Approximation%20on%20Finite%20Elements

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Approximation on Finite Elements

• Bruce A. Finlayson
• Professor Emeritus of
• Chemical Engineering

Outline Approximation on finite elements Mesh
refinement Calculus of variations Galerkin method
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The function x2 exp(y-0.5)looks like this when
plotted
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Approximation on finite elements

• Break the region into small blocks, and color
  each block according to an average value in the
  block.
• The approximation depends on the number of blocks.

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Here is what we expect in a contour plot of the
function
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Divide the domain into blocks (finite elements)
and assign an average value to each block.
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Write the approximation as a linear combination
of trial functions, each of which takes the value
one in one block and zero in all the other blocks.
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Using color to represent the value, this is the
solution with N x N blocks, N4
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N 8
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N 16
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N 32
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N 64
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N 128
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This is mesh refinement.

• Notice how the picture got better and better the
  more squares we took.
• We approximated the function on each block - a
  finite element approximation.
• We get a better approximation when we use small
  finite elements.
• As the number of blocks increases, the picture
  approaches that of a continuous function.

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To Review N 4, 8, 16, and 32
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Let functions in the block be bilinear functions
of u and v, 0 u,v 1.

• N1 (1 - u) (1 - v)
• N2 u (1 - v)
• N3 u v
• N4 (1 - u) v
• The value of each Ni is 1.0 at one corner and
  zero at the other three corners.
• ExampleN3(1,1) 1 N3(0,1) N3(1,0) N3(0,0)0

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N 4, bilinear interpolation
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N 8, bilinear interpolation
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N 16, bilinear interpolation
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Compare constant interpolation on finite elements
with bilinear interpolation on finite elements. A
better approximation is achieved using fewer
blocks when the trial function is a higher degree
polynomial.
Constant interpolation with 32x32 1024 blocks.
Bilinear interpolation with 4x4 16 blocks.
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Instead of matching the function at the
block-corners, find the best interpolant by
minimizing the mean square difference between the
approximation and the exact function. Still use
finite elements, but bilinear approximations.
The approximation - Minimization - Equations to
solve -
Test function
Function wed like to be zero
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What do you do if you dont know the function?
When the function is the solution to a
differential equation, for example, the Calculus
of Variations can be used to find the
approximation, as follows.
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Calculus of Variations
The function that satisfies this differential
equation
minimizes this integral (this must be proved for
each equation)
The same approach can be taken to satisfy the
differential equation, one approximates the
integral on the finite element blocks and finds
the minimum.
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We choose finite element functions which satisfy
the boundary conditions, and then find the values
of the parameters that make the integral a
minimum.
Minimize this integral with respect to the
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The solution with linear elements on 312
triangles (177 nodes) is
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The solution with linear elements on 1248
triangles (665 nodes) is
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Finite Element Variational Method

• Divide the domain into small regions.
• Write a low degree polynomial on each small
  region constant, bilinear, biquadratic. These
  are the trial functions.
• Write the solution as a series of trial
  functions.
• Determine the coefficients by minimizing an
  integral. (The trick is to know what integral to
  use.)

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Galerkin Finite Element Method

• If a variational principle exists, the Galerkin
  method is the same as the variational method. It
  applies when there is no integral to be minimized
  or made stationary.
• The same finite elements can be used.
• The finite element approximation is put into
  the differential equation, and this is called the
  residual. It needs to be zero.
• Now the residual is made orthogonal to each trial
  function if this is done for infinitely many
  trial functions, or test functions, then the
  residual is zero. In practice the approximation
  is better the smaller the residual, and the
  approximation converges to the solution as the
  number of trial functions increases.

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Conclusion - Three Basic Ideas

• Write the solution in a series of functions, each
  of which is defined over small elements, using
  low-order polynomials.
• Minimize some integral to solve a differential
  equation (or use Galerkin or the Method of
  Weighted Residuals, MWR).
• Increase the number of basis functions in order
  to demonstrate convergence.