Bayesian optimization of discrete dislocation plasticity of two-dimensional precipitation hardened crystals Mika Sarvilahtiand Lasse Laurson

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Bayesian optimization of discrete dislocation plasticity

of two-dimensional precipitation hardened crystals

Mika Sarvilahti∗and Lasse Laurson

Computational Physics Laboratory, Tampere University,

P.O. Box 692, FI-33014 Tampere, Finland

(Dated: November 24, 2022)

Discovering relationships between materials’ microstructures and mechanical properties is a key goal of ma-

terials science. Here, we outline a strategy exploiting Bayesian optimization to efﬁciently search the multi-

dimensional space of microstructures, deﬁned here by the size distribution of precipitates (ﬁxed impurities or

inclusions acting as obstacles for dislocation motion) within a simple two-dimensional discrete dislocation dy-

namics model. The aim is to design a microstructure optimizing a given mechanical property, e.g., maximizing

the expected value of shear stress for a given strain. The problem of ﬁnding the optimal discretized shape for

a distribution involves a norm constraint, and we ﬁnd that sampling the space of possible solutions should be

done in a speciﬁc way in order to avoid convergence problems. To this end, we propose a general mathematical

approach that can be used to generate trial solutions uniformly at random while enforcing an Euclidean norm

constraint. Both equality and inequality constraints are considered. A simple technique can then be used to

convert between Euclidean and other Lebesgue p-norm (the 1-norm in particular) constrained representations.

Considering different dislocation-precipitate interaction potentials, we demonstrate the convergence of the al-

gorithm to the optimal solution and discuss its possible extensions to the more complex and realistic case of

three-dimensional dislocation systems where also the optimization of precipitate shapes could be considered.

I. INTRODUCTION

The need to develop novel materials with desired properties

for applications is the driving force behind much of mate-

rials science. One way of framing the problem is in terms

of structure-property relationships [1], where one aims at es-

tablishing relations between, say, the microstructural features

of a solid material and its mechanical properties [2]. In gen-

eral, the problem is very challenging due to the combination

of high dimensionality of microstructural descriptors (due to

the very large number of different microstructural features [3])

and non-linearities and statistical ﬂuctuations in the material

response to external stimuli [4, 5]. For these reasons, con-

ventional materials design strategies relying essentially on a

combination of educated guesses and trial and error are sub-

optimal and constitute a bottleneck for discovery of novel ma-

terials.

Recent years have witnessed the emergence of “smart”

methods in the toolbox of materials scientists, such as ma-

chine learning (ML) and optimization algorithms, which are

used to discover previously unknown dependencies of ma-

terial properties on a wide range of microstructural parame-

ters [6, 7]. Indeed, such developments are currently spawn-

ing a new research ﬁeld sometimes referred to as materials

informatics [8]. However, a large fraction of applications of

“ML for materials” are currently limited to considering atom-

istic and molecular properties of materials in ﬁelds such as

quantum chemistry [9]. Yet, the macroscopic mechanical

properties of realistic crystalline materials are largely con-

trolled by the microstructural features on scales well above

the atomic/molecular scale, calling for novel applications of

ML to discover novel microstructure-property relations using

microstructural features on a coarse-grained scale as input.

∗mika.sarvilahti@tuni.ﬁ

Here, we present an approach based on Bayesian optimiza-

tion to ﬁnd optimal values for the properties, such as the size

distribution, of precipitate particles within crystals resulting

in desired mechanical properties, such as maximal stress at

a given strain. Bayesian optimization [10–12] is known to

be an excellent choice for optimization problems such as the

present one in which evaluating a data point (here, performing

a discrete dislocation dynamics (DDD) simulation) is com-

putationally expensive and the outcome is stochastic (noisy).

The method can be applied to various problems, and there has

already been success in the context of optimization of, e.g.,

atomistic structures [13] and metamaterials [14]. Bayesian

optimization has also been used to calibrate the parameters of

a gradient plasticity model [15] that predicts the plastic size

effects of micropillars.

In this work, we apply Bayesian optimization to 2D disloca-

tion systems. The stress-strain response of such pure disloca-

tion systems has been studied both using stress-controlled [16,

17] and strain-controlled [18, 19] loading, along with recent

attempts to predict the response to external stresses by using

machine learning techniques [20, 21]. Like in these works, we

impose periodic boundary conditions for simplicity, although

it should be mentioned that recently, various methods have

been developed for simulating ﬁnite systems [22–24]. Our

system also contains ﬁxed round precipitates (or solute clus-

ters) acting as obstacles for the moving dislocations. Precip-

itation is known to cause various pinning effects, which have

been studied with 2D DDD simulations in the case of identi-

cal precipitates or pinning centres [25]. Here, we go further

and allow the precipitates to be of varying sizes with the aim

of employing Bayesian optimization to ﬁnd the optimal shape

for a discretized size distribution, subject to the constraint of

a ﬁxed volume fraction (area fraction in 2D) of precipitates,

resulting in a designed mechanical response of the material.

The design objective in our case is to maximize the average

shear stress required to produce a certain value of strain.

arXiv:2210.02242v2 [cond-mat.mtrl-sci] 23 Nov 2022

An alternative 2D dislocation modelling choice would be

to represent dislocations as lines on a single glide plane [26],

leading to a more complicated dislocation-precipitation inter-

action, but this would also make the set of dislocations form

an effectively one-dimensional pileup system [27, 28], which

is unable to capture some of the phenomena happening in sys-

tems with multiple glide planes, typically related to the com-

petition between dislocation jamming [18] and pinning due to

obstacles [25]. Certain cross-section models called the 2.5D

models [29] have also been developed with the aim of incor-

porating some of these effects into 2D simulations by utilizing

statistics related to such phenomena collected from 3D simu-

lations.

Realistic but computationally more demanding 3D DDD

models have also been considered, both without [30] and with

precipitates [31, 32]. We intend to extend our optimization

study to such systems after the approach has been tested and

polished for the 2D case, which is the focus of this work. In

the 3D case, bypassing mechanisms between dislocations and

solute clusters have been observed to change considerably de-

pending on the cluster size [33, 34], which motivates our at-

tempts to ﬁnd ways of taking advantage of such mechanisms

when designing materials.

Our work starts by explaining the speciﬁcs of the 2D DDD

model to be studied in Section II. Section III introduces the

Bayesian optimization method. In Section IV, we investigate

the speciﬁc problem of generating feasible points, which turns

out to be a critical point for ensuring optimization conver-

gence. The proof-of-concept test results are presented in Sec-

tion V, followed by discussion and conclusions in Section VI.

II. THE 2D DDD MODEL

The discrete dislocation model in two dimensions starts with

N=64 dislocation points, representing cross-sections of edge

dislocation lines, placed on a square simulation box. The dis-

locations have their Burgers vector along the x-axis of the xy-

plane, and the direction of the Burgers vector is ±ˆ

xwith even

portions of both signs. Each dislocation generates a shear

stress ﬁeld [25, 35]

σyx(r,sb) = µsb

2π(1−ν)

x(x2−y2)

(x2+y2)2,(1)

where r=x

yis position with respect to the dislocation with

Burger’s vector sign sand magnitude b, in a material with

shear modulus µand Poisson’s ratio ν. The dislocations are

allowed to move only in the x-direction.

The system also contains spherical precipitates which oc-

cupy 3% of the total area. An example of a relaxed conﬁg-

uration is presented in Figure 1. The precipitates are ﬁxed

in place and interact with dislocations through a spherically

symmetric Gaussian potential U,

U(r,R) = ARexp −1

2r

R2!,(2)

0 10 20 30 40

Position x

Position y

Precipitates Dislocations+ Dislocations-

FIG. 1. An example of a relaxed dislocation conﬁguration with

N=64 dislocations and periodic unit cell side length L=40. The

randomly positioned precipitates are shown as circles, and the T-

symbols correspond to dislocations, with different orientation and

color for opposite Burgers vectors.

where ris the distance from the center of the precipitate, Ris

the radius (the size) of the precipitate, and A=0.5 is a con-

stant scale parameter. The interaction force is the negative

partial derivative of Uwith respect to the x-coordinate.

We also consider an alternative potential Ualt having a

stronger scaling for the force magnitude with respect to R:

Ualt (r,R) = BR2exp −1

2r

R2!,(3)

where B=5.0 is another constant. Changing to this potential

changes the expected location of the optimum as the scaling

of the interaction force with respect to the precipitate size is

stronger.

All dislocations in this model are chosen to have the same

Burger’s vector magnitude b. Then, the equation of mo-

tion [25] for a dislocation positioned at riis

dtxi=sib2χ

∑

j6=i

σyx(ri−rj,sjb) + σext 

−χ∂

∂xUt ot al (ri),

(4)

with dislocation mobility χ, external shear stress σext , and

Ut ot al (r) = ∑kU(||r−rk||2,Rk), where kiterates over every

precipitate. A simpliﬁed unit system is chosen by setting the

variables b,χand µ

2π(1−ν)equal to 1. The square-shaped sys-

tem’s side length is L=40. Furthermore, we impose periodic

boundary conditions and take all the periodic images of dis-

locations into account in a ﬁnite form by modifying [35] the

long-range shear stress ﬁeld formula of Eq. (1). Then, inte-

grating the equation of motion while slowly (quasistatically)

increasing the external stress from zero (after relaxing the sys-

tem without external stress) makes the dislocations move, and

the strain [25]

ε(t) = 1

∑

i=1

sibxi(t)−xi(0)(5)

is measured. Two oppositely signed dislocations are annihi-

lated if their distance becomes less than b(= 1 in the cho-

sen unit system). The simulation ends when the average

strain εreaches the value 0.2 (a typical value in some recent

works [19–21]).

The response of a dislocation system to external shear

stress is described by a stress-strain curve. Figure 2 shows

how the average stress-strain curve depends on the precip-

itate size and on the choice of the interaction potential for

the case where all precipitates have the same size. In con-

trast to the average response, an individual response typi-

cally alternates between jammed states (with slight elastic de-

formation) and strain bursts (plastic deformation caused by

dislocation avalanches), producing staircase-like stress-strain

curves [16, 20]. Avalanches become more prominent in the

response curve the smaller the system is, and there can be

signiﬁcant differences between the responses of systems built

from the same recipe but having different random conﬁgu-

rations. From the perspective of optimization, this can be

viewed as noise. Therefore, information should be collected

from multiple conﬁgurations when determining the links be-

tween recipes and responses.

In practice, the level of noise in the system response can

be decreased by taking the sample mean over Mrandom con-

ﬁgurations. An alternative way to reduce noise would be to

increase the size of the dislocation system. We know that

the computational cost of a DDD simulation scales as ∼N2,

where Nis the number of dislocations, whereas running M

multiple simulations (parallel runs possible) simply multiplies

the computational cost by M, which means linear scaling ∼N

with respect to the total number of dislocations over the M

systems. Both methods typically decrease noise as ∼1/√N

(may not be exact when increasing the system size [16], but

still close to). This suggests that the most efﬁcient way to

reduce noise is to run many systems in parallel and to have

the averaged results be the observations for the optimization.

However, the system size should be large enough so that all

relevant physical phenomena can be observed and appear as

they would in a bulk system without too much additional un-

wanted effects due to small system size. If the objective would

instead be to control the ﬂuctuations around the average re-

sponse, then the method of performing many simulations for

each observation becomes compulsory; multiple responses are

required to obtain such statistical information.

A. The precipitate size distribution

Sizes for the precipitates are generated from a continuous,

piecewise uniform size distribution that describes how the

area of the simulation box that is reserved for precipitates is

portioned among different precipitate sizes. There are nine ad-

jacent pieces, each having a locally constant probability den-

sity. Each piece covers a size interval of width 0.1, and the

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Strain

0.1

0.2

0.3

Average Stress

R = 1

R = 0.7

R = 0.5

R = 0.3

R = 0.2

R = 0.15

R = 0.1

R = 0.07

R = 0.05

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Strain

0.1

0.2

0.3

0.4

Average Stress

R = 1

R = 0.7

R = 0.5

R = 0.3

R = 0.2

R = 0.15

R = 0.1

R = 0.07

R = 0.05

(a)

(b)

FIG. 2. Mean stress–strain curves σ(ε)(averaged curves over 100

random conﬁgurations) for nine different precipitation sizes R(num-

ber of dislocations N=64, volume fraction of precipitates =0.03, δ

distribution for precipitate sizes), using the interaction potential (a)

Ufrom Eq. (2) or (b) Ualt from Eq. (3). The size that maximizes

stress depends on the chosen potential. Our aim in this work is to

maximize the average stress needed for strain ε=0.2 with respect to

a precipitate size distribution that can take any shape (given a ﬁnite

resolution) instead of assuming one (such as a δor a Gaussian shape)

for the distribution.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Precipitate Size R

Count Prob. Density

Source Distribution

Realized Histogram

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Precipitate Size R

Area Prob. Density

Source (Total Rel. Area 3.00%)

Realized (Total Rel. Area 2.69%)

(Bar Heights Relative to Intended Area)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5

(a)

(b)

FIG. 3. An example of a precipitate size distribution with respect

to (a) the number of precipitates, (b) the area that the precipitates

of each size cover. The staircase curve corresponds to a realized

histogram made from an example set of size values drawn from the

source distribution. This set corresponds to the sizes of the precip-

itates illustrated in Figure 1. In this study, we attempt to optimize

a nine-dimensional vector made of the relative piece heights (area

portions) of the area-proportional, piecewise uniform source distri-

bution. The objective is to maximize the average stress needed to

cause a certain amount of strain.

centers of the pieces are evenly spaced between 0.1 and 0.9.

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Bayesianoptimizationofdiscretedislocationplasticityoftwo-dimensionalprecipitationhardenedcrystalsMikaSarvilahtiandLasseLaursonComputationalPhysicsLaboratory,TampereUniversity,P.O.Box692,FI-33014Tampere,Finland(Dated:November24,2022)Discoveringrelationshipsbetweenmaterials'microstructuresandmechanic...

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Bayesian optimization of discrete dislocation plasticity of two-dimensional precipitation hardened crystals Mika Sarvilahtiand Lasse Laurson

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