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O metod ě konečných prvků Lect_02

O metod ě konečných prvků Lect_02.ppt. Principle of virtual work, a few simple elements. M. Okrouhlík Ústav termomechaniky, AV ČR , Praha Plzeň , 2010. Contents. Governing equations of solid continuum mechanics Fundamental ideas of finite element method (FEM)

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O metod ě konečných prvků Lect_02

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  1. O metodě konečných prvkůLect_02.ppt Principle of virtual work, a few simple elements M. Okrouhlík Ústav termomechaniky, AV ČR, Praha Plzeň, 2010

  2. Contents • Governing equations of solid continuum mechanics • Fundamental ideas of finite element method (FEM) • Principle of virtual displacements and work • Discretization of displacements and strains • Energy balance • Equations of equilibrium, equations of motion • Lagrangian interpolation– Lagrangian elements – generalized coordinates • Bar, beam, triangle, quadrilateral, tetrahedron and brick elements • Derivation by hand and by means of Matlab • Hermitian elements • Conditions of completeness and compatibility and convergence

  3. Governing equations of solid continuum mechanics • Cauchy equations of motion • Kinematic relations • Constitutive relations 3 equations 6 equations 6 equations

  4. Solution of above 15 partial differential equations • The mentioned system of partial differential equations could be analytically solved only for simple geometry and simple initial and boundary conditions. • For a long time there were attempts to solve it numerically. Historically, it was the method of finite differences which was used at first. • The solved area in space was covered by a regular mesh and the partial derivatives were replaced by a suitable difference formula at each node. • This way the partial differential equation were replaced by ordinary differential equations. • We say that the problem was discretized in space . • The resulting ordinary differential equations have (usually) to be discretized in time to find a transient solution.

  5. Today, approximate methods of solution prevail • They are based on discretization in space and time and have numerous variants • Finite difference method • Transfer matrix method • Matrix methods • Finite element method • Displacement formulation • Force formulation • Hybrid formulation • Boundary element method • Meshless element method

  6. Finite element method (FEM) In FEM we "fill" the structure in question by a lot of small geometrically simple parts (elements) that are connected only by their corner points (nodes).

  7. FEM • For these elements we will derive their inertia and stiffness (damping) properties - in matrix form and will find a way how equilibrium conditions, boundary and initial conditions, and constitutive relations are satisfied. • So instead of knowing the state of stress and strain at each material point (particle) we will find a solution in nodes only.

  8. There are many ways how the FE theory could be presented. The one, I like best, is based on the principle of virtual work.

  9. Virtual displacements and work Stejskal, V., Okrouhlík, M.: Kmitání s Matlabem, Vydavatelství ČVUT, 2002

  10. Práce osamělých sil Práce objemových sil Práce povrchových sil Práce vnitřních sil

  11. Posuvy v uzlech Zatím neznámý operátor

  12. Assembling – k tomu se ještě vrátíme

  13. Lagrangian interpolation– Lagrangian elements

  14. Tak a ještě jednou - stručně

  15. … lagrangeovská Do hry vstupují pouze hodnoty funkce v uzlech Později se též zmíníme o hermiteovské polynomialní aproximaci – kromě hodnot funkce v uzlech uvažujeme navíc i hodnoty derivací v uzlech

  16. Lagrangian elementsMethods of generalized coordinatesLater, we will explain another approach, namelyIsoparametric elements

  17. (konsistentní) Say a few words about the diagonal mass matrix

  18. 1D Hermitian element

  19. For more details see: Okrouhlík, M.: Aplikovaná mechanika kontinua II, Ediční středisko ČVUT, Praha, 1989.

  20. Summary for 1D elements • L1 … lagrangian, linear approximation function • L2 … lagrangian, quadratic • L3 … lagrangian, cubic • H3 … hermitian, cubic approximation function • H5 … hermitian, quintic See: Okrouhlík, M. – Hoeschl, C.: A contribution to the study of dispersive properties of 1D and 3D Lagrangian and Hermitian elements, Computers and structures, Vol. 49, pp. 779 – 795, 1993

  21. C stands for consistent mass matrix

  22. How does dispersion for L1C and L1D elements depend on the mass matrix formulation The subject will be treated in more detail later. See dp_part_1.ppt

  23. Say a few words about alternative numbering

  24. Linear displacement distribution … … constant strain element … … discontinuity at element boundaries

  25. 4-node plane elementwith bilinear displacement approximation

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