Advanced Computational Tools for Structural Mechanics and ... · Advanced Computational Tools for Structural Mechanics and ... transformation is obtained from control points affine

Advanced Computational Tools for Structural Mechanics and Earthquake Engineering

Alessandro Reali Dipartimento di Meccanica Strutturale, Università degli Studi di Pavia, Italy

European Centre for Training and Research in Earthquake Engineering (EUCENTRE), Pavia, Italy Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), CNR, Pavia, Italy

Centro di Simulazione Numerica Avanzata (CeSNA), IUSS, Pavia, Italy

[email protected] http://www-1.unipv.it/alereali

Reasearch Group Webpage: http://www-1.unipv.it/dms/compmech

Earthquake Engineering by the beach - Capri, 2-4 July 2009

Earthquake Engineering by the beach - Capri, 2-4 July 2009

Acknowledgements

in alphabetical order:

Jamal Arghavani (Sharif University of Technology); Domenico Asprone (University of Naples “Federico II”); Gabriele Attanasi (ROSE School, Pavia); Ferdinando Auricchio (University of Pavia); Giuseppe Balduzzi (University of Pavia); Lourenco Beirao da Veiga (University of Milan); Annalisa Buffa (IMATI/CNR, Pavia); Thomas JR Hughes (University of Texas at Austin); Carlo Lovadina (University of Pavia); Gaetano Manfredi (University of Naples “Federico II”); Andrea Prota (University of Naples “Federico II”); Giancarlo Sangalli (University of Pavia); Ulisse Stefanelli (IMATI/CNR, Pavia);

as well as the whole research groups led by Prof. Auricchio and Prof. Hughes.

Presentation outline

Isogeometric Analysis Shape Memory Alloy Modeling and Application Other Interesting Research Works in Progress

• Meshless Methods • Beam Finite Elements

Conclusions




Conclusions

CAD (1970’s - 1980’s) – Engineering Design Process:

FEM analysis and CAD geometry

analysis framework based on functions – e.g., NURBS – capable of representing exact geometry; one, and only one, geometrical description; mesh refinement vastly simplified

engineering designs are encapsulated in CAD systems; CAD geometry is exact; more than 100,000 analyses of CAD designs are performed

in engineering offices throughout the world every day

CAD geometry is replaced by FEM geometry (“mesh”); mesh generation accounts for more than 80% of overall

analysis time and is the major bottleneck; mesh refinement requires interaction with CAD geometry; the mesh is an approximate geometry

FEM (1950’s - 1960’s) – Engineering Analysis Process:

IDEA: Isogeometric Analysis

NURBS

Main properties of Non-Uniform Rational B-Splines:

basis functions from an open knot vector constitute a partition of unity;

the support of each basis function is compact;

basis functions are point-wise non-negative;

basis functions possess high continuity;

NURBS enjoy the affine covariance property (i.e., NURBS affine transformation is obtained from control points affine transformation);

knot insertion and degree elevation are simple operations

References:

Rogers. An Introduction to NURBS with Historical Perspective. Academic Press, San Diego, CA, 2001

Piegl and Tiller. The NURBS Book (Monographs in Visual Communication), 2nd ed. Springer-Verlag, New York, 1997.

Isogeometric Analysis

Isogeometric Analysis:

exact geometry alternative to standard FE analysis (based, e.g., on NURBS), including FEA as a special case, but offering other possibilities:

precise and efficient geometric modeling simplified mesh refinement superior approximation properties integration of design and analysis

Isogeometric analysis

Main features:

geometry defined by control points (associated with basis functions)

isoparametric concept invoked: unknown variables (dof’s or control variables) represented in terms of the basis functions defining the geometry

three refinement strategies available: -“h-refinement” (by knot insertions) -“p-refinement” (by degree elevation) -“k-refinement” (sort of high-order/high-continuity h-refinement)

array assembly strategy same as in FEM; Dirichlet b.c. applied to control variables, Neumann b.c. satisfied naturally, as in FEM

structural analysis: all rigid body motions and constant strain states represented exactly (i.e., standard “patch tests” passed)

Hughes, Cottrell, and Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005.

Reference:

Structural vibrations

Isogeometric analysis for structural vibrations: main advantages with respect to standard FEM

basis function high continuity;

basis function point-wise non-negativity, which implies element-wise non-negativity of the consistent mass matrix;

exact geometry

Focus on spectrum analysis, i.e. analysis of the error for the natural frequencies in structural vibration problems


Longitudinal vibrations of a fixed-fixed rod

Numerical results:

discrete spectra normalized to the exact solution

(quadratic NURBS versus quadratic FE)

6th eigenmode 9th eigenmode

NURBS (top) and FEM (bottom) [21 c.p.] vs. exact (red-dashed)

12th eigenmode 15th eigenmode 18th eigenmode


high order NURBS versus high order FE (i.e.: k-method versus p-method)

Longitudinal vibrations of a fixed-fixed rod

NURBS spectra show a nice convergence to the exact solution increasing the order p, while higher-order FEM have no approximability for higher modes


2D problem (membrane)

Transversal vibrations of a clamped elastic membrane: NURBS normalized discrete spectrum

high order NURBS vs high order FE (i.e.: k-method vs p-method)


Fourth order problems (i.e., beams and plates)

2D: Transversal vibrations of a simply-supported Kirchhoff plate: NURBS normalized discrete spectrum

1D: Transversal vibrations of a simply-supported Euler-Bernoulli beam: NURBS normalized discrete spectrum


Vibrations of a clamped thin circular plate using 3D solid elements

exact solution:

where:

and: 8-element mesh

first three frequencies obtained using different order NURBS basis functions

first three eigenmodes (p = 4, q = 5, r = 2)


Study of the NASA Aluminum Testbed Cylinder (ATC) structure

Experimental modal results available for the whole structure as well as for single components

NURBS (as-drawn) model of the whole structure (frame + skin and frame only):

Model details:


Study of the NASA Aluminum Testbed Cylinder (ATC) structure

Whole structure frequencies and first Rayleigh and Love modes

Stringer frequencies and first three x-z modes

Main rib frequencies and first three out-of-plane modes


References:

AR. An isogeometric analysis approach for the study of structural vibrations. Journal of Earthquake Engineering, 10(s.i.1):1–30, 2006.

Cottrell, AR, Bazilevs, and Hughes. Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering, 195:5257–5296, 2006.

Hughes, AR, and Sangalli. Duality and Unified Analysis of Discrete Approximations in Structural Dynamics and Wave Propagation: Comparison of p-method Finite Elements with k-method NURBS. Computer Methods in Applied Mechanics and Engineering, 197:4104–4124, 2008.

Wave propagation

Dispersion analysis

Focus on dispersion analysis, i.e. analysis of the error for the wave-number in (time-harmonic) wave propagation problems described by the Helmholtz equation

Duality principle

Spectrum analysis is equivalent to dispersion analysis in the regime where kh is real

Linear case:

Resolution limit:

Wave propagation

p-method (FEM, p=2): acoustical and optical branches separated by a stopping band

In general (p > 2): p branches separated by p-1 stopping bands

k-method (NURBS, p=2): one single branch with no stopping bands

In general (p > 2): again, one single branch with no stopping bands

Stopping bands (bands with complex discrete wave-number,

present before the resolution limit)

Wave propagation

Stopping bands

No stopping bands

Stopping bands (bands with complex discrete wave-number,

present before the resolution limit)

1D (steady-state) problem: frequency response spectra for p = 2

Wave propagation

computed at x = L/10, L/2, and 9L/10

1D (steady-state) problem: frequency response spectra for p = 3

Wave propagation

computed at x = L/10, L/2, and 9L/10

1D (steady-state) problem: b.v.p. solution for p = 2

Wave propagation

NOTE: k = 33 is within the p-method stopping band → unphysical attenuation

Wave propagation

NOTE: k = 31.5 is within the 1st p-method stopping band but no evident attenuation is observed since the imaginary part is very small; k = 31.5 is however close to a resonance peak not well reproduced by p-method k = 71 is within the 2nd p-method stopping band → unphysical attenuation

1D (steady-state) problem: b.v.p. solution for p = 3

Wave propagation

Reference:

Hughes, AR, and Sangalli. Duality and Unified Analysis of Discrete Approximations in Structural Dynamics and Wave Propagation: Comparison of p-method Finite Elements with k-method NURBS. Computer Methods in Applied Mechanics and Engineering, 197:4104–4124 2008.

Summary and future developments

Isogeometric analysis for structural vibrations and wave propagation: main advantages with respect to p-FEM

no optical branches

better approximation properties on a per-d.o.f. basis

exact geometry

main drawback with respect to p-FEM high number of (gaussian) quadrature points

solution: new quadrature rules taking into account inter-element regularity

main future development isogeometric collocation techniques for structural dynamics

(work in progress…)

[Preliminary results presented by Tom Hughes during the opening plenary lecture of the COMPDYN 2009 Conference (Rodhes, June 2009)]

Hughes, AR, and Sangalli. Efficient Quadrature for NURBS-based Isogeometric Analysis. Computer Methods in Applied Mechanics and Engineering, doi:10.1016/j.cma.2008.12.004

Current isogeometric projects

Isogeometric Analysis for Naval Ship Structures Coordinator: Tom Hughes Funded by: Office of Naval Research Duration: January 2006 - December 2009

GeoPDEs - Innovative compatible discretization techniques for Partial Differential Equations

Coordinator: Annalisa Buffa Funded by: FP7 Ideas ERC Starting Grant Duration: July 2008 - June 2012




Conclusions

Shape memory alloy modeling

Superelastic effect

(mechanical recovery)

Shape-memory effect

(thermal recovery)

SMAs: materials with an intrinsic capability of recovering their initial “shape” even after undergoing severe deformations

From a macroscopic point of view they show the following two effects not available, in general, in traditional materials:

“Background” SMA models:

Souza et al. (1998);

Auricchio&Petrini (2004)


include and control cyclic permanent inelasticity and degradation

include and control tension/compression asymmetries

in the yield stress level in loop length in loop width

include a dependence of the elastic properties on the phase transformation level

Experimental tests show that, besides modeling superelastic and shape-memory effects, other secondary effects should be taken into account. In particular, in some cases it is important to:


Therefore, enhanced SMA models able to include such secondary effects have been designed, mathematically analyzed and numerically tested:

a) Model including permanent inelasticity and degradation effects: [based on a new (extra) tensorial variable describing permanent inelasticity]

Auricchio and AR. A Phenomenological One-dimensional Model Describing Stress-induced Solid Phase Transformation with Permanent Inelasticity, Mechanics of Advanced Materials and Structures, 14:43-55, 2007.

Auricchio, AR, Stefanelli. A Three-dimensional Model Describing Stress-induced Solid Phase Transformation with Permanent Inelasticity, International Journal of Plasticity, 23:207-226, 2007.

References:


Auricchio, AR, Stefanelli. A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Computer Methods in Applied Mechanics and Engineering, 23:207-226, 2009.

Reference:

Therefore, enhanced SMA models able to include such secondary effects have been designed, mathematically analyzed and numerically tested:

b) Model including tension/compression asymmetry and elastic properties depending on the phase transformation level:

SMA modeling validation

Collaboration on SMA modeling and validation with SAES Getters, an Italian company recently become the world-leading SMA producer:


Collaboration on SMA modeling and validation with SAES Getters, an Italian company recently become the world-leading SMA producer:


Within the collaboration with SAES Getters: development of a reliable parameter identification technique given two strain-temperature curves at two different constant stresses (typical curves characterizing SMAs for actuators)

Auricchio, Coda, AR, Urbano. SMA numerical modeling versus experimental results: parameter identification and model prediction capabilities, Journal of Materials Engineering and Performance, doi:10.1007s11665-009-9409-7, 2009.

Reference:

Design and FE simulation of biomedical devices

Many devices, in particular for biomedical applications (e.g., stents, spinal spacers, micro-actuators, etc.), have been simulated using SMA constituive models implemented within commercial FE codes (e.g., Abaqus). Such a tool allows also the development of new design strategies.

Auricchio and AR, Shape Memory Alloys: material modeling and device finite element simulations, Materials Science Forum, 583:257-275, 2009

+ references therein reported

References:

SMA applications in Earthquake Engineering

Our research group is active in SMA applications for Earthquake Engineering [cf. also recent collaborations with the group of Prof. R. DesRoches]

Recently the focus is on SMA Isolation Devices [work developed within G. Attanasi’s PhD thesis at ROSE School] Comparison of a SMA isolation system with a Lead Rubber Bearing (LRB):

LRB – SMA system comparison:

same secant stiffness to the design displacement (equivalence)

different theoretical hysteretic dissipation


Summary of time-history analysis results:

Superelastic hysteresis eliminates the residual displacements

Overall flag-shape isolation system behavior close to elastoplastic response in: controlling maximum resisting force; limiting maximum displacement demand; dissipating the absorbed energy

Results valid for many values of dissipation capability and rigid&flexible superstructure condition


Work in progress:

Main issues are related to: provide enough displacement capacity

validate the dynamic behavior of the material investigate on the multidirectional response of the device

Design and development of a real device able to provide the horizontal displacement – base shear relation like the one assumed:

Attanasi, Auricchio, Fenves. Feasibility Assessment of an Innovative Isolation Bearing System with Shape Memory Alloys, Journal of Earthquake Engineering, 13(s.i.1):18-39, 2009.

Attanasi, Auricchio, Fenves. “Feasibility Investigation of Superelastic Effect Devices for Seismic Isolation Applications Journal of Materials Engineering and Performance, doi:10.1007/s11665-009-9372-3, 2009.

References:

Current SMA projects

SMARTeR: Shape Memory Alloys to Regulate Transient Responses in civil engineering

Coordinator: Michel Fremond Funded by: ESF Eurocores Programme Duration: December 2006 - December 2009

BioSMA - Mathematics for Shape Memory Technologies in Biomechanics

Coordinator: Ulisse Stefanelli Funded by: FP7 Ideas ERC Starting Grant Duration: September 2008 - October 2013




Conclusions

Work in progress: meshless methods

Besides the aforementioned isogeometric collocation techniques, we are currently focusing on evolutions of Smoothed Particle Hydrodynamics (SPH).

Aim: reliable simulation of fast dynamics, impact problems and blasts. Tool: particle methods (particularly suitable for explicit solvers)

But: classical SPH-based methods show many problems in approximating Mechanics of Solids equations [lack of or low order h-convergence; bad or tricky imposition of boundary conditions; etc.]

Idea: going back to the basis of SPH [on simple examples!] removing unnecessary hypotheses and approximations [many!], focusing on obtaining higher orders of convergence.

Asprone, Auricchio, Manfredi, Prota, AR, Sangalli. SPH methods for a 1D elastic model problem: error analysis and development of a second-order accurate formulation, submitted, 2009. [also higher dimension static results are now available; next step: dynamics]

Reference:

Work in progress: beam finite elements

We also deal with the study and development of advanced finite element beam formulations.

In this context, we recently devoted a good deal of attention to the study of geometrically nonlinear beam elements.

Currently the focus is on the development of advanced theories for homogeneous and multi-layered beams, for which some good preliminary results have been obtained [work developed within G. Balduzzi’s research period to be completed within his (future) PhD thesis].

Auricchio, Carotenuto, AR. On the geometrically exact beam model: a consistent, effective and simple derivation from three-dimensional finite elasticity. International Journal of Solids and Structures, 45:4766-4781, 2008.

Reference:




Conclusions

Conclusions

Final message: before running an analysis, always keep in mind the two Basic Principles of (good) Engineering [Prof. MP Collins]:

You can’t push on a rope!

[never ask a model to do something that it simply can’t do]

To find an answer, you must know the answer!

[never use numerical analysis as a “black-box” but, if you have to, use your engineering judgment before relying on the results]

Idea: introducing some Computational Mechanics topics we are currently working on, i.e.

Isogeometric analysis

Shape memory alloys

Meshless methods

Advanced beam models

Aim: collaborating with interested structural engineers!

Conclusions

Thank you all for your attention!

Thanks to Iunio and Paolo for their kind invitation (despite I am a Computational Mechanics guy…)

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Advanced Computational Tools for Structural Mechanics and ... · Advanced Computational Tools for Structural Mechanics and ... transformation is obtained from control points affine