Multiple Spatial and Temporal ScalesMultiple Spatial and Temporal ScalesThe ChallengeThe Challenge

For optimal product design which spatial and temporal scales should be resolved?

Adaptive Multiscale ModelingAdaptive Multiscale ModelingThe Vision of EinsteinThe Vision of Einstein

EinsteinEinstein: “: “The model used should The model used should be the simplest one possible, but be the simplest one possible, but not simpler.not simpler.””

Adaptive Multiscale Modeling:Adaptive Multiscale Modeling: “Start with “Start with a a simpler modelsimpler model, based on a single scale , based on a single scale and uncoupled physical processes, and then and uncoupled physical processes, and then adaptively introduce additional scales to adaptively introduce additional scales to permit coupled multiscale-multiphysics permit coupled multiscale-multiphysics considerations, whenever and wherever these considerations, whenever and wherever these are needed, until the are needed, until the simplest possible simplest possible modelmodel is obtained.” is obtained.”

Adaptive Multiscale ModelingAdaptive Multiscale Modeling

Is additional scale necessary?Is additional scale necessary? Model error estimatorsModel error estimators Phenomenological modelingPhenomenological modeling

YesYesNo No Does additional scale provide cost Does additional scale provide cost

savings?savings? Bulk vs. constituents-based modelingBulk vs. constituents-based modeling Key Issue: Does scale refinement reduce Key Issue: Does scale refinement reduce

experimentation (probability, potentials)?experimentation (probability, potentials)?

*

10 ( , , ,) )( , ( , )u x t u u x t yx t

*

Adaptive Multiscale SchemeAdaptive Multiscale Scheme

How to introduce additional scales (discrete How to introduce additional scales (discrete and/or continuum)?and/or continuum)?

Embedded schemesEmbedded schemes– Multilevel (multigrid) methodsMultilevel (multigrid) methods– Space/time decomposition methodsSpace/time decomposition methods

Sequential schemesSequential schemes– Multiple space/time asymptotic methodsMultiple space/time asymptotic methods– Variational multiscale methodsVariational multiscale methods– Sequential phenomenological methodsSequential phenomenological methods

Adaptive Multiscale ModelingAdaptive Multiscale Modeling

Model transition schemesModel transition schemes– Pollution errors at the interface for continuous-Pollution errors at the interface for continuous-

continuous and continuous-discrete transitionscontinuous and continuous-discrete transitions– Mathematically consistent discrete-continuum Mathematically consistent discrete-continuum

transitiontransition

Is stochastic modeling required?Is stochastic modeling required? Probability error estimatorsProbability error estimators Multiscale sensitivity analysisMultiscale sensitivity analysis

Variational Multiscale ApproachVariational Multiscale Approach Biological systems

0 1 2u u u u

0 1u u

10-100 nm

Protein-protein, protein-surface interactions

Protein Structure RVE

5 nm

Hybrid Quantum RVE

0.5 nm

0u

Coarse Grained Discrete

Asymptotic Multiscale ApproachAsymptotic Multiscale ApproachNanocomposites ApplicationNanocomposites Application

xy

y

z

0 1 2 2( , ) ( , , , ) ( , , , , , )u u x t u x y t u x y z t

y1

y2

z1

z2

x1

x2

Continuous RVE

Discrete RVE

Component

DiscreteContinuous

RPI: Consistent Discrete-to-Continuum Space-Time Homogenization

Embedded Multiscale SchemeEmbedded Multiscale SchemeMultigrid ApproachMultigrid Approach

0 1

multiscalecorrectionclassical

loc loc locQ Q Q

Discrete restriction

0 1glob glob glob

multiscaleclassicalcorrection

Q Q Q

Restriction

Pr

TglobQ

olongation

TlocQ

SmoothingSmoothing

Examples of Mono-scale ModelsExamples of Mono-scale Models

Enzyme in non-biological media

Examples of Multiscale ModelsExamples of Multiscale Models

Zoom

Stochastic Nature of Multiscale Stochastic Nature of Multiscale ProblemProblem

•Physical uncertainties (loads, domain, material properties)

•Statistical uncertainties (amount of data available, probability fields such as correlations)

•Model uncertainties (mathematical modeling of physical behavior)