functions are chosen in such a way that the coefficients defining the unknown quantity are precisely the value of the unknown quantity at the nodes.
1( )
0j i
i jx
i j
1( )
0 at other nodesj
j
x xx
There are two ways to find the basis functions over anarbitrary element. We will call them Method 1 and Method 2
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 1 for Triangles
Consider a 3-node triangle. With 3 nodes, one has 3 degreesof freedom for the basis functions,
( , )mj j j jx y a b x c y
An obvious choice for the placement of the nodes is at the vertices of the triangle.
The basis functions will be continuous along the edges ofthe triangle as long as the unknowns at the nodes associatedtith that edge are single valued.
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 1 for Triangles (cont’d 2)
1
3
21 1( , )x y
2 2( , )x y
3 3( , )x y1 1 1 1 1 1 1 1( , ) 1m x y a b x c y
1 2 2 1 1 2 1 2( , ) 0m x y a b x c y
1 3 3 1 1 3 1 3( , ) 0m x y a b x c y
1 1 1
2 2 1
3 3 1
1 1
1 0
1 0
x y a
x y b
x y c
1 2 3 3 2 2 3 3 2
1( , )
2m
e
x y x y x y y y x x x yA
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 1 for Triangles (cont’d 3)
is the area of the triangle and is given by
1/2 times the determinant of the matrix:eA
12 3 1 2 3 1 1 2 2 3 3 12eA x y x y x y y x y x y x
Note: is positive if nodes numbered clockwiseeA
2 3 1 1 3 3 1 1 3
1( , )
2m
e
x y x y x y y y x x x yA
3 1 2 2 1 1 2 2 1
1( , )
2m
e
x y x y x y y y x x x yA
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Higher Order Triangles
Pascal’s triangle: 1x y
2 2x xy y3 2 2 3x x y xy y
Examples:
1x y
1x y
2 2x xy y
does not
fit
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Quadrilateral Elements
( , )mj j j j jx y a b x c y d xy
The 4-node quadrilateral has bilinear behavior,
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Concept of Master Element
Let us consider the concept of master element for the 1D case.
ˆThe master element defined on the coordinate system,
1 1
1 2
11 2
12 2
1
1
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Mapping from Master Element
1 1
m
1mx 2
mx
2
2 1 2 11
1
2m m m m mj j
j
x x x x x x
1mx
2mx
1 1
x
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Mapping from Master Element (cont’d 2)
( ) ( ( ))m mj j jx x
e.g., if 2 , ( ) sinmjx x x then 2( ( )) sinm
j jx
2
Note and are not the same functions
( ) sin , sin
mj j
mj jx x x x
m mj j jd d dd d
dx d dx d dx
2 1
2 1
2 m m
m m
x x x
x x
2 1
2m m
d
dx x x
2 1
2
m mx xdxdx d d
d
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 2 for Quadrilaterals
The elements formed by method 2 are called isoparametric elements,
m
12
3
4
1 2
34
1 1,x y
4 4,x y 3 3,x y
2 2,x y 1, 1
1,1
1, 1
1,1
4 4
1 1
( , ) ( , ) ( , ) ( , )j j j jj j
x x y y
Note: In general, the inverse map ( ( , ), ( , )) is nonlinearx y x y
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 2 for Quadrilaterals (cont’d 2)
The basis functions ( , ) can be written in terms of
the , system,
mj x y
( , ) ( ( , ), ( , )) ,m mj j jx y x y
1 at node ( , )
0 at other nodesj
j
11 4
12 4
13 4
14 4
, 1 1
, 1 1
, 1 1
, 1 1
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 2 for Quadrilaterals (cont’d 3)
Finding the derivatives of the basis functions is not
as easy as in the 1-D case. We can write,m m mj j j j jd d d d dd d d d
dx d dx d dx d dx d dx
m m mj j j j jd d d d dd d d d
dy d dy d dy d dy d dy
d dd dx dy
dx dy
d dd dx dy
dx dy
d d
d dxdx dy
d d d dy
dx dy
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 2 for Quadrilaterals (cont’d 4)
dx dxdx d d
d d
dy dydy d d
d d
dx dx
dx dd d
dy dy dy d
d d
The 2x2 matrix is referred to as the Jacobian matrix [J]. Thedeterminant of the Jacobian matrix is
dx dy dx dy
d d d d J
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 2 for Quadrilaterals (cont’d 5)
1
1
dx dx dy dx
d dx dxd d d d
d dy dy dy dy dx dy
d d d d
J
1 1 1 1d dy d dx d dy d dx
dx d dy d dx d dy d
J J J J
4 4
1 1
4 4
1 1
i ii i
i i
i ii i
i i
d ddx dxx x
d d d d
d ddy dyy y
d d d d
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Method 2 for Quadrilaterals (cont’d 6)
4 4
1 1
4 4
1 1
1
1
mj j ji i
i ii i
mj j ji i
i ii i
d d dd dy y
dx d d d d
d d dd dx x
dy d d d d
J
J
dxdy d d J1 1
1 1
( , ) ( ( , ), ( , ))m
f x y dxdy f x y d d
J
Applying Gauss Quadrature,1 1
1 11 1
( , ) ( , )I J
i j i ji j
g d d g ww
CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering
Accuracy of Mapping
The accuracy of the FEM solution is highly dependent on theelement shape. The best solutions are produced in those gridswhere the element is well-shaped (the best shape being a square)For quadrilaterals with inner angles greater than 180 degrees, themapping may be outside the quadrilateral.
Typically, we would likeno inner angles greaterthan 150 degrees.
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