1. Multiple Spatial and Temporal Scales The Challenge For optimal product design which spatial and temporal scales should be resolved?

2. Adaptive Multiscale Modeling The Vision of Einstein Einstein: “The model used should be the simplest one possible, but not simpler.” Adaptive Multiscale Modeling: “Start with a simpler model, based on a single scale and uncoupled physical processes, and then adaptively introduce additional scales to permit coupled multiscale-multiphysics considerations, whenever and wherever these are needed, until the simplest possible model is obtained.”

3. Adaptive Multiscale Modeling Is additional scale necessary?  Model error estimators  Phenomenological modeling Yes  No  Does additional scale provide cost savings?  Bulk vs. constituents-based modeling  Key Issue: Does scale refinement reduce experimentation (probability, potentials)?

4. Adaptive Multiscale Scheme How to introduce additional scales (discrete and/or continuum)?  Embedded schemes Embedded schemes Embedded schemes –Multilevel (multigrid) methods –Space/time decomposition methods  Sequential schemes Sequential schemes Sequential schemes –Multiple space/time asymptotic methods –Variational multiscale methods –Sequential phenomenological methods

5. Adaptive Multiscale Modeling  Model transition schemes –Pollution errors at the interface for continuous- continuous and continuous-discrete transitions –Mathematically consistent discrete-continuum transition Is stochastic modeling required? Is stochastic modeling required?  Probability error estimators  Multiscale sensitivity analysis

6. Variational Multiscale Approach Variational Multiscale Approach Biological systems 10-100 nm Protein-protein, protein- surface interactions Protein Structure RVE 5 nm Hybrid Quantum RVE 0.5 nm Coarse Grained Discrete

7. Asymptotic Multiscale Approach Nanocomposites Application y 1 y 2 z 1 z 2 x 1 x 2 Continuous RVE Discrete RVE Component DiscreteContinuous RPI: Consistent Discrete- to-Continuum Space-Time Homogenization

8. Embedded Multiscale Scheme Multigrid Approach Smoothing

9. Examples of Mono-scale Models Enzyme in non-biological media

10. Examples of Multiscale Models Zoom

11. Stochastic Nature of Multiscale Problem Physical uncertainties (loads, domain, material properties) Statistical uncertainties (amount of data available, probability fields such as correlations) Model uncertainties (mathematical modeling of physical behavior)