Higher-Order Localization Relationships Using the MKS Approach
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Presentation on a parallelizable, effective models technique to replace costly simulation techniques (e.g. Finite Element Models or Phase Field Simulation). This presentation was given at the ASME 2011 Applied Mechanics and Materials Conference In Chicago, IL.
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Higher-Order Localization Relationships Using the MKS Approach
- 1. Higher-Order Localization Relationships Using the MKS Approach Tony Fast and Surya R. Kalidindi Department of Materials Science and Engineering Drexel University Philadelphia,PA
- 2. Material Hierarchy LOCALIZATION ~ TOP-DOWN HOMOGENIZATION ~ BOTTOM-UP Localization is important to track microstructure evolution at lower length scales in concurrent multiscale modeling McDowell DL. A perspective on trends in multiscale plasticity. International Journal of Plasticity 2010;26:1280.c
- 3. Concurrent Multi-scaling 1. Homogenize coarse scale $ $ $ $ 2. Isolate region where homogenization fails 3. Implement lower scale numerical simulations 4. Isolate region where homogenization fails $$ $ 5. Implement fine scale numerical simulations ….. Quasicontinuum, Bridging Scale Method Develop efficient, effective localization models for DNS to reduce cost Liu W, McVeigh C. Computational Mechanics 2008;42:147
- 4. DATA VS. KNOWLEDGE Stress, strain, evolution • Many Inputs and Many Outputs • Repetitive simulation is demanding • Simulation produces a lot of data, but what information is determined about the system? How can information about new structures be extracted? What knowledge is gained?
- 5. DATA VS. KNOWLEDGE • DSP representation of local structure-local response • Localization relationship and its influence coefficients • Extended from Kroner’s Green’s function • Influence coefficients capture the combined point effects of the MS configuration on the local response
- 6. DATA VS. KNOWLEDGE In the Materials Knowledge System, influence coefficients capture knowledge (physics) of the system as a convolution filter with the microstructure. They provide a database to efficiently extract, store, and recall local microstructure-processing-property linkages. Facilitates exploration of the local response of other MS configurations
- 7. DISCRETE MICROSTUCTURE TO DIGITAL SIGNAL Microstructure is inherently discrete because of probe size and resolution limits of physical model or characterization method Microstructure Position (s) FIRST-ORDER FILTER Decompose MS into salient features
- 8. Materials Knowledge Systems Effective Localization Models Elastic and Thermo-Elastic Spinodal decomposition of binary Response of Dual Phase Composites alloy – Finite Element Analysis – Phase field modeling Evolution Field Accuracy E1 1 .5 E2 Landi, G., S. R. Niezgoda, et al. (2009). Acta Materialia 58(7): 2716-2725 T. Fast, S. R. Niezgoda, S. R. Kalidindi, Acta Materialia 59, 699 (2011)..
- 9. Establishing Knowledge Databases • Coefficients calibrated to validated direct numerical simulations • Calibration using OLSF facilitated by decoupling spatial components using Fourier transform – Drastic reduction in complexity and parallelizable • Limited to weakly nonlinear systems linkages assumed to be linear • Further extension relies on nonlinear system identification methods
- 10. Higher-Order Microstructure Signals First Order Filter <s+1> Higher-Order Filter <s+2> Higher-Order Filter Number of HO signals grows rapidly
- 11. Higher-Order Coefficients (Convolution Filters) HO Strain H H S 1 S 1 h h h h h h ps a t11t 2 t N N m s 1 t1 m s 2 t1 2 t2 m s N t1 tN h1 1 h N 1 t1 0 tN 0 • IC relating to Higher-Order Signals are Volterra kernels that capture strong nonlinear interactions
- 12. Finite Element Simulation of Dual Phase Composite FEM ε=5e-4 E1 E2 • Contrast (nonlinearity) – Young’s modulus ratio • Uniaxial 1-1 strain • Random distribution of phases in microstructure
- 13. Protocols for Establishing Higher-Order Terms • Redundant signals translate to linear relationships in spectral domain n m6 • Assumptions of HOIC n m1 n ms n m3 – Nearest vectors (neighbors) contribute most n m2 n m5 – Higher-Order coefficients capture nonlinearity best Case Combination of Coefficients Selected 1 First Order Coefficients 2 Second Order Coefficients up to first neighbors 3 Second Order Coefficients up to second neighbors … 7 Second Order Coefficients up to sixth neighbors 8 Seventh Order Coefficients up to first neighbors Seventh Order Coefficients up to first neighbors plus Second 9 Order Coefficients from second neighbors to sixth neighbors
- 14. Stronger Contrast Knowledge Systems • Different HOICs calibrated from 400 FEM simulations • Training (calibration) set vs. Validation set describes how well knowledge is captured E1 E1 5 10 E2 E2 • Improvement in accuracy decreases with distant neighbors • Combining coefficients improves accuracy and precision while maintaining tractability Case 1: First Order Case 2 – 7: Second Order Case 8-9: Seventh Order
- 15. Accurate Strain Localization • HOIC of increasing order captures local information better E1 E1 5 10 E2 E2 • Drastic improvement of linkages of FOIC • Accuracy has a strong dependence on nonlinearity Case 1: First Order Case 2 – 7: Second Order Case 8-9: Seventh Order
- 16. Accurate Strain Distribution Characteristics of distribution are improved particularly in the tails with HOIC E1 E1 5 10 E2 E2 Case 1: First Order Case 2 – 7: Second Order Case 8-9: Seventh Order
- 17. Extension to larger domains drastic time savings 153 influences coefficients have finite memory and decay to zero at larger distances FEM required 45 min on supercomputer MKS required 15 seconds on a desktop computer MKS – NlogN(N) Case 9: Seventh-Order to First Neighborhood and Second-Order to Sixth Neighborhoods
- 18. Conclusions • Higher-order coefficients are crucial in developing effective localization models for strongly nonlinearity systems • Systematic selection of higher-order neighborhoods facilitated the development of this work – These concepts hinge off the finite-memory of the physical interactions • MKS provides drastic time savings over DNS particularly in the extension to larger spatial domains
Editor's Notes
- SFHIT FROM DETERMINISTIC TO STOCHASTIC TOOLS TO DEAL THE LARGE SCALE DATA
- The main trhust of materials science and engineering rests on establishing the linkages between microstructure property and processing. Under the umbrella of materials by design rigorously esxtablishing the connections between disparate hierarchies is key in understanding the effect of the lower length scales upon the higher length scalesconverselyShifting of atoms>dislocations>dislocation densities forrestgnb>crystal rotation>plasticityHow for example the response of different crystal orientation under evolution even with the same bound conditions are differentImportant to track with localization extract failure criteriaEvolution in different grains may vary drastically for the same macroscale boundary conditionsConcurrent multiscale simulation presents a unique challenge. Atoms shift creating dislocations these dislocations get tangled up and combine into dislocation patterns (what are these called Necessary DD? dislocation motion causes grain fragmentation and facilitates movement on the polycrystalline scale which ultimately effects the plastic properties of the material.This type of communication between length scales is a classically researched method that has made many strides since the 1970s called bottom-up multi-scale simulation. This technique is performed using homogenization relationships that take the fine scale heterogeneous features and describes them as a coarse scale homogeneous medium. In aggregating the fine scale information in this manner the localized features are lost. In most real applications, the engineer is most concerned with failure, damage, corrosion, or fatigue properties of the material. Thus we must apply localization relations to pass information in the reverse direction where coarse scale information is communicated to the finer scale as boundary conditions. Simulation of a complete material volume in this manner poses a problem of scale.Track the local evolution of microstructure features at each length scale in particular for failure and fraction. ?Mean field theories? Need something better if this concept falls apart
- Concurrent multiscale methods are used to resolve the effect of macrscopic loading conditions upon the lower length scalesTHE CURRENT STATE OF THE ART IN MULTISCALE SIMULATION IS TO ….ESTABLISH EFFICIENT EFECTIVE LOCALIZATION MODELS TO SUPPLANT TEDIOUS AND COSTLY DIRECT NUMERICAL SIMULATIONS.Multiscale modeling addresses these challenges by using bottom-up homogenization methodsMathematical tools have been developed to interrogate the specific areas of interest.Concurrent multi-scalingPoint out how small area is explored
- TO ADDRESS THE CHALLENGE OF SIMULATING LOCALIZED MATERIAL PROPERTIES WE MUST ADDRESS THE IDEA OF WHAT IS PRODUCED BY WAY OF DIRECT NUMERICAL SIMULATIONModel is independent of MSUnderstand physics in a way that they are independent of MSOnly knw information about simulated structures, what do yhou know about new structures wahat you can you tell about structures that havent been simulated
- More information about what knowledge is. In our framework knowledge is captured by influence coefficients could be other fitting parameters. This work was built off of Kroner’s statistical mechanics and extended to a digital signal processing framework which facilitated the coefficients to be calibrated to physics based modelsWhat are influence coeefficients
- More information about what knowledge is. In our framework knowledge is captured by influence coefficients could be other fitting parameters. This work was built off of Kroner’s statistical mechanics and extended to a digital signal processing framework which facilitated the coefficients to be calibrated to physics based modelsWhat are influence coeefficients
- BEFORE WE CONTINUE OUR DISCUSSION OF THE EFFECTIVE LOCALIZATION SIMULATIONS THAT THIS WORK IS MOTIVATED UPON BUILDING WE MUST ESTABLISH SOME CONCEPTS PRIOR. THE METHODS DESCRIBED IN THIS WORK HINGE ON THE DIGITAL SIGNAL PROCESSING AND OUR ABILITY TO DEFINE A MICROSTRUCTURE EXTACTED FROM PBM OR CHARACTERIZATION ROUTINES INTO A DISCRETE MULTICHANNEL SIGNAL. THE MULTICHANNEL SIGNAL IS DELINEATED BY A ROVING FILTER OF THE SALIENT MICROSTRUCTURE FEATURES. IN THIS PARTICULAR EXAMPLE WE ARE FOCUSED UPON DUAL PHASE COMPOSITES.
- THE CONCEPT OF EFFICIENT LOCALIZATION MODELS HAS BE ENCAPSULATED IN A NOVEL MATHEMATICAL FRAMEWORK CALLED THE MATERIALS KNOWLEDGE SYSTEM WHICH EXTRACTS KNOWLEDGE OR THE UNDERLYING PHYSICS CONTAINED IN A DIRECT NUMERICAL SIMULATIONS
- Basic idea has been validated in 3-D for reasonably large datasets of basic phenomenaEnd with materials knowledge system then say that this work is motivated at extending the approach.These concepts proved to be extremely accurate and efficient, but to harness the power of the method so critical features components must be developed. Extending the MKS to a broader range of material systems or material responses is the focus of this work.Point out localization different from previous slideFourier introduction – makes it easier to get constants, decouples, observed before
- Nonlinearfeautres of the system are evaluated by exploring the higher order moments of the microstructure signal. This entails …See how this blows up very fastInsert text about scalingNot just two point associations but can extend third point assoication
- The Materials Knowledge System hinges on a convolution filter that conveys the physical influence of the selected filtered signals upon the response. This approach is robust in the fact that the response being captured can be practically any response like local stress or strain meanwhile being amenable to evolution responses like the time derivative of concentration. The filters are applied to every spatial cell in the microstructure signal to resolve the local response over the entire domain. This process allows the local response to be rapidly and accurately capture.Furthermore, this translates to a linear causal system where the microstructure is convolved with the influence coefficient to reproduce the response. The influence coefficients are the central of the MKS. They comprise an N-D array where is N is dependent upon the dimensionality of the spatial domain and there is a set of coefficients defined for each local state signal. When the filtered signal is delineated by higher-order filters then the corresponding coefficients are effectively Volterra kernels otherwise they are linear kernels. The Volterra kernel is effective in capturing strongly nonlinear interactions. The higher-order the order of the filtered signal the more nonlinearity is expected to be captured.Lastly, an important feature of the influence coefficients that will be exploited later is that their magnitudes or influence are expected to decay with increasing values of t. This concept will be important in developing protocols to establish MKS databasesHuge challenge to calibrate kernels
- It is possible to see that each signal is not unique
- It is possible to see that each signal is not unique