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Development of Computational Algorithms for Materials
Simulation and Design
Qiang Du Penn State University
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Outline • An unauthorized story about “Materials Genome”
• A brief review of our recent algorithmic development works supported by: – NSF-DMR ITR: Computational Tools for Multicomponent
Materials Design 2002-2007
– NSF-IIP Center for Computational Materials Design (CCMD) Phase I, 2005-2010
– NSF-IIP I/UCRC CGI: Center for Computational Materials Design (CCMD), Phase II, 2010-2015
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A Glimpse of Materials Genome from Happy Valley
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Materials Genome as seen from outside
Ceder group, MIT
Zikui Liu, PSU
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General Manager: Dr.Zikui Liu, Prof of MSE Penn State
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Media
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Fun activities at Penn State
Working with State College High Math Club: outreach activities of NSF-CCMD 2010
NSF-ITR project group photo 2005
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Story as told on Wiki
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Performance
Property
Structure
Processing
Chemistry
Performance
Defects
Crystallo-graphy
Kinetics
Thermo-Dynamics
Materials Engineering and Science
Top-Down, Inverse Design Bottom-Up, Forward Simulation Courtesy: Zikui Liu, Penn State/Materials Genome
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MatCASE Project • An Information Technology Research (ITR) Project
supported by NSF-DMR (2002 to 2007) Computational Tools for Multicomponent Materials Design • PI Zikui Liu (MSE) • Co-PIs: Chen (MSE), Du (MATH), Raghavan (CSE) • http://www.matcase.psu.edu (Materials Computation And Simulation Environment)
• It involved researchers from the Pennsylvania State University, Ford Motor Company, and National Institute of Standard and Technology.
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Four Major Steps 1. First principles calculations to determine thermodynamic
properties, lattice parameters, and kinetic data of unary, binary and ternary compounds
2. CALPHAD data optimization to extract thermodynamic properties, lattice parameters, and kinetic data of multicomponent systems combining results from the first-principle calculations and experimental data
3. Multicomponent phase-field modeling to produce a microstructure in one to three space dimensions
4. Finite element analysis to generate the mechanical response from the simulated microstructure
Liu, Z.K., Chen, L.Q., Raghavan, P., Du, Q., Sofo, J., Langer, S. & Wolverton C., An integrated framework for multi-scale materials simulation and design, J. Comput-Aided Mater. Des., 11, 2004, 183–199.
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NSF ITR: MatCASE Project
Experimental data
Mechanical responses of simulated
microstructures
Interfacial energies, lattice parameters and elastic constants
Kinetic data
Bulk thermodynamic data
Database for lattice parameters, elastic constants and interfacial energies
Kinetic database
Bulk thermodynamic database
Plasticity of phases Microstructure in 2D and 3D Elasticity of phases
First-principles calculations
CALPHAD
OOF: Object-oriented finite element
analysis
Phase-field simulation
Our focus
Our focus
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Going beyond ITR: IUCRC CCMD
Director: Zikui Liu (Penn State) Co-Director: David McDowell (GaTech)
Phase I: 2005-2009 Phase II: 2011-2015 NSF IIP PI: Zikui Liu, Co-PI: Chen, Du, Raghavan
NSF IUCRC: CCMD
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Algorithmic development 1. Efficient phase field simulation codes + Mathematical
and numerical analysis – Iterative solver for inhomogeneous elasticity, variable mobility – Exponential time integrator for higher order time marching – Adaptive resolution via moving spectral methods – Saddle point search for nucleation (a fundamental problem)
Du-Chen-Yu-Zhu-Feng-Hu-Dai-J.Zhang-L.Zhang-JY.Zhang 1. Robust codes for phase diagram computation
– Automated phase diagram calculation – Statistical analysis of input date, uncertainty quantification (critical for making computation predictive tools)
Du-Liu-Emelianenko-Saal-JY.Zhang
Joint work with Maria Emelianenko and Zikui Liu 16
Automated computation of phase diagrams
Some typical binary and ternary phase diagrams
Phase diagrams are maps of the equilibrium phases associated with various combinations of temperature and composition
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Gibbs energy minimization
is the number sites in phase i, is volume fraction of k particles
in phase i, is the Gibbs energy of phase i
if!ik
iG
min(fi,!i
k ){G = fi Gi(!i
k )}i=1
n
!
fi =i=1
n
! f0
fi!ik = f0 !0
k
i=1
n
! , k =1,…,K
! (user-dependence) rely on prior knowledge
! (stability, reliability) failure in identifying miscibility gap, lack quality assessment
Mathematically, equilibrium analysis of a K-component system with n phases leads to a minimization problem:
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Algorithms for automation and statistical analysis of phase diagram computation
Goals: " Calculate equilibria in multicomponent multiphase systems " Minimize the number of trials, get comparable accuracy of
solution with lower complexity
" Provide quantitative assessment of data quality and reliability of resulting phase diagrams, essential for predictive simulations
Ideas: " Rely on the geometrical properties of the Gibbs energies to find
better starting points, automatically identify miscibility gaps
" Use adaptive approach with effective sampling techniques
" Perform statistical data analysis and error quantification Emelianenko M., Liu Z.K., Du Q., Computational Materials Science, 2005
(with illustrations for ternary Ca-Li-Na system) J. Saal, J.Y. Zhang, V. Manga, M. Carolan, Q. Du and Z.-K. Liu, Review and statistical assessment of La1-xSrxCoO3 ! oxygen nonstoichiometry, 2012
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Critical nuclei morphology in solid state phase transformations
• Predicting nucleation rate and its dependence on composition, temperature under various conditions is a basic problem in materials design and is of broad interests
• Recently we studied the effect of elastic energy contribution in nucleation; critical nucleus may exhibit non-convex shapes and/or may have lower symmetry, leading to various crystallographic orientations (structure domains)
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Nucleation and growth with diffuse-interface nuclei
0 50000 100000 150000 2000000.0
0.2
0.4
0.6
0.8
1.0
1.2
Are
a fra
ctio
n (f(i))
Time step (i)
Diffuse-interface nucleation and growth KJMA's interpretation (fixed growth rate) KJMA's interpretation (varying growth rate)
(b)
Comparing to kinetics using the classical nucleation and normal growth theory and the JMAK equation
),3
exp(1)( 32
tuItf !!""=#
),exp( *0 GII !"=
We developed a diffuse-interface model to identify critical nuclei offline and combined with a KMC scheme to study nucleation and growth. L. Zhang-L.Q. Chen-Q. Du, Phy. Rev. Lett, 2007, Acta Mater., 2008, Comm. Comp. Phys., 2010, J. Comp. Phys, 2010; Heo,-Zhang-Du-Chen, Scripta Mater. 2011
Algorithmic development • Robust algorithms for computing saddle points and
minimum energy paths – Minimax algorithms – Constrained string methods – Shrinking dimer dynamics (SDD, a recently proposed
dynamic system variant of the popular dimer method). E.g.,
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Algorithmic development • SDD: search index-1 saddle in an extended space
– Efficiency: uses only force, just like a gradient system, – Robustness: enjoys local stability, with guaranteed
convergence for some special potential energy
• SDD can handle high dimensional complex energy landscape – Rigorous mathematical/numerical analysis – Demonstration for free energy surfaces associated with
the diffuse interface model – Demonstration for potential energy surfaces associated
with geometrically constrained particles
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L.Zhang-L.Q. Chen-Q. Du, J. Sci. Compt., 2008; Q. Du-L.Zhang, Comm. Math. Sci., 2009; J.Zhang-Q,Du, J. Comp. Phys. 2012, SIAM Num Anal 2012
Main Collaborators • Maria Emelianenko, GMU • Lei Zhang, UCI • Jingyan Zhang, Penn State • Zikui Liu, James Saal, Penn State • Long-Qing Chen, Tae-woo Heo, Penn State
Publications available at: http://www.math.psu.edu/qdu Thank you!
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