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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