HIGHER-ORDER FAR-FIELD BOUNDARY CONDITIONS FOR CRYSTALLINE DEFECTS JULIAN BRAUN CHRISTOPH ORTNER YANGSHUAI WANG AND LEI ZHANG

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HIGHER-ORDER FAR-FIELD BOUNDARY CONDITIONS

FOR CRYSTALLINE DEFECTS

JULIAN BRAUN, CHRISTOPH ORTNER, YANGSHUAI WANG, AND LEI ZHANG

Abstract. Crystalline materials exhibit long-range elastic ﬁelds due to the presence of defects,

leading to signiﬁcant domain size eﬀects in atomistic simulations. A rigorous far-ﬁeld expansion

of these long-range ﬁelds identiﬁes low-rank structure in the form of a sum of discrete multipole

terms and continuum correctors [3]. We propose a novel numerical scheme that exploits this low-

rank structure to accelerate material defect simulations by minimizing the domain size eﬀects. Our

approach iteratively improves the boundary condition, systematically following the asymptotic

expansion of the far ﬁeld. We provide both rigorous error estimates for the method and a range of

empirical numerical tests, to assess it’s convergence and robustness.

1. Introduction

This work is concerned with the precise characterization of geometric and energetic proper-

ties of defects within crystalline materials [13, 14, 24, 27]. Practical simulation schemes operate

within ﬁnite domains and hence cannot fully resolve the long-ranged elastic far-ﬁeld, hence the

computation of these properties requires the careful consideration of artiﬁcial (e.g., clamped or

periodic) boundary conditions [4, 11]. The choice of boundary condition signiﬁcantly inﬂuences

the emergence of cell-size eﬀects, an issue that will be studied in detail in this paper.

Standard supercell methods for modeling defect equilibration are recognized for their relatively

slow convergence concerning cell size [7, 11]. Addressing this limitation systematically necessitates

the development of higher-order boundary conditions, aiming to improve the convergence rates in

terms of cell size. An important step in this endeavor is the careful analysis of the elastic far-ﬁelds

induced by the presence of defects.

A common approach to characterize the elastic far-ﬁeld behavior is to leverage the low-rank

structure of defect conﬁgurations. This involves modeling defects using continuum linear elasticity

and the defect dipole tensor, ﬁrst introduced in [12, 19]. This concept was later applied to atomistic

models of fracture [22, 23]. More recent related works utilize lattice Green’s functions to improve

accuracy in defect computations [25, 26]. However, these contributions tend to be application-

focused, lacking a rigorous mathematical foundation and framework for systematically designing

and improving the models and numerical algorithms.

Recently, Braun et al. [2, 3] developed a uniﬁed mathematical framework that exploits the low-

rank defect structure to characterize the elastic far-ﬁelds. In this framework, the defect equilibrium

is decomposed into a sum of continuum correctors and discrete multipole terms. This novel for-

mulation exposes avenues for improved convergence rates in cell problems concerning cell size, as

demonstrated theoretically. However, a notable challenge arises in the practical implementation

of multipole expansions for simulating crystalline defects, given that the terms associated with

multipole moments are deﬁned on an inﬁnite lattice, rendering their direct evaluation unfeasible

within ﬁnite computational domains.

The purpose of the present paper is to propose a novel numerical framework that utilizes the

multipole expansions of Braun et al. [2, 3] to accelerate the simulation of crystalline defects. We

Date: April 2, 2024.

arXiv:2210.05573v2 [math.NA] 31 Mar 2024

2 JULIAN BRAUN, CHRISTOPH ORTNER, YANGSHUAI WANG, AND LEI ZHANG

design an iterative method that systematically enhances the accuracy of approximate multipole

tensor evaluations, accompanied by rigorous error estimates. Additionally, we leverage a contin-

uous version of multipole expansions, employing continuous Green’s functions to further enhance

computational eﬃciency. To evaluate the eﬀectiveness of our approach, we present numerical

examples for a range of point defects, assessing both the geometry error and energy error conver-

gence. Our numerical results demonstrate that the proposed framework for higher-order boundary

conditions eﬀectively achieves accelerated convergence rates with respect to computational cell

size.

This paper concentrates on point defects to provide a comprehensive understanding of key

concepts. Future research will delve into broader applications of this approach. However, extending

the method to complex scenarios like edge dislocations, cracks or grain boundary structures poses

fundamental challenges beyond the scope of the current analysis, necessitating also the development

of new theory.

Outline. The paper is organized as follows: In Section 2, we provide an overview of the variational

formulation for the equilibration of crystalline defects and review the result on multipole expansions

(cf. [3, Theorem 3.1]) that provides a general characterization of the discrete elastic far-ﬁelds

surrounding point defects. In Section 3, we propose a numerical framework that leverage the

multipole expansions to accelerate the simulation of crystalline defects. We utilize continuous

multipole expansions instead of discrete ones to obtain an eﬃcient implementation. In Section 4,

we apply our main algorithm (cf. Algorithm 3.3) to various prototypical numerical examples of

point defects. Section 5 presents a summary of our work and future directions. The proofs as

well as additional analysis that can aid in understanding the main idea of this paper, are given in

Section 6.

Notation. We use the symbol ⟨·,·⟩ to denote an abstract duality pairing between a Banach space

and its dual. The symbol |·| normally denotes the Euclidean or Frobenius norm, while ∥·∥ denotes

an operator norm. We denote A\{a}by A\a, and {b−a|b∈A}by A−a. For E∈C2(X), the

ﬁrst and second variations are denoted by ⟨δE(u), v⟩and ⟨δ2E(u)v, w⟩for u, v, w ∈X.

We write |A|≲Bif there exists a constant Csuch that |A| ≤ CB, where Cmay change from

one line of an estimate to the next. When estimating rates of decay or convergence, Cwill always

remain independent of the system size, the conﬁguration of the lattice and the the test functions.

The dependence of Cwill be normally clear from the context or stated explicitly.

Given a k-tuple of vectors in Rd,σ= (σ(1), . . . , σ(k))∈(Rd)k. We write the k-fold product

σ⊗∈(Rd)⊗kas

σ⊗:=

i=1

σ(i):= σ(1) ⊗ · · · ⊗ σ(k).

Similarly, we write v⊗k:= v⊗... ⊗v∈(Rd)⊗kfor v∈Rd.

For any σ∈(Rd)k, we deﬁne the symmetric tensor product by

σ⊙:= σ(1) ⊙ · · · ⊙ σ(k):= sym σ⊗:= 1

k!X

g∈Sk

g(σ)⊗,

where Skis the usual symmetric group of all permutations acting on the integers {1, . . . , k}and

g(σ) := (σ(g(1)), . . . , σ(g(k))) for any g∈Skand σ∈(Rd)k.

HIGHER-ORDER FAR-FIELD BOUNDARY CONDITIONS 3

The natural scalar product on (Rd)⊗kis denoted by A:Bfor A, B ∈(Rd)⊗k, which is deﬁned

to be the linear extension of

σ⊗:ρ⊗:=

i=1

σ(i)·ρ(i).

For two second-order tensors C,U∈(RA)⊗k, given speciﬁcally as a sum C=Pρ∈AkCρEρwith

Eρthe natural basis of the space Rdk, we can then write

C:U=X

ρ∈Ak

CρUρ.

2. Background: Equilibrium of crystalline defects and its multipole expansions

Our work concerns the modeling of crystalline defects, with particular emphasis on single point

defects embedded within a homogeneous crystalline bulk, a setting that allows a detailed and

rigorous analysis of our approach. To motivate the formulation of our main results in this context,

we ﬁrst review and adapt the framework introduced in [7, 11, 16] in Section 2.1. Subsequently, in

Section 2.2, we provide a brief summary of the multipole expansion of equilibrium conﬁgurations

proposed in [3], which serves as the cornerstone for this work.

2.1. Equilibrium of crystalline defects. Let d∈ {2,3}be the dimension of the system. A

homogeneous crystal reference conﬁguration is given by the Bravais lattice Λ = AZd, for some non-

singular matrix A∈Rd×d. We admit only single-species Bravais lattices. There are no conceptual

obstacles to generalising our work to multi-lattices, however, the technical details become more

involved. The reference conﬁguration with defects is a set Λdef ⊂Rd. The mismatch between Λdef

and Λ represents possible defected conﬁgurations. We assume that the defect cores are localized,

that is, there exists Rdef >0, such that Λdef \BRdef = Λ\BRdef . We refer to Figure 2.1 for a two

dimensional example with Adeﬁned by [20, Eq. (4.3)] specifying a triangular lattice.

Rdef

Figure 2.1. 2D triangular lattice: Reference lattice Λ (left); Defective lattice Λdef

with one self-interstitial atom inside BRdef (right).

The displacement of the inﬁnite lattice Λ is a map u: Λ →RN. Typically, N=d, but both N < d

and N > d arise e.g. in dimension-reduced models. For ℓ, ρ ∈Λ, we denote discrete diﬀerences by

Dρu(ℓ) := u(ℓ+ρ)−u(ℓ). To introduce higher discrete diﬀerences we denote by Dρ=Dρ1· · · Dρj

for a ρ= (ρ1, ..., ρj)∈Λj. For a subset Rℓ⊂Λ−ℓ, we deﬁne Du(ℓ) := DRℓu(ℓ) := Dρu(ℓ)ρ∈R.

For the sake of simplicity, we assume throughout that Rℓis ﬁnite for each ℓ∈Λ. An extension to

inﬁnite interaction range is possible but involves additional technical complexities [7, 8].

4 JULIAN BRAUN, CHRISTOPH ORTNER, YANGSHUAI WANG, AND LEI ZHANG

We deﬁne two useful discrete energy space by

H1(Λdef ) := u: Λdef →RN∥Du∥ℓ2<∞,

Hc(Λdef ) := u: Λdef →RNsupp(Du) bounded ,

where Hcis a dense subspace of H1with compact support. Analogously, H1(Λ) and Hc(Λ) can

be deﬁned for displacements on homogeneous lattice.

We consider the site potential to be a collection of mappings Vℓ: (RN)Rℓ→R, which represent

the energy distributed to each atomic site. We make the following assumption on regularity and

symmetry: Vℓ∈CK((RN)Rℓ) for some Kand Vℓis homogeneous outside the defect region BRdef ,

namely, Vℓ=Vand Rℓ=Rfor ℓ∈Λ\BRdef . Furthermore, Vand Rhave the following point

symmetry: R=−R, it spans the lattice spanZR= Λ, and V({−A−ρ}ρ∈R) = V(A). We refer

to [7, §2.3 and §4] for a detailed discussion of those assumptions and symmetry.

The potential energy under the displacements ﬁeld are given by

Edef (u) := X

ℓ∈Λdef hVℓDRℓu(ℓ)−Vℓ0i,E(u) := X

ℓ∈ΛhVDu(ℓ)−V0i.(2.1)

It is shown in [11, Lemma 1] that Edef (resp. E) is well deﬁned on Hc(Λdef ) (resp. Hc(Λ)) and has

a unique continuous extension to H1(Λdef ) (resp. H1(Λ)).

The main objective in this work is the characterisation of the far-ﬁeld behaviour of lattice

displacements u: Λ →RNthat are close to equilibrium. Following the works [3, 11], it is important

to characterise the linearised residual forces f(ℓ) := H[u](ℓ), where

(2.2) H[u](ℓ) := −Div∇2V(0)[Du],

with Div A=−Pρ∈R D−ρA·ρthe discrete divergence for a matrix ﬁeld A: Λ →RN×R. Given

i∈N, if ℓ7→ H[u](ℓ)⊗ℓ⊗i∈ℓ1(Λ), we deﬁne the i-th force moment

(2.3) Ii[u] = X

ℓ∈Λ

H[u](ℓ)⊗ℓ⊗i,

which serves as a key concept in the following analysis and algorithms.

We assume throughout that the Hamiltonian H=δ2E(0) is stable (cf. [3, Eq. (4)]). For stable

operator Hthere exists a lattice Green’s function (inverse of H)G: Λ →RN×Nsuch that

H[Gek](ℓ) = ekδℓ,0,for 1 ≤k≤N.(2.4)

We write Gk:= Gekfor simplicity.

The equilibrium displacement ¯udef ∈ H1(Λdef ) satisﬁes

(2.5) δEdef (¯udef )[v] = 0 ∀v∈ Hc(Λdef ).

For the purpose of the following analysis, it is advantageous to project ¯udef onto the homogeneous

lattice Λ denoted by ¯u. The possible projections are not unique and will not inﬂuence the main

results; we therefore refer to [3, Section 3.1] for details.

It was shown [7, 11] that the equilibrium displacement has a generic decay |D¯udef (ℓ)| ≤ C|ℓ|−d,

which gives rise to a slow convergence of standard supercell methods with cell size [7, 11]. The

multipole expansion we introduce next gives additional information about the equilibrium far ﬁeld

and allows us to construct improved boundary conditions.

HIGHER-ORDER FAR-FIELD BOUNDARY CONDITIONS 5

2.2. Multipole expansion of equilibrium ﬁelds. In this section, we brieﬂy review the results

on the multipole expansion of the equilibrium ¯ufor point defects [3], which provides the general

structure for characterising the discrete elastic far-ﬁelds induced by defects.

Since ¯u= ¯udef outside of the defect core, for |ℓ|large enough, we can obtain

δE(¯u)(ℓ) = δE(¯u)[δℓ]=0,

where δℓ(ℓ′) := δℓℓ′. As a matter of fact, it is shown in [11] that

δE(¯u)[v] = (g, Dv)ℓ2∀v∈ H1(Λ),

where g: Λ →Rd×R with supp(g)⊂BRdef . The following Theorem taken from [3, Theorem 3.1] is

provided here for the sake of completeness.

Theorem 2.1. Choose p≥0, J ≥0and suppose that V∈CK(Rd×R), such that K≥J+ 2 +

max{0,⌊p−1

d⌋}. Let g: Λ →Rd×R with compact support, and let ¯u∈ H1(Λ) such that

δE(¯u)[v]=(g, Dv)ℓ2∀v∈ Hc(Λ).

Furthermore, let S ⊂ Λbe linearly independent with spanZS= Λ and G: Λ →RN×Nbe a lattice

Green’s function deﬁned by (2.4). Then, there exist uC

i∈C∞and coeﬃcients b(i,k)

exact ∈(RS)⊙isuch

that

¯u=

i=d+1

i=1

k=1

b(i,k)

exact :Di

SGk+rp+1,(2.6)

where uC

isatisﬁes the PDEs in [3, Eq.(59)] for d+ 1 ≤i≤pwhile uC

i= 0 for 0≤i≤d.

Furthermore, for j= 1, . . . , J, the remainder term rp+1 satisﬁes the estimate

|Djrp+1|≲|ℓ|1−d−j−plogp+1(|ℓ|).(2.7)

Remark 2.1. Since uC

i= 0 for 0≤i≤d, one can obtain the pure multipole expansion up to the

order p=d

¯u=

i=1

k=1

b(i,k)

exact :Di

SGk+rd+1,where |Djrd+1|≲|ℓ|1−2d−jlogd+1(|ℓ|).(2.8)

Moreover, as discussed in [3, Remark 3.4], it is possible to reduce the number of logarithms in the

estimate somewhat for all orders in (2.7) and (2.8). Despite this, we choose to include them as

they do not have a signiﬁcant impact on the core algorithms presented in this work.

The foregoing theorem oﬀers a framework for characterizing the discrete elastic far-ﬁelds encom-

passing crystalline point defects. This characterization relies on the decomposition (2.8), with a

particular focus on the multipole terms comprising the coeﬃcients b(i,k)

exact and the lattice Green’s

function G. By determining these two crucial components, it becomes theoretically feasible to

attain the desired regularity and decay of the remaining term rd+1. This, in turn, facilitates the

establishment of higher-order boundary conditions, which constitutes the primary objective of our

present work.

As demonstrated in [3, Lemma 5.6], the coeﬃcients b(i,k)

exact are theoretically obtainable through a

linear transformation Ii(¯u)·k= (−1)ii!X

ρ∈Si

(b(i,k)

exact)ρ·ρ⊙,(2.9)

where the force moments are deﬁned by (2.3). For a detailed derivation of (2.9) especially when

i= 1,2,3, we refer to Section 6.2 (cf. (6.45)). It is worth noting, however, that both the force

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HIGHER-ORDERFAR-FIELDBOUNDARYCONDITIONSFORCRYSTALLINEDEFECTSJULIANBRAUN,CHRISTOPHORTNER,YANGSHUAIWANG,ANDLEIZHANGAbstract.Crystallinematerialsexhibitlong-rangeelasticfieldsduetothepresenceofdefects,leadingtosignificantdomainsizeeffectsinatomisticsimulations.Arigorousfar-fieldexpansionoftheselong-ran...

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HIGHER-ORDER FAR-FIELD BOUNDARY CONDITIONS FOR CRYSTALLINE DEFECTS JULIAN BRAUN CHRISTOPH ORTNER YANGSHUAI WANG AND LEI ZHANG

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