1. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Carlos A. Felippa Recent Advances in Finite Element Templates Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado at Boulder Boulder, CO 80309, USA Presentation to the CST 2000 September 8, 2000, Leuven, Belgium

2. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu High Performance (HP) elements Templates: concept, “genetics” Constraints on template parameters + families Kirchhoff Plate Triangle (KPT) template benchmarks Conclusions Outline

3. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Evolution of FEM

4. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu High Performance Elements - Definition See written paper for discussion of “simple”, “engineering accuracy” “arbitrary” and “coarse” Simple elements that deliver results of engineering accuracy with arbitrary coarse meshes (Felippa & Militello, 1989)

5. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu 1965-date: Old Tricks –Incompatible shape functions, reduced & selective integration 1968-date: More Scientific –Hybrid/mixed elements, enhanced strain, stabilized elements 1975-date: Physically Based –Free Formulation, Assumed Natural Strain Author’s Approach –A mixture of above, ending with templates (next slide) Approaches to the Construction of HP Elements

6. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu The Road to Templates

7. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Templates are: parametrized forms of element level FEM equations that satisfy: (C) Consistency –The Individual Element Test (IET) of Bergan and Hanssen (1975) is identically passed (S) Stability –Element operators (stiffness, mass, etc) have correct rank (I) Observer invariance (P) Contain free parameters Template Definition

8. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Example: Stiffness of BE Plane Beam Element Rank 1 +

9. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Fundamental Decomposition of Template Stiffness Matrix

10. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu 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 Template “Genetics”

11. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Kirchhoff Plate Bending Triangle (KPT) Template

12. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Identifiers ofExisting KPT Elements

13. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Signatures of Existing KPT Elements

14. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Linear Constraints on Template Parameters Observer Invariance –Equations invariant wrt node numbering; symmetries preserved Aspect Ratio Insensitivity –Energy ratio remains bounded as element aspect ratio(s) goes to infinity –Avoids “aspect ratio locking” Energy orthogonality –Always used in older work, nowadays optional

15. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Aspect Ratio Insensitivity for KPTs

16. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Configurations to Apply ARI Constraints

17. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Linear Constraints for KPT Template

18. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Three KPT Families Appear

19. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Quadratic Constraints on Template Parameters Morphing –Next slides Higher order patch tests Mesh distortion insensitivity Others under study –Lack of directionality in wave propagation

20. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Morphing Concept

21. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Morphing DOF Matching

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

23. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu F-16 Aeroelastic Structural Model Present model: 150000 Nodes, 6 DOFs/Node

24. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu F-16 Exterior Surface Zoom 95% of elements are HPSHEL3 18 DOF shells

25. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu F-16 Internal Structure Zoom Some solid elements (bricks & tetrahedra) used for “wing fingers”

26. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Interesting New KPT Element Instances

27. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Signatures of Interesting New KPT Elements

28. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu “Genealogy” of Existing & New Elements

29. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Benchmark: SS Square Plate Under Central Load

30. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Benchmark: Clamped Square Plate Under Central Load

31. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Benchmark: Uniformly Loaded Cantilever

32. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Benchmark: End Shear Loaded Cantilever

33. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Benchmark: Twisted Ribbon (Robinson’s Test)

34. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu The KPT Benchmark Score So Far No “best instance for all seasons” has emerged HCTS (new), MDIT1 (new), AQR1 (old) are best overall performers Bending- and Twist-Exact instances outperform others in cases favoring beam and twib morphing Mesh distortion insensitity associated with  = 1

35. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Conclusions 1: Advantages of Template Approach One routine does all possible elements –Advantageous in benchmarking HP families emerge naturally Can be customized to problem at hand –Static, vibration, buckling, wave propagation... Signatures detect clones

36. University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu Conclusions 2: Difficulties of Template Approach Heavy symbolic manipulations required –On present computers, restricted to 1D and simple 2D configurations Mathematical framework needed –In particular, precise connection between template constraints and global errors