MECH300H Introduction to Finite Element Methods

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MECH300H Introduction to Finite Element Methods
Lecture 10 Time-Dependent Problems
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Time-Dependent Problems
In general,
Key question How to choose approximate
functions?
Two approaches
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Model Problem I Transient Heat Conduction
Weak form
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Transient Heat Conduction
and
let
ODE!
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Time Approximation First Order ODE
Forward difference approximation - explicit
Backward difference approximation - implicit
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Time Approximation First Order ODE
a - family formula
Equation
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Time Approximation First Order ODE
Finite Element Approximation
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Stability of Family Approximation
a
Example
Stability
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FEA of Transient Heat Conduction
a - family formula for vector
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Stability Requirment
where
Note One must use the same discretization for
solving the eigenvalue problem.
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Transient Heat Conduction - Example
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Transient Heat Conduction - Example
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Transient Heat Conduction - Example
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Transient Heat Conduction - Example
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Transient Heat Conduction - Example
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Transient Heat Conduction - Example
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Transient Heat Conduction - Example
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Model Problem II Transverse Motion of
Euler-Bernoulli Beam
Weak form
Where
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Transverse Motion of Euler-Bernoulli Beam
and
let
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Transverse Motion of Euler-Bernoulli Beam
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ODE Solver Newmarks Scheme
where
Stability requirement
where
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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)
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Fully Discretized Finite Element Equations
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Transverse Motion of Euler-Bernoulli Beam
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Transverse Motion of Euler-Bernoulli Beam
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Transverse Motion of Euler-Bernoulli Beam
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Transverse Motion of Euler-Bernoulli Beam