MECH593 Introduction to Finite Element Methods Eigenvalue Problems and Time-dependent Problems

Eigenvalue Problems Definition: Examples: Eigenvalue problems refer to the problems with the following type: where A, B are operators.

Eigenvalue Problems – Engineering Applications Example: Vibration of an axial bar General solution scheme: let

Eigenvalue Problems – A Model Problem Model problem: Weak form: Assume: Assembling:

Eigenvalue Problems – Heat Conduction Non-dimensional variables/parameters: 0L x Initial condition: Boundary conditions:

Eigenvalue Problems – Heat Conduction Let 0L x Initial and boundary conditions: Element equation for linear element:

Eigenvalue Problems – Heat Conduction Element equation for linear element: 1 2 h h Assembled system: Boundary conditions:

Eigenvalue Problems – Heat Conduction 1 2 h h Condensed system: Eigenvalues: Eigenvectors: Mode shapes:

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

Transient Heat Conduction - Example Element equation for linear element  - family formula : Initial condition: Boundary conditions: for One element mesh: and

Transient Heat Conduction - Example Element equation: for Stability requirement:

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 One needs: Element equation

Transverse Motion of Euler-Bernoulli Beam Element equation of one element:

Transverse Motion of Euler-Bernoulli Beam Symmetry consider only half the beam Boundary conditions: Initial conditions: Imposing bc and ic:

Transverse Motion of Euler-Bernoulli Beam