MatOpt: Nanomaterials Optimization

The MatOpt module provides tools for nanomaterials design using Mathematical Optimization. MatOpt can be used to design crystalline nanostructured materials, including but not limited to particles, wires, surfaces, and periodic bulk structures.

The main goals of this package are as follows:

• To automate many of the steps that are necessary for utilizing mathematical optimization to design materials, speeding up the development of new mathematical models and accelerating new materials discovery.

• To simplify the representation of nanostructured materials and their structure-function relationships as Pyomo objects, streamlining the creation of materials optimization problems in the Pyomo modeling language.

• To provide a simple interface so that users need not handle the details of casting efficient mathematical optimization models, invoking mathematical optimization solvers, or utilizing specialized Pyomo syntax to do this.

Thank you for your interest in MatOpt. We would love to hear your feedback! Please report any thoughts, questions or bugs to: gounaris@cmu.edu

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 discrete optimization. Journal of Chemical Information and Modeling, 62(2), pp.295-308.

MatOpt Examples

Several case studies are provided to illustrate the detailed usage of MatOpt. In each case, a Jupyter notebook with explanations as well as an equivalent Python script is provided.

References

• 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, 5, pp.232-244.

• Yin, X., Isenberg, N.M., Hanselman, C.L., Dean, J.R., Mpourmpakis, G. and Gounaris, C.E., 2021. A mathematical optimization-based design framework for identifying stable bimetallic nanoclusters. 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 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.opt module contains objects and methods for speeding up the casting of a Mixed-integer Linear Programming (MILP) model with simplified modeling syntax and automatic model formulation.

Dependencies

User access to the MILP 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.

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:

class idaes.apps.matopt.materials.lattices.lattice.Lattice

A class used to represent crystal lattice locations.

The class encodes methods for determining which Cartesian coordinates to consider as sites on an infinite crystal lattice. A Lattice can be constructed from a point on the lattice (i.e., a shift from the origin), an alignment (i.e., rotation from a nominal orientation), and appropriate scaling factors. With these attributes, we generally support the translation, rotation, and rescaling of lattices. Additionally, Lattice objects include a method for determining which sites should be considered neighbors.

class idaes.apps.matopt.materials.canvas.Canvas(Points=None, NeighborhoodIndexes=None, DefaultNN=0)

A class for combining geometric points and neighbors.

This class contains a list of Cartesian points coupled with a graph of nodes for sites and arcs for bonds. A Canvas object establishes a mapping from the abstract, mathematical modeling of materials as graphs to the geometry of the material lattice. The list of points and neighbor connections necessary to create a Canvas object can be obtained from the combination of Lattice, Shape, and Tiling objects.

class idaes.apps.matopt.materials.design.Design(Canvas_=None, Contents=None)

A class used to represent material designs.

This class combines a Canvas objects and a list of contents. It assigns an element (possibly None) to each point in the Canvas. This generally works for any type of content, but it is intended to work with Atom objects and can be used to generate CFG, PDB, POSCAR, and XYZ files.

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.

class idaes.apps.matopt.opt.mat_modeling.MaterialDescriptor(name, canv=None, atoms=None, confDs=None, bounds=(None, None), integer=False, binary=False, rules=None, **kwargs)

A class to represent material geometric and energetic descriptors.

This class holds the information to define mathematical optimization variables for the properties of materials. Additionally, each descriptor has a ‘rules’ list to which the user can append rules defining the descriptor and constraining the design space.

name

A unique (otherwise Pyomo will complain) name

Type:

string

canv

The canvas that the descriptor will be indexed over

Type:

Canvas

atoms

The building blocks to index the descriptor over.

Type:

list<BBlock>

confDs

The designs for conformations to index over.

Type:

list<Design>

integer

Flag to indicate if the descriptor takes integer values.

Type:

bool

binary

Flag to indicate if the descriptor takes boolean values.

Type:

bool

rules

List of rules to define and constrain the material descriptor design space.

Type:

list<DescriptorRules>

bounds

If tuple, the lower and upper bounds on the descriptor values across all indices. If dict, the bounds can be individually set for each index.

Type:

tuple/dict/func

See IndexedElem for more information on indexing. See DescriptorRule for information on defining descriptors.

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.

class idaes.apps.matopt.opt.mat_modeling.MatOptModel(canv, atoms=None, confDs=None)

A class for the specification of a materials optimization problem.

Once all the material information is specified, we use this class to specify the material design problem of interest. This class is intended to be interpretable without mathematical optimization background while the conversion to Pyomo optimization models happens automatically.

canv

The canvas of the material design space

Type:

Canvas

atoms

The list of building blocks to consider. Note: This list does not need to include a void-atom type. We use ‘None’ to represent the absence of any building block at a given site.

Type:

list<BBlock>

confDs

The list of conformations to consider.

Type:

list<Design>

maximize(func, **kwargs)

Method to maximize a target functionality of the material model.

Parameters:

• func (MaterialDescriptor/Expr) – Material functionality to optimize.

• **kwargs – Arguments to MatOptModel.optimize

Returns:

(Design/list<Design>) Optimal designs.

Raises:

• pyomo.common.errors.ApplicationError` if MatOpt can not fin –

• usable solver (CPLEX or NEOS-CPLEX) –

See MatOptModel.optimize method for details.

minimize(func, **kwargs)

Method to minimize a target functionality of the material model.

Parameters:

• func (MaterialDescriptor/Expr) – Material functionality to optimize.

• **kwargs – Arguments to MatOptModel.optimize

Returns:

(Design/list<Design>) Optimal designs.

Raises:

• pyomo.common.errors.ApplicationError` if MatOpt can not find usabl –

• solver (CPLEX or NEOS-CPLEX) –

See MatOptModel.optimize method for details.

optimize(func, sense, nSolns=1, tee=True, disp=1, keepfiles=False, tilim=3600, trelim=None, solver='cplex')

Method to create and optimize the materials design problem.

This method automatically creates a new optimization model every time it is called. Then, it solves the model via Pyomo with the CPLEX solver.

If multiple solutions (called a ‘solution pool’) are desired, then the nSolns argument can be provided and the populate method will be called instead.

Parameters:

• func (MaterialDescriptor/Expr) – Material functionality to optimize.

• sense (int) – flag to indicate the choice to minimize or maximize the functionality of interest. Choices: minimize/maximize (Pyomo constants 1,-1 respectively)

• nSolns (int) – Optional, number of Design objects to return. Default: 1 (See MatOptModel.populate for more information)

• tee (bool) – Optional, flag to turn on solver output. Default: True

• disp (int) – Optional, flag to control level of MatOpt output. Choices: 0: No MatOpt output (other than solver tee) 1: MatOpt output for outer level method 2: MatOpt output for solution pool & individual solns. Default: 1

• keepfiles (bool) – Optional, flag to save temporary pyomo files. Default: True

• tilim (float) – Optional, solver time limit (in seconds). Default: 3600

• trelim (float) – Optional, solver tree memory limit (in MB). Default: None (i.e., Pyomo/CPLEX default)

• solver (str) – Solver choice. Currently only cplex or neos-cplex are supported Default: cplex

Returns:

(Design/list<Design>) Optimal design or designs, depending on the number of solutions requested by argument nSolns.

Raises:

• pyomo.common.errors.ApplicationError` if MatOpt can not fin –

• usable solver (CPLEX or NEOS-CPLEX) –

populate(func, sense, nSolns, tee=True, disp=1, keepfiles=False, tilim=3600, trelim=None, solver='cplex')

Method to a pool of solutions that optimize the material model.

This method automatically creates a new optimization model every time it is called. Then, it solves the model via Pyomo with the CPLEX solver.

The populate method iteratively solves the model, interprets the solution as a Design object, creates a constraint to disallow that design, and resolves to find the next best design. We build a pool of Designs that are guaranteed to be the nSolns-best solutions in the material design space.

Parameters:

• func (MaterialDescriptor/Expr) – Material functionality to optimize.

• sense (int) – flag to indicate the choice to minimize or maximize the functionality of interest. Choices: minimize/maximize (Pyomo constants 1,-1 respectively)

• nSolns (int) – Optional, number of Design objects to return. Default: 1 (See MatOptModel.populate for more information)

• tee (bool) – Optional, flag to turn on solver output. Default: True

• disp (int) – Optional, flag to control level of MatOpt output. Choices: 0: No MatOpt output (other than solver tee) 1: MatOpt output for outer level method 2: MatOpt output for solution pool & individual solns. Default: 1

• keepfiles (bool) – Optional, flag to save temporary pyomo files. Default: True

• tilim (float) – Optional, solver time limit (in seconds). Default: 3600

• trelim (float) – Optional, solver tree memory limit (in MB). Default: None (i.e., Pyomo/CPLEX default)

• solver (str) – Solver choice. Currently only cplex or neos-cplex are supported Default: cplex

Returns:

(list<Design>) A list of optimal Designs in order of decreasing optimality.

Raises:

• pyomo.common.errors.ApplicationError` if MatOpt can not fin –

• usable solver (CPLEX or NEOS-CPLEX) –

MatOpt Output

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.