MECH300H Introduction to Finite Element Methods Lecture 10 Time-Dependent Problems

In general, Key question: How to choose approximate functions? Two approaches:

Model Problem I – Transient Heat Conduction Weak form:

Transient Heat Conduction let: and ODE!

Time Approximation – First Order ODE Forward difference approximation - explicit Backward difference approximation - implicit

Time Approximation – First Order ODE  - family formula: Equation

Time Approximation – First Order ODE Finite Element Approximation

Stability of – Family Approximation Stability  Example

FEA of Transient Heat Conduction  - family formula for vector:

Stability Requirment where Note: One must use the same discretization for solving the eigenvalue problem.

Transient Heat Conduction - Example

Model Problem II – Transverse Motion of Euler- Bernoulli Beam Weak form: Where:

Transverse Motion of Euler-Bernoulli Beam let: and

Transverse Motion of Euler-Bernoulli Beam

ODE Solver – Newmark’s Scheme where Stability requirement: where

ODE Solver – Newmark’s Scheme Constant-average acceleration method (stable) Linear acceleration method (conditional stable) Central difference method (conditional stable) Galerkin method (stable) Backward difference method (stable)

Fully Discretized Finite Element Equations

Transverse Motion of Euler-Bernoulli Beam