## The program of the short course

#### Monday

**1. Basics of dynamics and introduction with motivation (K.C. Park)**

Introduction of the course

Historical background of dynamics

Newtonian, Lagrangian and Hamiltonian mechanics

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

Kinematics of deformation

Strains and stresses

Governing equations, Strong form

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

Constitutive equations for small strains - elasticity, hyper-elasticity, plasticity

Numerical integration of constitutive equations

**4. Continuum mechanics III (A. Tkachuk)**

Variational formulations in dynamics Mixed formulations, Tonti diagram, Hamilton’s principle, Weak forms

**5. Dynamics of multibody systems (J. Kim)**

Governing equations, Constrains

Lagrange equations and Lagrange multipliers

Numerical methods in multibody dynamics

#### Tuesday

**6. Finite element method I - Basics (J. Gonzalez)**

Principle of virtual work

Finite Element Formulation

Assembly of global matrices

Convergence properties

**7. Finite element method II (J. Gonzalez)**

Shape functions and higher order FEM

Isoparametric formulation

Numerical integration

Hybrid and mixed formulation, inf-sup condition

**8. ****Finite element method III - Mass matrices (A. Tkachuk)**

Properties of mass matrix

Consistent and lumped mass matrices

Higher-order mass matrix

Direct inversion of mass matrix

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

Implementation of FE codes for linear dynamics (Matlab)

**10. Poster section of participants**

#### Wednesday

**11. Finite element method V – Beams and Plates (A. Combescure)**

Basics of beam theory – Euler-Bernoulli and Timoshenko theory

Basics of plate theory - Kirchoff-Love and Mindlin theory

FEM for beams and plates

Free vibration of beams and plates

**12. Finite element method VI – Shells (A. Combescure)**

Basics of shell theory

FEM shell models

Shells in dynamics

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

Locking phenomena and hourglass effect

Assumed strain, enhanced strain FEM, B-bar formulation

Reduced integration and stabilization

**14. Finite element method VIII– Linear Solvers (J. Kruis)**

Linear solvers in FEM

Matrix factorization

Sparse solvers, Krylov methods (especially conjugate gradient method)

**15. Finite element method IX (J. Kruis)**

FEM in vibration problems

Spectral and modal analysis

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

Convergence of FEM in eigen-value problem

Dynamic steady state response

#### Thursday

**16. Finite element method X (A. Popp)**

Basics of nonlinear continuum mechanics, FEM for nonlinear problems, Total Lagrangian formulation, Nonlinear solvers - Newton-Raphson method

**17. Finite element method XI - Direct time integration in dynamics (R. Kolman)**

FEM in dynamics, formulation of dynamic problems

Introduction into direct time integration

Basic methods (Newmark method and central difference method)

Solving of nonlinear time-depend problems

Time step size estimates, mass scaling

**18. Buckling phenomena (A. Combescure)**

Linear theory of stability

Solution methods, path following techniques

Identification of critical points

Pre-buckling analysis and nonlinear stability analysis

**19. Wave propagation (R. Kolman)**

Theory of wave propagation in elastic solids, Wave speeds in solids

Dispersion and frequency analysis of FEM,

**20. Partitioned analysis (K.C. Park)**

Theory of Lagrange multipliers

Basic theory of partitioned analysis

Equations of motion for partitioned systems, Domain decomposition methods and FETI

#### Friday

**21. Model reduction in dynamics (J. Kim)**

Variational analysis of dynamic sub-structuring, Hurty and Craig-Bampton methods, dynamic reduction, mode selection, error estimation,

**22. Contact problems I (A. Popp)**

Basics of contact mechanics, FEM for contact problems

Penalty, Lagrange multiplier and Augmented Lagrangian methods

**23. Contact problems II (A. Popp) **

Modeling of friction

Advanced discretization techniques and solution algorithms

Mortar methods, Semi-smooth Newton methods

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

Variational formulation

Methods of discretizations

Staggered analysis

**25. Closing and discussion**