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Chapter 8. Rayleigh-Ritz Method RRM

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Chapter 8. Rayleigh-Ritz Method RRM

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    1. 1 Chapter 8. Rayleigh-Ritz Method (RRM) Basic idea: we assume that we know the spatial variation of the displacement field, except for some unknown coefficients to be determined with the aid of the PMPE

    2. 2 Rayleigh-Ritz Method Parenthesis : what is an essential boundary condition?

    3. 3 Rayleigh-Ritz Method

    4. 4 Rayleigh-Ritz Method Substitute the approximate displacement field into the total potential energy to get Then apply the PMPE (or PVW) Thus we end up with 3n equations for the 3n unknown coefficients (qi,ri,si). This is an approximate method since (unless you are very lucky), the basis functions are not correct, thus u, v and w will be approximate. The closer the basis functions are to the exact spatial variation of the displacements, the better the approximation.

    5. 5 RRM: applications

    6. 6 Application 1 - Exact Solution

    7. 7 Application 1 - First attempt

    8. 8 Application 1 - Second attempt

    9. 9 Application 2 - First attempt

    10. 10 Application 2 - Second attempt

    11. 11 RRM: final notes Conclusions Advantages Simplicity “keep old terms” when adding new ones Disadvantages Basis functions are hard to find for complicated geometries (especially in 2-D and 3-D cases) qi have no physical significance Convergence is hard to quantify

    12. 12 3. Basic concepts of FEM: solution of 1-D bar problem Table of contents 3.1 Basic concepts : mesh, nodes, elements, interpolation, ... 3.2 FEA of axially loaded bar 3.3 Notes : direct method, higher-order elements, … 3.4 Principle of Virtual Work (PVW) approach 3.5 Galerkin Weighted Residual (GWR) method

    13. 13 3.1 Basic concepts The FEM is also based on the RRM, but the basis functions are easy to find : interpolation the qi have a physical significance : nodal displacements Basic idea Discretize the domain with a finite element mesh composed of nodes and elements Compute the “best values” of the nodal displacements (based, for example, on the PMPE) Use interpolation to find the solution everywhere else in the discretized domain There are many elements of different types and geometries : 1-D, 2-D, 3-D, plane stress, plane strain, plates, shells, structural, thermal, fluid mechanics, electromagnetic, elastic, plastic, static, dynamic, …

    14. 14 3.2 FEA of axially loaded bar In this section, we introduce the 6 basic steps of a FEA by solving the following simple structural problem

    15. 15 FEA of axially loaded bar Finite element solution We will use the PMPE. The total potential energy P for this problem is

    16. 16 FEA of axially loaded bar

    17. 17 Na(s) and Nb(s) play an important role in FEA and are called interpolation or shape functions FEA of axially loaded bar

    18. 18 FEA of axially loaded bar Expand the square term :

    19. 19 Since we know that

    20. 20 FEA of axially loaded bar

    21. 21 FEA of axially loaded bar Note : the approximate displacement solution is continuous within and between elements Within an element :why? Between elements : why?

    22. 22 FEA of axially loaded bar Let’s add the contribution of all three elements to the total potential energy

    23. 23 FEA of axially loaded bar

    24. 24

    25. 25 FEA of axially loaded bar

    26. 26 FEA of axially loaded bar

    27. 27 FEA of axially loaded bar

    28. 28 FEA of axially loaded bar

    29. 29 FEA of axially loaded bar

    30. 30 FEA of axially loaded bar

    31. 31

    32. 32 FEA of axially loaded bar

    33. 33 FEA of axially loaded bar

    34. 34 3.3 Notes

    35. 35

    36. 36

    37. 37 Notes (Cont’d)

    38. 38 Notes (Cont’d)

    39. 39 Notes (Cont’d)

    40. 40 Notes (Cont’d)

    41. 41 Two applications

    42. 42 Notes (Cont’d)

    43. 43 3.4 PVW approach

    44. 44

    45. 45 3.5 Galerkin Weighted Residual approach

    46. 46

    47. 47

    48. 48 WRM: application

    49. 49 WRM: application

    50. 50 WRM: application

    51. 51

    52. 52

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