multiscalesolution of deformable solids in linear fe time ... · abengoa(mce: abg) is an...

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Innovative Solutions for Sustainability

3 junio 2012 1Industrial and Environmental Mathematics Day

Multiscale Solution of Deformable Solids in Linear FE Time Order by Coupling Proper Generalized Decomposition and VariationalMultiscale Methods

M. Doblaré

Abengoa Research and

Group of Structural Mechanics and Materials Modelling

Aragón Institute of Engineering Research (I3A). University of Zaragoza,

Motivation2

3 Brief Review of Multiscale Techniques

4

Examples and Discussion5

Concluding Remarks and Future Challenges6

Overview

1 Abengoa and Abengoa Research


Application of PGD to Multiscale Problems

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3 33 junio 2012 Industrial and Environmental Mathematics Day

Abengoa

Revenues FY 2011

7,089 M€

Abengoa (MCE: ABG) is an international company that applies innovative technology solutions for sustainability in the energy and environment sectors, generating energy from

the sun, producing biofuels, desalinating sea water and recycling industrial waste

Brazil

Spain

Rest of Europe

US

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Africa

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Ebitda FY 2011

1,103 M€

Concession-typeinfrastructures

E&C

Industrial production

7,089 M€

Revenues

↑↑↑↑ 46 % (4,860 M€ FY 2010)

1,103 M€

Ebitda

↑↑↑↑ 36 % (812 M€ FY 2010)

257 M€

Net income

↑↑↑↑ 24 % (207 M€ FY 2010 figure)

2,1x

Corporate net debt to corporate ebitda

↓↓↓↓ From 3.8x at FY 2010

Traditional business

Growth

Future options

R&D ProgramsAbengoa’s strategy R&D Projects

R&D in Abengoa

R&D investment in 2011 amounted to 90.6 M€

Total of 190 patents, 43 granted and the rest pending

Team of 682 people devoted to R&D under the direction of Abengoa Research


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• Radical innovation

• Options for the future

• New processes and products

• Researchers training

• Differential innovation

• Technology development

• Cost reduction, efficiency and reliability increase

• Need for innovation

• Strategy

Universities and Research Centers

ABENGOA RESEARCH

New Technologies Companies

Business Units

AR in Abengoa


Mission: To become an international reference on R&D for knowledge generation applied to energy and sustainability

Mission and objectives

Specific Objectives

� To create knowledge and to apply it to energy and sustainability

� To help a scientific and technological basis to all of Abengoa’s current and future businesses in those fields

� To generate competitive advantages for Abengoa through R&D

� To support Abengoa’s business units from a high technical perspective.

� To contribute to the development of new technologies, and calculation, design and verification tools in Abengoa

� To increase the scientific and technical level of Abengoa’s R&D projects and companies (units)

� To educate highly qualified people within the scientific and technical areas related to Abengoa’s business units

� To disseminate scientific and technical progress that might be relevant to Abengoa’sactivity


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1 group

7 areas of knowledge.

Between 50 and 60 researchers

Between 10 and 12 technicians

Between 8 and 10 managers andstaff

40 Internal PhDs

25 associated groups with around50 PhD co-supervised

ABENGOA RESEARCH

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Figures

Fluid Mechanics

Structural and Functional Materials

Biotechnology and Biomass

Solid Mechanics and

Structures

Thermal Engineering

Electrical Engineering

Process Engineering

Motivation2


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Overview




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� Complexity in simulating industrial problems may be due to themultiphysical fields, material microstructure, non-linear response,number of independent variables (dimensions), etc.

� For models with complex microstructural geometry and/or highlylocalized heterogeneous behavior, calculations must be performed on afinely discretized model of the structure associated to the “micro”characteristic length, leading to problems with very high number ofdegrees of freedom and computational cost O(Nd), prohibitive even withthe current (and sometimes foreseen) hardware and softwarecapabilities ("curse of dimensionality”).

� These kind of problems are often described by partial differentialequations with highly oscillatory coefficients.

Motivation

Simulation of Complex Multidimensional Problems


Motivation

Simulation of Complex Multidimensional Problems. Heart Electrophysiology


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intermediate-scale physico-chemical theories depending on thespecific field (molecular dynamics, statistical mechanics, coarse grainapproaches, continuum thermodynamics).

� Those simplifications are only valid for a certain range of length andtime scales, while, in many cases, we are interested in modeling thebehavior in several of these ranges simultaneously, or getting thesolution of problems close to the scale limits of one theory.

� When using several scales simultaneously, a problem arises inidentifying the interacting variables. A large variety of procedureshave been proposed to address this problem.

Motivation


Multiscale Problems

� Using only first principles of quantum mechanics andthermodynamics at the human scale is not possibledue to available computer power. Therefore,simplifying assumptions are used giving rise to

3 junio 2012 Industrial and Environmental Mathematics Day

( a )

( e )( d )

( c )

( b )Macro Nano10-3 m 10-6 m 10-9 mMeso Micro10-1 m

Motivation

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� The microstructure of a material defines its macroscopic mechanicalproperties. Many times, the microstructure also evolves upon the valuesof macroscopic variables (constitutive non-linear problems). This isespecially important in cases such as phase change materials, growingand adaptive living tissues, local microdamage and self healingmaterials.

Multiscale Problems

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Motivation2


4



Overview




Some techniques for solving multiscale problems

Multiscale Techniques


� Submodelling: The displacement distribution on the boundaryof submodel, computed in the global mode) is used as inputfor the lower scale to compute the rest of variables.Macroscopic properties are phenomenologically set in theglobal model. Convenient in non-evolving microstructures.

� Substructuring: The macroscopic stiffness of eachsubstructure is "condensed" into its boundary and assembledwith the rest of substructures. This process can not beconsidered as a real multiscale approach, but the actual lowerscale is solved in a two-step solving process.

� Multimeshing: Damping of the high-frequency errors in thefiner grid. It may be understood as a numerical multiscalemodeling approach. A clear definition of the transferringoperators between scales (meshes) is made (“restriction” and“prolongation” operators).

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enabling obtaining both the local state and the global solution bycomputing effective macroscopic properties from the microstructure.

� Hypotheses:

• Length scale separation, i.e., microscale << macroscale.

• Each macro point is statistically represented by an RVE at the microstructrelevel and its properties obtained by an averaging procedure

• BCs in the RVE have to be assumed (usually periodic BCs are used)

� Main drawbacks:• Accuracy reduction when actual periodicity of the microstructure is lost.• Boundary regions require special attention.• Microstructure cannot be easily updated.



Homogenization

� Asymptotic homogenization with periodicboundary conditions has become into themost used and successful approach

Sponceram® carriers

Sponceram scaffold (Kasper et al., 2008)


An example of asymptotic homogenization. Effective properties of a scaffold


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Sanz-Herrera et al., (2008a)



An example of asymptotic homogenization. Effective properties of a scaffold

� Extending the last theory to microstructure evolving,nonlinear and non-periodic heterogeneity problems,implies modeling the whole domain and each RVE foreach load (or time step, “FE2”).

Multilevel FE (FEn)



� The macromechanical behavior arises directly from what happens at themicroscopic scale. “Localization” and “Homogenization” operators areused to compute the interfacing variables between scales.

� Linearization of the homogenization operator is needed when using NRapproaches in the macro scale.

� Cost is strongly reduced when using parallel computation for whichthis approach is especially well suited.

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MACRO

Bone organ: • Mechanics (solid elastic material)• Bone remodelling

Scaffold: • Mechanics (solid elastic material) • Cell migration (permeability-dependent)

MICRO

Scaffold microstructure:• Mechanics (multiphasic solid material and fluid flow)• Microscopic model of bone growth

• Scaffold resorption

Babuska (1978), Sánchez-Palencia (1980), Suquet (1983), Bakhvalov

(1984), Nemat-Nasser (1993), Terada and Kikuchi (1998)...

Homogenization (fluid and solid phases)

(stiffneses)(permeability)

Localization problem (solid and fluid phases)

(deformation, cell density)


An example of multilevel FE. Bone growth in a scaffold

Non-mature bone

Mature bone

Scaffold

Sanz-Herrera et al., (2008c)




� FE2 global approach:

� Standard FEs at themacroscale (bone and scaffold domains)

� Voxel-FEM approach at the microscale (scaffolddomain)

Scaffolding (Pothuaud et al., 2005)

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� Multiscale results,




(Sanz-Herrera et al., 2008d)




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Motivation2


4



Overview





PGD

� PGD is an «a priori» reduction order approach useful inmultiparametric problems (high dimensional spaces).

� It has been used in many problems including parametric ODEs, randomproblems, optimization, etc.

� K. Garikipati, T. J. R. Hughes, A variational multiscale approach to strain localization -formulation for multidimensional problems, Computer Methods In Applied Mechanics andEngineering 188 (1-3) (2000).

� A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some, classes ofmultidimensional partial differential equations encountered in kinetic theory modeling ofcomplex fluids, Journal of Non-newtonian Fluid Mechanics 139 (3) (2006).

� F. Chinesta, A. Ammar, E. Cueto, Recent advances and new challenges in the use of theproper generalized decomposition for solving multidimensional models, Archives ofComputational Methods In Engineering 17 (4) (2010).

PGD Fundamentals

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Application of PGD to Multiscale problems

� In the variational multiscale approach, the macroscopic solution(displacements) is enriched (enhanced) in the regions of interest by afunction defined at the microscopic scale (Hughes,1998).

� � �� This enriching function fulfills homogeneous boundary conditions in the

regions of interest, respecting internal compatibility.

� In our case, we identify each of these regions with the macroscopicindividual finite elements (RVEs). Therefore, �� may be interpreted asgeneralized bubble functions, different in each macroelement.

Variational Multiscale Method and PGD


PGD Approximation

� �� is then represented as a function of a set of parameters thatinclude all possible variables defining geometry, properties and statein the RVE (micro and macro spatial coordinates, BCs, material

properties, loads, time, etc…) �� , ��, ��, �, �, �, � .� The multidimensional dependent variable is expressed in terms of a

finite sum of products of separated functions of each independentvariable following the well-known Fourier approximation scheme. Let�� be a scalar function of � � ��, ��, … , �� ∈ Ω � ∏ !" , #"�"$� .

�� %&'�"��"��

"$�

(

$�

� Piecewise uni-dimensional polynomial functions are usually used for

�" �" .


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Microscopic step solution

� This multiparametric problem is then solved in a previous "off-line"step within the whole multidimensional space of parameters (in theexpected range of parameter values) using PGD.

� The resulting �� includes all possible solutions for any RVE andstate. Then, homogenized values (e.g. internal stresses and stiffness atthe macro level) and possible local updates of the state variables arecomputed.

� Once the value of the parameters at each RVE are solved in the macrostep, local microstructural state is then updated directly (whenneeded), particularizing again the multiparametric solution.


� The macro solution �� is solved in an iterative process (Chinesta etal., 2006) after substituting the approximation and its variation intothe weak form of the problem to be solved.

� Alternating directions (fixed-point) and Newton-type schemes havebeen used. In the framework of alternating directions technique, thenon-linear multidimensional system is transformed into a set of Dindependent linear systems, solved cyclically until convergence.

� At each iteration, the element variables are computed directly, with nomore than particularizing �� to the specific element (RVE), that is,location, geometry, material properties and state (e.g. BCs, loads).

� All variables in the problem are parameterized in the same way usingSVD schemes when needed. Then, the resulting integrals are productsof uni-dimensional integrals, thus strongly reducing cost.


Macroscopic step solution


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Motivation2

3 Brief Review of Multiscale Techniquess

4



Overview




� Solve the equation)�)* � +*,-.+ /* 0 +/*+,-. /* .12 /* with / � 34

� The micro problem is solved by parameterizing the problem variable interms of the discrete macro time (T), the micro time (5 � � 0 6), the initialcondition at 5 � 0, (�8) and (eventually other independent variable like thesource term, state variables, etc.).

� � �( � �9 �&�:( 5 ;:( 6<

:$�=:( �8 �&�(�5�;(�6�

(

$�=(��8�

� After solving this problem for any set of parameters by using PGD, themacroproblem is immediately solved for any source term byparticularizing the time interval (starting in t=T= 5 � 0) with �8 known andcontinuing along time in a simple post-processing step that only impliesalgebraic operations.


Examples and Discussion

Time multiscale. An example

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Time multiscale. An example


� Resolution equal to the fine scale is got, although with much lesscomputer cost (besides the initial costly off-line microscalemultiparametric problem).

� Solve the 1-D elasticity problem))* > �

)�)* � ? � � 4 for a

heterogeneous material with spatial variation of the elastic modulus(functionally gradient material), defined as andexternal body forces


Spatial multiscale. 1D example


� The microscale problem is parameterized interms of:@ : Micro spatial coordinate.@� : Macro spatial coordinate.�� : Essential boundary condition at @ � 0.�A�� : Essential boundary condition at @ � ℓ.

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� A six-term PGD approximation in the microscale problem was needed to getconvergence.

� Differences in displacements between the results from the multiscaleapproach and a 100 elements traditional FEM for different micro and macrodiscretizations resulted < 1%.


Spatial multiscale. 1D example


� Problem definition:

CCD � @, �

CECD � 0f with f � 100 0 50 sin�2M��NONP � �Q with � @, � � 0 � �8 1 � D

R

� Variables approximation:

� � �@, @�, �� , �A�� , 5, 6, ��, �� 5, 6, �SP� Total spatial length: L=5mm; Total time length: tmax=3s; �8=100MPa;

Spatial domain: 10 macro and 5 micro elements; Time domain: 6 macroand 10 micro elements.

� 5 approximation terms where needed in the PGD approximation of thetime microscale problem to get convergence.



Time and 1D spatial multiscale example

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� Displacement field over time

� Capture load and elastic modulus variation due to the microscaleinformation when solving macroscale.



Fast localizationprocedure bypost-processingalgebraicaloperations.


� Error in comparison with a 100 spatial element FE analysis with a 0.001stime discretization. Errors below 1% in the total microscalediscretization (time and spatial).




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� Solve the 2D elasticity with spatial distribution of the elastic modulus

TSP�UV � & W X(YZ[

\[

S$�]^_ΩS

^ � X �� &X`:'a"( �" b:"�

"$�

<

:$�

� Bilinear square elements were used in both scales. This implies linearvariation of displacements over the RVE boundary.

� The dimension of the PGD approximation is reduced from 11 to 9 bytaking relative displacements with respect to one vertex.

Spatial multiscale. 2D example with continuous varying elastic modulus


� Convergence analysis associated to the number of terms in the PGDapproximation (5 × 5 microelements in each RVE)




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� Two RVEdiscretizations:(a) 5x5(b) 3x3

Reference FE mesh:4900 elements.

X direction Y direction








� Stress comparison:

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� Two RVE discreti-zations (30 PGD terms): (a) 263 elements (b) 63 elements

� Reference FE mesh:

� 15439 quadrilateral elements.



Spatial multiscale. 2D example with periodic distribution of pores



� Computer time for solving standard finite element analysis for the first2D problem with 4900 elements took about 55s. For the multiscale casewith 192 elements in the macroscale, the computer time was about 3s,despite the much worse software performance.

� For the porous case, the macroscale step took about 10s for 63microelements. This means a strong reduction when compared to the320s required to solve the problem with a standard FE procedure(15439 elements) and the much higher computer memory needed.

� In the microscale domain, a 40 terms approximation with 16 nodes inthe microscale took about 140s, while 121 nodes needed 425s. This isnot much important since this is an off-line computation.

� For the porous case, the microscale step computer times for this casewas about 145s to achieve a 30-terms approximation

Spatial multiscale. Computer cost


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Motivation2

3 Brief Review of Multiscale Techniquess

4



Overview




� The presented approach allows solving multiscale problems withgood accuracy and a strong reduction in computer time (FE linearorder) assuming previously solved (off line) the multiparametric RVEneeded.

� Microscale computer time depends on the discretization of eachindependent variable, the number of terms for the PGDapproximation, problem complexity and non-linear solverconvergence. This is costly problem but is carried out off line.

� Computer cost in the microscale does not increase very much whenadding additional independent parameters since generally, theoriginal equation does not contain derivatives with respect to manyof those parameters.


Concluding remarks

Concluding remarks

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� Higher-order variation (e.g. quadratic) in the RVE boundary.

� Geometrical parameterization of the microscale (arbitrary hexaedrals).

� Addition of state variables in the RVE

� 3-D problems problems

� Optimization, random and inverse problems problems

� Multiple scales

� Towards real-time


Future challenges

Future challenges

� Mathematical modelling and simulation are essential in industry(mathematicians are needed everywhere)

� Problems in industry are very complex, subject to uncertainties, lack ofknown parameters, solving time requirements, etc. Accuracy is notalways the main question. Approximate, simplified and reduced ordermodels are many times useful enough.

� Taking right decisions is the most important. Modeling and algorithmsare only tools (although essential) that have to be combined with manyother ones (labs, full scale tests, human experience, …). A strong effortis needed to understand the real problem and to combine quantitativean qualititative fuzzy knowledge.


Final comments

Final comments

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Innovative Solutions for Sustainability


Multiscale Solution of Deformable Solids in Linear FE Time Order by Coupling Proper Generalized Decomposition and VariationalMultiscale Methods

M. Doblaré

Abengoa Research and

Group of Structural Mechanics and Materials Modelling

Aragón Institute of Engineering Research (I3A). Unive rsity of Zaragoza,

multiscalesolution of deformable solids in linear fe time ... · abengoa(mce: abg) is an...

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