Multiple Spatial and Multiple Spatial and Temporal Scales Temporal Scales The Challenge The Challenge For optimal product design which spatial and temporal scales should be resolved?
Jan 21, 2016
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)