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FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD

FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD. FINITE ELEMENT METHOD DISCRETIZATION. The Finite Element Method (FEM) consists in construction of the finite dimensional sub-space. We seek the approximate solution as a linear combination of the basis functions. HP ADAPTATION.

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FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD

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  1. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD

  2. FINITE ELEMENT METHOD DISCRETIZATION The Finite Element Method (FEM) consists in construction of the finite dimensional sub-space We seek the approximate solution as a linear combination of the basis functions

  3. HP ADAPTATION Goal: increase the number of the basis functions in order to increase the accuracy of the approximate solution hp adaptation consists in breaking selected finite elements into smaller elements and increasing polynomial order of approximation on selected finite elements.

  4. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Coarse mesh

  5. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Coarse mesh solution

  6. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Fine mesh

  7. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Fine mesh solution

  8. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Optimal mesh is constracted based on comparison of coarse and fine mesh solutions

  9. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Final mesh delivering solution with 0.001 relative error

  10. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Zoom 10 X

  11. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Zoom 100 X

  12. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Zoom 1000 X

  13. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Zoom 10000 X

  14. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Zoom 100000 X

  15. FULLY AUTOMATIC HP ADAPTIVE FINITE ELEMENT METHOD Zoom 1000000 X

  16. LOKALNY WYBÓR OPTYMALNEJ STRATEGII ADAPTACJI Element siatki gęstej Element siatki rzadkiej Lokalne rozwiązanie na elemencie siatki rzadkiej Lokalne rozwiązanie na elemencie siatki gęstej Lokalnie, dla każdego elementu siatki rzadkiej rozważane są różne strategie adaptacji Dla proponowanych strategii adaptacji obliczam lokalne rozwiązanie poprzez mechanizm projekcji z rozwiązania na siatce gęstej ??? (mechanizm projekcji) (dla proponowanych strategii adaptacji) Lokalnie, dla każdego elementu siatki rzadkiej, wybierana jest taka strategia, która daje nam największy spadek błędu a jednocześnie najmniejszy przyrost rozmiaru zadania (ilości niewiadomych)

  17. HP ADAPTATION PROVIDES EXPONENTIAL CONVERGENCE RATE

  18. 3D Fichera problem Laplace equation

  19. 3D Fichera problem

  20. 3D Fichera problem

  21. 3D Fichera problem + => + => Solution over the coarse grid Solution over the fine grid Optimal grid Relative error estimation (energy norm) Decisions about optimal h, p or hp refinementsover each coarse grid finite element

  22. 3D Fichera problem

  23. 3D Fichera problem

  24. 3D Fichera problem + => + => Solution over the coarse grid Solution over the fine grid Optimal grid Relative error estimation (energy norm) Decisions about optimal h, p or hp refinementsover each coarse grid finite element

  25. 3D Fichera problem

  26. 3D Fichera problem

  27. 3D Fichera problem

  28. 3D Fichera problem

  29. Results for 3D Fichera problem

  30. Convergence curve for the 3D Fichera problem Exponential convergence delivered by parallel code Corresponding fine mesh solution has relative error below 1%

  31. EXEMPLARY BOUNDARY VALUE PROBLEMHEAT TRANSFER OVER THE L-SHAPE DOMAIN Strong formulation (Partial Differential Equations) Find temperature scalar field, such that where is the L-shape domain boundary On part of the boundary we define the Dirichlet boundary condition (zero temperature) On part of the boundary we define the Neumanna boundary condition (heat transfer rate)

  32. STRONG AND WEAK (VARIATIONAL) FORMULATIONS Strong formulation temperature scalar field, of the order of such that Find Weak (variational) formulation Find temperature scalar field such that

  33. FINITE ELEMENT METHOD DISCRETIZATION The Finite Element Method (FEM) consists in construction of the finite dimensional sub-space We seek the approximate solution as a linear combination of the basis functions

  34. Finite element discretization global degrees of freedom

  35. MESH STRUCTURE • Meshisbased on Euler’s model: • finite elementiscomposed of nodes • nodesare: vertices, edges, faces, interiors etc. • edgeconsists of 2 vertices • face consists of 4 edges • interior consists of (i.e. isdelimited by) 6 faces • … face interior edge vertex

  36. SHAPE FUNCTIONS The base of approximation space is composed of global base functions. These are splines, made of multi-dimensional polynomials. Global base functions are connected with a node and have supports of one or several neighbor elements. (2D example)

  37. SHAPE FUNCTIONS Shape function is a restriction of a GBF into a single finite element. Shape functions are connected with elements, and are just single multi-dimensional polynomials (2D example) For example: global edge base function consists of two local shape functions: and Typically in 2D we use shape functions of orders up to (9, 9).Bilinear local shape functions are obligatory for each element. (not all functions are shown in the pictures for clarity)

  38. Relations between 1D 2D and 3D shape functions Higher dimension shape functions are constructed as tensor products of several 1D shape functions

  39. 1D Hp Finite Element 1D hierarchical shape functions:

  40. 2D Hp Finite Element 4 vertices 4 mid-edge nodes 1mid-face nodes The reference element shape functions are created as tensor products of 1D hierarchical shape functions

  41. 2D Hp Finite Element Vertex shape functions and second order edge and interior shape functions

  42. 2D Hp Finite Element One bilinear shape function for each of 8 vertices (order of approximation equal to 1 in each vertex)

  43. 2D Hp Finite Element shape functions for each of 4 mid-edge nodes (various orders of approximation)

  44. 2D Hp Finite Element face bubble shape functions for interior node

  45. Relations between 1D and 2D Higher dimension shape functions are constructed as tensor products of several 1D shape functions

  46. 3D Fichera problem Laplace equation

  47. 3D Hp Finite Element 8 vertices 12 mid-edge nodes 6 mid-face nodes 1 middle node The reference element shape functions are created as tensor products of 1D hierarchical shape functions

  48. 3D Hp Finite Element One trilinear shape function for each of 8 vertices (order of approximation equal to 1 in each vertex)

  49. 3D Hp Finite Element shape functions for each of 12 mid-edge nodes (various orders of approximation)

  50. 3D Hp Finite Element face bubble shape functions for each of 6 mid-face nodes

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