Outline Why drilling freedoms


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Here is a Well Known Distortion Benchmark



Outline

  • Why drilling freedoms

  • Element development

  • Finding the best

  • Templates

  • Conclusions



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



High Performance Elements - Definition



(Personal) Preferences for HP Elements (1)



(Personal) Preferences for HP Elements (2)



Membrane Elements with Drilling DOFs



18 DOF Thin Shell Triangle Template



F-16 Aeroelastic Structural Model



F-16 Internal Structure Zoom



Customization Example: Aerospace Structures



A Brief History of Drilling DOFs



Template Approach for Element Stiffness



The Basic Stiffness (disc. 1984, Pal Bergan)



HO Stiffness by ANDES: (1) Choosing Natural Strains



HO Stiffness by ANDES: (2) Define Hierarchical Rotations



HO Stiffness by ANDES: (3) Pick Hierarchical Strain Modes



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



Mathematica Implementation



Fortran Version



Bending Optimality Criterion



Confirmation: Morphing to Beam



ANDES Template Instances



ANDES Template Signatures



ANDES Bending Energy Test



ANDES Thin Cantilever Example



Berkeley Cantilever



Berkeley Cantilever Results



Cook’s Trapezoidal Wing



Cook’s Trapezoidal Wing Results



Constructing HP Elements - 3 Approaches



Constructing HP Elements - 3 Approaches



The Template Approach

  • Element equations (stiffness, mass, etc) are directly constructed as parametrized algebraic forms

  • The construction is done in two stages: basic and higher 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



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



Advantages of Template Approach

  • One form generates an infinite # of instances

  • Instances are not necessarily obtainable by conventional (variational based) formulations

  • Unified implementation (input: signature) weeds out clones, simplifies benchmarking

  • Can be customized, or optimized, for specific configurations or needs



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 bending combined for shell elements with drilling freedoms)


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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