matopt 0.0.1

Materials Design via Mathematical Optimization

[!IMPORTANT] This is a standalone version adapted from the original implementation

MatOpt (Materials Design via Mathematical Optimization) is an Algebraic Modeling Language (AML) system specifically design to create Pyomo objects for optimization-based nanomaterials design. MatOpt can be used to model crystalline nanostructured materials, including but not limited to particles, surfaces, and periodic bulk structures.

The main goals of this package are as follows:

• To simplify the representation of nanostructured materials, streamlining the creation of materials optimization problems.
• To automate many of the necessary steps of materials optimization, speeding up the development of new models and accelerating new materials discovery.
• To provide a simple interface so that users do not need to handle the details of building efficient mathematical optimization models or the specific Pyomo syntax to do this.

If you are using MatOpt, please consider citing:

• Hanselman, C.L., Yin, X., Miller, D.C. and Gounaris, C.E., 2022. MatOpt: A Python Package for Nanomaterials Design Using Discrete Optimization. Journal of Chemical Information and Modeling, 62(2), pp.295-308.

Basic Usage

There are two main sub-modules contained in the package serving two distinct purposes:

• The matopt.materials module contains objects and methods for efficiently representing and manipulating a nanomaterial and its design space.
• The matopt.aml module contains objects and methods for speeding up the casting of a pyomo model with simplified modeling syntax and automatic model formulation tools.

Dependencies

User access to the solver CPLEX through Pyomo is assumed. For users who do not have access to CPLEX, the use of NEOS-CPLEX is suggested as an alternative (Users need to setup a NEOS_EMAIL environment variable).

Define design canvas

Several pieces of information about the material and design space need to be specified in order to formulate a materials optimization problem. To fulfill this need, the matopt.materials module defines generic and simple objects for describing the type of material to be designed and its design space, also referred to as a "canvas".

Some key objects are listed as follows:

Build model via descriptors

The material type and design space specified provide indices, sets, and parameters for the optimization model. Using simple syntax, inspired by materials-related terminology, MatOpt users define a MatOptModel object, which will be translated into a Pyomo ConcreteModel object automatically.

MatOpt uses MaterialDescriptor objects to represent variables, constraints, and objectives. A MatOptModel object holds lists of MaterialDescriptor objects. By default, several universal site descriptors are pre-defined in the model.

Descriptor	Explanation
Yik	Presence of a building block of type k at site i
Yi	Presence of any type of building block at site i
Xijkl	Presence of a building block of type k at site i and a building block of type l at site j
Xij	Presence of any building block at site i and any building block at site j
Cikl	Count of neighbors of type l next to a building block of type k at site i
Ci	Count of any type of neighbors next to a building block at site i

User-specified descriptors are defined by DescriptorRule objects in conjunction with Expr expression objects. Available expressions include:

Expression	Explanation
LinearExpr	Multiplication and addition of coefficients to distinct descriptors
SiteCombination	Summation of site contributions from two sites
SumNeighborSites	Summation of site contributions from all neighboring sites
SumNeighborBonds	Summation of bond contributions to all neighboring sites
SumSites	Summation across sites
SumBonds	Summation across bonds
SumSiteTypes	Summation across site types
SumBondTypes	Summation across bond types
SumSitesAndTypes	Summation across sites and site types
SumBondsAndTypes	Summation across bonds and bond types
SumConfs	Summation across conformation types
SumSitesAndConfs	Summation across sites and conformation types

Several types of DescriptorRules are available.

Rule	Explanation
LessThan	Descriptor less than or equal to an expression
EqualTo	Descriptor equal to an expression
GreaterThan	Descriptor greater than or equal to an expression
FixedTo	Descriptor fixed to a scalar value
PiecewiseLinear	Descriptor equal to the evaluation of a piecewise linear function
Implies	Indicator descriptor that imposes other constraints if equal to 1
NegImplies	Indicator descriptor that imposes other constraints if equal to 0
ImpliesSiteCombination	Indicator bond-indexed descriptor that imposes constraints on the two sites
ImpliesNeighbors	Indicator site-indexed descriptor that imposes constraints on neighboring sites

From the combination of the above pre-defined descriptors, expressions, and rules, a user can specify a wide variety of other descriptors, as necessary.

Solve optimization model

Once the model is fully specified, the user can optimize it in light of a chosen descriptor to serve as the objective to be maximized or minimized, as appropriate. Several functions are provided for users to choose from. The results of the optimization process will be loaded into Design objects automatically. Users can then save material design(s) into files for further analysis and visualization using suitable functions provided. MatOpt provides interfaces to several standard crystal structure file formats, including CFG, PDB, POSCAR, and XYZ.

Examples

Below is a basic example of designing stable monometallic Pt nanoclusters given the size constraint. User can specify the problem using intuitive syntax and solve it using free NEOS-CPLEX solver (need to specify an email using NEOS_EMAIL environment variable)

import numpy as np
from math import sqrt
from matopt.materials.atom import Atom
from matopt.materials.lattices.fcc_lattice import FCCLattice
from matopt.materials.canvas import Canvas
from matopt.aml.expr import SumSites
from matopt.aml.rule import PiecewiseLinear, EqualTo
from matopt.aml.model import MatOptModel

Lat = FCCLattice(IAD=2.7704443686888935)
Canv = Canvas()
Canv.addLocation(np.array([0, 0, 0], dtype=float))
Canv.addShells(2, Lat.getNeighbors)
Canv.setNeighborsFromFunc(Lat.getNeighbors)
Atoms = [Atom("Pt")]

N = 22
m = MatOptModel(Canv, Atoms)
m.addSitesDescriptor(
    "CNRi",
    bounds=(0, sqrt(12)),
    integer=False,
    rules=PiecewiseLinear(
        values=[sqrt(CN) for CN in range(13)],
        breakpoints=[CN for CN in range(13)],
        input_desc=m.Ci,
        con_type="UB",
    ),
)
m.addGlobalDescriptor(
    "Ecoh", rules=EqualTo(SumSites(desc=m.CNRi, coefs=(1.0 / sqrt(12) * 1.0 / N)))
)
m.addGlobalDescriptor("Size", bounds=(N, N), rules=EqualTo(SumSites(desc=m.Yi)))

D = None
try:
    D = m.maximize(m.Ecoh, tilim=100, solver="neos-cplex")
except:
    print("MaOpt can not find usable solver (CPLEX or NEOS-CPLEX)")
if D is not None:
    D.toPDB("result.pdb")

For more usage cases and detailed explanation and walkthrough, please check the original repository. In each case, a Jupyter notebook with explanations as well as an equivalent Python script is provided.

Applications

• Hanselman, C.L. and Gounaris, C.E., 2016. A mathematical optimization framework for the design of nanopatterned surfaces. AIChE Journal, 62(9), pp.3250-3263.
• Hanselman, C.L., Alfonso, D.R., Lekse, J.W., Matranga, C., Miller, D.C. and Gounaris, C.E., 2019. A framework for optimizing oxygen vacancy formation in doped perovskites. Computers & Chemical Engineering, 126, pp.168-177.
• Hanselman, C.L., Zhong, W., Tran, K., Ulissi, Z.W. and Gounaris, C.E., 2019. Optimization-based design of active and stable nanostructured surfaces. The Journal of Physical Chemistry C, 123(48), pp.29209-29218.
• Isenberg, N.M., Taylor, M.G., Yan, Z., Hanselman, C.L., Mpourmpakis, G. and Gounaris, C.E., 2020. Identification of optimally stable nanocluster geometries via mathematical optimization and density-functional theory. Molecular Systems Design & Engineering.
• Yin, X., Isenberg, N.M., Hanselman, C.L., Dean, J.R., Mpourmpakis, G. and Gounaris, C.E., 2021. Designing stable bimetallic nanoclusters via an iterative two-step optimization approach. Molecular Systems Design & Engineering, 6(7), pp.545-557.
• Yin, X. and Gounaris, C.E., 2022. Search methods for inorganic materials crystal structure prediction. Current Opinion in Chemical Engineering, 35, p.100726.
• Hanselman, C.L., Yin, X., Miller, D.C. and Gounaris, C.E., 2022. MatOpt: A Python Package for Nanomaterials Design Using Discrete Optimization. Journal of Chemical Information and Modeling, 62(2), pp.295-308.