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## MECH300H Introduction to Finite Element Methods

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### Note: One must use the same discretization for solving. the eigenvalue problem. ... Central difference method (conditional stable) Galerkin method (stable) ... – PowerPoint PPT presentation

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Title: MECH300H Introduction to Finite Element Methods

1
MECH300H Introduction to Finite Element Methods
Lecture 10 Time-Dependent Problems
2
Time-Dependent Problems
In general,
Key question How to choose approximate
functions?
Two approaches
3
Model Problem I Transient Heat Conduction
Weak form
4
Transient Heat Conduction
and
let
ODE!
5
Time Approximation First Order ODE
Forward difference approximation - explicit
Backward difference approximation - implicit
6
Time Approximation First Order ODE
a - family formula
Equation
7
Time Approximation First Order ODE
Finite Element Approximation
8
Stability of Family Approximation
a
Example
Stability
9
FEA of Transient Heat Conduction
a - family formula for vector
10
Stability Requirment
where
Note One must use the same discretization for
solving the eigenvalue problem.
11
Transient Heat Conduction - Example
12
Transient Heat Conduction - Example
13
Transient Heat Conduction - Example
14
Transient Heat Conduction - Example
15
Transient Heat Conduction - Example
16
Transient Heat Conduction - Example
17
Transient Heat Conduction - Example
18
Model Problem II Transverse Motion of
Euler-Bernoulli Beam
Weak form
Where
19
Transverse Motion of Euler-Bernoulli Beam
and
let
20
Transverse Motion of Euler-Bernoulli Beam
21
ODE Solver Newmarks Scheme
where
Stability requirement
where
22
ODE Solver Newmarks Scheme
Constant-average acceleration method (stable)
Linear acceleration method (conditional stable)
Central difference method (conditional stable)
Galerkin method (stable)
Backward difference method (stable)
23
Fully Discretized Finite Element Equations
24
Transverse Motion of Euler-Bernoulli Beam
25
Transverse Motion of Euler-Bernoulli Beam
26
Transverse Motion of Euler-Bernoulli Beam
27
Transverse Motion of Euler-Bernoulli Beam