charilaos mylonas - personal introduction · 2017. 2. 27. · homogenization for composites -...

Charilaos Mylonas - Personal Introduction

PhD Candidate - Charilaos Mylonas

Institute of Structural Engineering Method of Finite Elements I 1

Personal Introduction

Studies

Civil Engineering - Aristotle University of Thessaloniki

Computational Science and Engineering - ETH Zurich

Research interests/research focus

Computational techniques for uncertainty quantification

Numerical solution of PDEs (Including Finite Elements)

Current research focus: uncertainty propagation for Wind Turbinecomposite blade fatigue analysis


Previous FE-related projects - motivation

Homogenization for Composites

Composite elastic response cannot be treated in thehomogeneous elasticity framework (”Lamé” parameters, λ, µ).

Anisotropic elastic properties can be approximated from thesolution of elasticity-like problems in the microscale.

PDE Boundary Value Problem using elastic constitutiverelations and assuming,

ui = u(0)i + �u

(1)i + �

2u(2) + · · ·

where �→ 0 1. Displacement ends up being decomposed in afast spatially varying periodic component u(1) that we cananalyse in the microscale, induced by the periodicity of themicrostructure, and a ”slow” component that we can analyse inthe macroscopic scale.

1This is a technique called perturbation method. Introduced for the analysisof periodic structures (composites) in [BLP78]



Homogenization for Composites

Weak form (aka - Variational form) 2:

∫ΩDijkl

∂χmnk∂yl

∂νi∂yj

dy =

∫Ωνj∂Dijmn∂yi

dy

where Ω our microscopic periodically repeating cell, Dijklconstitutive stiffness tensor3 and χ a function that that satisfies

u(1)i = χ

kli

∂u(0)k

∂xl+ ũ

(1)i (x)

where x = {x1, x2, x3} the coordinates in the macro-scale.2An expression between integrals that when it is satisfied, the PDE boundary

value problem is satisfied. In practice it’s minimized (we search for a set of aithat make this expression stationary) not ”satisfied”.

3For homogeneous materials can be represented by Lamé parametersInstitute of Structural Engineering Method of Finite Elements I 4


Homogenization for Composites - Ritz-Galerkin method

We want to approximate χmni .

We take νi to be a linear combination of functions(Ritz/Galerkin idea ) νi(y) =

∑Nm=1 a

(i)m Ψ

(i)m (y) and the same

for χmni .

By passing these approximations in the variational form, a linearsystem that we can solve with standard linear algebra arises

however, as you will learn, by a special choice of the Ψfunctions the linear system has some nice properties.



Homogenization for Composites - Finite Element solution

This system is sparse (most of the elements are zero) whenusing functions that overlap only at a small region. Ideaattributed to Courant - That’s what we call today FiniteElements.

Solution with FE:

The solution for one component of χ consists of approx. 70000local and piecewise linear functions (aka - hat functions).

Try inv(rand(70000,70000)) on Matlab... Due to Courant andsparse linear algebra nobody ever needs to do that for solvingPDEs.



Homogenization for Composites - Finite Element solution

Why homogenization? Great computational savings, while beingable to retrieve information about the micro-stresses in thecomposite.

Nothing more than the mathematically consistent way tocompute macroscopic properties for periodic structures.

For more details on the computational treatment of thehomogenization problem refer to [CTN01].

Many big names introduced here (Ritz, Galerkin, Courant). Fora historical review, [GW12] is highly recommended.

For students interested in computational methods in general 4

[Str08]

4not strictly oriented to this course but for general education on numericalmethods in engineering



Hyperelastic Materials

Architects consider soft, air-pressure actuated structures forfacades

In order to quickly test new designs, and assess their mechanicalbehavior, FE models were developed:


Current project: Fatigue Assessment of Wind TurbineBlades

Composite materials - thin cross-sections

Special purpose tools for the estimation of warping and shearflows are used. 5

Time-dependent FE analysis for aerodynamic loads on beams

Cross-section analysis on beam resultants:

5Recall theory of thin-walled beam analysis. Here slightly different due tocomposite anisotropy. You may refer to [GBM+83] and [YHH12] for moreinformation on the treatment.


Current project: Fatigue Assessment of Wind TurbineBlades

Issues with simulation:

A problem where ”less is more”

a complete 3D FE treatment practically unfeasible (highcomputational times - time dependent simulations do notscale6)

So-called mixed7 FE-formulations have found application forincreased accuracy in beam-type modelling.

6Solving with 10 CPUs does not make (in general) the solution 10 timesfaster. Cannot straightforwardly benefit from advances in hardware.

7variables of different ”nature” enter the weak formulation - such as velocitiesand rotations, instead of displacements


A step back

For analysis of wind turbine blades back to Ritz/Galerkin:

Displacements on beams, are generally smooth

a piecewise linear approximation may not be the best approachfor a basis function set...

In the context of the ”Geometrically Exact Beam Theory” otherchoices for the basis functions may be investigated.

For the intrigued students, please do refer to [PA11].

In Wind-turbine simulation codes Legendre polynomials alreadyconsidered.


Other interests

Uncertainty Quantification:

Probabilistic treatment8 of uncertainty with non-intrusive 9

methods

Optimization:

Partial Differential Equation constrained shape optimization

Genetic algorithms and Structural optimization

8Meaning doing something more efficient than running many times withdifferent parameters (Monte-Carlo)

9meaning not changing existing FE codeInstitute of Structural Engineering Method of Finite Elements I 12

Thank you for your attention!

Thank you for your attention! - we wish you a productive andcreative semester!


References I

Alain Bensoussan, Jacques-Louis Lions, and GeorgePapanicolaou.Asymptotic analysis for periodic structures, volume 5.North-Holland Publishing Company Amsterdam, 1978.

Peter W Chung, Kumar K Tamma, and Raju R Namburu.Asymptotic expansion homogenization for heterogeneous media:computational issues and applications.Composites Part A: Applied Science and Manufacturing,32(9):1291–1301, 2001.

Vittorio Giavotto, Marco Borri, Paolo Mantegazza,G Ghiringhelli, V Carmaschi, GC Maffioli, and F Mussi.Anisotropic beam theory and applications.Computers & Structures, 16(1-4):403–413, 1983.


References II

Martin J Gander and Gerhard Wanner.From euler, ritz, and galerkin to modern computing.Siam Review, 54(4):627–666, 2012.

Mayuresh J Patil and Matthias Althoff.Energy-consistent, galerkin approach for the nonlinear dynamicsof beams using intrinsic equations.Journal of Vibration and Control, 17(11):1748–1758, 2011.

Gilbert Strang.Mit ocw 18.085- computational science and engineering i.MIT OpenCourseWare: Massachusetts Institute of Technology,2008.


References III

Wenbin Yu, Dewey H Hodges, and Jimmy C Ho.Variational asymptotic beam sectional analysis–an updatedversion.International Journal of Engineering Science, 59:40–64, 2012.


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