Curricula

The basics of nonlinear continuum mechanics
  1. Introduction to matrix calculus (the summation convention, eigenvalues and eigenvectors, the Cayley-Hamilton theorem, the polar decomposition theorem)
  2. Vectors and Cartesian tensors (vectors, coordinate transformations, the dyadic product, Cartesian tensors, isotropic tensors, multiplication of tensors, tensor and matrix notation, invariants of a second-order tensor, deviatoric tensors, vector and tensor calculus)
  3. Particle kinematics (bodies and their configurations, displacement and velocity, time rates of changes, acceleration, steady motion, particle paths and streamlines)
  4. Stress (surface traction, components of stress, transformation of stress components, equations of equilibrium, principal stress components, principal axes of stress and stress invariants, the stress deviator tensor, shear stress, some simple states of stress)
  5. Motions and deformations (rigid-body motions, extension of a material line element, the deformation gradient tensor, some simple finite deformations, infinitesimal strain, infinitesimal rotation, the rate-of-deformation tensor, the velocity gradient and spin tensors, some simple flows)
  6. Conservation laws (conservation laws of physics, conservation of mass, the material time derivative of a volume integral, conservation of linear momentum, conservation of angular momentum, conservation of energy, the principle of virtual work)
  7. Linear constitutive equations (constitutive equation and the ideal material, material symmetry, linear elasticity, Newtonian viscous fluids, linear viscoelasticity)
  8. Further analysis of finite deformation (deformation of a surface element, decomposition of a deformation, principal stretches and principal axes of deformation, strain invariants, alternative stress measures)
  9. Non-linear constitutive equations (non-linear theories, the theory of finite elastic deformations, a non-linear viscous fluid, nonlinear viscoelasticity, plasticity)
  10. Cylindrical and spherical polar coordinates (curvilinear coordinates, cylindrical polar coordinates, spherical polar coordinates)
Nonlinear continuum mechanics
  1. Introduction to vector and tensor calculus (algebra of vectors, algebra of tensors, higher-order tensors, eigenvalues ​​and eigenvectors of tensors, transformation of vectors and vector components, general bases, scalar, vector and tensor functions, gradients and associated operators, integral theorems)
  2. Kinematics of motion (motions and configurations of bodies, displacement, velocity and acceleration fields, material and spatial derivatives, deformation gradient tensor, rotation and stretch tensors, Lie-time derivatives)
  3. The concept of stress (surface traction vectors and stress tensors, principal stresses, examples of states of stress, alternative stress tensors)
  4. Balance principles (conservation of mass, Reynolds theorem transmission, conservation of linear and angular momentum, conservation of energy in continuum thermodynamics, entropy inequity principle, master balance principle)
  5. Some aspects of objectivity (change of observer and objective tensor fields, superimposed rigid-body motions, objective rates, invariants elastic material response)
  6. Hyperelastic materials (principles of constitutive modelling, isotropic hyperelestic materials, incompressible hyperelestic materials, compressible hyperelestic materials, some forms of strain-energy functions, elasticity tensors, transversely isotropic materials, composite materials with two families of fibres, constitutive models with internal variables, viscoelastic materials at large strains, hyperelastic materials with isotropic damage)
  7. Thermodynamics of materials (thermoelasticity of macroscopic networks, thermodynamic potentials, calorimetry, isothermal and isothermal elasticity tensors, entropic elastic materials, thermodynamics extension of Ogden’s material model, thermodynamics with internal variables)
  8. Variational principles (virtual displacement, principle of virtual work, principle of stationary potential energy, linearization of the principle of virtual work, variational principles)
Linear finite elements
  1. Introduction to the finite element method (FEM) (structural modelling and FE analysis, discrete systems and bar structures, calculation of the local stiffness matrix, direct assembly of the global stiffness matrix, derivation of the matrix equilibrium equations for the bars using the principle of virtual work and the minimum of total potential energy, plane frameworks)
  2. One-dimensional (1D) finite elements (axially loaded rod of constant cross-section, derivation of the discretized equations from the global displacement interpolation field, extrapolation of the solutions from two different meshes, advanced rod elements, twisting, isoparametric formulation, interpolation of the displacement field, numerical integration, selection of finite element (FE) type, requirements for convergence of the solution, assessment of convergence, the patch test, error types in the  FE solution)
  3. Slender plane beams based on Euler-Bernoulli theory (classical beam theory, the 2-noded Euler-Bernoulli beam element, rotation-free Euler-Bernoulli beam elements)
  4. Thick plane beams based on Timoshenko theory (Timoshenko plane beam theory, he 2-noded Timoshenko beam element, locking of the numerical solution, quadratic Timoshenko beam element, rotation-free beam element accounting for transverse shear deformation effects, beams on elastic foundation)
  5. Thin plates based on Kirchhoff theory (Kirchhoff plate theory, rectangular thin plate elements, 12 degrees of freedoms (DOF) plate rectangle proposed by Melosh, conforming BFS plate rectangle, triangular thin plate element, non-conforming thin plate triangles, conforming thin plate elements derived from Reissner-Mindlin formulation, rotation-free thin plate elements, patch tests for Kirchhoff plate elements)
  6. Thick and thin plates based on Reissner-Mindlin theory (Reissner-Mindlin plate theory, Reissner-Mindlin plate elements, performance of Reissner-Mindlin plate elements, selected Reissner-Mindlin  plate elements, four-noded plate quadrilaterals (Q4), eight-noded Serendipity plate quadrilaterals (QS8), nine-noded hierarchical plate quadrilaterals (QH9), generalization of nine-noded hierarchical plate quadrilaterals (QHG), nine-noded heterosis plate quadrilaterals (QHET), Higher order Reissner-Mindlin plate quadrilaterals with 12 and 16 nodes, etc., Reissner-Mindlin triangular plate elements etc.)
  7. Composite and laminated plates
  8. Flat shell elements
  9. Axisymmetric shell elements
  10. Two dimensional (2D) spatial triangular and rectangular elements (three-noded triangular elements, four-noded rectangular elements, reduced integration, addition of internal nodes)
  11. Higher order 2D spatial triangular and rectangular elements (derivation of the shape functions, Lagrage rectangular elements, serendipity rectangular elements, 2D isoparametric elements 2D)
  12. Axisymmetric solids (three-noded axisymmetric triangular elements, other 3 or 4 noded axisymmetric elements)
  13. Three dimensional (3D) solids (four-noded tetrahedron, other 3D solid elements, right prisms of the Lagrange family, Serendipity prisms, numerical integration, performance of 3D solid elements)
  14. Additional topic (boundary conditions in inclined supports, displacement constraints, error estimation, mesh adaptivity (spatial discretization), elimination of constrained DOFs, mesh refinement, submodeling, structure on elastic foundation, etc.)
Nonlinear finite elements
  1. Lagrangian and Eulerian finite elements in a one dimensional (1D) space (total Lagrangian formulation, updated Lagrangian formulation, Eulerian formulation, generalized (weak) forms of the total Lagrangin, updated Lagrangian and Eulerian equations, approximation of primary variables, matrix equilibrium equations)
  2. Summary of the basic equations of continuum mechanics for a three dimensional continuous media (total Lagrangian formulation, updated Lagrangian formulation, Eulerian formulation, generalized (weak) forms of the total Lagrangin, updated Lagrangian and Eulerian equations, approximation of primary variables, matrix equilibrium equations)
  3. Summary of the major types of constitutive models (non-linear elasticity, multiaxial plasticity, hyperelastic-plastic materials, viscoelasticity, stress calculation)
  4. Solution methods and their stability (explicit time integration, implicit time integration, linearization, stability, numerical stability and material stability)
  5. Arbitrary Lagrangian Eulerian (ALE) formulation (governing equations, generalized (weak) forms, approximation of primary variables, matrix equilibrium equations)
  6. Element technology (element performance, patch test, rectangular (Q4) elements and volumetric locking, generalized (weak) forms, stability)
  7. Beams and shells (Timoshenko beams, Euler-Bernoulli beams, Kirchhoff and Mindlin-Reissner beams, continuum-based beams (CB), continuum-based shells, membrane and shear locking)
  8. Contact and impact (one-dimensional (1D) rheological model of contact, contact interface equations and alternative forms of contact, contact models with friction, contact models with adhesion, flow functions, material parameters of contact models, contact stiffness, contact damping, contact detection, generalized (weak) forms of contact, spatial discretization of contact interface with finite elements, implicit and explicit solution methods of contact problems)
  9. Extended finite element method (X-FEM)
  10. Introduction to the theory of multiscale problems
  11. The theory of crystal plasticity
  12. Linearization of the equations of equilibrium, solution methods, aspects of spatial and time discretization
Material models
  1. Introduction to material modeling using small strain formulation (primary variables, small strain tensor and its time derivatives, stress tensor and its time derivatives for small strain formulation, introduction to constitutive modeling)
  2. Material models in elasticity (one dimensional (1D) rheological model of an elastic material, 1D constitutive equations, 3D constitutive equations of an elastic material, isotropic material models, anisotropic material models, constraints originating from material symmetry, material parameters and their determination using material tests, alternative forms of material models: material model formulation in the ​​principal stress space, material models based on the decomposition of the stress tensor into a deviatoric and spherical parts)
  3. Elastoplastic material models (1D rheological model of an elastic-plastic material, 1D constitutive equations, 1D plastic flow functions, 3D constitutive equations of elastic-plastic materials, plastic flow functions for a 3D continuum, anisotropic and isotropic material models, constraints coming from material symmetry, material parameters and their determination using material tests, alternative forms of material models: material model formulation in the ​​principal stress space, material models based on the decomposition of the stress tensor into a deviatoric and spherical parts, isotropic and kinematic hardening, softening, the most widespread elastoplastic material models, calculation of the stress tensor)
  4. Viscoelastic material models (1D rheological model of a visco-elastic material, 1D constitutive equations, 3D constitutive equations of visco-elastic materials, anisotropic and isotropic material models, constraints coming from the material symmetry, materials parameters and their determination using material tests, alternative forms of material models: material model formulation in the ​​principal stress space, material models based on the decomposition of the stress tensor into a deviatoric and spherical parts, the most widespread viscoelastic material models, Maxwell fluid, creep, Newtonian fluids, calculation of the stress tensor)
  5. Viscoplastic material models (1D rheological model of a viscoplastic material, 1D constitutive equations, 1D plastic flow functions, 3D constitutive equations of visco-plastic materials, plastic flow functions for a 3D continuum, anisotropic and isotropic material models, constraints coming from material symmetry, material parameters and their determination using material tests, the most widespread visco-plastic material models, calculation of the stress tensor)
  6. Hypoelastic material models (1D rheological model of a hypoelastic material, 1D constitutive equations, 3D constitutive equations of hypoelastic materials, anisotropic and isotropic material models, constraints coming from material symmetry, material parameters and their determination using  material tests)
  7. Hypoelastic-plastic material models (1D rheological model of a hypoelastic-plastic material, 1D constitutive equations, 1D flow functions, 3D constitutive equations of hypoelastic-plastic materials, flow functions for a 3D continuum, isotropic and anisotropic material models, constraints coming from material symmetry, material parameters and their determination using material tests, the most widespread hypoelastic-plastic material models, calculation of the stress tensor)
  8. Introduction to material models using large strain formulation (primarily variables, deformation and strain measures, rates of deformation and strain rates, stress measures and rates of stress, Lagrangian and Eulerian formulation, summary of the basic equations for total Lagrangian, updated Lagrangian and Eulerian formulation, introduction to constitutive modeling, requirements of objectivity, material objectivity and symmetry)
  9. Extension of the viscoelastic, viscoplastic, hypoelastic and hypoelastic-plastic material models for large strains
  10. Hyperelastic material models (summary of the basic equations of hyperelasticity materials, compressible and incompressible hyperelastic materials, Mooney-Rivlin model, Ogden model, Hencky model, Neo-Hook model, Blatz-Ko's material)
  11. Modelling of damage of materials (continuum damage theory, Lemaitre’s damage model, Gurson’s damage model, damage in rubber and rubbery polymers)
Selected topics of nonlinear finite elements
  1. Finite elements in multiphysics (multiple physical models, weak coupling, strong coupling, summary of the basic equations in multiphysics, generalized (weak) forms of the basic equations)
  2. Coupled thermal-structural analyses using an additive decomposition of the strain rate tensor into an elastic part, a plastic part and a thermal part (Summary of the basic equations using large strain formulation, a summary of the basic equations of objectivity, higher objective time derivatives of strains and stresses, total Lagrangian formulation, updated Lagrangian formulation, small strain formulation, generalized (weak) forms of the basic equations)
  3. Coupled thermal-structural analyses using a multiplicative decomposition of the deformation gradient into an elastic part, a plastic part and a thermal part (the theory of multiplicative decomposition, initial, intermediate and final configurations of a body, summary of the basic equations using large strain formulation, summary of the basic equations of objectivity, total Lagrangian formulation, updated Lagrangian formulation, generalized (weak) forms of the basic equations)
  4. Thermal analyses (summary of the basic equations using large strain formulation, summary of the basic equations of objectivity, higher order objective time derivatives of strain and stress measures, total Lagrangian formulation, updated Lagrangian formulation, Eulerian formulation, small strain formulation, generalized (weak) forms of the basic equations)
  5. Computational fluid mechanics (summary of the basic equations, constitutive equations of fluids: Newtonian viscous fluids / linear viscous fluids, non-Newtonian fluids / nonlinear viscous fluids, Reiner-Rivlin fluids, compressible and incompressible fluids, the most widespread turbulence models, generalized (weak) forms of the basic equations)
  6. Vibration and structure dynamics (summary of the basic equations of linear and nonlinear dynamics, generalized (weak) forms of the basic equations, mass matrix, stiffness matrix and damping matrix, reduction of degrees of freedom, modal methods, harmonic excitation, spectral methods)
  7. Buckling (linear and nonlinear buckling)
  8. Electrical analyses
  9. Electromagnetic analyses
  10. Submodelling (aspects of spatial discretization, structured and free meshing, types of error and error estimation, meshing, h / p / r-  mesh refinement, adaptive mesh refinement)
  11. Solution methods (direct solvers, iterative solvers, calculation of eigenvalues and eigenvectors, explicit and implicit time integration, arc-length methods)