Transcript Finite-Element Method Chapter 31

Chapter 31
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Finite-Element Method
Chapter 31
• Finite element method provides an alternative to
finite-difference methods, especially for systems with
irregular geometry, unusual boundary conditions, or
heterogeneous composition.
• This method divides the solution domain into simply
shaped regions or elements. An approximate solution
for the PDE can be developed for each element.
• The total solution is generated by linking together, or
“assembling,” the individual solutions taking care to
ensure continuity at the interelement boundaries.
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Figure 31.1
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The General Approach
Figure 31.2
Discretization/
• First step is
dividing the
solution domain
into finite
elements.
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Element Equations/
• Next step is develop equations to approximate the
solution for each element.
– Must choose an appropriate function with unknown
coefficients that will be used to approximate the solution.
– Evaluation of the coefficients so that the function
approximates the solution in an optimal fashion.
Choice of Approximation Functions:
For one dimensional case the simplest case is a firstorder polynomial;
u( x)  a0  a1x
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u1  a0  a1 x1
u 2  a0  a1 x2
u1  u ( x1 )
u 2  u ( x2 )
Using Cramer's rule
u1 x2 -u2 x1
a0 
x2 -x1
u  N1u1  N 2u 2
x2x
N1 
x2  x1
u 2 -u1
a1 
x2 -x1
Approximation or shape
function
x  x1
N2 
x2  x1
Interpolation functions
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Figure 31.3
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• The fact that we are dealing with linear equations facilitates
operations such as differentiation and integration:
du dN1
dN2

u1 
u2
dx dx
dx
or
x2
x2
 u dx   N u
1 1
x1
 N 2u 2  dx
x1
Obtaining an Optimal Fit of the Function to the Solution:
• Most common approaches are the direct approach, the method
of weighted residuals, and the variational approach.
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• Mathematically, the resulting element equations will
often consists of a set of linear algebraic equations
that can be expressed in matrix form:
k u  F
[k]=an element property or stiffness matrix
{u}=a column vector of unknowns at the nodes
{F}=a column vector reflecting the effect of any
external influences applied at the nodes.
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Assembly/
• The assembly process is governed by the concept of continuity.
• The solutions for contiguous elements are matched so that the
unknown values (and sometimes the derivatives) at their
common nodes are equivalent.
• When all the individual versions of the matrix equation are
finally assembled:
K u  F 
[K] = assemblage property matrix
{u´} and {F´}= assemblage of the vectors {u} and {F}
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Boundary Conditions/
• Matrix equation when modified to account for
system’s boundary conditions:
k u  F 
Solution/
• In many cases the elements can be configured so that
the resulting equations are banded. Highly efficient
solution schemes are available for such systems (Part
Three).
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Postprocessing/
• Upon obtaining solution, it can be displayed in
tabular form or graphically.
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Two-dimensional Problems
• Although the mathematical “bookkeeping”
increases significantly, the extension of the
finite element approach to two dimensions is
conceptually similar to one-dimensional
applications. It follows the same steps.
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Figure 31.9
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Figure 31.10
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