## The program of the short course

#### Monday

**1. Basics of dynamics (M. Okrouhlík)**

- Historical background

- Newton’s laws

- Newtonian, Lagrangian and Hamiltonian mechanics

**2. Continuum mechanics I (J. Plešek)**

- Kinematics of deformation

- Strains and stresses

- Governing equations

**3. Continuum mechanics II (J. Plešek)**

- Convexity condition

- Basics of thermodynamics

- Constitutive equations for small and large strains

**4. Continuum mechanics III (R. Ohayon)**

- Variational formulations in linear dynamics and vibrations

- Modal analysis

- Rayleigh quotient, Hamilton’s principle

**5. Dynamics of multibody systems (A. Ibrahimbegovic)**

- Governing equations

- Constrains

- Lagrange equations and Lagrange multipliers

- Numerical methods in multibody dynamics

#### Tuesday

**6. Finite element method I (J. González)**

- Principle of virtual work

- Finite Element Formulation

- Assembly of global matrices

- Convergence properties

- Examples

**7. Finite element method II (A. Ibrahimbegovic)**

- Shape functions and higher order FEM

- Isoparametric formulation

- Numerical integration

- Hybrid and mixed formulation, inf-sup condition

**8. Finite element method III (A. Tkachuk)**

- Locking phenomena and hourglass effect

- Assumed strain, enhanced strain FEM, B-bar formulation

- Reduced integration and stabilization

**9. Finite element method IV (J. Kruis)**

- Linear solvers in FEM

- Matrix factorization

- Sparse solvers, Krylov methods (especially conjugate gradient method)

**10. Finite element method V (A. Ibrahimbegovic)**

- FEM for nonlinear problems

- Solvers for nonlinear static problems - NR, BGFS, semi-Newton methods, etc.

- Convergence criteria

#### Wednesday

**11. Finite element method VI (J. Kruis)**

- FEM in vibration problems, mass matrix

- Spectral and modal analysis

- Numerical methods for eigen-value problem (subspace iteration, etc)

- Convergence of FEM in eigen-value problem

- Dynamic steady state response

**12. Direct time integration in dynamics I (R. Kolman)**

- FEM in linear dynamics, formulation of dynamic problems

- Introduction into direct time integration

- Basic methods (Newmark method and central difference method)

- Lumping techniques for mass matricesStability of time schemes

- Stability of time schemes

**13. Finite element method VII (A. Ibrahimbegovic)**

- Dynamic problems

- Solving of nonlinear time-depend problems

**14. Finite element method VIII (A. Combescure)**

- Basics of shell theory

- FEM shell models

- FEM for shells in dynamics

- Mass matrices for shells

**15. Direct time integration in dynamics II (A. Tkachuk)**

- Time step size estimates – global/local estimate in FEM

- Treatment of time step size – mass scaling, bi-penalty, etc

- Application in crash problems

#### Thursday

**16. Modal reduction and reduction methods in dynamics (R. Ohayon)**

- Variational analysis of dynamic sub-structuring

- Substructuring analysis in discretized (finite element) case

- Hurty and Craig-Bampton methods

**17. Partitioned analysis I – basic theory (K.C. Park)**

- Theory of Lagrange multipliers

- Basic theory of partitioned analysis

- Equations of motion for partitioned systems

**18. Partitioned analysis II (K.C. Park)**

- Domain decomposition methods

- Finite element tearing and interconnect (FETI)

- Coupling of FEM/FEM

**19. Dynamic contact problems (A. Tkachuk)**

- FEM in contact problems

- Penalty method

- Augmented Lagrangian method

- Mortar methods

**20. Finite element method - wave propagation (R. Kolman)**

- Theory of wave propagation in elastic solids

- Wave speeds in solids

- Dispersion and frequency analysis of FEM

- Numerical benchmarks

#### Friday

**21. Modern methods of direct time integration (A. Combescure)**

- Generalized time schemes

- Asynchronous and variational schemes

- Sub-cycling methods, coupling of different time schemes

**22. Coupled problems – Fluid-structures interactions (K.C. Park)**

- Variational formulation

- Methods of discretizations

- Staggered analysis

**23. Boundary element method (J. González)**

- Introduction into Boundary Element Method

- Linear Acoustics

- Coupling FEM/BEM

**24. Numerical methods for dynamic crack propagation (A. Combescure)**

- Extended finite element method (XFEM)

- Meshless methods and Level-set methods

- Cohesive FEM

**25. Vibroacoustic/elastoacoustic modelling (R. Ohayon)**

- General local equations of vibroacoustic problem

- Choice of unknown fields and variational formulations of the problem

- Finite element discretization and Reduced Order Models for the interior problem