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

Carlos A. Felippa

Recent Advances in Finite Recent Advances in Finite Element TemplatesElement Templates

Recent Advances in Finite Recent Advances in Finite Element TemplatesElement Templates

Department of Aerospace Engineering Sciences and Center for Aerospace Structures

University of Colorado at BoulderBoulder, CO 80309, USA

Presentation to the CST 2000September 8, 2000, Leuven, Belgium

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

OutlineOutline

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Evolution of FEMEvolution of FEM

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

High Performance Elements - Definition

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)

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

Approaches to the Construction of HP Elements

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The Road to TemplatesThe Road to Templates

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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 DefinitionTemplate Definition

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Example: Stiffness of BE Plane Beam Element

Example: Stiffness of BE Plane Beam Element

Rank 1 Rank 1+

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Fundamental Decomposition of Template Stiffness Matrix

Fundamental Decomposition of Template Stiffness Matrix

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• 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”Template “Genetics”

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Kirchhoff Plate Bending Triangle (KPT) Template

Kirchhoff Plate Bending Triangle (KPT) Template

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Identifiers ofExisting KPT Elements

Identifiers ofExisting KPT Elements

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Signatures of Existing KPT Elements

Signatures of Existing KPT Elements

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Linear Constraints on Template Parameters

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

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Aspect Ratio Insensitivity for KPTs

Aspect Ratio Insensitivity for KPTs

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Configurations to Apply ARI Constraints

Configurations to Apply ARI Constraints

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Linear Constraints for KPT Template

Linear Constraints for KPT Template

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Three KPT Families AppearThree KPT Families Appear

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Quadratic Constraints on Template Parameters

Quadratic Constraints on Template Parameters

• Morphing– Next slides

• Higher order patch tests

• Mesh distortion insensitivity

• Others under study– Lack of directionality in wave propagation

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Morphing ConceptMorphing Concept

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Morphing DOF MatchingMorphing DOF Matching

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Importance of Morphing in Aerospace Structures

Importance of Morphing in Aerospace Structures

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F-16 Aeroelastic Structural ModelF-16 Aeroelastic Structural Model

Present model:150000 Nodes,6 DOFs/Node

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F-16 Exterior Surface ZoomF-16 Exterior Surface Zoom

95% ofelements areHPSHEL318 DOF shells

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F-16 Internal Structure ZoomF-16 Internal Structure Zoom

Some solid elements (bricks & tetrahedra) used for “wing fingers”

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Interesting New KPT Element Instances

Interesting New KPT Element Instances

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Signatures of Interesting New KPT Elements

Signatures of Interesting New KPT Elements

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“Genealogy” of Existing & New Elements

“Genealogy” of Existing & New Elements

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Benchmark: SS Square Plate Under Central Load

Benchmark: SS Square Plate Under Central Load

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Benchmark: Clamped Square Plate Under Central Load

Benchmark: Clamped Square Plate Under Central Load

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Benchmark: Uniformly Loaded CantileverBenchmark: Uniformly Loaded Cantilever

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Benchmark: End Shear Loaded Cantilever

Benchmark: End Shear Loaded Cantilever

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Benchmark: Twisted Ribbon (Robinson’s Test)

Benchmark: Twisted Ribbon (Robinson’s Test)

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The KPT Benchmark Score So FarThe 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

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Conclusions 1: Advantages of Template Approach

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

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Conclusions 2: Difficulties of Template Approach

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