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

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

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

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

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

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

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

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

9. Time-Dependent Problems In general, Key question: How to choose approximate functions? Two approaches:

10. Model Problem I – Transient Heat Conduction Weak form:

11. Transient Heat Conduction let: and ODE!

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

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

14. Time Approximation – First Order ODE Finite Element Approximation

15. Stability of – Family Approximation Stability  Example

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

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

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

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

20. Transient Heat Conduction - Example

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

26. Transverse Motion of Euler-Bernoulli Beam let: and

27. Transverse Motion of Euler-Bernoulli Beam

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

29. 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)

30. Fully Discretized Finite Element Equations One needs: Element equation

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

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

33. Transverse Motion of Euler-Bernoulli Beam