Outline Why drilling freedoms

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Here is a Well Known Distortion Benchmark


  • 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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