outline - purdue engineeringdjiao/ee618/classnote/lect16-fem...outline a simple example – the ritz...
TRANSCRIPT
Outline
�A Simple Example– The Ritz Method– Galerkin’s Method– The Finite-Element Method
�FEM Definition�Basic FEM Steps
Example
�Problem Statement
x=0 x=1m
Φ=0 Φ=1
ε
ρ(x) = −(x+1)ε C/m3
Example
Differential equation:
Boundary condition:
Solution:
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ExampleA. The Ritz method
Vartiational solution:
= 0
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Proof:
= 0
ExampleExpand
Apply BC:
Functional:
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ExampleB. Galerkin’s method
Weighted residual:
Expansion:
Weighting:
Solution:
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ExampleC. Another method
x = 0 x = 1
1 2 3 4
Linear interpolation:
Boundary condition:
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Example
Integration:
Apply the Ritz method:
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ExampleApply Galerkin’s method:
Integration by parts:
Weighting:
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Example
Numerical solution:
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FEM Definition
The above solution procedure
The finite element method
The Ritz variationalFEM
The GalerkinFEM
Equivalent for self-adjoint problems
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Basic FEM Steps
1. Discretization/subdivision of solution domain
2. Selection of interpolation schemes
3. Formulation of the system of equations
4. Solution of the system of equations
1-D: 2-D: 3-D:
Linear or higher-order polynomials
Using either the Ritz or Galerkin method
Using either a direct or iterative method
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