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Carlos A. Felippa

A Study of Optimal Membrane Triangles with Drilling Freedoms. Carlos A. Felippa. Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado at Boulder Boulder, CO 80309-0429, USA. Presentation to FEM Minisymposium USNCCM7, Albuquerque, July 2003.

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Carlos A. Felippa

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  1. A Study of Optimal Membrane Triangles with Drilling Freedoms Carlos A. Felippa Department of Aerospace Engineering Sciences and Center for Aerospace StructuresUniversity of Colorado at Boulder Boulder, CO 80309-0429, USA Presentation to FEM Minisymposium USNCCM7, Albuquerque, July 2003

  2. Here is a Well Known Distortion Benchmark • UBOTP (UniDirectional Bending-Optimal Trapezoidal Panel) • Passes the patch test on any mesh • Retains a bounded condition number, even for a triangle!

  3. Outline • Why drilling freedoms • Element development • Finding the best • Templates • Conclusions

  4. Presentation Source • Technical details in expository paper: • C. A. Felippa, A study of optimal membrane triangles with drilling freedoms, Comp. Meth. Appl. Mech. Engrg., 192, 2125-2168, 2003 • Reprints available here

  5. High Performance Elements - Definition Simple elements that deliver results of engineering accuracy with arbitrary coarse meshes

  6. (Personal) Preferences for HP Elements (1) - only corner nodes - only physical freedoms - passes patch test on any plane mesh - no condensation: EAS, incompatible modes etc, are waste of time

  7. (Personal) Preferences for HP Elements (2) - do not assume you can refine the mesh in real engineering systems - robust when used by inexperienced modelers - can be customized to do multiple duties - can be optimized for specific criteria - develop by symbolic computation

  8. Membrane Elements with Drilling DOFs Interesting as HPFEM because - Performance improved without adding side DOFs - Rotational DOF “free of charge” - Simplifies shell modeling: intersections, stiffeners, ..

  9. 18 DOF Thin Shell Triangle Template

  10. F-16 Aeroelastic Structural Model Finest model: 250000 Nodes, 1.5 M DOF

  11. F-16 Internal Structure Zoom Over 95% ofelements are18-DOF triangleshell elements,including stiffeners

  12. Customization Example: Aerospace Structures

  13. A Brief History of Drilling DOFs Late 1960s: rectangles developed @ Berkeley (by Scordelis’ students) 1979: quadrilateral shell element @ Lockheed for STAGS 1970-1984: many efforts to construct satisfactory triangles fail 1984: Allman triangle, works but rank deficient 1985: Bergan-Felippa FF triangle, rank sufficient but complicated 1988: Allman’s rank-sufficient triangle 1992: Felippa-Militello ANDES triangle, bending optimal Since 1992: many more models 2002: placed in template framework

  14. Template Approach for Element Stiffness

  15. The Basic Stiffness (disc. 1984, Pal Bergan)

  16. HO Stiffness by ANDES: (1) Choosing Natural Strains

  17. HO Stiffness by ANDES: (2) Define Hierarchical Rotations

  18. HO Stiffness by ANDES: (3) Pick Hierarchical Strain Modes Filter torsion mode:

  19. HO Stiffness by ANDES: (4) Parametrize and Finish Up

  20. Mathematica Implementation

  21. Fortran Version Freely available: contact author by e-mail ~350000 elements/sec on a 3GHz P4 (DP float) Eventual goal: 1 million elements/sec Membrane takes about 35% of formation time of 18 DOF linear shell element, 25% of corotational shell

  22. Bending Optimality Criterion

  23. Confirmation: Morphing to Beam

  24. ANDES Template Instances

  25. ANDES Template Signatures

  26. ANDES Bending Energy Test

  27. ANDES Thin Cantilever Example

  28. Berkeley Cantilever

  29. Berkeley Cantilever Results

  30. Cook’s Trapezoidal Wing

  31. Cook’s Trapezoidal Wing Results

  32. Constructing HP Elements - 3 Approaches Fig from CMAME 192, 2125-2168, 2003 Templates

  33. Constructing HP Elements - 3 Approaches

  34. The Template Approach • Element equations (stiffness, mass, etc) are directlyconstructed as parametrized algebraic forms • The construction is done in two stages: basic andhigher order components • Constraints (e.g., orthogonality) are enforced between the components to insure a priori satisfaction of the patch test without need of a posteriori checks

  35. 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

  36. Advantages of Template Approach • One form generates an infinite # of instances • Instances are not necessarily obtainable byconventional (variational based) formulations • Unified implementation (input: signature) weeds out clones, simplifies benchmarking • Can be customized, or optimized, for specific configurations or needs

  37. Technical Difficulties • Limited theory available • Symbolic manipulations beyond human endurance. Only possible via computer algebra systems.Memory and CPU power limitations have so far restricted template development to • 1D: 2-node beams • 2D: 3-node triangles (membrane + plate bendingcombined for shell elements with drilling freedoms)

  38. Web Resources • If interested in FE templates, see material posted at • http://caswww.colorado.edu/Felippa.d/Home.html -> Papers and Reports

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