**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)**
**18 DOF Thin Shell Triangle Template**
**F-16 Aeroelastic Structural Model**
**Customization Example: Aerospace Structures**
**A Brief History of Drilling DOFs**
**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: (4) Parametrize and Finish Up**
**Mathematica Implementation**
**Fortran Version**
**Bending Optimality Criterion**
**ANDES Template Instances**
**ANDES Template Signatures**
**ANDES Thin Cantilever Example**
**Berkeley Cantilever**
**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**
**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** ## http://caswww.colorado.edu/Felippa.d/Home.html -> Papers and Reports
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