1. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Carlos A. Felippa A Template Tutorial Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado at Boulder Boulder, CO 80309-0429, USA Dedicated to Pål Bergan on his 60th birthday

2. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Bit of history How I learned the Free Formulation How templates came about Conclusions Outline

3. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu A Brief History of “Finitelementology ”: 52-75

4. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Historically, Stress Elements were the First From Clough (1960) explaining the TCMT rectangular panel (JAS, 1956)

5. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu A Brief History of Elementology: 75- date

6. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu We did not overlap at Berkeley. After Pål’s thesis (1972), Ray Clough gave up finite elements. So did I after 1968. My work at Lockheed Palo Alto (1970-85) was heavily software oriented, with some research breaks. One came in 1984 Paths

7. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu The MAFELAP II paper (1975) by Bergan & Hanssen: A New Approach for Deriving ‘Good’ Finite Elements contains two key contributions: A direct algebraic approach to the element stiffness equations The Individual Element Test (IET) - a strong, a priori form of Irons’ patch test applicable to a single element - and hence ideal for design Detour: Seeds of the Free Formulation

8. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu “An important observation is that each element is, in fact, only represented by the numbers in its stiffness matrix during the analysis of the assembled system. The origin of these stiffness coefficients is unimportant to this part of the solution process... The present approach is in a sense the opposite of that normally used in that the starting point is a generally formulated convergence condition and from there the stiffness matrix is derived... The patch test is particularly attractive [as such a condition] for the present investigation in that it is a direct test on the element stiffness matrix and requires no prior knowledge of interpolation functions, variational principles, etc.” Quote from B&H ‘75

9. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu MAFELAP 2 : innovative approach, but difficult to realize. The direct computation of stiffness entries becomes a problem in nonlinear constrained optimization A first realization came in the early 1980: the Free Formulation (FF) published by Bergan & Nygård in IJNME (1984) A Realization: the Free Formulation

10. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu FF Idea: Divide & Conquer K = K + K hb Basic stiffness Higher order (HO) stiffness Basic stiffness: stress based (a constant-stress hybrid) Consistency constraint: must pass Bergan-Hanssen IET HO stiffness: based on nonconforming displacement modes Stability constraint: positivity and correct rank Energy orthogonality constraints: to avoid HO-to-basic pollution

11. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu In 1983-84 Pål Bergan, then a faculty at NIT- Trondheim, spent a sabbatical at Stanford. He was then involved with the FENRIS project: the implementation of the Free Formulation, with Magne Nygård and others, for nonlinear analysis We worked together during Jan-April 1984 in a project supported by the Lockheed IR Program. Our FF Collaboration

12. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu A Short Bike Ride...

13. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Using the FF we developed a 9-DOF membrane triangle with drilling freedoms. It was the first one to pass the patch test (actually the IET) while being rank sufficient. Tom Hughes kindly expedited publication in CMAME, where it came out in 1985. We also developed a 9-DOF plate bending triangle, which appeared in 1987 Our FF Collaboration

14. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Conventional formulations lead to a trial and error process: implement, test and fix - or discard FF focused on a priori element design, a quality retained by templates Technical problem: picking the HO displacement modes requires experience and judgement as the element gets complicated FF vs. Conventional Element Formulations

15. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu The Road to Templates - Upon Moving to CU Boulder

16. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Template Stiffness Decomposition K = K (  ) + K (  ) h jb i Basic stiffness: takes care of consistency and mixability HO stiffness: takes care of stability (rank sufficiency) and accuracy Free parameters:  take care of optimality and customability ij

17. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu The set of free parameters is the signature The # of free parameters can be reduced by applying behavioral constraints to produce element families 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 Template “Genetics”

18. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Ex 1: 3-Node Trigs with Drilling DOFs

19. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Signatures for Ex 1

20. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Ex 2: Some 3-Node C1 Plate Bending Triangles

21. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Signatures for Ex 2

22. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Customizing

23. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Macroelement Morphing as Customizing Tool

24. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Customizing

25. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Customizing

26. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Templates: Advantages One parametrized form generates an infinite # of instances Instances are not necessarily obtainable by conventional (variational-based) formulations Unified computer implementation (input: signature) weeds out clones, simplifies benchmarking Can be customized, or optimized, for specific configurations or needs

27. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Technical Obstacles - 1 Limited theory available. Some missing knowledge: - global optimization procedures - can all template instances be constructed variationally? - if not, will variational methods go the way of Fortran?

28. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Technical Obstacles - 2 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, shells) 4-node quads (in progress)

29. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Recent Result: A Bending-Optimal 4-Node Quad

30. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Web Resources If interested in FE templates, see reports posted at http://caswww.colorado.edu/Felippa.d/Home.html -> Papers and Reports Recent expository paper: C. A. Felippa, A study of optimal membrane triangles with drilling freedoms, Comp. Meth. Appl. Mech. Engrg., 192, 2125-2168, 2003

31. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu In Conclusion My warm thanks and best wishes to Pål for an enjoyable and productive collaboration that eventually inspired the research in templates, and for hosting my 1993 sabbatical at DNV