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Carlos A. Felippa

Recent Advances in Finite Element Templates. Carlos A. Felippa. Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado at Boulder Boulder, CO 80309, USA. Presentation to the CST 2000 September 8, 2000, Leuven, Belgium. Outline.

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Carlos A. Felippa

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  1. Recent Advances in Finite Element Templates Carlos A. Felippa Department of Aerospace Engineering Sciences and Center for Aerospace StructuresUniversity of Colorado at Boulder Boulder, CO 80309, USA Presentation to the CST 2000 September 8, 2000, Leuven, Belgium

  2. Outline • High Performance (HP) elements • Templates: concept, “genetics” • Constraints on template parameters + families • Kirchhoff Plate Triangle (KPT) template benchmarks • Conclusions

  3. Evolution of FEM

  4. High Performance Elements - Definition Simple elements that deliver results of engineering accuracy with arbitrary coarse meshes (Felippa & Militello, 1989) See written paper for discussion of “simple”, “engineering accuracy” “arbitrary” and “coarse”

  5. Approaches to the Construction of HP Elements • 1965-date: Old Tricks • Incompatible shape functions, reduced & selective integration • 1968-date: More Scientific • Hybrid/mixed elements, enhanced strain, stabilized elements • 1975-date: Physically Based • Free Formulation, Assumed Natural Strain • Author’s Approach • A mixture of above, ending with templates (next slide)

  6. The Road to Templates

  7. Template Definition • Templates are: parametrized forms of element level FEM equations that satisfy: • (C) Consistency • The Individual Element Test (IET) of Bergan and Hanssen (1975) is identically passed • (S) Stability • Element operators (stiffness, mass, etc) have correct rank • (I) Observer invariance • (P) Contain free parameters

  8. Example: Stiffness of BE Plane Beam Element Rank 1 + Rank 1

  9. Fundamental Decomposition of Template Stiffness Matrix

  10. Template “Genetics” • The set of free parameters is the template signature • The number of free parameters can be reduced by applying behavioral constraints to produce element families • Specific elements instances are obtained by assigning numerical values to the free parameters of a family • Elements with the same signature, possibly derived through different methods, are called clones

  11. Kirchhoff Plate Bending Triangle (KPT) Template

  12. Identifiers ofExisting KPT Elements

  13. Signatures of Existing KPT Elements

  14. Linear Constraints on Template Parameters • Observer Invariance • Equations invariant wrt node numbering; symmetries preserved • Aspect Ratio Insensitivity • Energy ratio remains bounded as element aspect ratio(s) goes to infinity • Avoids “aspect ratio locking” • Energy orthogonality • Always used in older work, nowadays optional

  15. Aspect Ratio Insensitivity for KPTs

  16. Configurations to Apply ARI Constraints

  17. Linear Constraints for KPT Template

  18. Three KPT Families Appear

  19. Quadratic Constraints on Template Parameters • Morphing • Next slides • Higher order patch tests • Mesh distortion insensitivity • Others under study • Lack of directionality in wave propagation

  20. Morphing Concept

  21. Morphing DOF Matching

  22. Importance of Morphing in Aerospace Structures

  23. F-16 Aeroelastic Structural Model Present model: 150000 Nodes, 6 DOFs/Node

  24. F-16 Exterior Surface Zoom 95% of elements are HPSHEL3 18 DOF shells

  25. F-16 Internal Structure Zoom Some solid elements (bricks & tetrahedra) used for “wing fingers”

  26. Interesting New KPT Element Instances

  27. Signatures of Interesting New KPT Elements

  28. “Genealogy” of Existing & New Elements

  29. Benchmark: SS Square Plate Under Central Load

  30. Benchmark: Clamped Square Plate Under Central Load

  31. Benchmark: Uniformly Loaded Cantilever

  32. Benchmark: End Shear Loaded Cantilever

  33. Benchmark: Twisted Ribbon (Robinson’s Test)

  34. The KPT Benchmark Score So Far • No “best instance for all seasons” has emerged • HCTS (new), MDIT1 (new), AQR1 (old) are best overall performers • Bending- and Twist-Exact instances outperform others in cases favoring beam and twib morphing • Mesh distortion insensitity associated with a = 1

  35. Conclusions 1: Advantages of Template Approach • One routine does all possible elements • Advantageous in benchmarking • HP families emerge naturally • Can be customized to problem at hand • Static, vibration, buckling, wave propagation ... • Signatures detect clones

  36. Conclusions 2: Difficulties of Template Approach • Heavy symbolic manipulations required • On present computers, restricted to 1D and simple 2D configurations • Mathematical framework needed • In particular, precise connection between template constraints and global errors

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