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Penalty-Based Solution for the Interval Finite Element Methods

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Penalty-Based Solution for the Interval Finite Element Methods

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    1. Penalty-Based Solution for the Interval Finite Element Methods Rafi L. Muhanna Georgia Institute of Technology

    2. Outline Interval Finite Elements Element-By-Element Penalty Approach Examples Conclusions

    3. Center for Reliable Engineering Computing (REC)

    4. Interval Calculator

    5. Outline Interval Finite Elements Element-by-Element Penalty Approach Examples Conclusions

    6. Follows conventional FEM Loads, nodal geometry and element materials are expressed as interval quantities Element-by-element method to avoid element stiffness coupling Lagrange Multiplier and Penalty function to impose compatibility Iterative approach to get enclosure Non-iterative approach to get exact hull for statically determinate structure Interval Finite Elements

    7. Interval Finite Elements

    8. = Interval element stiffness matrix B = Interval strain-displacement matrix C = Interval elasticity matrix F = [F1, ... Fi, ... Fn] = Interval element load vector (traction) Interval Finite Elements

    9. Finite Element Load Dependency Stiffness Dependency

    10. Load Dependency The global load vector Pb can be written as Pb = M q where q is the vector of interval coefficients of the load approximating polynomial Finite Element Load Dependency

    11. Sharp solution for the interval displacement can be written as: U = (K -1 M) q Finite Element Load Dependency

    12. Outline Interval Finite Elements Element-by-Element Penalty Approach Examples Conclusions

    13. Stiffness Dependency Finite Element Element-by-Element Approach

    14. Finite Element Element-by-Element Approach

    15. Finite Element Element-by-Element Approach Element by Element to construct global stiffness

    16. K: block-diagonal matrix Finite Element Element-by-Element Approach

    17. Element-by-Element Finite Element Element-by-Element Approach

    18. Outline Interval Finite Elements Element-by-Element Penalty Approach Examples Conclusions

    19. Finite Element Present Formulation In steady-state analysis-variational formulation With the constraints

    20. Finite Element Present Formulation Invoking the stationarity of ?*, that is ??* = 0

    21. The physical meaning of Q is an addition of a large spring stiffness Finite Element Penalty Approach

    22. Interval system of equations where and Finite Element Penalty Approach

    23. Finite Element Penalty Approach

    24. Finite Element Penalty Approach

    25. Finite Element Penalty Approach

    26. Outline Interval Finite Elements Element-by-Element Penalty Approach Examples Conclusions

    27. Statically indeterminate (general case) Two-bay truss Three-bay truss Four-bay truss Statically indeterminate beam Statically determinate Three-step bar Examples

    28. Two-bay truss Three-bay truss A = 0.01 m2 E (nominal) = 200 GPa Examples Stiffness Uncertainty

    29. Four-bay truss Examples Stiffness Uncertainty

    30. Examples Stiffness Uncertainty 1% Two-bay truss

    31. Examples Stiffness Uncertainty 1% Three-bay truss

    32. Examples Stiffness Uncertainty 1% Four-bay truss

    33. Examples Stiffness Uncertainty 5% Two-bay truss

    34. Examples Stiffness Uncertainty 5% Three-bay truss

    35. Examples Stiffness Uncertainty 10% Two-bay truss

    36. Examples Stiffness Uncertainty 10% Three-bay truss

    37. Examples Stiffness and Load Uncertainty Statically indeterminate beam

    38. Examples Stiffness and Load Uncertainty Statically indeterminate beam

    39. Examples Stiffness and Load Uncertainty Statically indeterminate beam

    40. Examples Stiffness and Load Uncertainty Statically indeterminate beam

    41. Examples Stiffness and Load Uncertainty Three-bay truss

    42. Examples Stiffness and Load Uncertainty Three-bay truss

    43. Examples Stiffness and Load Uncertainty Three-bay truss

    44. Three-step bar E1 = [18.5, 21.5]MPa (15% uncertainty) E2 = [21.875,28.125]MPa (25% uncertainty) E3 = [24, 36]MPa (40% uncertainty) P1 = [? 9, 9]kN P2 = [? 15,15]kN P3 = [2, 18]kN Examples Statically determinate

    45. Examples Statically determinate Statically determinate 3-step bar

    46. Conclusions Formulation of interval finite element methods (IFEM) is introduced EBE approach was used to avoid overestimation Penalty approach for IFEM Enclosure was obtained with few iterations Problem size does not affect results accuracy For small stiffness uncertainty, the accuracy does not deteriorate with the increase of load uncertainty In statically determinate case, exact hull was obtained by non-iterative approach

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