computational multiscale modeling · here and in the following, we use dots to denote rates, i.e.,...

Computational Multiscale Modeling (151-0520-00L) May 22, 2018Spring 2018 Prof. Dennis M. Kochmann, ETH Zurich

Computational Multiscale Modeling

(Spring 2018)

Dennis M. Kochmann

Mechanics & MaterialsDepartment of Mechanical and Process Engineering

ETH Zurich

course website:www.mm.ethz.ch/teaching.html

Copyright © 2018 by Dennis M. Kochmann

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http://www.mm.ethz.ch/teaching.html


These lecture notes are a concise collection of equations and comments. They are by nomeans a complete textbook or set of class notes that could replace lectures.

Therefore, you are strongly encouraged to take your own notes during lectures and to use thisset of notes rather for reference.

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Contents

0 Review: Continuum Mechanics and Notation 6

0.1 An introductory example: heat conduction . . . . . . . . . . . . . . . . . . . . . . . 6

0.2 Mechanical equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

0.3 Linearized kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

0.4 Initial boundary value problems (IBVPs) . . . . . . . . . . . . . . . . . . . . . . . . 13

1 Microstructure & Unit Cells 15

1.1 Ensembles and Representative Volume Elements . . . . . . . . . . . . . . . . . . . . 15

2 Averaging Theorems 17

2.1 Averaging Theorems in Linearized Kinematics . . . . . . . . . . . . . . . . . . . . . 17

2.2 Averaging Theorems in Finite Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Numerical Evaluation of RVE Averages 19

3.1 Average Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Average Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Linearized Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Homogenization Problem 21

5 Micro-to-Macro Transition 23

5.1 Effective Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 The Hill-Mandel Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3 Example: Affine Deformation of the RVE . . . . . . . . . . . . . . . . . . . . . . . . 25

6 RVE Boundary Conditions 27

6.1 Uniform Traction BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.2 Affine Displacement BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.3 Periodic BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.4 Enforcing RVE Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Numerical Aspects of Periodic Boundary Conditions 32

7.1 Enforcing Average Deformation Gradients: . . . . . . . . . . . . . . . . . . . . . . . 33

7.2 Enforcing average stress components: . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.3 Enforcing mixed averages: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.4 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.5 Linearized Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 The FE2 Method 36

8.1 Affine BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9 Analytical Techniques: Cauchy-Born Kinematics 38

9.1 Local and nonlocal expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9.2 Applications and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 Spectral Techniques 42

10.1 Reciprocal Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10.2 Fourier Spectral Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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10.3 Example: Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10.4 Mechanical Problem using Fourier Spectral Techniques . . . . . . . . . . . . . . . . 46

10.4.1 Finite Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.4.2 Linearized Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.5 Fourier-Related Stability Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11 Bloch Wave Analysis 55

12 From Quantum Mechanics to Atomistics 56

12.1 A Cartoon Introduction to Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 56

12.2 Density Functional Theory in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . 60

12.3 Hamiltonian Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13 Interatomic Potentials 63

13.1 Pair Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13.2 (An)Harmonicity and the Quasiharmonic Approximation . . . . . . . . . . . . . . . 66

13.3 Multi-Body Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13.4 Centrosymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

13.4.1 Infinite vs. finite-size samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14 Cauchy-Born Approximation and Effective Properties at Zero Temperature 70

15 Molecular Dynamics 73

15.1 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

15.2 Center of mass coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

15.3 The microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

16 Atomistic Solution Algorithms 80

16.1 Zero-Temperature Molecular Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

16.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

16.3 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

16.4 Thermostats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

17 Multiscale Modeling Techniques 85

18 The Quasicontinuum Method 89

18.1 Representative atoms and the quasicontinuum approximation . . . . . . . . . . . . 89

18.2 Summation rules and spurious force artifacts . . . . . . . . . . . . . . . . . . . . . . 92

18.2.1 Energy-based QC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

18.2.2 Force artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

18.2.3 Force-based QC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

18.2.4 Local/nonlocal QC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

18.2.5 Repatom masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

18.2.6 Example: Embedded Atom Method . . . . . . . . . . . . . . . . . . . . . . . 96

18.3 Adaptive refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

18.4 Features and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

18.4.1 Multilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

18.4.2 Finite temperature and dynamics . . . . . . . . . . . . . . . . . . . . . . . . 98

18.4.3 Heat and mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

18.4.4 Ionic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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18.4.5 Beyond atomistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

19 Applications 102

19.1 Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

19.2 Surface effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

19.3 Truss networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

19.4 Summary and open challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

20 Concurrent Multiscale Challenges at Finite Temperature 111

20.1 Finite-Temperature Quasicontinuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

20.1.1 Maximum-Entropy HotQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

20.1.2 Quasiharmonic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 113

20.2 Irving-Kirkwood Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

21 Discrete Dislocation Dynamics 115

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0 Review: Continuum Mechanics and Notation

0.1 An introductory example: heat conduction

We describe a body Ω ⊂ Rd with boundary ∂Ω as a collection of material points. Each pointhas a position x in a Cartesian coordinate system (x1, . . . , xd) in d dimensions with origin O.

Points are described by vectors defined by components in the Cartesian reference frame:

x =d

∑i=1

xigi = xigi. (0.1)

Here and in the following we use Einstein’s summation convention which implies summationover repeated indices. The usual index notation rules apply; e.g., the inner product is writtenas

a ⋅ b = aigi ⋅ bjgj = aibjδij = aibi with Kronecker’s delta δij =⎧⎪⎪⎨⎪⎪⎩

1 if i = j,0 else.

(0.2)

This is used to define the length of a vector as

∥a∥ =√a ⋅ a =

√aiai =

¿ÁÁÀ d

∑i=1

aiai. (0.3)

Matrix-vector multiplication becomes

a =Mb ⇔ ai =Mijbj and (MT)ij =Mji. (0.4)

We use mappings to denote fields. For example, the temperature field in a static problem isdescribed by a mapping

T (x) ∶ Ω→ R, (0.5)

which assigns to each point x ∈ Ω a real number, the temperature. If the field is differentiable,one often introduces kinematic variables such as the temperature gradient field:

β ∶ Ω→ Rd and β = gradT = ∇T ⇔ βi =∂T

∂xi= T,i. (0.6)

Here and in the following, we use comma indices to denote partial derivatives.

For every kinematic variable, there is conjugate field (often called flux) like the heat flux q inthis thermal problem, which is also a mapping:

q ∶ Ω→ Rd. (0.7)

The heat flux vector q assigns to each point in Ω a heat flux direction and magnitude. If weare interested, e.g., in the loss of heat through a point x ∈ ∂Ω on the surface of Ω with outwardunit normal n(x), then that amount of heat leaving Ω is the projection q(x) ⋅ n(x). Clearly,the components of q = (q1, . . . , qd)T imply the heat flux through surfaces perpendicular to eachof the d Cartesian coordinate directions.

Next, constitutive relations link kinematic quantities to fluxes. For example, define the heatflux vector as q = q(β). Fourier’s law of heat conduction states, e.g.,

q = −Kβ ⇔ qi = −Kijβ,j , (0.8)

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whereK denotes a conductivity tensor. Kij are the components of the conductivity tensor; theyform a d × d matrix. Such a second-order tensor is a convenient way to store the conductivityproperties in any arbitrary orientation in the form of a matrix, along with the coordinate basisin which it is defined. K provides for each direction of the temperature gradient β = αn theresulting normalized heat flux q. To this end, one defines

K =Kijgi ⊗ gj ⇒ Kn = (Kijgi ⊗ gj)n =Kijgi(gj ⋅n) =Kijnjgi. (0.9)

Such a second-order tensor is hence a linear mapping of vectors onto vectors. Here, we definedthe dyadic product (or tensor product), which produces a tensor according to

M = a⊗ b ⇔ Mij = aibj . (0.10)

A special tensor is the identity, which maps vectors onto themselves:

I = δij gi ⊗ gj . (0.11)

To solve a thermal problem, we also need balance laws. Here, conservation of energy may beused, which states the change of internal energy E in a body over time t balances the inwardand outward flux of energy and the energy being produced inside the body (by some heat sourcedensity ρs). Mathematically, this implies

d

dtE = ∫

Ωρs dV − ∫

∂Ωq ⋅n dS. (0.12)

We can use the divergence theorem to write

∫∂Ωq ⋅n dS = ∫

∂Ωqini dS = ∫

Ωqi,i dS = ∫

Ωdivq dS, (0.13)

which defines the divergence of a vector as the scalar quantity

div(⋅) = (⋅)i,i. (0.14)

Energy is an extensive variable, i.e., the total energy doubles when adding to bodies of thesame energy. This is in contrast to intensive variables, such as temperature or pressure, whenadding to bosides having the same, e.g., temperature. Since energy is an extensive variable, wecan introduce an energy density e and write

E = ∫Ωe dV ⇒ d

dtE = ∫

Ωe dV . (0.15)

Here and in the following, we use dots to denote rates, i.e., time derivatives. Note that we usea Lagrangian description and that, so far, there is no motion or deformation involved in ourdiscussion, so the time derivative does not affect the volume integral. Rewriting the conservationof energy now yields

∫Ωe dV = ∫

Ωρs dV − ∫

Ωdivq dV . (0.16)

This can be rewritten as

∫Ω(e − ρs + divq) dV = 0. (0.17)

Since conservation of energy does not only have to hold for Ω but for any subbody ω ⊂ Ω, wemay conclude that the local energy balance equation is

e = ρs − divq (0.18)

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This is the local (i.e., pointwise) counterpart to the macroscopic energy balance and states thatat each point x ∈ Ω the rate of energy change (e) is given by the local production of heat (ρs)minus the heat lost by outward fluxes q away from the point.

Finally, exploiting that thermally stored energy gives e = ρcvT (with constant mass density ρand specific heat capacity cv), and we insert Fourier’s law of heat conduction to overall arriveat

∫ΩρcvT dV = ∫

Ωρs dV − ∫

Ωdiv(−Kβ) dS = ∫

Ωρs dV − ∫

Ω(−KijT,j),i dS. (0.19)

Note that the final term requires the use of the product rule since

(−KijT,j),i = −Kij,iT,j −KijT,ij . (0.20)

Let us assume a homogeneous body with K(x) =K = const. Further, let us rewrite the aboveunder a single integral:

∫Ω(ρcvT − ρs −KijT,ij) dV = 0. (0.21)

Again, by extension of energy conservation to arbitrary subbodies, we conclude the local energybalance equation

ρcvT =KijT,ij + ρs (0.22)

This is the heat equation in its anisotropic form. In the special case of isotropy (i.e., theconductivity is the same in all directions), we obtain the Laplacian since

Kij = κδij ⇒ KijT,ij = κδijT,ij = κT,ii = κ∆T with ∆(⋅) = (⋅),ii, (0.23)

and we arrive at the well-known heat equation with sources:

ρcvT = κ∆T + ρs (0.24)

Whenever we consider a static problem, we assume that the body is in equilibrium and thetemperature field is constant, which reduces the above to Poisson’s equation, viz.

κ∆T = −ρs. (0.25)

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0.2 Mechanical equilibrium

The mechanics of solids (and fluids) generally describes deformable bodies. To this end, we labeleach material point by its position X in a reference configuration (e.g., the configuration attime t = 0). The current position x, by contrast, is a function of X and of time t: x = x(X, t).Fields in the reference and current configuration are generally referred to by upper- andlower-case characters (and the same applies to indices), e.g.,

x = xigi, X =XIGI . (0.26)

Note that, to avoid complication, we will work with Cartesian coordinate systems only (see thetensor notes, e.g., for curvilinear coordinates).

The mechanics of a deformable body undergoing finite deformations is generally described bya deformation mapping

ϕ(X, t) ∶ Ω ×R→ Rd such that x = ϕ(X, t). (0.27)

Since it depends on time, we can take time derivatives to arrive at the velocity and acceler-ation fields, respectively:

V (X, t) = d

dtx(X, t) = d

dtϕ(X, t), A(X, t) = d

dtV (X, t) = d2

dt2ϕ(X, t). (0.28)

Note that those are Lagrangian fields (one could also write those as functions of the currentposition x, which results in the Eulerian counterparts; this is usually done in fluid mechanics).

Like in the thermal problem, we introduce kinematics by defining the deformation gradient

F = Gradϕ ⇔ FiJ =∂ϕi∂XJ

= ϕi,J . (0.29)

Note that this is a second-order, two-point tensor defined across both configurations. If one usesthe same coordinate frame for the undeformed and deformed configurations, one may alternativeintroduce the displacement field

u(X) = x −X and x =X +u(X), (0.30)

so that

F = Gradϕ = Grad(X +u) = I +Gradu ⇔ FiJ = δiJ + ui,J . (0.31)

Note that in case of no deformation, we have x =X so that F = I (and not F = 0).

F contains plenty of information about the local deformation. For example, the volume changeat a point is given by

dv

dV= J = detF , (0.32)

and for physical reasons we must have J > 0 (this ensures that the deformation mapping is injec-tive; i.e., no two material points are mapped onto the same point in the current configuration).No volume change implies J = 1.

Similarly, the stretch in a direction defined by a unit vector N is determined by

λ(N) = ds

dS=√N ⋅CN with C = FTF (0.33)

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the right Cauchy-Green tensor. As for F , an undeformed point has C = I.

Next, we need a constitutive law that links the deformation gradient to a “flux”, which inmechanical problems we refer to as the stress. Heat flux may be defined as energy flow peroriented area,

qi =dQ

dAiwith dA =N dA and ∥N∥ = 1. (0.34)

Analogously, stresses are force vector per oriented area:

PiJ =dFidAJ

, (0.35)

which defines the so-called First Piola-Kirchhoff (1st PK) stress tensor. This is a second-order tensor that captures the force components on any oriented area into all d coordinatedirections. The resulting traction vector on a particular infinitesimal area with unit normalvector N is

T = PN ⇔ Ti = PiJNJ . (0.36)

Notice that the thus defined tractions satisfy Newton’s law of action and reaction since

T (N) = PN = −P (−N) = −T (−N), (0.37)

which we know well from inner forces in undergraduate mechanics. The total force acting on asurface A is hence

Ftot = ∫AT dS. (0.38)

For a mechanical problem, the relevant balance laws are conservation of linear momentumand of anguluar momentum. As before, one can formulate those as macroscopic balance laws.For example, macroscopic linear momentum balance is nothing but the well-known equationFtot =MA (sum of all forces equals mass times mean acceleration). To derive the local balancelaw of a continuous body, note that external forces include both surface tractions T and bodyforces ρ0B – using ρ0 to denote the reference mass density. Overall, we obtain

∫∂ΩT dS+∫

Ωρ0B dV = ∫

Ωρ0A dV ⇔ ∫

∂ΩPN dS+∫

Ωρ0B dV = ∫

Ωρ0A dV . (0.39)

Practicing the divergence theorem once more, we see that

∫∂ΩPiJNJ dS = ∫

ΩPiJ,J dV ⇒ (DivP )i = PiJ,J , (0.40)

which defines the divergence of a second-order tensor, which is a vector. Note that we usea capitol operator “Div” as opposed to “div” to indicate differentiation with respect to theundeformed coordinates.

When we again exploit that the above balance law must hold for all subbodies ω ⊂ Ω, we arriveat the local statement of linear momentum balance:

DivP + ρ0B = ρ0A ⇔ PiJ,J + ρ0Bi = ρ0Ai (0.41)

Note that the special case of quasistatics assumes that inertial effects are negligible, so onesolves the quasistatic linear momentum balance DivP + ρ0B = 0. Except for gravity or elec-tro/magnetomechanics, body forces also vanish in most cases, so that one simply arrives atDivP = 0.

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It is important to recall that stresses in finite deformations are not unique but we generallyhave different types of stress tensors. Above, we introduced the first Piola-Kirchhoff stresstensor P , which implies actual force per undeformed area. Similarly, one can define the Cauchystress tensor σ, which denotes actual force per deformed area. The definition and link aregiven by

σij =dFidaj

, σ = 1

JPFT. (0.42)

Later on, it will be helpful to link stresses and deformation to energy. If the stored mechanicalenergy is characterized by the strain energy density W =W (F ), then one can show that

P = ∂W∂F

⇔ PiJ =∂W

∂FiJ. (0.43)

Without knowing much about tensor analysis, we may interpret the above as a derivative ofthe energy density with respect to each component of F , yielding the corresponding componentof P . This concept can also be extended to introduce a fourth-order tensor, the incrementaltangent modulus tensor

C = ∂P∂F

⇔ CiJkL = ∂PiJ∂FkL

, (0.44)

for which each component of P is differentiated with respect to each component of F . Forfurther information on tensor analysis, see the tensor notes. Without further discussion, noticethat a stress-free state implies that the energy attains an extremum.

Note that the dependence of W on F (and consequently the consitutive law between P and F )is generally strongly nonlinear, which limits opportunities for closed-form analytical solutions.Also, for material frame indifference we must in fact have W =W (C), but that is a technicaldetail of minor importance here.

0.3 Linearized kinematics

Whenever only small deformation is expected, the above framework can be significantly sim-plified by using linearized kinematics. To this end, we assume that ∥Gradu∥ ≪ 1 (“smallstrains”). Note that in this case it does not make a significant difference if we differentiate withrespect to xi or XI , so that one generally uses only lower-case indices for simplicity.

In small strains, the displacement field is the key field to be determined (rather than thedeformation mapping), i.e., we seek u = u(x, t).

Recall that

C = FFT = (I +Gradu)(I +GraduT) = I +Gradu+(Gradu)T+(Gradu)(Gradu)T. (0.45)

Now, the final term is dropped by a scaling argument (∥Gradu∥ ≪ 1). Therefore, we mayintroduce a kinematic relation like in the thermal problem:

β = Gradu, (0.46)

and all important local deformation information is encoded in β. Like a temperature gradientcauses heat flux, a displacement gradient causes stresses (if displacements are constant every-where, the body is undergoing rigid body translation and does not produce any stresses).

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To make sure we also do not pick up rigid body rotation, one introduces the infinitesimalstrain tensor

ε = 1

2[Gradu + (Gradu)T] = 1

2(ε + εT) ⇔ εij =

1

2(ui,j + uj,i) . (0.47)

Notice that, unlike in finite deformations, no deformation implies ε = 0. Furthermore, bydefinition ε is symmetric since ε = εT (not like F which is asymmetric). The same applies forσ and P which are, respectively, symmetric and asymmetric.

As before, local deformation metrics are encoded into ε. For example, volumetric deformationis characterized by the trace of ε, viz.

dv

dV= 1 + trε = 1 + εii, (0.48)

while stretches in the three coordinate directions are given by ε(ii) (parentheses implying nosummation over i) and angle changes are identified as γij = 2εij with i ≠ j.

In linearized kinematics, all three stress tensors coincide and one commonly uses only theCauchy stress tensor σ to define the constitutive relation σ = σ(ε). In the simplest case oflinear elasticity, those are linearly linked via

σ = Cε ⇔ σij = Cijklεkl (0.49)

with a fourth-order elasticity tensor C linking each component of σ to those of ε. Alter-natively, we can again encode the constitutive response in a strain energy density W = W (ε),which, e.g., for the case of linear elasticity reads

W = 1

2ε ⋅Cε ⇔ W = 1

2εijCijklεkl, (0.50)

so that

σ = ∂W∂ε

, C = ∂σ∂ε

= ∂2W

∂ε∂ε. (0.51)

When taking derivatives with respect to ε, caution is required since ε is symmetric, so εij = εkland, consequently, derivatives with respect to εij must also take into account those termscontaining εji (for i ≠ j). Therefore, the derivative should always be computed as

∂

∂εij= 1

2( ∂

∂εij+ ∂

∂εji) . (0.52)

Alternatively, one may also use ε = εT and simply replace ε = 12(ε + ε

T) before differentiating.

The traction vector on a surface with unit normal n now becomes t = σn.

The local statement of linear momentum balance in linearized kinematics is

divσ + ρb = ρa ⇔ σij,j + ρbi = ρai (0.53)

where the small-strain versions of density, body force density and acceleration field were intro-duced as ρ, b and a = u, respectively.

In small strains, we can insert the kinematic and linear elastic constitutive relations as well asthe definition of the acceleration field into linear momentum balance to obtain

(Cijklεkl),j + ρbi = ρui ⇔ (Cijkluk,l),j + ρbi = ρui (0.54)

12


and in case of a homogeneous body with C(x) = C = const. we finally arrive at

Cijkluk,lj + ρbi = ρui, (0.55)

which is known as Navier’s equation to be solved for the unknown field u(x, t).

Finally, the following will be helpful when implementing material models in our code. Note thatwe may use the relations

εij =1

2(ui,j + uj,i) and FiJ = δiJ + ui,J (0.56)

to write (using the chain rule)

∂W

∂ui,J= ∂W

∂FkL

∂FkL∂ui,J

= PkL∂

∂ui,J(δkL + uk,L) = PkLδikδJL = PiJ (0.57)

and (exploiting the symmetry of σ)

∂W

∂ui,j= ∂W∂εkl

∂εkl∂ui,j

= σkl∂

∂ui,j

1

2(uk,l + ul,k) =

1

2σkl(δikδjl + δilδjk) =

1

2(σij + σji) = σij . (0.58)

Therefore, we can use the alternative relations for our stresses

PiJ =∂W

∂ui,Jand σij =

∂W

∂ui,j. (0.59)

The beauty in those relations is that the stress tensor definition is now identical irrespective ofwhether we are working in linearized or finite kinematics.

0.4 Initial boundary value problems (IBVPs)

So far, we have seen how partial differential equations govern the thermal and mechanicalbehavior of solid bodies (and, of course, those two can be coupled as well to describe the thermo-mechanical behavior of deformable bodies). In order to solve a problem, we need an initialboundary value problem (IBVP), which furnishes the above equations with appropriateboundary conditions (BCs) and initial conditions (ICs).

To this end, we subdivide the boundary ∂Ω of a body Ω into

∂ΩD ≡ Dirichlet boundary, prescribing the primary field (ϕ, T , etc.):

e.g. u(x, t) = u(x, t) on ∂ΩD or T (x, t) = T (x, t) on ∂ΩD. (0.60)

∂ΩN ≡ Neumann boundary, prescribing derivatives of the primary field (F , β, etc.):

e.g. t(x, t) = σ(x, t)n(x, t) = t(x, t) on ∂ΩN or q(x, t) = q(x, t) on ∂ΩN ,

(0.61)

Note that we may generally assume that

∂ΩD ∪ ∂ΩN = ∂Ω and in most problems also ∂ΩD ∩ ∂ΩN = ∅. (0.62)

In addition, all time-dependent problems require initial conditions, e.g.,

T (x,0) = T0(x) ∀ X ∈ Ω,

or ϕ(X,0) = x0(X) and V (X,0) = V0(X) ∀ X ∈ Ω.(0.63)

13


The number of required BCs/ICs depends on the order of a PDE, e.g.,

ρcvT = div(K gradT ) + ρs (0.64)

is first-order in time and therefore requires one IC, e.g., T (x,0) = T0(x). It is second-order inspace and hence requires BCs along all ∂Ω (e.g., two conditions per x and y coordinates).

In summary, we have governing PDEs supplemented by ICs and BCs, as required (e.g., qua-sistatic problems, of course, do not require any initial conditions). Those are to be solved forthe primary fields (e.g., temperature T or displacements u or the deformation mapping ϕ). Un-fortunately, analytical solutions are hardly ever available – except for relatively simple problemsinvolving simple geometries, simple material behavior, simple ICs/BCs. Therefore, realistic ge-ometries, material models and/or ICs/BCs call fornumerical techniques to obtain approximatesolutions.

All of the above was discussed in the course Computational Solid Mechanics. Here, weassume that those methods of finite elements and related approximation schemes at the continu-ums scale are available, and we will discuss how those techniques (and various other approaches)can be utilized to model material behavior that bridges across scales – from the electronic andatomic structure of matter all the way up to the macroscopic, device-level scale. Exactly thatis the purpose of Computational Multiscale Modeling.

14


1 Microstructure & Unit Cells

We will consider problems in which the macroscale response of a body depends on featureson the microscale of the material the body is made of. Here and in the following, macroand micro do not refer to particular length scales but solely to the bigger and smaller scale ofconcern.

A key assumption in most models is that material is statistically homogeneous, i.e., thestatistical properties of the microstructure do not change from point to point of the materialat the macroscale. Statistical properties include, e.g., averages such as volume averages butalso higher-order statistical data such as two-point correlation or n-point correlation data. Ifa material is not statistically homogeneous, it is called statistically inhomogeneous (e.g.,consider a composite with a concentration gradient).

When discussing microstructures, we often consider some notion of a unit cell (UC) and we willneed average quantities computed by averaging over the volume V = ∣Ω∣ of a UC Ω. Thus, wewill define

⟨⋅⟩Ω = 1

V∫

Ω(⋅)dV. (1.1)

Notice that we define the average with respect to the undeformed configuration (and thataverage may not agree with its deformed counterpart since volumes change with deformation).To this end, observe that

⟨⋅⟩ϕ(Ω) =1

v∫ϕ(Ω)

(⋅)dv =∫ϕ(Ω)

(⋅)dv

∫ϕ(Ω)

dv=∫

Ω(⋅) ϕJ dV

∫ΩJ dV

= ⟨⋅⟩Ω

⟨J⟩Ω≠ ⟨⋅⟩Ω (1.2)

in general (unless the volume does not change). In the following, we will always average withrespect to the undeformed configuration and thus omit the subscript Ω (in linearized kinematicsthere is no difference anyways).

1.1 Ensembles and Representative Volume Elements

Microstructures often display either randomness or periodicity, or combinations thereof.

A periodic microstructure admits the identification of a unit cell, i.e., the simplest (smallest)repeating substructure so that the complete microstructure results from periodic repetition ofthe unit cell. The size of the unit cell matches the length scale of periodicity. For periodicsystems, the choice of the unit cell is never unique; i.e., the size is but exact location within themicrostructure is not since the system displays translational invariance.

If we generate random microstructures (e.g., by randomly arranging spherical particles of a givensize and volume fraction into a matrix material), then each random microstructure creationresults in a different configuration or realization. A set of multiple realizations is called anensemble of realizations. Assume that each configuration produces a response which generallydiffers between realizations (e.g., some averaged quantity).

The ensemble average of a response over a set of N realizations is defined by

⟪⋅⟫N = 1

N

N

∑i=1

(⋅)i. (1.3)

15


For an increasing number of realizations, the central limit theorem from probability theorytells that this sequence converges for a truly random generation process, i.e.,

limN→∞

⟪⋅⟫N = ⟪⋅⟫∞. (1.4)

Also, the individual realizations follow a Gaussian distribution.

An alternative strategy to reaching a statistical limit is by sample enlargement, i.e., byincreasing the size of the unit cell while keeping microstructural feature sizes constant (e.g., fora constant particle size and volume fraction increase the unit cell size). With enlargement theresponse of the unit cell, typically also exhibits convergence. Let us denote by L the unit cellsize and by l a characteristic microstructural size (e.g., particle diameter). Then we can defineanother limit by

⟨⋅⟩∞ = limL/l→∞

⟨⋅⟩Ω(L). (1.5)

Note that there is no reason for the unit cell to be cuboidal; it can have in principle any shape,and the shape loses significance as the sample size increases, i.e., in the limit L/l →∞.

Note that the convergence by sample enlargement vs. ensemble averaging at fixed size is notequally uniform (ensemble average typically leads to smoother convergence). However, weshould have that

⟨⋅⟩∞ = ⟪ ⟨⋅⟩∞ ⟫∞. (1.6)

Of course, we would ideally want an infinitely large sample, for which averaged quantities donot differ anymore between realizations. Such a sample is called statistically representative,and a single analysis would be sufficient with such a sample to reveal the effective behavior ofthe microstructure. This is generally called a representative volume element (RVE). Forall practical purposes, we cannot work with infinite RVE sizes but will choose a finite RVE sizewhose response is as close to the infinite limit as needed. For example, one can find the requirednumber of realizations N or the size L of an RVE by checking for a specific quantity if

⟪⋅⟫N+1 − ⟪⋅⟫N⟪⋅⟫N

≤ ε, ⟨⋅⟩i+1 − ⟨⋅⟩i⟨⋅⟩i

≤ ε (1.7)

where ⟨⋅⟩i = ⟨⋅⟩Ω(Li) denotes a sequence of increasing sample sizes with limi→∞Li =∞, and ε > 0is a tolerance. That is, one finds a sample size that is sufficiently large for accuracy, so that asingle calculation based on the identified RVE is sufficient to extract effective material behavior.

If the microstructure is periodic, then a single unit cell is sufficient in most cases to serve asRVE (unless in case of, e.g., bifurcation or instabilities of any kind).

In the following, we will assume that a suitable RVE for computational purposes has beenidentified and is used for subsequent calculations and analysis. For mechanical problems, thekey fields to be averaged are stresses and strains, for which we can derive specific averagingtheorems.

16


2 Averaging Theorems

For mechanical problems, we will have to compute average mechanical quantities from RVEs.Therefore, the following averaging theorems will be most helpful.

2.1 Averaging Theorems in Linearized Kinematics

Consider an RVE Ω with volume V = ∣Ω∣. The average infinitesimal strain tensor is computedas

⟨εij⟩ =1

V∫

Ωεij dV = 1

2V∫

Ω(ui,j + uj,i) dV = 1

2V∫∂Ω

(uinj + ujni) dS (2.1)

or, symbolically,

⟨ε⟩ = 1

V∫∂Ω

sym (u⊗n) dS or shorter ⟨ε⟩ = 1

V∫∂Ωu⊙ndS, (2.2)

where we adopted the mathematical notation

a⊙ b = sym(a⊗ b) = 1

2(a⊗ b + b⊗ a) . (2.3)

Note that in case of discontinuous displacement fields, the boundary integral must accountfor displacement jumps across interfaces. If no such discontinuous exist, we say the compositephases are perfectly bonded (this also implies that tractions are continuous across interfaces).Otherwise, we have

⟨ε⟩ = 1

V∫∂Ω

sym (u⊗n) dS + 1

V∫C

sym ([[u]]⊗n) dS, (2.4)

where C is an interior discontinuous interface, [[⋅]] = (⋅)+ − (⋅)− denotes the jump across C, andn is chosen to point from the + to the − side of the interface.

Analogously, the average stress tensor is derived by exploiting the relation

σij = σikδjk = σikxj,k, (2.5)

so that (by using linear momentum balance and tractions ti = σiknk)

⟨σij⟩ =1

V∫

Ωσikxj,kdV = 1

V[∫

∂ΩσikxjnkdS − ∫

Ωσik,kxj dV ]

= 1

V[∫

∂Ωtixj dS − ∫

Ωρ(ai − bi)xj dV ] .

(2.6)

This is equivalent to saying

⟨σ⟩ = 1

V[∫

∂Ωt⊗xdS − ∫

Ωρ(a − b)⊗xdV ] . (2.7)

In case of quasistatic processes and in the absence of body forces we thus have

⟨σ⟩ = 1

V∫∂Ωt⊗xdS = 1

V∫∂Ωx⊗ tdS (2.8)

where the latter form results from the fact that σ is by definition symmetric (note that [[t]] = 0in equilibrium in case of internal interfaces).

17


2.2 Averaging Theorems in Finite Kinematics

Analogously, in finite deformations the average deformation gradient tensor in an RVE Ω is

⟨FiJ⟩ =1

V∫

Ωϕi,J dV = 1

V∫∂ΩϕiNJ dS (2.9)

or

⟨F ⟩ = 1

V∫∂Ωϕ⊗N dS (2.10)

Analogous to the linearized setting, the average first Piola Kirchhoff stress tensor is derived byexploiting the relation

PiJ = PiKδJK = PiKXJ,K , (2.11)

so that (by using linear momentum balance and tractions Ti = PiKNK)

⟨PiJ⟩ =1

V∫

ΩPiKXJ,K dV = 1

V[∫

∂ΩPiKXJNK dS − ∫

ΩPiK,KXJ dV ]

= 1

V[∫

∂ΩTiXJ dS − ∫

ΩR(Ai −Bi)XJ dV ] .

(2.12)

Using symbolic notation, the above is equivalent to

⟨P ⟩ = 1

V[∫

∂ΩT ⊗X dS − ∫

ΩR(A −B)⊗X dV ] . (2.13)

Under quasistatic conditions and in the absence of body forces, we thus have

⟨P ⟩ = 1

V∫∂ΩT ⊗X dS (2.14)

18


3 Numerical Evaluation of RVE Averages

3.1 Average Deformation Gradient

When the above averaging theorems are used in the FE context, it is convenient to find theirapproximate analogues within the discrete FE representation. For example, let us begin withthe average deformation gradient in a RUC, for which we showed that

⟨F ⟩ = 1

V∫∂Ωϕ⊗N dS. (3.1)

The approximate numerical counterpart reads

⟨F h⟩ = 1

V∑e∫∂Ω′

e

n

∑a=1

ϕaeNa(X)⊗N dS

= 1

V∑e

n

∑a=1

ϕae ∫∂Ω′

e

Na(X)⊗N dS,

(3.2)

where ∂Ω′e ⊂ ∂Ω denotes the external part of the boundary associated with element e. Consider,

e.g., elements that interpolate linearly on the boundary (e.g., simplicial elements or (bi/tri)linearelements) and an RVE geometry with planar faces. In this case N =Ne = const. across elementson the same face, so that

⟨F h⟩ = 1

V∑e

n

∑a=1

ϕae ⊗Ne∫∂Ω′

e

Na(X)dS = 1

V

n

∑a=1

ϕa ⊗ 1

d∑e

NaeAe, (3.3)

where Nae denotes the outward unit normal at node a as seen from element e (for corner or

edge nodes, two or three such normals exist per node, respectively, and all must be accountedfor within all adjacent elements). The factor 1/d in d dimensions stems from the integration ofthe shape function on the boundary. Ae denotes the element external boundary area (in 3D)or external edge length (in 2D). If we introduce an effective nodal normal Na, then

⟨F h⟩ = 1

V

n

∑a=1

ϕa ⊗ Na with Na = 1

d∑e

NaeAe. (3.4)

Thus, (3.1) has been transformed into a discrete sum over the deformed nodal positions of allsurface nodes, which can easily be evaluated for a given RVE and state of deformation.

3.2 Average Stress Tensor

Analogous to the previous section, we can derive a discrete version of the averaging theorem forthe stress tensor, starting with

⟨P ⟩ = 1

V∫∂ΩT ⊗X dS. (3.5)

Recall that surface tractions were related to external forces applied to nodes on the surface via

F a = ∫∂ΩTNadS. (3.6)

Also, for isoparametric elements we may write for each element (including those on the surface)

X = ∑a=1

XaeN

ae (X). (3.7)

19


Applying both to (3.5) yields

⟨P h⟩ = 1

V∑e∫∂Ω′

e

T ⊗∑a=1

XaeN

ae (X)dS = 1

V∑a=1∑e∫∂Ω′

e

TNae dS ⊗Xa (3.8)

If we now introduce the total force F a applied to each boudnary node, then

⟨P h⟩ = 1

V∑a

F a ⊗Xa with F a =∑e∑a=1∫∂Ω′

e

TNae dS. (3.9)

Note that nodal force F a contains contributions from all adjacent boundary edges/surfaces(therefore the sum over all elements). Again, we have transformed the continuous averagingtheorem, (3.5), into a discretized version to be evaluated in FE simulations.

3.3 Linearized Kinematics

The analogous relations can be derived for linearized kinematics. Without derivation, we men-tion here that the averaging theorem for the infinitesimal strain tensor,

⟨ε⟩ = 1

V∫∂Ω

sym (u⊗n) dS, (3.10)

can be treated analogously to (3.4), which leads to

⟨εh⟩ = 1

V

n

∑a=1

sym (ua ⊗ na) with na = 1

d∑e

naeAe. (3.11)

Similarly, starting with

⟨σ⟩ = 1

V∫∂Ωt⊗xdS, (3.12)

by analogy to (3.9) we arrive at the discrete version of the average stress theorem

⟨σh⟩ = 1

V∑a

F a ⊗xa with F a =∑e∑a=1∫∂Ω′

e

tNae dS. (3.13)

20


4 Homogenization Problem

Assuming a statistically homogeneous microstructure, the mechanical homogenization problemcan be summarized as follows. We formulate the problem in finite kinematics and can easilydeduce the linearized version later if needed.

Let us first consider a heterogeneous body with microstructure so the constitutive relationchanges from point to point, i.e., P = P (X,F ), as does the mass density R(X). Therefore,the heterogeneous problem reads

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

find ϕ(X, t) s.t.

divP (X,F (X)) +R(X)B(X) = R(X)A(X) in Ω,

ϕ(X) = ϕ(X) on ∂ΩD,

T (X) = P (X,F (X))N(X) = T (X) on ∂ΩN ,

P = P (X,F ).

(4.1)

Because of the small-scale variations in material properties, stress and strain fields are highlyoscillatory, so the solution is challenging to find.

Assume that the fields of interest (e.g., stresses and strains) on the microscale vary over acharacteristic length scale l and the macroscale features (e.g., dimensions of body Ω, variationof boundary conditions T and ϕ and body forces B) are of characteristic length scale L. If

ε = l

L≪ 1, (4.2)

then we may assume what is known as a separation of scales.

If a separation of scales applies, we may take the limit ε → 0. For a statistically homogeneousmicrostructure, this results in an effective material with constant, homogeneous materialproperties such as constant mass density R(X) = R∗ and a constant constitutive responseP (X,F ) = P ∗(F ). We generally label all effective quantities by an asterisk (⋅)∗. This definesthe homogenized problem:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

find ϕ∗(X, t) s.t.

divP ∗(F (X)) +R∗B(X) = R∗A∗(X) in Ω,

ϕ∗(X) = ϕ(X) on ∂ΩD,

T ∗(X) = P ∗(F (X))N(X) = T (X) on ∂ΩN ,

P ∗ = P ∗(F ).

(4.3)

The solution ϕ∗ is smooth compared to the solution ϕ of the heterogeneous problem. Note thatwe assume a local form of the constitutive relations which may not be possible in general dueto, e.g., discontinuous stress or strain fields at material interfaces at the micro-level. However,when using the FE method, we solve the problem with relaxed continuity requirements anywaysand we assume that the possibly discontinuous micro-fields translate into smooth macro-fields.

Note that one can extend the above homogenized problem to statistically inhomogneous mi-crostructures, in which case we have an inhomogeneous effective material with P ∗ = P ∗(X,F ∗)but the variation in X is of O(L) and not O(l).

Solving the above homogenized problem can be accomplished by the traditional FE techniquesestablished previously in this course. However, in order to solve the homogenized problem,we first need to find the effective density R∗ as well as the effective constitutive relation P ∗ =

21


P ∗(F ∗). We generally identify the effective response from an RVE that was deemed sufficientto find (an approximation of) the limit ⟨⋅⟩∞ by a single test, independent of the particularmicrostructural realization. Precisely that is the objective of computational homogenizationto be discussed in the following.

22


5 Micro-to-Macro Transition

5.1 Effective Material Properties

By conservation of mass, we can find the effective mass density via

R∗V = ∫ΩR(X)dV ⇒ R∗ = ⟨R⟩ (5.1)

Analogously, we define the effective stress and strain tensors as the RVE-averaged quantities,viz.

P ∗ = ⟨P ⟩ and F ∗ = ⟨F ⟩ (5.2)

In order to find an effective constitutive relation we must relate these two averages. That is,we must apply some boundary conditions to the RVE to induce stress and strain fields. Forgiven BCs, one can solve the equilibrium equations within the RVE and use the above averagingtheorems to identify the average quantities. Then, one defines the effective constitutive relationas the best functional form that relates P ∗ and F ∗.

It is important to note this choice of relating P ∗ to F ∗ is not unique. One could alternatively useother stress or strain measures, which results in different effective relations because generally,e.g.,

⟨P ⟩ = ⟨FS⟩ ≠ ⟨F ⟩ ⟨S⟩. (5.3)

In the following, we stick to our choice of P ∗ and F ∗ as stress and strain measures in order toextract the effective response. In linearized kinematics this if, of course, not an issue and onecan uniquely use ε∗ and σ∗.

In order to avoid the influence of body forces (which is legitimate if body forces vary on themacroscale), we will solve the RVE problem in the absence of body forces. Also, we will assumethat the effective constitutive behavior is an intrinsic material property that is not affected bythe boundary value problem to be solved. This implies that, if dynamic behavior is present, thewavelength λ of any dynamic behavior is on the order of the macroscale, λ ∼ L (and not λ ∼ l).This allows us to include inertial effects on the macroscale, so that the RVE problem to besolved does not include inertial effects. Overall, this implies that the homogenization problemto be solved is the quasistatic one,

divP (X,F ) = 0 in Ω or divσ(x,ε) = 0 in Ω (5.4)

In order to identify suitable BCs to be applied to the RVE, we need the following concepts.

5.2 The Hill-Mandel Condition

Instead of solving the strong form, computational techniques usually prefer to solve the weakform which derives from the first variation of the associated variational problem, as discussed inPart I. Since we seek an effective macroscale formulation that is equivalent to the full-resolutionmicroscale problem, we postulate that the averaged first variation on the microscale,

δI

V= 1

V∫

ΩδW dV = 1

V∫

Ω

∂W

∂F⋅ δF (X)dV = ⟨P (X) ⋅ δF (X)⟩, (5.5)

23


equals the effective pointwise variation on the macroscale, which is

δW ∗ = V P ∗ ⋅ δF ∗. (5.6)

Equating the two yields the so-called Hill-Mandel condition for micro-macro equivalence:

P ∗ ⋅ δF ∗ = ⟨P (X) ⋅ δF (X)⟩ (5.7)

The above can also be interpreted as the equivalence of micro- and macro-power by writingalternatively

P ∗ ⋅ F ∗ = ⟨P (X) ⋅ F (X)⟩ (5.8)

For simplicity of notation, we will omit the variation in the following. Thus, we write in finiteand infinitesimal kinematics, respectively,

P ∗ ⋅F ∗ = ⟨P (X) ⋅F (X)⟩ and σ∗ ⋅ ε∗ = ⟨σ(x) ⋅ ε(x)⟩. (5.9)

Note that in linear elasticity, the relation admits a clear energetic interpretation as it relatesthe microscale and macroscale strain energy densities (otherwise, the equivalence must be in-terpreted with regards to the weak form or the first variation or virtual work).

We now must find BCs that ensure the above energetic equivalence. There are three possibletypes of BCs that do exactly that:

(i) affine displacement BCs which enforce Dirichlet boundary conditions

x = F0X on ∂Ω. (5.10)

(ii) uniform traction BCs which enforce Neumann boundary conditions

T = P0N on ∂Ω. (5.11)

(iii) periodic BCs which enforce

x+ = x− +F0(X+ −X−) and T + = −T − on ∂Ω. (5.12)

In order to verify that those boundary conditions satisfy the Hill-Mandel condition, let us writethe equivalence criterion by expanding stresses and strains into averages plus fluctuation fields:

P = ⟨P ⟩ + P , F = ⟨F ⟩ + F . (5.13)

Taking averages of these two forms indicates that

⟨P ⟩ = 0, ⟨F ⟩ = 0, (5.14)

i.e., the fluctuation fields must have zero averages (this is what defines a fluctuation field).

This leads to

⟨P ⋅F ⟩ = ⟨(⟨P ⟩ + P ) ⋅ (⟨F ⟩ + F )⟩= ⟨⟨P ⟩ ⋅ ⟨F ⟩⟩ + ⟨P ⋅ ⟨F ⟩⟩ + ⟨⟨P ⟩ ⋅ F ⟩ + ⟨P ⋅ F ⟩= ⟨P ⟩ ⋅ ⟨F ⟩ + ⟨P ⟩ ⋅ ⟨F ⟩ + ⟨P ⟩ ⋅ ⟨F ⟩ + ⟨P ⋅ F ⟩= ⟨P ⟩ ⋅ ⟨F ⟩ + ⟨P ⋅ F ⟩.

(5.15)

24


Hill-Mandel requires that ⟨P ⋅F ⟩ = ⟨P ⟩ ⋅ ⟨F ⟩, which implies that the boundary conditions mustguarantee that the fluctuations satisfy

⟨P ⋅ F ⟩ = 0. (5.16)

The following relation will be helpful in analyzing the different boundary condition cases:

⟨P ⋅F ⟩ = 1

V∫

ΩPiJϕi,J dV = 1

V[∫

∂ΩPiJNJϕidS − ∫

ΩPiJ,JϕidV ]

= 1

V[∫

∂ΩT ⋅ϕdS − ∫

ΩDivP ⋅ϕdV ]

= 1

V[∫

∂ΩT ⋅ϕdS − ∫

ΩR(A −B) ⋅ϕdV ] .

(5.17)

Again, we are assuming that phases are perfectly bonded; otherwise, discontinuities must beaccounted for by additional interface integrals. Also, as discussed above, the RVE problem issolved without body forces and inertial effects, so that

⟨P ⋅F ⟩ = 1

V∫∂ΩT ⋅ϕdS (5.18)

5.3 Example: Affine Deformation of the RVE

As a naive example that implements average kinematics, let us consider affine deformationimposed onto the entire RVE, i.e., for each point inside the RVE we postulate

ϕ = F ∗X or u = ε∗x. (5.19)

In this case it is obvious that the correct averages are imposed because

F (X) = ∂ϕ

∂X= F ∗ = const. or ε(x) = 1

2(ε∗ + ε∗,T) = ε∗ = const. (5.20)

so that

⟨F ⟩ = 1

V∫

ΩF (X)dV = 1

V∫

ΩF ∗dV = F ∗ or ⟨ε⟩ = 1

V∫

Ωε(x)dv = ε∗. (5.21)

Such affine deformation applied to, e.g., a composite of i homogeneous phases leads to an averagestress response of

⟨P ⟩ = 1

V∫

ΩP (X,F ∗) = 1

V∑i

ViPi(F ∗), (5.22)

where Vi is the volume fraction of phase i, and Pi is obtained from the constitutive law of phase i.That is, affine deformation results in stress volume-averaged over the various composite phases.Such a construction of the average response is known as the Taylor model (frequently used,e.g., in plasticity to extract the effective response of polycrystals with homogeneous constitutivelaws assumed across each grain but different from grain to grain because of varying orientations).

One can also show that the affine deformation trivially satisfies the Hill-Mandel condition since

⟨P ⋅F ⟩ = ⟨P ⋅F ∗⟩ = ⟨P ⟩ ⋅F ∗ = ⟨P ⟩ ⋅ ⟨F ⟩. (5.23)

25


The problem of the above construction is that it generally leads to a response that is too stiff.Imagine, e.g., a composite made of one very compliant and one very stiff phase. When only anaverage deformation is to be imposed, it is irrealistic to assume that both phases are deformingin exactly the same way (it is much more likely that the compliant phase absorbs significantlymore deformation than the stiff phase).

For the simple example of linear elasticity, one can even extract the effective material modelfrom the above construction. To this end, note that in linear elasticity

σ(x,ε(x)) = C(x)ε(x), (5.24)

so that, for a composite made of i phases with constant moduli Ci within each phase,

⟨σ(x)⟩ = 1

V∑i

Ciε∗ = C∗Voigtε

∗ with C∗Voigt =

1

V∑i

ViCi. (5.25)

That is, the Taylor construction results in the so-called Voigt bound on the elastic moduli ofthe effective, homogenized linear elastic medium. The Voigt bound presents an upper boundon the linear elastic effective composite responds.

Similarly, if a constant stress σ(x) = σ∗ is imposed, we use the inversion ε(x) = C−1(x)σ(x)to arrive at

⟨ε(x)⟩ = 1

V∑i

C−1i σ

∗ = (C∗Reuss)

−1σ∗ with CReuss = ( 1

V∑i

C−1i )

−1

. (5.26)

This latter is the so-called Sachs model and it results in the Reuss bound of the effectivelinear elastic moduli. The Reuss bound presents a lower bound on the linear elastic effectivecomposite responds.

26


6 RVE Boundary Conditions

Next, we consider the three types of BCs listed above and verify that they satisfy the Hill-Mandelequivalence condition.

6.1 Uniform Traction BCs

Let us first consider uniform tractions applied to the entire boundary of the RVE, i.e.,

T = P0N on ∂Ω, (6.1)

where P0 is a yet unknown constant stress tensor. Using the average stress theorem for qua-sistatic problems and in the absence of body forces and discontinuities gives

⟨P ⟩ = 1

V∫∂ΩT⊗X dS = 1

V∫∂ΩP0N⊗X dS = P0

V∫∂ΩN⊗X dS = P0

V∫

ΩGradX dV = P0, (6.2)

where we used that

∫ΩXI,J dV = δIJ ∫

ΩdV so that ∫

ΩGradX dV = V I. (6.3)

This indicates that P0 is nothing but the average stress inside the RVE, which offers a convenientway to impose average RVE stresses via uniform traction BCs.

Now, let us turn to the Hill-Mandel condition and observe that, using (5.17),

⟨P ⋅F ⟩ = 1

V∫∂ΩT ⋅ϕdS = 1

V∫∂ΩP0N ⋅ϕdS = 1

V∫∂ΩP 0iJNJϕidS

=P 0iJ

V∫∂ΩNJϕidS = P0 ⋅

1

V∫∂Ωϕ⊗N dS = P0 ⋅ ⟨F ⟩,

(6.4)

where we used identity (2.10) to arrive at the last expression. This implies that uniform tractionBCs with P0 = ⟨P ⟩ guarantee that

⟨P ⋅F ⟩ = P0 ⋅ ⟨F ⟩ = ⟨P ⟩ ⋅ ⟨F ⟩, (6.5)

which confirms that the Hill-Mandel condition is satisfied.

6.2 Affine Displacement BCs

Next, consider the case of affine displacements imposed across the entire RVE boundary, i.e.,

ϕ = F0X on ∂Ω (6.6)

with some yet unknown constant tensor F0. Using the average strain theorem (2.10) shows that

⟨F ⟩ = 1

V∫∂Ωϕ⊗N dS = 1

V∫∂ΩF0X ⊗N dS = F0

V∫∂ΩX ⊗N dS = F0. (6.7)

In analogy to the above uniform traction case, this result indicates that F0 is the averagedeformation gradient inside the RVE, which offers a convenient way to impose average RVEdeformation gradients via affine displacement BCs.

27


Hill-Mandel can be verified by

⟨P ⋅F ⟩ = 1

V∫∂ΩT ⋅ϕdS = 1

V∫∂ΩT ⋅F0X dS = 1

V∫∂ΩTiF

0iJXJ dS

= F 0iJ

1

V∫∂ΩTiXJ dS = F0 ⋅

1

V∫∂ΩT ⊗X dS = F0 ⋅ ⟨P ⟩,

(6.8)

where the last step resulted from the average stress theorem (2.13) (for quasistatic problemsand in the absence of body forces and discontinuities).

Therefore, affine displacement BCs with F0 = ⟨F ⟩ guarantee that

⟨P ⋅F ⟩ = F0 ⋅ ⟨P ⟩ = ⟨F ⟩ ⋅ ⟨P ⟩, (6.9)

which confirms that the Hill-Mandel condition is satisfied.

6.3 Periodic BCs

So far we have seen that uniform traction BCs offer a convenient way to impose average RVEstresses, whereas affine displacement BCs can conveniently be used to enforce average RVEdeformation gradients. Both types of BCs have shortcomings when considering the physicsof the problem. Uniform tractions result in a response that is generally too compliant (theyproduce a lower bound), while affine displacements result in a response that is too stiff (theyproduce an upper bound, to be discussed later). Also, uniform traction BCs do not impose anydeformation on the RVE so that the displacements on the RVE boundary will not be matchingnor linearly distributed; by contrast, affine displacement boundary conditions result in a “tile-able” deformation but the tractions on opposite RVE boundaries are not matching and tractionsare not uniform.

As an alternative, periodic BCs offer a compromise. We decompose the boundary into twoparts ∂Ω+ and ∂Ω− so that ∂Ω = ∂Ω+ ∪∂Ω−. Each point X+ on ∂Ω+ is linked to a unique pointX− on ∂Ω−, and the outward normal vectors at those points satisfy N− = −N+. Periodic BCsare then defined by

ϕ+ −ϕ− = F0(X+ −X−) and T + = −T − on ∂Ω. (6.10)

That is, displacements are forced to be periodic and tractions are forced to be anti-periodic.By comparison, affine displacement BCs only enforce periodic displacements, whereas uniformtraction BCs only enforce anti-periodic tractions. Notice that for pratical purposes we may alsowrite u+ −u− = (F0 − I)(X+ −X−).

Periodic BCs ensure that, if the RVE is in equilibrium with BCs (6.10), the “tiled” solutionof a periodic microstructure is also in equilibrium (since tractions are anti-periodic) and geo-metrically matching (since displacements are periodic); this also implies that stress and strainfields are continuous and periodic. This reveals that periodic BCs are ideally suited to simulatebodies with periodic microstructure whose unit cell coincides with the RVE. In practice, oneoften uses periodic BCs also for problems whose microstructure is not periodic.

In order to identify F0, let us use the average strain theorem (2.10) and make use of the boundary

28


decomposition and N− = −N+:

V ⟨F ⟩ = ∫∂Ωϕ⊗N dS = ∫

∂Ω+

ϕ+ ⊗N+dS + ∫∂Ω−

ϕ− ⊗N−dS

= ∫∂Ω+

[ϕ− +F0(X+ −X−)]⊗N+dS + ∫∂Ω−

ϕ− ⊗N−dS

= ∫∂Ω+

[ϕ− +F0(X+ −X−)]⊗N+dS − ∫∂Ω+

ϕ− ⊗N+dS

= ∫∂Ω+

F0(X+ −X−)⊗N+dS

= ∫∂Ω+

F0X+ ⊗N+dS − ∫

∂Ω+

F0X− ⊗N+dS

= ∫∂Ω+

F0X+ ⊗N+dS + ∫

∂Ω−

F0X− ⊗N−dS

= ∫∂ΩF0X ⊗N dS = F0∫

∂ΩX ⊗N dS = V F0.

(6.11)

This proves that ⟨F ⟩ = F0 so that we have identified another way to impose average deformationgradients.

Next, we will verify that the periodi BCs (6.10) satisfy the Hill-Mandel condition:

V ⟨P ⋅F ⟩ = ∫∂ΩT ⋅ϕdS = ∫

∂Ω+

T + ⋅ϕ+dS + ∫∂Ω−

T − ⋅ϕ−dS

= ∫∂Ω+

T + ⋅ [ϕ− +F0(X+ −X−)] dS − ∫∂Ω+

T + ⋅ϕ−dS

= ∫∂Ω+

T + ⋅F0(X+ −X−)dS

= ∫∂Ω+

T + ⋅F0X+dS − ∫

∂Ω+

T + ⋅F0X−dS

= ∫∂Ω+

T + ⋅F0X+dS + ∫

∂Ω−

T − ⋅F0X−dS

= ∫∂ΩT ⋅F0X dS = ∫

∂ΩTiF

0iJXJ dS = F 0

iJ ∫∂ΩTiXJ dS

= F0 ⋅ ∫∂ΩT ⊗X dS = F0 ⋅ V ⟨P ⟩,

(6.12)

where we used the average stress theorm (2.13) to arrive at the final form (for quasistaticproblems and in the absence of body forces and discontinuities). Therefore, since F0 = ⟨F ⟩ (asshown above), we conclude that

⟨P ⋅F ⟩ = F0 ⋅ ⟨P ⟩ = ⟨F ⟩ ⋅ ⟨P ⟩, (6.13)

which proves the Hill-Mandel equivalence condition for periodic boundary conditions.

29


6.4 Enforcing RVE Averages

So far we have only discussed different types of boundary conditions but not under what cir-cumstances they should be used and, especially, how to numerically enforce them. We generallydifferentiate three types of RVE formulations:

(i) average strain-driven BCs enforce ⟨F ⟩ = F ∗. As discussed above, it is natural to useeither affine displacement BCs or periodic BCs for this case.

(ii) average stress-driven BCs enforce ⟨P ⟩ = P ∗. As discussed above, it is natural to useuniform traction BCs for this case.

(iii) mixed stress/strain-driven BCs enforce some components of ⟨F ⟩ = F ∗ and somecomponents of ⟨P ⟩ = P ∗ (i.e., each component enforces either a stress or a strain).

Note that the above exmaples are not exclusive; e.g., one can also use periodic BCs to enforceaverage stresses – the formulation is just not as simple as for the cases mentioned above.

Numerical Aspects:

The three most popular choices in computational homogenization are the following:

From an implementation viewpoint, strain-driven affine displacement BCs are simplest toenforce since all they require is to enforce (6.6) with known average F0 = ⟨F ⟩ everywhereon the RVE boundary ∂Ω.

Stress-driven uniform traction BCs are, in principle, simple to enforce since they requirethe application of tractions (6.1) with known P0 = ⟨P ⟩ everywhere on the RVE boundary∂Ω. The devil is in the detail though since any numerical implementation requires thesuppression of rigid body motion and fixing individual nodes on ∂Ω leads to uncontrollablereaction forces that may break the applied tractions. Therefore, special caution is required.

Strain-driven periodic BCs can be implemented in a number of ways. The next sectionwill discuss a particular numerical formulation to implement those in an elegant practicalfashion. Alternatives are, e.g., penalty methods which add to the total potential energyto be minimized a constraint potential of the form

IC[u] = 1

2ε∫∂Ω+

∥(F0 − I)(X+ −X−) − (u+ −u−)∥2dS, (6.14)

where 0 < ε≪ 1. Taking variations yields

δIC[u] = −1

ε∫∂Ω+

[(F0 − I)(X+ −X−) − (u+ −u−)] ⋅ δu+dS

+ 1

ε∫∂Ω−

[(F0 − I)(X+ −X−) − (u+ −u−)] ⋅ δu−dS = 0.(6.15)

Therefore, if one considers the total potential energy

I[u] = ∫ΩW (F )dV + IC[u], (6.16)

then the first variation becomes

δI[u] = 0 = ∫ΩP ⋅ δF dV − 1

ε∫∂Ω+

[(F0 − I)(X+ −X−) − (u+ −u−)] ⋅ δu+dS

+ 1

ε∫∂Ω−

[(F0 − I)(X+ −X−) − (u+ −u−)] ⋅ δu−dS.(6.17)

30


Note that we can identify those two terms arising from the constraint potential as tractionsapplied on the boundary, viz.

T + = 1

ε[(F0 − I)(X+ −X−) − (u+ −u−)] on ∂Ω+,

T − = −1

ε[(F0 − I)(X+ −X−) − (u+ −u−)] on ∂Ω−.

(6.18)

This shows that tractions are indeed anti-periodic, and parameter ε controls the enforce-ment of periodicity. The drawback of this method is that for finite ε > 0, the solution isonly approximately periodic (and for ε ≪ 1 conditioning issues can affect the numericalsolution process).

Similar constraint potentials can be introduced to enforce, e.g., average strain-drivenuniform traction BCs but will not be discussed here.

31


7 Numerical Aspects of Periodic Boundary Conditions

There are many different ways of implementing periodic BCs numerically. This section willoutline one possible, practical solution that avoids penalty constraints and instead enforces theperiodicity directly by modifying the linear systems to be solved.

To begin, let us review linear and more general constraints within the FE context. We showedthat the solution to quasistatic BVPs is obtained by solving a (non)linear system

f(Uh) = Fint(Uh) −Fext = 0, (7.1)

which includes one equation for each node. Any constraint can be formulated as

fc(Uh) = 0. (7.2)

In our finite element code, we implement the constraint by replacing the equation for one dofby fc = 0. The equation to be replaced must involve the node (or one of the nodes) to beconstrained.

The simplest case is that of essential BCs. Consider node a, dof i is to be set to δ; this isequivalent to

fc(Uh) = uai − δ = 0. (7.3)

More generally, we write any linear constraint as

fc(Uh) =n

∑a=1

d

∑i=1

αai uai − δ = 0. (7.4)

Now, consider periodic boundary conditions applied to a pair of (±)-nodes, which must satisfythe following d constraints:

fc(Uh) = u+ −u− − (F ∗ − I)(X+ −X−). (7.5)

Let us consider a general four-sided RVE in 2D; the generalization to 3D is straight-forwardbut more cumbersome to formulate. The corners be denoted as nodes 1 through 4 (starting inthe bottom left corner), and the two lattice vectors are defined as

L21 =X2 −X1 =X3 −X4, L41 =X4 −X1 =X3 −X2. (7.6)

To prevent rigid body motion, we fix the bottom left node, i.e.,

ϕ1 =X1 or u1 = 0. (7.7)

As a consequence, the displacements at the remaining corner nodes must satisfy

u2 = u1 + (F ∗ − I)(X2 −X1) = (F ∗ − I)L21,

u4 = u1 + (F ∗ − I)(X4 −X1) = (F ∗ − I)L41,

u3 = u1 + (F ∗ − I)(X3 −X1) = (F ∗ − I)(L21 +L41) = u2 +u4.

(7.8)

32


7.1 Enforcing Average Deformation Gradients:

Let us consider strain-driven periodic BCs. Consider a pair of (±)-points located on the leftand right edges. We have by the periodicity of the unit cell

u+ = u− + (F ∗ − I)(X+ −X−) = u− + (F ∗ − I)L21 = u− +u2. (7.9)

Analogously, a node pair on the top and bottom edges is constrained by

u+ = u− + (F ∗ − I)(X+ −X−) = u− + (F ∗ − I)L41 = u− +u4. (7.10)

This implies that the displacements of all boundary nodes can be traced back to u2 and u4. Insummary, the constraints to be imposed are

f1(Uh) = u1,

f3(Uh) = u3 −u2 −u4,

f+(Uh) = u+ −u− −u2 for each left/right pair,

f+(Uh) = u+ −u− −u4 for each top/bottom pair

(7.11)

along with

u2 = (F ∗ − I)L21,

u4 = (F ∗ − I)L41.(7.12)

If the average F ∗ to be enforced is known, all we need to do is implement constraints (7.11)and (7.12). Notice that we formulated (7.11) free of the average deformation gradient, whichwill become very helpful in a moment.

7.2 Enforcing average stress components:

If average stresses P ∗ are to be imposed in conjunction with periodic BCs, we must imposetractions on the boundary ∂Ω of the RVE such that T = P ∗N , as shown in Section 6.1. Thus,we have

I[ϕ] = ∫ΩW (F )dV − ∫

∂ΩT ⋅ϕ dS

= ∫ΩW (F )dV − ∫

∂ΩP ∗N ⋅ϕ dS

= ∫ΩW (F )dV −P ∗ ⋅ ∫

∂Ωϕ⊗N dS,

(7.13)

which by the averaging theorem (2.10) is equivalent to

I[ϕ] = ∫ΩW (F )dV − V P ∗ ⋅ ⟨F ⟩. (7.14)

The boundary integral can be decomposed into bottom (b), top (t), left (l) and right (r) edgesin 2D:

∫∂Ωϕ⊗N dS = V I + ∫

∂Ωb

ub ⊗Nb dS + ∫∂Ωt

ut ⊗Nt dS + ∫∂Ωl

ul ⊗Nl dS + ∫∂Ωr

ur ⊗Nr dS

= V I − ∫∂Ωt

ub ⊗Nt dS + ∫∂Ωt

(ub +u4)⊗Nt dS

− ∫∂Ωr

ul ⊗Nr dS + ∫∂Ωr

(ul +u2)⊗Nr dS

= V I +u4 ⊗ ∫∂Ωt

Nt dS +u2 ⊗ ∫∂Ωr

Nr dS,

33


(7.15)

where we used that

∫∂ΩX ⊗N dS = ∫

ΩGradX dV = V I. (7.16)

Therefore,

P ∗ ⋅ ∫∂Ωϕ⊗N dS = P ∗ ⋅ [V I +u4 ⊗ ∫

∂Ωt

Nt dS +u2 ⊗ ∫∂Ωr

Nr dS]

= V P ∗ ⋅ I +u4 ⋅P ∗∫∂Ωt

Nt dS +u2 ⋅P ∗∫∂Ωr

Nr dS.(7.17)

In summary, the total potential energy becomes (dropping the term independent of displace-ments)

I[ϕ] = ∫ΩW (F )dV −u4 ⋅P ∗∫

∂Ωt

Nt dS −u2 ⋅P ∗∫∂Ωr

Nr dS. (7.18)

If the average stress tensor P ∗ is known and to be imposed, than the above potential energycan be reinterpreted as a BVP that applies constant external forces

F4 = P ∗∫∂Ωt

Nt dS, F2 = P ∗∫∂Ωr

Nr dS (7.19)

to nodes 4 and 2, respectively. Thus, we can enforce an average stress tensor P ∗ by imposingconstraints (7.11) along with applying external forces (7.19). That is, nodes 2 and 4 are free todisplace while external forces are being applied to those two nodes.

7.3 Enforcing mixed averages:

Next, assume that some of the components of F ∗ and some of P ∗ are known. For example,consider a uniaxial extension test enforcing a stretch λ in the X1-direction, while leaving thetransverse X2-direction stress-free; this means

F ∗ = ( λ F12

F21 F22) , P ∗ = (P11 0

0 0) , (7.20)

where the four unknowns are F12, F21, F22, P11.

Enorcing such mixed averages is challenging in general. However, if the RVE is rectangular,then note that (7.19) reduces to

F2 = P ∗N1L1 = L1 = (P∗11

P ∗12) , F4 = P ∗N2L2 = L2 (

P ∗21

P ∗22) , (7.21)

while (7.12) becomes

u2 = (F ∗ − I)N1L1 = (F∗11 − 1F ∗

12)L1, u4 = (F ∗ − I)N2L2 = ( F ∗

21

F ∗22 − 1

)L2. (7.22)

Notice that the equations decouple, so that we may enforce individual components of P ∗ andF ∗ simultaneously by enforcing selected components from those two sets of equations. That is,we need to impose periodic BCs (7.11) complemented by whichever components of (7.22) apply.

For example, for the above uniaxial extension test we would enforce (7.11) along with

u21 = (λ − 1)L1, F2 = F4 = (0

0) , (7.23)

and u22 and u4 are to be determined.

34


7.4 Three Dimensions

Periodic BCs can be enforced analogously in 3D where instead of u2 and u4 we must formulateBCs by working with three corner nodes to enforce average stresses and strains. All relationsfor enforcing average deformation gradients and stress tensors apply analogously.

7.5 Linearized Kinematics

The above concepts apply equally in linearized kinematics. As a key difference, periodicityinvolves the infinitesimal strain tensor ε instead of deformation gradient F . That is, the corre-sponding linearized expressions are obtained by replacing (F −I) by ε (as well as their averagesand effective values) and using normal vectors ni, positions xi, etc. in the linearized framework.

35


8 The FE2 Method

Since the above computational homogenization procedure provided a methodology to computeeffective stress-strain relations P ∗ = P ∗(F ∗), or analogously σ∗ = σ∗(ε∗), it has become con-venient to use such effective constitutive laws within macroscale FE calculations. That is, theconstitutive response at each quadrature point is computed on-the-fly by solving the BVP atthe RVE-level. As this approach couples to FE BVPs (at the macro- and microscales) it isknown as the FE2 method.

One additional complication arises here since solving the macro-problem iteratively may requirethe tangent matrix of the material model to be used at the quadrature point-level, i.e., we need

C∗ = ∂σ∗

∂ε∗or C∗ = ∂P

∗

∂F ∗ . (8.1)

One possibility is to compute C∗ as the numerical tangent, which is prone to causing numericalerrors and possibly slow but generally easy to implement and (almost) always applicable.

As an alternative, a general strategy is to decompose all nodes into interior and boundary nodes,so that displacements and corresponding nodal forces and tangent matrix are written as

Uh = (Ui

Ub) , Fext = (Fi

Fb) , T = ∂Fint

∂Uh= (Kii Kib

Kbi Kbb) . (8.2)

Since no interior forces are being applied, we have Fi = 0. Also, by definition we must havesymmetry so that Kib =Kbi.

8.1 Affine BCs

Consider affine BCs applied to the RVE, so that the boundary displacements can be formulatedas

Ub =D(F ∗ − I), (8.3)

where D is a matrix that depends on the boundary node locations and implements ϕ = F ∗X.The governing equations are thus summarized as

⎡⎢⎢⎢⎢⎢⎢⎣

Fi(Uh) = 0,

Fb(Uh) −Ξ = 0,

Ub −D(F ∗ − I) = 0

(8.4)

with unknown boundary forces Ξ. Linearization of the system with respect to Uh,Ξ aboutan equilibrium solution Uh

0 ,Ξ0 leads to

⎡⎢⎢⎢⎢⎢⎢⎣

Fi(Uh0 ) +Kii(Uh

0 )∆Ui +Kib(Uh0 )∆Ub = 0,

Fb(Uh0 ) −Ξ +Kbi(Uh

0 )∆Ui +Kbb(Uh0 )∆Ub −∆Ξ = 0,

Ub,0 −D(F ∗ − I) +∆Ub −D∆F ∗ = 0.

(8.5)

Since we start with an equilibrium solution Uh0 ,Ξ0 satisfying (8.4), perturbations must satisfy

∆Ub −D∆F ∗ = 0. (8.6)

36


Moreover, in equilibrium we know that Fi(Uh0 ) = 0 and Fb(Uh

0 ) = Ξ, so that

∆Ui = −K−1ii (Uh

0 )Kib(Uh0 )∆Ub (8.7)

and

Kbi(Uh0 )∆Ui +Kbb(Uh

0 )∆Ub −∆Ξ = 0 (8.8)

or, altogether,

∆Ξ = [Kbb(Uh0 ) −Kbi(Uh

0 )K−1ii (Uh

0 )Kib(Uh0 )]∆Ub

= [Kbb(Uh0 ) −Kbi(Uh

0 )K−1ii (Uh

0 )Kib(Uh0 )]D∆F ∗.

(8.9)

Finally, recall that the average stresses were obtained from

P ∗ = 1

V∑a

F a ⊗Xa = 1

VDTFi, (8.10)

where we reused the transformation matrix D introduced above (which depends on referencenodal locations on the boundary). Linearization results in

P ∗ +C∗∆F ∗ = 1

V[DTFi(Uh) +DT∆Ξ] , (8.11)

so that in equilibrium the perturbations are linked via

C∗∆F ∗ = 1

VDT∆Ξ (8.12)

Insertion of (8.9) finally admits the conclusion

C∗ = 1

VDT [Kbb(Uh

0 ) −Kbi(Uh0 )K−1

ii (Uh0 )Kib(Uh

0 )]D (8.13)

That is, the consistent tangent matrix can be obtained from the element tangent matrices onceone load step has been equilibrated. Note that because Kii is constructed from only the innernodes, it is invertible by definition.

Similar procedures can be applied to uniform traction BCs and periodic BCs. For uniformtraction BCs, the procedure is analogous but may require iterations. For periodic BCs, theprocedure is more involved but in principle analogous. Details will be skipped here and can befound, e.g., in (Miehe and Koch, 2002).

37


9 Analytical Techniques: Cauchy-Born Kinematics

9.1 Local and nonlocal expansions

The above multiscale techniques involved solving a BVP on the microscale to extract the effectivematerial behavior on the macroscale. Under certain conditions, one may want to formulate asimpler microscale problem that does not necessarily require solving a microscale BVP butadmit analytical solutions for average microscale quantities.

As before, we assume a clear spatial separation of scales between micro- and macroscale features.Assume the macroscale deformation mapping ϕ ∶ Ω → Rd is known, then we may expand thedeformation within an infinitesimal neighborhood of a point X0 as

ϕi(X) = ϕi(X0) +∂ϕi∂XJ

(X0)dXJ +1

2

∂2ϕi∂XJ ∂XK

(X0)dXJ dXk + h.o.t.

= ϕi(X0) + FiJ(X0)dXJ +1

2FiJ,K(X0)dXJ dXk + h.o.t.

(9.1)

with deformation gradient F , second gradient GradF , and higher-order terms continued anal-ogously. Hence, if the deformation at a point – characterized by ϕ(X0), F (X0), GradF (X0),etc. – is known, the deformation in its infinitesimal neighborhood is obtained from the aboveTaylor expansion. Dropping higher-order terms results in an approximation.

Now, consider a microstructure that admits a separation of scales from the macroscale. In thiscase, one may regard (9.1) as an approximation of the deformation across the RVE, so thatinfinitesimal distances dX are replaced by finite but small microscale distances ∆X =X −X0:

ϕi(X) ≈ ϕi(X0) + FiJ(X0)∆XJ +1

2FiJ,K(X0)∆XJ ∆Xk + h.o.t. (9.2)

The simplest approximation is obtained by dropping all but the linear term, which results inthe so-called (historically not quite correct) local Cauchy-Born rule

ϕi(X) = ϕi(X0) + FiJ(X0)∆XJ , ∆X =X −X0, (9.3)

where X0 is some reference point on the RVE-level (e.g., the center of mass of the RVE).Obviously, (9.3) prescribes an affine deformation with constant deformation gradient F ∗ =F (X0) across the entire RVE (not only on the boundary as in the affine-displacement casediscussed before). Note that for quasistatic problems the constant term ϕi(X0) constitutesrigid body motion of the RVE, so that omitting this term (or, equivalently, arbitrarily choosingX0) does not affect the deformation of the RVE.

Assume now that the material in the RVE has an energy density W (F ,X), then the aboveapproximation results in the average energy density of the RVE being (for simplicity choosingX0 = 0)

W ∗(F ∗) = 1

V∫

ΩW (F ∗,X)dV = ⟨W (F ∗,X)⟩. (9.4)

The average first Piola-Kirchhoff stress tensor can be obtained in closed form as

P ∗ = ∂W∗

∂F ∗ (F ∗) = ⟨∂W∂F

(F ∗,X)⟩ = ⟨P (F ∗,X)⟩. (9.5)

38


Analogously, the incremental stiffness tensor C∗ follows from averaging. Note that the obtainedbehavior agrees with the Taylor construction discussed in Section 5.3. Consequently, onemay expect the effective response to be overly stiff.

One example is the so-called Arruda-Boyce model named after its inventors (Arruda andBoyce, 1993). This is a hyperelastic constitutive model commonly used to describe the mechan-ical response of rubber and polymers. Leaving all statistical-mechanics complications aside, themodel is based on a cubic RVE containing eight polymer chains along the diagonal directionswhose mechanical stretches are related to stored energy. Application of affine deformation isexploited to extract the effective material response.

The so-called extended Cauchy-Born rule (also known as nonlocal Cauchy-Born rule)retains one more term in the above expansion, resulting in

ϕi(X) = ϕi(X0) + FiJ(X0)∆XJ +1

2FiJ,K(X0)∆XJ∆XK , ∆X =X −X0. (9.6)

Here, the RVE deformation is no longer affine but follows the linearly-varying deformationgradient

FiJ = FiJ(X0) + FiJ,K(X0)∆XK , (9.7)

where we exploited that FiJ,K = ϕi,JK = ϕi,KJ = FiK,J .

Again, if the RVE’s material has an energy density W (F ,X), then the average is

W ∗(F ∗,GradF ∗) = 1

V∫

ΩW (F ∗ +GradF ∗X,X)dV = ⟨W (F ∗ +GradF ∗X,X)⟩. (9.8)

The result is an effective nonlocal model whose energy density depends on both F ∗ andGradF ∗ on the macroscale. The resulting first Piola-Kirchhoff stress tensor is by definition

P ∗iJ =

∂W ∗

∂F ∗iJ

(F ∗) = ⟨PiJ(F ∗ +GradF ∗X,X)⟩ , (9.9)

and there is a third-order couple stress tensor

Y ∗iJK = ∂W ∗

∂F ∗iJ,K

(F ∗) = 1

V∫

Ω

∂

∂FkLW (F ∗ +GradF ∗X,X)

∂F ∗kL + F

∗kL,MXM

∂F ∗iJ,K

dV

= ⟨PiJ(F ∗ +GradF ∗X,X)XK⟩.(9.10)

Note that now the macroscale variational problem must be revised as the energy density isno longer local. This results in added complexity and requires reformulation in particular ofboundary conditions and variations. For example, consider a total potential energy (droppingthe asterisks for convenience) in its most general form

I[ϕ] = ∫ΩW (ϕ,F ,GradF ,Grad GradF , . . .) dV (9.11)

whose first variation is

δI[ϕ] = ∫Ω(∂W∂ϕi

δϕi +∂W

∂FiJδϕi,J +

∂W

∂FiJ,Kδϕi,JK + . . .) dV. (9.12)

Using our usual discretization Bubnov-Galerkin scheme, the internal force vector becomes

F aint,i = ∫Ω(∂W∂ϕi

Na + PiJNa,J + YiJKNa

,JK + . . .) dV. (9.13)

This requires that we need to use higher-order interpolation. E.g., using the above formulationwith W = W (F ,GradF ), the need at least quadratic interpolation. We will not discuss thisproblem here in detail (see, e.g., the rich literature on gradient elasticity theories). Alterna-tively, one may apply the divergence theorem in order to successively remove the higher-ordervariational terms. This, however, introduces intricate boundary terms to account for.

39


9.2 Applications and extensions

The above formulation can be extended in various directions, the following gives a few examples.

First, it can be applied to discrete RVE descriptions (e.g., in case of atomic crystals orstructural trusses). In this case, (9.2) is applied to the position of each discrete point in the RVE.In this case, the unit cell is defined by a point set X1, . . . ,Xn and a suitable connectivity inbetween points. For example, in case of stretching-dominated trusses (where each strut behaveslike a geometrically nonlinear spring) the energy may be formulated via a pair potential V (r)as

W ∗(F ∗) = 1

V

N

∑i=1∑j≠iV (∥xi −xj∥) with xi = F ∗Xi. (9.14)

The first Piola-Kirchhoff stress tensor then follows as

P ∗(F ∗) = 1

V

n

∑i=1∑j≠iV ′ (rij)

∂ ∥F ∗Xi −F ∗Xj∥∂F ∗

= 1

V

n

∑i=1∑j≠iV ′ (rij)

xi −xj∥F ∗Xi −F ∗Xj∥

⊗ (Xi −Xj)

= 1

V

n

∑i=1∑j≠iV ′ (rij) rij ⊗Rij

= 1

V

n

∑i=1∑j≠ifij ⊗Rij

(9.15)

where the deformed unit direction of a bond ij and the undeformed distance vector are intro-duced as

rij =xi −xj

∥xi −xj∥, Rij =Xi −Xj (9.16)

and

fij = V ′ (rij) rij (9.17)

is the force acting between nodes i and j. Note that such an extension can also be appliedto discrete granular RVEs where the interaction between static particles is described by thepotential V (as written here in the special case without rotational/frictional contact forces).

In case of truss lattices, one may also wish to introduce rotational degrees of freedom to describethe rotation of individual nodes. To this end, consider a macroscale problem described by adeformation mapping ϕ∗ = ϕ(X0) and a rotational field θ∗ = θ(X0) whose gradients are thedeformation gradient F ∗ = F (X0) and the curvature tensor κ∗ = κ(X0). Even though thereis no clear physical interpretation of the rotation field on the macroscopic continuum level, itsmeaning becomes clear from the microscale.

Application of the Cauchy-Born rule to both fields at the microlevel yields

ϕ(X) = ϕ∗ +F ∗X + h.o.t., θ(X) = θ∗ +κ∗X + h.o.t.. (9.18)

By again defining the effective energy density as the sum over all beams inside the RVE, wearrive at an effective, homogenized macroscale problem

I[ϕ,θ] = ∫ΩW ∗(ϕ∗,F ∗,θ∗,κ∗)dV − ∫

∂ΩT

T ⋅ϕ∗dS − ∫∂Ωm

m ⋅ θ∗dS − ∫∂ΩM

M ⋅κ∗N dS

40


(9.19)

with the effective stresses

τ∗i = ∂W∗

∂ϕ∗i, P ∗

iJ =∂W ∗

∂F ∗iJ

, m∗i =

∂W ∗

∂θ∗i, M∗

iJ =∂W ∗

∂κ∗iJ. (9.20)

Notice that the dependence on ϕ∗ can be dropped since it implies rigid-body translation of theRVE, which should not contribute to the stored energy for physical reasons (unless dynamicsare to be considered). By contrast, θ∗ does not drop out since θ∗ implies a uniform rotation ofall nodes inside the RVE, which does not correspond to rigid body motion.

Also, note the higher-order boundary terms in (9.19), which must be handled with caution astheir physical interpretation may be intricate.

As a further extension, consider a discrete RVE in which only some points are constraint bythe Cauchy-Born rule while others are free to displace (e.g., in complex atomic crystals inwhich the RVE corners displace in an affine manner while interior atoms show relative motion(so-called shuffles) with respect to the corners). Here, the procedure is analogous to that ofFE2: the constrained nodes are deformed according to the macroscale deformation while thedisplacements of the remaining nodes is determined from solving the microscale equilibriumequations.

An alternative perspective is the so-called multi-lattice formulation for a periodic medium.Consider a periodic material or structure composed of repeating unit cells that lie on a latticedefined by the translation vector basis B = a1, . . . ,ad. Let us identify each unique set of points(related by the aforementioned translation vectors) by a lattice, so that the entire discrete pointset follows from a superposition of the various lattices. Then, we may assume that each latticeundergoes a Cauchy-Born-type deformation, while the lattices can move relative to each other,i.e.,

ϕ(X) = ϕ∗+F ∗X+∑a

δϕa∆a(X)+h.o.t., θ(X) = θ∗+κ∗X+∑a

δθa∆a(X)+h.o.t., (9.21)

where (δϕa, δθa) are shifts of lattice a with respect to the static frame of reference, and

∆a(X) =⎧⎪⎪⎨⎪⎪⎩

1 if node X lies in lattice a,

0 else.(9.22)

In this case, we still satisfy the average deformation since the deformation at a node i is char-acterized by

∂

∂X(ϕ∗ +F ∗Xi + δϕi) = F ∗,

∂

∂X(θ∗ +κ∗Xi + δθi) = κ∗, (9.23)

if we further constrain that the sums over all nodes to give

1

n

n

∑i=1

δϕi = 0,1

n

n

∑i=1

δθi = 0, (9.24)

where (δϕi, δθi) denotes the shifts of node i.

41


10 Spectral Techniques

10.1 Reciprocal Bravais Lattices

Consider an RVE Ω discretized by a regular grid of n nodes located at lattice sites X =X1, . . . ,Xn defined by a Bravais basis B = A1, . . . ,Ad, such that every point in the RVEcan be decomposed into a linear combination of the Bravais vectors:

Xi =d

∑i=1

ciAi, ci ∈ Z s.t. Xi ∈ Ω. (10.1)

The infinite point set spanned by B is called a Bravais lattice.

By assuming periodicity, we can tessellate Rd by periodically repeating the RVE. Thus, we havea second set of Bravais lattice vectors L = L1, . . . ,Ld that define the corners of all RVEs inRd and we choose the vectors such that

Li = LiAi and (L1 − 1) ⋅ . . . ⋅ (Ld − 1) = n, (10.2)

where Li defines the number of grid points within the RVE in the Ai-direction.

Periodicity implies that all field quantities u must also periodic across RVEs, i.e., we mustensure

u(X) = u(X +d

∑i=1

ciLi) ∀ ci ∈ Z. (10.3)

To this end, note that we express the field in its discrete Fourier representation, where weinterpolate the field u by using the inverse Fourier transform, viz. we write

u(X) = ∑K∈T

u(K) exp (−ihK ⋅X) , h = 2π

n. (10.4)

We introduced the variable h for convenience and for simple redefinitions of the transform. Thechoice of vectors K must ensure (10.3), which implies that

exp (−ihK ⋅X) = exp [−ihK ⋅ (X +d

∑i=1

ciLi)] ∀ ci ∈ Z

= exp (−ihK ⋅X) ⋅ exp(−ihK ⋅d

∑i=1

ciLi) ∀ ci ∈ Z.(10.5)

This is turn requires that

hK ⋅d

∑i=1

ciLi =2π

n

d

∑i=1

ciK ⋅Li ∈ 2πZ ⇔ 1

n

d

∑i=1

ciK ⋅Li ∈ Z ∀ ci ∈ Z. (10.6)

Therefore (choose, e.g., all ci = 0 except one cj = 1), we conclude that each K-vector mustsatisfy

1

nK ⋅Li ∈ Z ∀ i = 1, . . . , d, (10.7)

which defines a reciprocal lattice of grid points K, which is also a Bravais lattice. To realizethis, consider, e.g., the 2D case and draw rays perpendicular to each Li at distances nZ/Li; this

42


results in a new periodic lattice with some basis K = Z1, . . . ,Zd so we may writeK = ∑di=1 kiZiwith ki ∈ Z. We notice that the new basis must be related to the original basis via

Zi ⋅Lj = nδij (10.8)

Note that the conventional definition of reciprocal lattices uses n = 1.

As an important feature, the reciprocal lattice of the reciprocal lattice is the original latticeitself, which is known as the Pontryagin duality of the associated vector spaces.

Examples:

Consider a 2D lattice with Bravais basis L = L1,L2. The reciprocal lattice has Bravais basisvectors (with R ∈ SO(2) denoting a rotation matrix by 90)

Z1 = nRL2

L1 ⋅RL2, Z2 = n

RL1

L1 ⋅RL2. (10.9)

We can easily verify the reciprocity relation (10.8) by noticing that

Z1 ⋅L1 = nRL2

L1 ⋅RL2⋅L1 = n, Z2 ⋅L1 = n

RL1

L1 ⋅RL2⋅L1 = 0,

Z1 ⋅L2 = nRL2

L1 ⋅RL2⋅L2 = 0, Z2 ⋅L2 = n

RL1

L1 ⋅RL2⋅L2 = n,

(10.10)

so that indeed Zi ⋅Lj = nδij .

Similarly, a 3D Bravais basis L = L1,L2,L3 has reciprocal Bravais basis vectors

Z1 = nL2 ×L3

L1 ⋅ (L2 ×L3), Z2 = n

L3 ×L1

L1 ⋅ (L2 ×L3), Z3 = n

L1 ×L2

L1 ⋅ (L2 ×L3), (10.11)

which satisfy the reciprocity relation (10.8) since Zi ⋅Lj = nδij .

10.2 Fourier Spectral Solution Techniques

For a field variable u, we use the discrete Fourier transform F whose inversion is defined as

u(X) = F−1 (u) = ∑K∈T

u(K) exp (−ihK ⋅X) , h = 2π

n. (10.12)

The complete discretized information of u is thus contained either in its values at the grid pointsin real space, u = u(x1), . . . , u(xn), or in Fourier space, u = u(k1), . . . , u(kn). By usingthe above transform (and its inverse), both sets of data contain equivalent information.

All FE equations (mechanical, thermal, electromagnetic, etc.) involve computing derivatives ofthe primary fields, e.g., to obtain strains from displacements, heat flux from temperature, etc.

The fundamental idea of Fourier spectral techniques is to not introduce shape functions butinstead

use Fourier transforms to compute derivatives and

solve the governing equations in Fourier space.

43


This not only ensures higher accuracy in computing derivatives but also provides a simple wayto enforce of average field quantities, as we will show in the following.

First, notice that out of all Fourier coefficients u(k), one is special as the integral over the RVEreveals:

∫Ωu(X)dV = ∑

K∈Tu(K)∫

Ωexp(−ihK ⋅X)dV

= V u(0).(10.13)

Here, we used that the integrals vanish unless K = 0 because

∫Ω

exp(−ihK ⋅X)dV = ∫L1

0∫

L2

0∫

L3

0exp(−i2π

n

d

∑i=1

kiZi ⋅X) dX1 dX2 dX3

= ∫1

0∫

1

0∫

1

0exp(−i2π

n

d

∑i=1

kiZi ⋅d

∑a=1

ξaLa)L1 dξ1L2 dξ2L3 dξ3

= V ∫1

0∫

1

0∫

1

0exp(−i2π

d

∑i=1

kiξi) dξ1 dξ2 dξ3

= Vd

∏i=1∫

1

0exp (−i2πkiξi) dξi where ki ∈ Z,

= V⎧⎪⎪⎨⎪⎪⎩

1, if k1 = . . . = kd = 0 i.e., K = 0,

0, else.

(10.14)

The latter identity can easily be verified by expanding the exponential into cosines and sinesusing Euler’s identity. Overall, we have thus shown the useful identity

⟨u⟩ = u(0) (10.15)

Next, notice that

∂

∂Xu(X) = − ∑

K∈TihKu(K) exp (−ihK ⋅X) , (10.16)

so that

F (u,J) = −ihKJF(u) (10.17)

The strategy is now to transform the governing ODEs/PDEs into Fourier space and solve theresulting equations in Fourier space, where possible.

10.3 Example: Heat Conduction

Consider a thermal problem that involves equilibrating an RVE by solving the static heat equa-tion with a known heat source distribution S(X) = R(X)Sh(X) and with periodic boundaryconditions, while enforcing an average temperature ⟨T ⟩.

Recall the static heat equation with a constant conductivity λ,

λT,II + S = 0 ∀ X ∈ Ω, (10.18)

44


which in Fourier space becomes

λ(−ihKI)(−ihKI)F(T ) +F(S) = 0 (10.19)

or

λ ∑K∈T

(−ihKI)(−ihKI)T (K) exp (−ihK ⋅X) + ∑K∈T

S(K) exp (−ihK ⋅X) = 0 (10.20)

For this to hold at all X ∈ Ω, we conclude that

−λh2KIKI T (K) + S(K) = 0 ∀ K ∈ T , (10.21)

which can be solved, unless K = 0, to yield

T (K) = S(K)λh2 ∥K∥2

∀ K ∈ T . (10.22)

Therefore, we can solve the problem by utilizing Fourier transforms as follows:

(i) transform the known source term S(X) into Fourier space to obtain S(K) for all K ∈ T ,

(ii) solve for T (K) in Fourier space:

T (K) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

⟨T ⟩ if K = 0,

S(K)h2λ ∥K∥2

else.(10.23)

(iii) apply the inverse transform to obtain T (X) from T (K) for all grid points Xi.

The advantages of this method are obvious:

No approximating shape functions must be introduced (and approximated derivatives areof significantly much higher order).

No iterative solver nor linear-system solver is required; the only numerical steps requiredare the (standard and inverse) Fourier transforms, which are numerically inexpensive.

No global stiffness matrix is required for the solution, which is a big memory advantage.

RVE-averages of the primary fields (and their derivatives) can be trivially enforced.

Next, consider the same problem with non-constant conductivity λ(X), so that

(λT,I),I + S = 0 ∀ X ∈ Ω (10.24)

with heat flux

QI(X) = −λ(X)T,I(X) ⇒ −QI,I + S = 0. (10.25)

Notice that Fourier transforming the above equation would turn the first term into a convolutionthat cannot be treated in the same simple fashion as above. To this end, we introduce a constantreference conductivity λ0 and define the perturbation P such that

QI = −(λ0T,I − PI) or PI = λ0T,I +QI . (10.26)

45


Insertion into (10.25) yields

(λ0T,I − PI),I + S = 0 ⇔ λ0T,II − PI,I + S = 0, (10.27)

which now can be Fourier-transformed without convolution:

−λ0h2KIKI T (K) + ihKI PI(K) + S(K) = 0 ∀ K ∈ T . (10.28)

Unless K = 0, we now obtain the solution

T (K) = S(K) + ihK ⋅ P (K)λ0h2 ∥K∥2

∀ K ∈ T . (10.29)

This form looks analogous to (10.22). Notice, however, that P = P (GradT ), so that the termK ⋅P (K) depends on temperature T . Therefore, one cannot expect the above equation to yieldthe correct solution. Yet, it provides the basis for an iterative solution scheme:

1. Choose a conductivity λ0 > 0.

2. Pick an initial guess for T (X) at all grid points Xi (in physical space).

3. Compute Q(GradT ) and P at all grid points (in physical space).

4. Transform Q(X) to Fourier space to obtain P (K) at all K-points.

5. Solve for all T (K) via (10.29) for a given average ⟨T ⟩ or ⟨GradT ⟩ (same treatment asbefore).

6. Transform P (K) back into physical space to obtain P (X) at all grid points.

7. Return to step 3. until the solution has converged for all grid points.

10.4 Mechanical Problem using Fourier Spectral Techniques

The quasistatic mechanical problem is, in principle, similar to the heat conduction scenariodiscussed above. The complication is, though, that the stress–strain relation is in generalnonlinear, so that the ODE to be solved is not necessarily linear. Let us first investigate thefinite-kinematics setting and then turn to linearized kinematics.

10.4.1 Finite Kinematics

Consider a quasistatic purely mechanical problem with linear momentum balance

DivP = 0. (10.30)

Since P = P (F ) is a generally nonlinear stress–strain relation, transforming the above ODEinto Fourier space does not present a solution. Therefore, let us introduce a perturbationstress tensor τ (x) such that

P (X) = C0F (X) − τ (X) (10.31)

with some constant fourth-order modulus tensor C0 (e.g., the RVE average ⟨C⟩).

Inserting the strain–displacement relation

F (X) = Gradϕ(X) (10.32)

46


into (10.30) gives

[C0iJkLϕk,L(X) − τiJ(X)]

,J= 0 (10.33)

or

C0iJkLϕk,LJ(X) = τiJ,J(X). (10.34)

This is a linear equation in the deformation mapping whose (inverse) Fourier transform heregives

ϕ(X) = ∑K∈T

ϕ(K) exp (−ihK ⋅X) , (10.35)

and analogously we can transform τ (X). This turns (10.34) into

−h2C0iJkL ϕk(K)KLKJ = −ihτiJ(K)KJ . (10.36)

Let us introduce the acoustic tensor

Aik(k) = C0iJkLKJKL (10.37)

to transform (10.36) into

−hAik(k)ϕk(k) = −iτiJ(K)KJ , (10.38)

which can be solved for the deformation mapping in Fourier space:

ϕk(K) = i

hA−1ik (K)τiJ(K)KJ (10.39)

Note that A(K) is invertible only if K ≠ 0 (and C0 is strongly elliptic).

Next, let us transform (10.32) into Fourier space, which gives

FkL(K) = −ihϕk(K)KL (10.40)

and insertion of (10.39) finally yields

FkL(K) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

⟨F ⟩ if K = 0,

A−1ik (K)τiJ(K)KJKL else,

(10.41)

where we exploited the fact that ⟨F ⟩ = F (0).

It is important to recall that τ depends on F , so that solving (10.41) for F can only be part ofan iterative solution scheme. For example, (10.41) can be solved by fixed-point iteration.

We note that even though (10.41) is formulated in terms of the deformation gradient (andthe deformation mapping is not even required), the solution is guaranteed to be compatible asfollows. Compatibility requires

CurlF = 0 ⇔ εKLIFkL,K = 0. (10.42)

or, in Fourier space,

−ihεKLI FkLKK = 0. (10.43)

47


Insertion of (10.41) shows that

−ihεKLI FkLKK = −ihεKLIA−1ik (K)τiJ(K)KJKKKL

= −ih1

2(εKLI + εLKI)A−1

ik (K)τiJ(K)KJKKKL = 0.(10.44)

Thus, the deformation gradient field resulting from (10.41) is per definition compatible.

The solution algorithm is now as follows:

1. Choose a stiffness tensor C0.

2. Pick an initial guess for F (X) at all grid points Xi (in physical space).

3. Compute P (X) and τ (X) at all grid points (in physical space).

4. Transform τ (X) Fourier space to obtain τ (K) at all K-points.

5. Solve for all F (K) via (10.41) for a given average ⟨F ⟩.

6. Transform F (K) back into physical space to obtain F (X) at all grid points.

7. Return to step 3. until the solution has converged for all grid points.

8. After convergence is reached, (10.39) can be used for postprocessing the deformationmapping.

Note that we can conveniently use the Fourier transform to also compute other RVE-averagessuch that, e.g., the average stress tensor needed for computational homogenization since

⟨P (X)⟩ = P (0) = 1

V∫

ΩP (X) dV = 1

n

n

∑i=1

P (Xi). (10.45)

Note that the effective incremental stiffness tensor is

C∗ = ∂P∗

∂F ∗ = 1

n

∂

∂F ∗

n

∑i=1

P (F (Xi)) =1

n

n

∑i=1

∂P

∂F(Xi) ⋅

∂F (Xi)∂F ∗

= 1

n

n

∑i=1

C(Xi) ⋅∂

∂F ∗ [F (0) + ∑K≠0

F (K) exp (−ihK ⋅Xi)]

= 1

n

n

∑i=1

C(Xi) ⋅ [I + ∑K≠0

exp (−ihK ⋅Xi)∂F (K)∂F ∗ ]

= 1

n

n

∑i=1

C(Xi) ⋅ [I + ∑K≠0

exp (−ihK ⋅Xi)∂

∂F ∗A−T(K)τ (K)K ⊗K]

= 1

n

n

∑i=1

C(Xi) ⋅ [I + ∑K≠0

exp (−ihK ⋅Xi)A−T(K)∂τ (K)∂F ∗ K ⊗K] ,

(10.46)

which unfortunately cannot be evaluated in closed form.

10.4.2 Linearized Kinematics

In linearized kinematics, the methodology is completely analogous and starts with introducinga perturbation stress tensor:

τ (x) = C0ε(x) −σ(x). (10.47)

48


Applying the same steps now aims at solving for the displacement field. In Fourier space weobtain the analogous relation to (10.39), viz.

uk(k) =i

hA−1ik (k)τij(k)kj . (10.48)

Here, the strain–displacement relation is ε = sym(gradu) which becomes in Fourier space

εkl(k) = −ih

2[ukkl + ulkk] . (10.49)

Insertion of (10.48) yields the analogous form of (10.41), viz.

εkl(k) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

⟨ε⟩ if k = 0,

12[A−1

ik (k)kl +A−1il (k)kk]τij(k)kj else.

(10.50)

The displacement field in Fourier space finally follows from (10.48).

The numerical algorithm is analogous to the finite-deformation formulation.

It is important to realize that, even though the stress strain relation in linear elasticity ispointwise linear (σij = Cijklεkl), the Fourier transform can only result in a linear system – likein the heat conduction problem above – if C is constant (which is a rather uninteresting case).Otherwise, we have

σij(x) = Cijkl(x)εkl(x), (10.51)

so that the Fourier transform involves a convolution, so that the equation in Fourier space isnot linear (which is why the perturbation stress tensor was introduced).

10.5 Fourier-Related Stability Issues

All of the above Fourier spectral formulations used a discrete grid with a finite set T of gridpoints to approximate a function f(x) by

fh(x) = ∑k∈T

f(k) exp(−ihk ⋅x), (10.52)

where T denotes the finite set of grid points in spectral space, while T∞ is the countably infiniteset of the corresponding exact Fourier representation, and we write T ∗ = T∞ ∖ T . Fouriercoefficients have pointwise convergence; however, the error of a truncated Fourier series, due tothe high-frequency terms, depends on the smoothness of the function.

To see this, note that we can bound the truncation error at any point x ∈ Ω by

∣fh − f ∣ = ∣ ∑k∈T ∗

f(k) exp(−ihk ⋅x)∣ = ∣ ∑k∈T ∗

a(k) sin(hk ⋅x) + b(k) cos(hk ⋅x)∣

≤ ∑k∈T ∗

∣a(k) + b(k)∣ ≤ ∑k∈T ∗

∣a(k)∣ + ∣b(k)∣,(10.53)

where we used the triangle inequality for the last step, and we defined a = −if , b = f .

By applying Poincare’s inequality to the transform and using Parseval’s identity, we obtain

∑k∈T ∗

∣a(k)∣ + ∣b(k)∣ ≤ ∑k∈T ∗

1

∣k∣(∣a′(k)∣ + ∣b′(k)∣)

≤√

2 [ ∑k∈T ∗

∣a′(k)∣2 + ∣b′(k)∣2]1/2

≤√

2

πN∥f ′∥1/2

L2(Ω).

(10.54)

49


Thus, we have derived the bound

supx∈Ω

∣fh − f ∣ ≤√

2

πN∥f ′∥1/2

L2(Ω)(10.55)

which depends on the smoothness of the function f . Most importantly, the decay rate of theFourier coefficients results in a non-uniform convergence. The consequences is a problem knownas ringing artifacts in case of sharp contrasts in local properties (such as jumps in elasticconstants, electric permittivity, or crystal orientation) due to high-frequency oscillations in thesolution near such discontinuities.

One can reduce such artifacts by ensuring that the derivative in a domain is bounded, which canbe accomplished, e.g., by approximating the differential operator by a discrete, finite-differenceoperator before the Fourier transform. While this ensures asymptotic consistency, the spectralaccuracy of the method is reduced to that of the finite-difference operator.

This works as follows. Instead of applying the Fourier transform to a spatial derivative of fdirectly, viz.

F−1 ( ∂f∂xi

) = −ihkiF−1 (f) , (10.56)

we first apply a central-difference approximation to the partial derivatives such that for a gridspacing ∣∆xi∣ ≪ 1 in the coordinate direction xi with Cartesian unit vector ei (no summationsover i implied)

∂f

∂xi(x) = f(x +∆xiei) − f(x −∆xiei)

2∆xi+O(∆x2

i ). (10.57)

Neglecting all higher-order terms and applying the discrete inverse Fourier transform yields

∂f

∂xi= ∂

∂xi∑k∈T

f(k) exp (−ihk ⋅x) ≈ ∑k∈T

f(k)exp [−ihk ⋅ (x +∆xiei)] − exp [−ihk ⋅ (x −∆xiei)]2∆xi

.

(10.58)

Note that

exp(−ihk ⋅ (x ±∆x)) = exp(−ihk ⋅x) exp(±ihk ⋅∆x), (10.59)

as well as Euler’s identity

exp(ihk ⋅∆x)) − exp(−ihk ⋅∆x)2i

= sin(hk ⋅∆x), (10.60)

which transforms the above into (no summations over i implied)

∂f

∂xi(x) ≈ − ∑

k∈Tf(k) exp (−ihk ⋅x) i sin(hki∆xi)

∆xi. (10.61)

For a regular grid with equal spacings ∆xi = ∆x, we thus obtain

∂f

∂xi≈ − ∑

k∈T

i sin(hki∆x)∆x

f(k) exp (−ihk ⋅x) . (10.62)

Finally, taking the Fourier transform leads to an approximate form of (10.56), viz.

F ( ∂f∂xi

) ≈ −isin(hki∆x)∆x

F(f) (10.63)

50


where the fractional term is related to the so-called Lanczos-σ factor.

In the limit of infinitely fine grids, notice that the above approximation converges to the exactrelation since

lim∆x→0

−isin(hki∆x)∆x

= −ihki. (10.64)

The truncation error of the central difference scheme creates numerical errors. These can bereduced by starting with higher-order finite difference stencils in (10.57). For example, considerthe fourth-order central difference approximation

∂f

∂xi(x) = −f(x + 2∆xei) + 8f(x +∆xei) − 8f(x −∆xei) + f(x − 2∆xei)

12∆x+O(∆x4). (10.65)

Proceeding with the same approach outlined above for the second-order accurate scheme, wearrive at

F ( ∂f∂xi

) ≈ −i [8 sin(hki∆x)6∆x

− sin(2hki∆x)6∆x

]F(f). (10.66)

Again, we recover the exact solution as the grid spacing tends to zero because

lim∆x→0

[8 sin(hki∆x)6∆x

− sin(2hki∆x)6∆x

] = hki. (10.67)

The same procedure can be applied to finite-difference approximations of arbitrary order toresult in increased accuracy.

Examples: benchmark tests

The effectiveness of the discrete derivative approximation is demonstrated in Fig. 1, whichillustrates the numerically-computed derivatives of two 1D examples, viz. a Heaviside stepfunction and a non-periodic half-sine wave. In both cases, we compare the exact (discontinu-ous) solution, the solution based on the standard Fourier transform, and the modified Fourier

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1500

-1000

-500

0

500

1000

1500

-600

-400

-200

0

200

400

600

800

1000

1200 modified Fourier transformanalytical derivative1st order correction

normalized length x /L

function

f

4th order correction

0.225 0.23 0.235 0.24 0.245 0.25 0.255 0.26 0.265 0.27 0.275normalized length x /L

(a) (b)

Figure 1: Illustration of the effectiveness of the first- and fourth-order finite-difference correctionon the spectral derivative (using FFT) of the double step function f(x) = δ(x− 1

4 +2−10)+ δ(x−34 − 2−10) periodically continued with period x ∈ [0,1). Shown are solutions obtained from theclassical Fourier spectral method, the modified Fourier transform (first-order and fourth-ordercorrection), and the exact analytical solution.

51


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

4

5

function

g

normalized length x /L(b)

0 0.005 0.01 0.015

modified Fourier transformanalytical derivative1st order correction4th order correction

normalized length x /L

5

4

3

2

1

4.5

3.5

2.5

1.5

(a)

Figure 2: Illustration of the effectiveness of the first- and fourth-order finite-difference correctionon the spectral derivative (using FFT) of the half-sine function f(x) = π cos(πx) periodicallycontinued with period x ∈ [0,1). Shown are solutions obtained from the classical Fourier spectralmethod, the modified Fourier transform (first-order and fourth-order correction), and the exactanalytical solution.

-30

-25

-20

-15

-10

-5

0

5

-10

0

10

20

30

40

50standard Fourier transform

1st order correction

2ndorder correction

3rdorder correction

4thorder correction

conv

erge

nce

orde

r

log (n)2log (n)2(a) (b)

log

(E )

2

1 2 3 4 5 6 7 1 2 3 4 5 6

Figure 3: (a) Mesh convergence indicating the loss of accuracy of the spectral method; shown isthe error E = ∣∣uh − u∣∣L2 vs. number of grid points n. (b) Comparison of the convergence orderof the Fourier spectral scheme for different orders of the finite-difference correction.

transform described above. Although they do not disappear entirely, high-frequency artifactsare considerably reduced by the finite-difference approximation as compared to the classicalFourier-transform. Besides numerical errors in the solution scheme, oscillations in shown resultsobtained from classical and modified Fourier transforms also stem from the Fourier interpola-tion after solving for function values at grid points (cf. the discrete solution at grid points alsoshown in Figs. 1 and 2).

To illustrate the effectiveness of higher-order corrections, we present in Fig. 3 the convergenceorder of the Euclidean error norm for corrections of varying order applied to the smooth function

f(x) =n=9

∑n=1

sin(2nπx). (10.68)

As can be expected for smooth functions, the spectral accuracy is degraded when using themodified Fourier transform with finite-difference approximation (see Fig. 3a). Using a higher-order correction is shown to be advantageous, as the error decreases significantly while resultingin lower convergence order (see Fig. 3b). Thus, the correction order can be chosen to controlthe competition between accuracy and convergence rate.

52


0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

analytical solution

standard spectral method

4th order correction

1st order correction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

normalized position x/L

axia

l st

raine x

Figure 4: Distribution of the axial strain in a composite bar obtained from solving linearmomentum balance by the iterative spectral method with first-order, fourth-order and withoutcorrection, compared to the exact piecewise-constant solution. Dots denote the converged gridpoint values, whereas lines follow from Fourier interpolation.

Next, we demonstrate the asymptotic consistency of the modified spectral scheme for a simpleelasticity problem. Consider a composite bar of length L having piecewise-constant cross-sections and Young’s moduli, as shown in Fig. 4, and subjected to uniaxial tension such thatthe average strain is 0.1. The outer two sections have Young’s modulus 2E and lengths 0.25L,while the inner section has Young’s modulus E and length 0.5L. The resulting axial straindistribution in the bar as obtained from the classical and modified Fourier spectral scheme withfirst- and fourth-order corrections is plotted in Fig. 4, demonstrating the effectiveness of themodified scheme in suppressing Gibbs effects.

As a representative 2D example of a high-contrast RVE, we compute the response of a linearelastic composite made of a circular inclusion (radius 0.25L; normalized Lame moduli λ = 1,µ = 1) embedded in a matrix (normalized Lame moduli λ = 0.6, µ = 0.6, outer side lengthL). Plotted in Fig. 5 is the normal stress distribution within the RVE for a biaxial tensiontest (for simplicity, shown is the affine interpolation of the 256 × 256 grid point values, i.e., the

(b)(a) (c)

position x/L position y/L

stre

sss

/m11

mat


stre

sss

/m11

mat


stre

sss

/m11

mat

s /m11 mat

stress

Figure 5: Uniaxial stress distribution in an elastic square-RVE with a circular inclusion using(a) the standard iterative spectral method and the modified spectral method with (b) fourth-order and (c) first-order correction. The first-order correction over-smooths, the fourth-ordercorrection still produces small oscillations, but note that no Fourier interpolation has beenperformed here (shown are the grid point values.)

53


shown solution is not affected by errors in the Fourier interpolation). Obviously, the classicalFourier spectral method results in strong artifacts near the interface between inclusion andmatrix (Fig. 5a), which is reduced significantly by using the first-and fourth-order corrections(see Fig. 5).

It is important to point out that several previous approaches circumvented ringing artifacts nearhigh-contrast interfaces by aborting the iterative solution scheme before oscillations appeared inthe solution. Unlike the corrections presented above, that does not guarantee convergence norbounded errors and, especially, is not suitable for a staggered, time-stepping solution procedureas that used for the dissipative processes in ferroelectrics. The numerical error of each truncatediteration can propagate and escalate over time, which is why the method presented here issuperior in the following scenarios.

54


11 Bloch Wave Analysis

So far, we have dealt with quasistatic homogenization problems. Bloch wave analysis dealswith the propagation of linear waves in periodic media and originated from harmonic atomiccrystals. For a given RVE, we study linearized waves of the form

u(x, t) = u(x) exp(−iωt) (11.1)

with propagation frequency ω ∈ R. For a periodic microstructure with RVE Ω, a Bloch waveis defined by

u(x) = u0(x0) exp [−ik ⋅ (x −x0)] , (11.2)

where k denotes the wave vector. x0 is located within the reference RVE is linked to any pointx by periodicity, i.e.,

x = x0 +d

∑a=1

caLa with x0 ∈ Ω. (11.3)

We now aim to solve a linearized wave propagation problem, for which the governing equationsare linearized about the current solution:

MU +KU = F . (11.4)

Application of

U(x, t) = U(x) exp(−iωt) (11.5)

yields

(K − ω2M)U = F . (11.6)

By exploiting the Bloch wave form, the RVE problem now becomes

(K − ω2M)U0 = F0 (11.7)

with

U+0 = U−

0 exp [−ik ⋅ (x+ −x−)] and F +0 = −F −

0 exp [ik ⋅ (x+ −x−)] , (11.8)

which can be solved in multiple ways. Note that because of inertial contributions of the movingwave, forces are no longer the same on opposite surfaces.

Note that we do not have to check all wave vectors. To notice this, recall that

exp [i(k ⋅x + 2πn)] = exp [ik ⋅x] ∀ n ∈ Z. (11.9)

This implies that we can restrict the choice of k to vectors lying in the First Brillouin Zone,i.e., the primitive cell in reciprocal space. By exploiting symmetry, one can often reduce thespace of k further to the so-called Irreducible Brillouin Zone.

55


12 From Quantum Mechanics to Atomistics

12.1 A Cartoon Introduction to Quantum Mechanics

At the level of elementary particles, matter is commonly described as waves rather than particles.

Consider a vibrating rod or string whose 1D equation of motion is

∂2

∂t2u(x, t) = c2 ∂

2

∂x2u(x, t) (12.1)

with wave speed c =√E/m. This is the approach of classical mechanics. At the quantum level,

the governing equations can be considered as follows.1

Consider a general form of a wave moving in one dimension, described by

u(x, t) = u(2πi(νt − kx)). (12.2)

One may conclude that

∂2u

∂x2= −4π2k2u, (12.3)

and

∂u

∂t= −i2πν u. (12.4)

Next, we need to consider the impulse and energy of a free particle, which behaves like a wave– unlike in classical mechanics. The great contributions of a series of physicists of the 20thcentury were given in the form of postulates.

De Broglie postulated that the wave length λ (and wave number k) and the particle’s impulsep are related via Planck’s constant2 h according to

λ = hp

and λ = 1/k ⇒ p = kh. (12.5)

Notice that by introducing for convenience

h = h

2π. (12.6)

and multiplying (12.3) by h/2m, we may reinterpret the impulse and kinetic energy as

h2

2m

∂2u

∂x2= − h2

2m4π24π2k2u = −(hk)2

2mu = − p

2

2mu ⇒ p2

2m→ − h

2

2m∇2 (12.7)

This makes for a convenient way to switch between particle and wave description.

Furthermore, the Einstein-Planck relation postulates that a particle’s energy is given by

E = hν. (12.8)

1Note that this is only a motivation and not a rigorous derivation of the governing equations of quantummechanics.

2h = 6.62607004 ⋅ 10−34kgm2/s.

56


Now, notice that multiplication by ih transforms (12.4) into

ih∂u

∂t= h2πν u = 2π(hν)u = 2πEu ⇔ ih

∂u

∂t= Eu, (12.9)

Finally, for a single particle of mass m and impulse p, exposed to an external potential V (e.g.,due to an electric field), the total energy at position x is

E(x, p) = p2

2m+ V (x). (12.10)

In summary, we have

h2

2m

∂2u

∂x2= − p

2

2mu and ih

∂u

∂t= Eu = ( p

2

2m+ V (x))u. (12.11)

Simple rearrangement and combination of the two equations yields Schrodinger’s equation

[− h2

2m

∂2

∂x2+ V (x)]Ψ(x, t) = ih ∂

∂tΨ(x, t) (12.12)

or

HΨ = ih∂Ψ

∂twith H = − h

2

2m

∂2

∂x2+ V (x) (12.13)

where we replaced the displacement u(x, t) by a general wave form Ψ(x, t) and introduced thedifferential operator H.

Note that like the wave equation, the solution to Schrodinger’s equation can be written in aseparable form

Ψ(x, t) =∞∑n=1

Ψn(x) exp(iEnt/h), (12.14)

whose insertion into Schrodinger’s equation yields an eigenvalue problem

HΨn = EnΨn (12.15)

with eigenvalues En (the energy levels) and associated (standing) wave forms Ψn. This impliesthat discrete, quantized energy levels En correspond to specific steady wave forms Ψn. (12.15) iscommonly referred to as the time-independent Schrodinger equation.

When considering elementary atomic particles, the dominant interactions are typically of Coulom-bic type. Recall that the interaction potential between two charged particles at a distance rhaving charges q1 and q2 is given by the Coulomb potential

V (r) = 1

4πε0

q1q2

r(12.16)

with ε0 denoting the electric permittivity of free space. Charges of the same sign repel eachother by minimizing the energy in the limit r → ∞, while opposite charges attract each otherby minimizing the energy as r → 0.

For a general system of multiple charged particles (electrons with charge −e and nuclei madeup of Z protons, each with charge +e), the potential energy can thus be written as

V = ∑α≠β

(Zαe)(Zβe)4πε0 ∣rαβ ∣

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶nuclei interactions

+ ∑i≠j

e2

4πε0 ∣rij ∣´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

electron interact.

− ∑i,α

(Zαe)e4πε0 ∣rαi∣

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶electron/nuclei interact.

(12.17)

57


where Zi denotes the number of protons in the respective nucleus.

Most important for the development of the concepts of atomistics is the Born-Oppenheimerapproximation: the relatively light electrons are assumed to be moving orders of magnitudefaster than the heavy nuclei (protons/neutrons are about 2,000 times heavier than electrons).Therefore, one may solve for the wave function of all electrons while assuming the nuclei remain(quasi)static at known positions.

Example: Hydrogen atom

Hydrogen is the simplest atom possible, having only one electron and one proton. Consequently,Schrodinger’s equation for the single electron – assuming the proton is fixed at x = 0 in 3D byBorn-Oppenheim – simplifies dramatically with the operator

H = −h2

2me∇

2 −e2

4πε0 ∥x∥, (12.18)

so that the resulting eigenvalue of the time-independent Schrodinger equation problem reads

(−h2

2me∇

2 −e2

4πε0 ∥x∥)Ψn(x) = EnΨn(x). (12.19)

In fact, the solution can be obtained analytically in spherical coordinates:

En = −mee

4

8ε20h

2n2, Ψ(n,l,m)(r, θ, φ) = cn,l,mRn,l(r)Pml (cos θ) exp(imφ), (12.20)

whose stationary states Ψ involve the Legendre polynomials P lm and Laguerre polynomialsRn,l, and cn,l,m are known, closed-form constants (omitted here for conciseness). Note that thethree principal quantum numbers3 are not independent: for 1 ≤ n one needs to have 0 ≤ l ≤ n−1and −l ≤m ≤ l. For example, the lowest-energy ground state corresponds to (n, l,m) = (1,0,0)and

Ψ(1,0,0)(r) =1

√πa

3/20

exp (−r/a0) with a0 =4πε0h

2

mee2(12.21)

begin Bohr’s radius (approximately the mean or most probable distance between proton andelectron in a hydrogen atom in its ground state). ε0 is the permittivity of free space.

The discrete energy levels are sometimes rewritten as

En = −e2

8πε0a0n2. (12.22)

For every wave function Ψ, one can define the spatial probability density ∣Ψ∣2. Plottingthe probability density for the various energy levels results in the well known orbitals. Forexample, ∣Ψ(1,0,0)∣2 from (12.21) corresponds to the 1s orbital.

3The fourth quantum number is a particle’s spin.

58


Next, let us consider more complex atomic systems composed of electrons at positions xe =x1, . . . ,xn and nuclei at positions qn = q1, . . . ,qn.

The time-independent Schrodinger equation for this scenario is

H(x,q)Ψ(x,q) = EΨ(x,q) (12.23)

with the combined operator (involving electron and nuclei momenta pe and pn, respectively)

H(x,q) =Ne

∑i=1

∣pei ∣2

2me+Nn

∑i=1

∣pni ∣2

2mn+∑α≠β

(Zαe)(Zβe)4πε0 ∣qα − qβ ∣

+∑i≠j

e2

4πε0 ∣xi −xj ∣−∑i,α

(Zαe)e4πε0 ∣qα −xi∣

, (12.24)

and recall that we may replace

Ne

∑i=1

∣pei ∣2

2me=Ne

∑i=1

− h2

2me∇2

xi. (12.25)

Using the Born-Oppenheimer approximation, one separates the (fast) electronic from the (slow)nuclei contributions by assuming a solution

Ψ(x,q) = Ψel(x;q)Ψnuc(q). (12.26)

Consequently, one solves a decoupled problem for the electron wave function Ψel (for givennuclei positions q), viz.

Hel(x,q)Ψel(x) = V (q)Ψel(x), (12.27)

with

Hel = −Ne

∑i=1

h2

2me∇2

xi+ ∑α≠β


+∑i≠j

e2

4πε0 ∣xi −xj ∣−∑i,α

(Zαe)e4πε0 ∣qα −xi∣

. (12.28)

This implies that for given nuclei positions, one solves for the electron eigenstates and theassociated energy V (q). Note that V (q) is a function of the nuclei positions only.

In a next, higher-level step, one then finds the solutions for the nuclei positions from

Hnuc(q)Ψnuc(q) = EΨnuc(q), (12.29)

with

Hnuc =N

∑i

∣pi∣2

2mi+ V (q), (12.30)

which can also be replaced by the time-dependent form. Note that, at this point, all electroncontributions have been condensed out; i.e., if V (q) is known, we can solve the above problemfor the nuclei positions only. This is the approach taken by atomistics – to be discussed below.Further, note that it is relatively arbitrary for the solution whether or not the second term in(12.28) is retained there or taken into (12.30).

One frequent assumption being made in solving the above equations is that of non-interactingparticles. As a simple example consider a hydrogen molecule, H2, which contains two electrons.A closed-form solution is feasible if these electrons are treated as non-interacting, so that thenthe wave function for each electron is obtained individually.

59


12.2 Density Functional Theory in a Nutshell

The problem with the equations of quantum mechanics is the extreme complexity involved whentreating systems with many particles (recall that the specifics of each electron and nucleus areto be solved for). Therefore, alternatives have been proposed to solve large systems in moreefficient ways, one such example being density functional theory (DFT).

Consider the Schrodinger equation for a system of Ne electrons with known nuclei positions:

H(x;q)Ψel(x) = EΨel(x), H(x;q) = −Ne

∑i=1

h2

2me∇2

xi+Ne

∑i=1

V (xi;q)+∑i≠j

e2

4πε0 ∣xi −xj ∣, (12.31)

where we abbreviated the interaction potential of each electron i with the surrounding protonsby V (xi;q). Note that for a given wave function Ψel the energy can be extracted as

E = ⟨Ψel∣H∣Ψel⟩⟨Ψel∣Ψel⟩

, (12.32)

where

⟨a∣b⟩ = ∫R3⋯∫

R3a∗(x1, . . . ,xNe) b(x1, . . . ,xN)dx1⋯dxNe (12.33)

and

⟨a∣A∣b⟩ = ∫R3⋯∫

R3a∗(x1, . . . ,xNe)A b(x1, . . . ,xNe)dx1⋯dxNe (12.34)

Density functional theory takes as its basis the minimization of E with respect to the wavefunctions to find the electron configurations. Note that the last term in the operator (12.31)complicates calculations, as it involves correlations between different electron wave functions.

Density functional theory introduces the electron density ρ(r) as the probability of findingan electron at a location r ∈ R3 (with all electrons being interchangeable), i.e.,

ρ(r) =Ne

∑i=1∫R3⋯∫

R3∣Ψ(x1, . . . ,xNe)∣

2 δ(r −xi) dx1⋯dxNe = ⟨Ψ∣Ne

∑i=1

δ(r −xi)∣Ψ⟩ (12.35)

such that

∫R3ρ(r) dr = Ne. (12.36)

When inserting into the governing equations, certain approximations are applied to make thevariational problem solvable for the electron density, resulting in the different flavors of DFT.

As a result, the energy functional is generally expressed in the form

E[ρ] = EZZ[ρ] + Vext[ρ] + Vee[ρ] +Ekin[ρ]. (12.37)

The individual contributions include the potential of the nuclei interactions,

EZZ[ρ] = ∑α≠β


, (12.38)

which is independent of the wave functions. The interaction of electrons with nuclei is capturedby the external interaction energy functional

Vext[ρ] = ∫R3Vqn(r)ρ(r) dr with Vqn(r) = −∑

β

(Zβe)e∣r − qβ ∣

. (12.39)

60


The electron interaction energy is usually approximated by the so-called Hartree energy,

Vee[ρ] ≈ VH[ρ] = 1

2∫R3∫R3

e2ρ(r)ρ(r′)∣r − r′∣

dr dr′, (12.40)

which is not exact because it includes purely Coulombic interactions and neglects quantummechanical effects. Since this introduces approximation errors, one usually adds a correctionterm, the so-called exchange-correlation functional Exc[ρ], so that

Vee[ρ] = VH[ρ] +Exc[ρ]. (12.41)

The split of the energy into that of non-interacting electrons and a correction goes back toKohn-Sham.

Finally, the kinetic energy is another critical case that can only be evaluated as a functionof the electron density under certain assumptions. For example, the Thomas-Fermi modelapproximates the kinetic energy by

Ekin[ρ] = cF ∫R3ρ5/3(r) dr, cF = 3h2

10me( 3

8π)

2/3. (12.42)

In order to solve for the electron density, one introduces approximations via basis functions. Tothis end, it is convenient to write

ρ(r) =N

∑i=1

∣ϕi(r)∣2 (12.43)

with to be determined orbitals ϕi(r).

12.3 Hamiltonian Description

Atomistic techniques such as molecular dynamics (MD) or molecular statics (MS) make effectiveuse of the Born-Oppenheimer assumption in that they separate the fast electronic configura-tional changes from the slow motion of the atomic nuclei. In particular, if the spacing betweenparticles is sufficiently large, then their wave functions are spatially localized and they can betreated as classical particles. In general, one introduces the de Broglie wavelength

Λ =√

h2

2πmkNT(12.44)

and treats atoms as classical particles if the average interatomic spacing a≫ Λ. By contrast, ifΛ is on the order of interatomic distances, then the wave-like behavior of atoms is non-negligible.Especially at high temperatures all atoms can be treated as classical particles.

The starting point of atomistics is (12.30) where, however, the nuclei wave functions are replacedby classical particle positions and momenta. To this end, an atomistic ensemble containing Natoms is uniquely described by how their nuclei positions

q(t) = q1(t), . . . ,qN(t) (12.45)

and the derived nuclei momenta

p(t) = p1(t), . . . ,pN(t) (12.46)

61


with pi =miqi evolve with time t. mi represents the mass of atom i, and dots denote materialtime derivatives. The ensemble’s total Hamiltonian is given by

H(q,p) =N

∑i=1

∣pi∣2

2mi+ V (q), (12.47)

where the first term accounts for the kinetic energy and the second term represents the potentialenergy of the ensemble with V denoting a suitable atomic interaction potential (which includesall information about electron interactions in between the atomic nuclei; it is thus a condensedform depending only on the nucleus or atomic positions).

Compared to the electron wave functions discussed above, atomic nuclei are moving relativelyslowly, which is why one assumes that the time evolution of an atomistic system is governed byHamilton’s equations,

q = ∂H∂p

, p = −∂H∂q

. (12.48)

The second equation yields Newton’s equations of motion for all atoms i = 1, . . . ,N :

miqi = fi(q) = −∂V

∂qi(q), (12.49)

where fi(q) represents the total (net) force acting on atom i. For molecular statics, theseequations reduce to

fi(q) = −∂V

∂qi(q) = 0, (12.50)

and they aim to minimize the total potential energy (at zero temperature).

The challenge is now to identify the interatomic potential V from either first principles (using,e.g., tight binding) or to introduce empirical potentials whose parameters are fitted to reproduceknown material properties (such as elastic constants, thermal expansion coefficients, etc.).

In most materials (including metals, ceramics and organic materials), the interaction potentialallows for an additive decomposition, i.e.,

V (q) =N

∑i=1

Vi(q) (12.51)

with Vi being the energy of atom i. Some authors in the scientific literature have preferredto distinguish between external and internal forces by introducing external forces fi,ext on allatoms i = 1, . . . ,N so that

fi(q) = fi,ext(qi) −∂V

∂qi(q). (12.52)

Ideally (and in any reasonable physical system), such external forces are conservative and derivefrom an external potential Vext(q) and we may combine internal and external forces into asingle potential, which is tacitly assumed in the following. Examples of conservative externalforces include gravitation, long-range Coulombic interactions, or multi-body interactions such asduring contact. For computational convenience, the latter is oftentimes realized by introducingartificial potentials.

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13 Interatomic Potentials

As we have seen above, the constitutive description of atomistic ensembles is provided by in-teratomic potentials. An interatomic potential describes the effective binding energy of anatom embedded into an ensemble of atoms whose nuclei are located at q = q1, . . . ,qN:

V = V (q). (13.1)

In the following, we only consider potentials that depend on atomic (nuclei) positions (whichcan be generalized to potentials that also depend on atomic momenta; this plays a role, e.g., infinite-temperature QC). The potential accounts for both repulsive and attractive forces.

In principle, every potential (written for an ensemble of N atoms) can be expanded as

V (q) =N

∑i=1

V1(qi) +1

2∑i≠jV2(qi,qj) +

1

3!∑

i≠j,i≠kV3(qi,qj ,qk) + . . . , (13.2)

where one usually avoids summation over identical indices (i.e., when writing i ≠ j we implysummation over i = 1, . . . ,N and j = 1, . . . ,N with the exclusion of the case where i = j).

The term involving V1 can be neglected, unless atoms are in an external field (e.g., ions in anelectric field). Otherwise, the potential should be invariant under rigid-body motion so V1 ≡ 0.Retaining only V2 and neglecting higher-order dependencies results in pair potentials (discussednext), whereas retaining V2 and V3 results in three-body potentials (discussed below), etc.

13.1 Pair Potentials

The specific class of two-body potentials or pair potentials assumes the form

V (q) =∑i≠jf(qi,qj) =∑

i≠jf(rij) with rij = ∥qi − qj∥ , (13.3)

where we made use of the fact that invariance under rigid body motion requires the potentialto only depend on relative distances between between pairs of atoms (lowest-order correlation).

Example I: Coulombic crystals

As a simple example, consider ions with charges q = q1, . . . , qN interacting through Coulom-bic interaction:

V (q) = 1

4πε0∑i≠j

qiqj

rijwith rij = ∥qi − qj∥ . (13.4)

The force between two isolated ions (N = 2) is thus obtained as

f1 = −f2 = −∂V

∂q1= − ∂

∂q1

1

4πε0

q1q2

r= 1

4πε0

q1q2

r2r with r = q1 − q2

∥q1 − q2∥. (13.5)

Consequently, equally-signed charges with sign(q1) = sign(q2) repel one another, and they pro-duce a positive binding energy V > 0. In contrast, charges of opposite signs, i.e., sign(q1) =

63


− sign(q2), attract each other, resulting in a negative binding energy V < 0. The force on anatom i in an ensemble of atoms is obtained by superposition:

fi = −∂V

∂qi= 1

4πε0∑j≠i

qiqj

r2ij

rij . (13.6)

Coulombic interactions are long-range because the 1/r-dependence decays slowly with distance.

Consider an infinite 1D chain of ions of opposite charges q1 = −q2 = q. This is the prototypeof an ionic crystal. Assume that the chain has equal interatomic spacings a. The potentialenergy of every atom in the chain, depending on distance a, then becomes

V (a) = 2

4πε0

∞∑i=1

[ q2

2ia− q2

(2i − 1)a] = − 2q2

4πε0a(1 − 1

2+ 1

3− 1

4+ . . .) , (13.7)

where we exploited symmetry (the factor of 2 results from symmetry of neighbors). Using theTaylor expansion

ln(1 + x) = x − x2

2+ x

3

3− x

4

4+ . . . (13.8)

allows us to write the potential as

V (a) = − q2

2πε0aln 2. (13.9)

We notice that this ionic crystal does not have an equilibrium spacing a0 = arg minV (a),since the minimum V is attained for a→ 0 (which could have been expected since the 1/r-scalingattracts nearest neighbors more strongly than next-to-nearest neighbors repel one another).

Example II: Lennard-Jones potential

Next, consider one of the simplest two-body interatomic potentials with an equilibrium distance,the Lennard-Jones potential (named after Sir John Lennard-Jones) which has the generalform

V (r) = − Arn

+ B

rm(13.10)

with integers exponents n,m > 0 and real-valued strength coefficients A,B > 0. A represents thestrength of the attractive contribution, whereas B dominates the repulsive contribution. For anequilibrium to exist, atoms should attract over large distances but repel over short distances;this implies that we must have m > n.

Note that two atoms at a distance r now have the binding energy (13.10), which has an extremumif

0 = ∂V∂r

= n A

rn+1−m B

rm+1= 1

rn+1(nA − mB

rm−n) ⇔ r0 = (mB

nA)

1/(m−n). (13.11)

Furthermore, notice that

∂2V

∂r2∣r=r0

= −n(n + 1) A

rn+20

+m(m + 1) B

rm+20

= 1

rn+20

[m(m + 1) B

rm−n0

− n(n + 1)A]

= 1

rn+20

[m(m + 1)BnAmB

− n(n + 1)A] = nA

rn+20

[(m + 1) − (n + 1)] = n(m − n)Arn+2

0

.

(13.12)

64


Consequently, the potential has a minimum at r = r0 if m > n (and A,B > 0), as expected.V (r0) is the maximum binding energy.

Next, let us compute the equilibrium spacing a0 of an infinite chain of identical atoms. To thisend, consider the energy of a single atom, which is:

V = 2∞∑i=1

(− A

(ia)n+ B

(ia)m) . (13.13)

Recall the definition of Riemann’s zeta-function,

ζ(n) =∞∑i=1

1

in, (13.14)

so that

V = 2(−Aζ(n)an

+Bζ(m)am

) . (13.15)

The minimizer is obtained from

0 = ∂V∂a

= 2(nAζ(n)an+1

−mBζ(m)am+1

) = 2

an+1(nAζ(n) −mBζ(m)

am−n) , (13.16)

so that

a0 = (mBζ(m)nAζ(n)

)1/(m−n)

. (13.17)

We may take, e.g., the most common form (known as the 6-12 potential) with n = 6 andm = 12. Then

a0 = (1382π6

675675

B

A)

1/6

≈ (1.96639B

A)

1/6. (13.18)

Note that for the same choice of n = 6 and m = 12 we have

r0 = (2B

A)

1/6. (13.19)

That is, the equilibrium distance r0 between two isolated atoms does not coincide with theequilibrium distance a0 between atoms in an infinite chain (and yet another distance would beobtained when considering only a finite number of neighboring atoms or 2D arrays of atoms).

Finally, note that choices such as m = 12 and n = 6 result in quickly decaying interaction forces,which is why such potentials are called short-range (in contrast to the Coulombic interactionsdiscussed above).

If the equilibrium spacing a0 and the maximum binding energy V (a0) are known (which canboth be determined experimentally), then the potential parameters can be identified by fitting.

Binding energies are typically on the order of electron-volts:

1eV = ∣qe∣ ⋅ 1V = 1.60217662 ⋅ 10−19J, (13.20)

where qe = 1.60217662 ⋅ 10−19C is the charge of a single electron. Therefore, 1eV corresponds tothe energy of a single electron that has passed through a potential of 1V.

Equilibrium spacings are on the order of a few Angstrom (named after Swedish physicistAnders Jonas Angstrom who studied the wavelengths of electromagnetic radiation, where thischaracteristic length emerges):

1A = 10−10m. (13.21)

65


Example III: Morse potential

A further example of a two-body potential of practical relevance is the Morse potential (namedafter American physicist Philip McCord Morse):

V (r) =D[1 − exp (−a(r − r0)) ]2

(13.22)

with equilibrium distance r0 (between two isolated atoms) and well depth

D = V∞ − V (r0), (13.23)

i.e., the energetic difference between the equilibrium and the energy at r →∞. The limit

V∞ = limr→∞

V (r) (13.24)

is known as the dissociation energy.

13.2 (An)Harmonicity and the Quasiharmonic Approximation

All of the above potentials have one thing in common: they are anharmonic, i.e., the potentialis not symmetric around the equilibrium point. Of importance later, one can always constructa harmonic potential by Taylor expansion:

V (r) = V (r0) +∂V

∂r∣r0

(r − r0) +1

2

∂2V

∂r2∣r0

(r − r0)2 + h.o.t. (13.25)

Note that, if the linear term vanishes by definition of equilibrium, then

V (r) = V (r0) +C

2(r − r0)2 + h.o.t. (13.26)

This expansion is known as the quasiharmonic approximation with a force constant

C = ∂2V

∂r2∣r0

> 0. (13.27)

This approximation is of particular importance when studying atomic vibrations, which are ofsmall amplitude so that the quasiharmonic approximation can be used to compute the lowest-order vibrational frequencies (phonon modes). Note that this assumes that each atom isvibrating individually. Therefore, the quasiharmonic approximation generally yields accuratepredictions at low absolute temperature, whereas it leads to significant errors at elevated tem-peratures where vibrational amplitudes become so large that the assumption no longer holds.

13.3 Multi-Body Potentials

Pair potentials are simple and efficient but not suitable to describe most material behaviorincluding, e.g., the behavior of metals. Here, multi-body potentials gain importance.

66


Example I: Stilinger-Weber potential

The Stilinger-Weber potential for silicon is an example for a three-body potential. In additionto pair interactions, the third-order potential term V3(qi,qj ,qk) is conveniently expressed interms of the two distances and the bond angle between triples of atoms. The Stilinger-Weberpotential thus has the form

V (q) = ∑i,j∈L

V2(rij) + ∑i,j,k∈L

V3(rij , rik, θijk), (13.28)

where θijk denotes the angle between rij and rik.

Example II: Embedded Atom Method

The most common example for metals is the so-called Embedded Atom Method whosepotential for an atom i surrounded by a set L of atoms is defined by

Vi(q) =1

2∑j∈L

Φ(rij) +U(ρi), with rij = ∥qi − qj∥ . (13.29)

Φ(r) is a pair potential representing nucleus-nucleus interactions. U(ρ) is called the embeddingfunction which denotes the access energy due to embedding atom i within the electron densityρi present at the location of atom i due to all other atoms. For simplicty, one commonly assumesthat ρi depends on the distances rij to all other atoms j:

ρi = ∑j∈L

f(rij). (13.30)

In order to compute the force acting on an atom k, we must consider the total potential energy

V (q) =∑i∈LVi(q). (13.31)

The force acting on an atom k is now derived as

fk = −∂V

∂qk= −

⎡⎢⎢⎢⎢⎣

1

2∑i∈L∑j∈L

Φ′(rij) +∑i∈LU ′(ρi) ∑

j∈Lf ′(rij)

⎤⎥⎥⎥⎥⎦

∂rij

∂qk. (13.32)

For atomic ensembles, the partial derivative can be computed exactly by using rij = ∥qi − qj∥so that

∑i,j

(⋅)ij∂rij

∂qk=∑i,j

(⋅)ijrij

rij(δik − δjk) =∑

i,j

(⋅)ij rij (δik − δjk) . (13.33)

Note that rij = rji and rij = −rji, so that the forces simplify to

fk = −⎡⎢⎢⎢⎢⎣

1

2∑i∈L∑j∈L

Φ′(rij) +∑i∈LU ′(ρi)∑

j∈Lf ′(rij)

⎤⎥⎥⎥⎥⎦

rij

rij(δik − δjk)

= −⎡⎢⎢⎢⎢⎣

1

2∑j∈L

Φ′(rjk) +U ′(ρk)∑j∈L

f ′(rjk)⎤⎥⎥⎥⎥⎦

rkj

rjk+ [1

2∑i∈L

Φ′(rik) +∑i∈LU ′(ρi)f ′(rik)]

rikrik

= ∑j∈L

[Φ′(rjk) + (U ′(ρj) +U ′(ρk)) f ′(rjk)] rjk,

(13.34)

67


where rjk = rjk/rjk is the unit vector between atoms j and k, and the amplitude of the forcebetween atoms j and k is

fjk = Φ′(rjk) + (U ′(ρj) +U ′(ρk)) f ′(rjk) (13.35)

so that

fk = ∑j∈L

fjkrjk. (13.36)

For most potentials, pair interactions and electron densities are short-range, so that summa-tions over L can usually be restricted to summations over a small number of neighboring atoms.To this end, one introduces a cut-off radius rcutoff which defines the cut-off neighborhoodaround an atom i as

Ci = j ∈ L ∶ ∥qj − qi∥ ≤ rcutoff, (13.37)

resulting in

fk = ∑j∈Ck

fjkrjk. (13.38)

Example III: Force Fields

Especially for organic materials such as, e.g., polymers, force fields have gained popularity.In a nutshell, a molecule is approximated by a collection of charged atoms linked by elasticconnections. Consequently, the potential decomposes into contributions from bonds, angles,dihedral angles, and long-range non-bond angles with the general form:

V (q) = ∑i,j∈bonds

kbondij r2

ij + ∑i∈angles

kanglei θ2

i + ∑i,j∈bonds

kdihedrali f(φi, θi)

+ ∑i,j∈L

4εij

⎡⎢⎢⎢⎢⎣(σij

rij)

12

− (σij

rij)

6⎤⎥⎥⎥⎥⎦+ ∑i,j∈L

qiqj

4πε0rij.

(13.39)

The Lennard-Jones term represents van der Waals interactions, while the Coulombic termstands for ionic interactions.

13.4 Centrosymmetry

Further, note that many crystals show centrosymmetry. Let us define the centrosymmetryparameter

CS =∑i

∥qi,1 − qi,2∥2 , (13.40)

where (qi,1,qi,2) are the positions of the ith pair of opposite neighbors of an atom. Note that –for centrosymmetric lattices (i.e., CS = 0) – the net force on every atom in an infinite, uniform,equally-spaced crystal vanishes always by symmetry, irrespective of the actual lattice spacing.This is not the case for atoms near defects or near boundaries like free surfaces.

68


13.4.1 Infinite vs. finite-size samples

It is important to note that the aforementioned construction of an equilibrium spacing a0

strictly applies in infinite crystals. Due to symmetry, the net force on every atom in an infinitecentrosymmetric lattice always vanishes for equally spaced atoms. The equilibrium spacing istherefore generally defined as the minimizer of the energy, a0 = arg minV (a), and not via solvingfor zero net forces.

By contrast, in finite-size samples with free surfaces, those atoms near the surface “feel” theirmissing neighbors and experience a non-zero net force in case of equally spaced atoms. Thus,energy minimization here leads to surface relaxation, i.e., changes in atomic spacing or evenfaceting (i.e., the formation of more complex surface facets) near the free surface. Note that sincethe free surfaces break the symmetry of the lattice, in this case the equilibrium configuration canbe obtained by energy minimization as well as by solving for zero net forces. For a sufficientlylarge sample, the interior atoms will obey the same equilibrium spacing a0 mentioned above.

69


14 Cauchy-Born Approximation and Effective Properties at ZeroTemperature

The Cauchy-Born rule(s) discussed before for structural lattices at the continuum scale canalso be applied to atomic ensembles, which is a popular technique to extract, e.g., the zero-temperature elastic moduli of crystalline solids. It also forms the basis for the quasicontinuummethod discussed later. Let us quickly review the underlying theory.

Consider an atomic ensemle of N atoms at positions Q = Q1, . . . ,QN. For crystals, thesepositions are usually given by a Bravais lattice with basis B = A1, . . . ,Ad and cj ∈ Z:

Qi =d

∑j=1

cjAj . (14.1)

When an average deformation gradient F is applied to a simple Bravais lattice of equal atoms(such as in fcc or bcc metals), then it is reasonable to assume under small deformation thatthe response is an affine deformation of the crystal lattice. Thus, we may apply the localCauchy-Born rule with average deformation gradient F to find the deformed atomic positionsas

qa = FQa, a = 1, . . . ,N. (14.2)

This allows us to define an effective energy density

W (F ) = 1

∣Ω∣V (q1, . . . ,qN), (14.3)

where V (q1, . . . ,qN) includes the energy of atoms in an atomic unit cell, and ∣Ω∣ denotes itsvolume. In the simplest case of simple Bravais lattices, each atom experiences the exact sameneighborhood changes (all atoms are equal). Then, V (q1, . . . ,qN) can be taken as the energyof a single atom only (and applying (14.2) to all of its neighbors within the cut-off radius).

The first Piola-Kirchhoff stress tensor components now follow as (writing qai for the ithcomponent of qa)

PiJ(F ) = 1

∣Ω∣∂

∂FiJW (F ) = 1

∣Ω∣

N

∑a=1

∂V

∂qak(q)

∂qak∂FiJ

= − 1

∣Ω∣

N

∑a=1

fak (q)∂

∂FiJFkLQ

aL = − 1

∣Ω∣

N

∑a=1

fai (q)QaJ

(14.4)

or

P (F ) = − 1

∣Ω∣

N

∑a=1

fa(q)⊗QJ , (14.5)

where

fa(q) = −∂V

∂qa. (14.6)

Analogously, we can derive the incremental stiffness tensor components as

CiJkL = ∂PiJ∂FkL

= − 1

∣Ω∣∂

∂FkL

N

∑a=1

fai (q)QaJ = −1

∣Ω∣

N

∑a,b=1

∂fai∂qbk

(q)QaJQbL. (14.7)

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The usual mapping relations apply to transform tensors between current and reference config-urations.

For simple crystals, this gives a convenient recipe to compute the zero-temperature elastic con-stants from molecular statics.

Example: Embedded Atom Method

For example, for the embedded atom method we have the atomic potential of a single atom,

V (q) = 1

2∑α∈L

Φ(rα) +U(ρ) with ρ = ∑α∈L

f(rα), (14.8)

where we sum over all neighboring atoms α ∈ L of an atom at q0 = FQ0, and we defined

rα = qα − q0 = FQα −FQ0 (14.9)

and rα = ∣rα∣. ρ is the effective electron density of the atom.

Without showing the full details (feel free to verify yourself), the first Piola-Kirchhoff stresstensor is obtained as

PiJ =1

∣Ω∣ ∑α∈L([U ′(ρ)f ′(rα) + 1

2Φ′(rα)]

rαi rαj

rα)F−1

Jj . (14.10)

The incremental modulus tensor in the current configuration follows as

cijkl =1

∣Ω∣

⎧⎪⎪⎨⎪⎪⎩U ′′(ρ) [∑

α∈Lf ′(rα)

rαi rαj

rα]⎡⎢⎢⎢⎢⎣∑β∈L

f ′(rβ)rβk r

βl

rβ

⎤⎥⎥⎥⎥⎦

+∑α∈L

[((U ′(ρ)f ′′(rα) + 1

2Φ′′(rα)) − 1

rα(U ′(ρ)f ′(rα) + 1

2Φ′(rα)))

rαi rαj r

αk r

αl

(rα)2] .

(14.11)

Thus, for given atomic reference positions Q in the crystal lattice and a given potential withknown functions Φ, U and f , the above can be used to compute the (zero-temperature) elasticmoduli of a single-crystal. Note that for fcc and bcc lattices, the resulting moduli will not beisotropic but exhibit cubic symmetry (the usual isotropic elastic moduli will have to be obtainedby averaging over many grains in a polycrystal).

Calculating specific elastic constants, such as effective Young, shear or bulk moduli, can alsobe obtained directly from the energy in a numerical fashion. For example, the effective bulkmodulus of a single crystal can be obtained as

K = 1

9Ω

∂2V (γFKQ)∂γ2

∣γ=0

with FK =⎡⎢⎢⎢⎢⎢⎣

1 0 00 1 00 0 1

⎤⎥⎥⎥⎥⎥⎦. (14.12)

Analogously, an effective shear modulus may be defined as

µ = C44 =1

4Ω

∂2V (γFµQ)∂γ2

∣γ=0

with Fµ =⎡⎢⎢⎢⎢⎢⎣

0 1 01 0 00 0 0

⎤⎥⎥⎥⎥⎥⎦, (14.13)

71


while the uniaxial elastic constant can be extracted via

C11 =1

Ω

∂2V (γF11Q)∂γ2

∣γ=0

with F11 =⎡⎢⎢⎢⎢⎢⎣

1 0 00 0 00 0 0

⎤⎥⎥⎥⎥⎥⎦, (14.14)

where Ω denotes the volume associated with a single atom, and V is the potential energy of asingle atom whose neighbors deform by the Cauchy-Born rule according to the given deformationmappings.

Note that the affine deformation mappings discussed here only apply in simple Bravais latticesand in infinite crystals. They are thus well-suited to describe elastic constants or crystallinebehavior under infinitesimal strains. They fail, however, when localization, pattern formation,or inhomogeneous deformation emerges.

Further note that in complex multi-species lattices (usually referred to as multi-lattices, eveninfinitesimal deformation may not be affine, so that a multi-lattice description with relativeshifts between the various lattices must be accounted for; cf. Section 9.2. This is especiallyrelevant in case of phase or structural transformations.

72


15 Molecular Dynamics

15.1 Statistical Mechanics

Recall that we describe the state of an atomistic system of N atoms by positions q = q1, . . . ,qNand momenta p = p1, . . . ,pN with pi =miqi. The total Hamiltonian yields Hamilton’s equa-tions of motion:

H(q,p) =N

∑i=1

∣pi∣2

2mi+ V (q), qi =

∂H∂pi

, pi = −∂H∂qi

. (15.1)

Every state of the system is thus described by a point in 2N -dimensional space, the so-calledphase space which is often denoted by Γ. Motion of atoms can be reinterpreted as a trajectory(q(t),p(t)) through phase space with some initial condition (q(0),p(0)).

Example: harmonic oscillator

As a simple example, consider a harmonic oscillator (i.e., a mass m attached to a linear springwith stiffness k) whose equation of motion mq + kq = 0 has the solution

q(t) = A sin(ωt) +B cos(ωt) with ω =√k/m. (15.2)

Also,

p(t) =mq(t) =mω [A cos(ωt) −B sin(ωt)] . (15.3)

Applying initial conditions q(0) = q0 and p(0) = p0 yields B = q0 and A = p0/(mω). Take, e.g.,the simple case p0 = 0 so that

q(t) = q0 cos(ωt), p(t) = −q0mω sin(ωt). (15.4)

Hence, the mass’ motion in phase space describes the ellipse

( qq0

)2

+ ( p

q0mω)

2

= 1 ⇔ q2 + ( p

mω)

2

= q20, (15.5)

with a unique ellipse for each initial condition q0 = q(0). Note that

H = p2

2m+ k

2q2 = 1

2mq2

0m2ω2 sin2(ωt) + k

2q2

0 cos2(ωt)

= k2q2

0 [sin2(ωt) + cos2(ωt)] = k2q2

0.

(15.6)

That is, without external forcing the Hamiltonian is constant along its trajectory and, moreover,the area enclosed by the ellipse grows with increasing energy. Specifically, the ellipse’s majorprincipal axis is

q0 =√

2H/k. (15.7)

This can be generalized for systems in classical mechanics: motion is unique so that trajectoriesdo not intersect and

dHdt

=N

∑i=1

(∂H∂qi

⋅ qi +∂H∂pi

⋅ pi) =N

∑i=1

(−pi ⋅ qi + qi ⋅ pi) = 0, (15.8)

73


i.e., energy is conserved and the Hamiltonian is constant along trajectories.

Similar in spirit to homogenization, we would like to define macroscopic, effective observableslike the state variables in continuum mechanics (e.g., energy, temperature, stress, etc.) Sincethe atomistic system is uniquely defined by (q,p), we may assume that each observable Acan be expressed as a function A = A (q,p). Unlike in computational homogenization discussedbefore, the key challenge here is to bridge across time scales, which is why one may define theeffective quantity A(t) at time t as the time average

A(t) = 1

∆t∫

t+∆t

tA (q(τ),p(τ)) dτ, (15.9)

where ∆t is large compared to the atomic fluctuations. Equilibrium in this context impliesthat A(t) = const. and hence independent of time t. For almost all initial conditions andwithout providing external power, the limit A obtained as ∆t → ∞ exists and represents thetrue macroscopic quantity:

A = lim∆t→∞

1

∆t∫

t+∆t

tA (q(τ),p(τ)) dτ. (15.10)

Numerically, such quantities can – more or less – easily be computed during MD simulationsby taking

A ≈ 1

n∞

n∞

∑i=1

A (q(ti),p(ti)) . (15.11)

The above time averaging is numerically simple but computationally expensive and theoreti-cally inconvenient. Therefore, one alternatively uses statistical approaches based on probabilitydensities.

In statistical mechanics, one defines macrostates (described by the macroscopic observables)and the associated microstates (describing all realizations in agreement with a particularmacrostate). For example, we may fix the total energy of a system, i.e.,

N

∑i=1

∣pi∣2

2mi+ V (q) = E = const. (15.12)

Here, energy E is a macroscopic observable and there are many microstates (i.e., particularatomic positions and momenta) that satisfy the contraint (15.12). All microstates in agree-ment with a given macrostate are collectively referred to as an ensemble. Depending on theparticular macro-constraints, one differentiates between the classical ensembles:

the microcanonical ensemble (NV E-ensemble) which describes an isolated system withconstant energy (particle number and volume),

the canonical ensemble (NV T -ensemble) which describes a system in contact with aheat bath, i.e., at constant temperature (and constant particle number and volume),

the grand canonical ensemble (µV T -ensemble) which is an extension of the canonicalensemble and describes a system in contact with both heat and particle baths (constanttemperature, volume and chemical potential).

There are further ensembles of practical relevance such as the isothermal-isobaric ensemble(NpT -ensemble) which implies constant temperature and constant pressure.

74


If only a macrostate such as E is prescribed, there will be large numbers of potential microstates(e.g., for the harmonic oscillator the entire ellipse presents admissible microstates for a given E).If one takes an ensemble with a macro-constraint, the full system will follow trajectories thatcomply with the imposed macrostate and we can define other macroscopic quantities A for thatsystem. The trajectory depends on the initial conditions, which means that in order to obtainthe true average A, one should ideally run very many, n to be specific, parallel simulations withthe same E but starting with different initial conditions so as to sample over as many points inphase space as possible. This yields the approximate ensemble average

⟨A⟩ ≈ 1

n

n

∑i=1

A(qn,pn). (15.13)

If we consider the limit n → ∞, we obtain the macroscopic observable A as the ensembleaverage or phase average

⟨A⟩ = ∫ΓA(q,p)ρ(q,p) dp dq (15.14)

where ρ(q,p) is a distribution function or probability density which satisfies

∫Γρ(q,p) dp dq = 1. (15.15)

Roughly speaking, the probability density ρ(q,p) defines the likelihood of finding an ensemblein the configuration (q,p).

For the phase average to make physical sense, Ludwig Boltzmann introduced the so-calledhypothesis of ergodicity: if one waits sufficiently long, any physical system will visit all possiblepoints in phase space consistent with a given macrostate. This admits replacing the discretesum over a countable number of measurements by the integral in phase space. Moreover, it alsoimplies that

A = ⟨A⟩. (15.16)

15.2 Center of mass coordinates

In the following, we will link thermal properties to atomic vibrations. Therefore, it is convenientto reformulate atomic positions and momenta as those relative to the center of mass and themean momentum of all N atoms:

Q = ∑Ni=1miqi

∑Ni=1mi

, P =N

∑i=1

miqi =MQ, (15.17)

with the total mass M = ∑Ni=1mi. Balance of linear momentum now becomes

MQ =N

∑i=1

fext = Fext. (15.18)

Now, we can express all atomic positions and momentum with respect to the average values:

qi =Q + δqi, pi =miQ + δpi such that δpi =miδqi. (15.19)

75


The total Hamiltonian becomes

H =N

∑i=1

∥miQ + δpi∥2

2mi+ V (Q + δq1, . . . ,Q + δqN)

=N

∑i=1

⎡⎢⎢⎢⎢⎣

m2i ∥Q∥2

2mi+ ∥δpi∥2

2mi+ miQ ⋅ δpi

2mi

⎤⎥⎥⎥⎥⎦+ V (Q + δq1, . . . ,Q + δqN)

= 1

2(N

∑i=1

mi)∥Q∥2 +N

∑i=1

∥δpi∥2

2mi+ 1

2Q ⋅

N

∑i=1

δpi + V (Q + δq1, . . . ,Q + δqN)

(15.20)

Note that

N

∑i=1

δpi =N

∑i=1

mi δqi =d

dt

N

∑i=1

mi δqi =d

dt

N

∑i=1

mi (qi −Q) = d

dt(MQ −MQ) = 0. (15.21)

Hence, the total Hamiltonian is

H = ∥P ∥2

2M+N

∑i=1

∥δpi∥2

2mi+ V (Q + δq1, . . . ,Q + δqN) (15.22)

and

Q = ∂H∂P

, P = −∂H∂Q

= −N

∑i=1

∂V

∂qi= −

N

∑i=1

(−fi) = Fext. (15.23)

Furthermore, notice that

δqi =∂H∂δpi

, δpi = pi − P = − ∂H∂δqi

− mi

MP = fi −

mi

MFext. (15.24)

In case no external forces act on the ensemble (Fext = 0), then the average motion decouplesfrom the atomic fluctuations and we may work with

H =N

∑i=1

∥δpi∥2

2mi+ V (Q + δq1, . . . ,Q + δqN) (15.25)

for which

δqi =∂H∂δpi

, δpi = −∂H∂δqi

. (15.26)

Since the mean motion is often of minor interest, we will – in the following – work only withthe perturbations (δq, δp) to describe atomic motion. For convenience, we will drop the δ inall perturbations and write (q,p).

15.3 The microcanonical ensemble

The microcanonical ensemble considers an isolated system containing a constant numberof N atoms, contained in a constant volume V and having a constant energy E. Because theHamiltonian of an isolated system is constant along trajectories in phase space, all microstatessatisfying H = E form a closed hypersurface which encloses a volume VR(E):

VR(E) = ∫ΓH (E −H(q,p)) dq dp = ∫

H<Edq dp (15.27)

76


with Heaviside jump function H(⋅). This allows us to define the density of states as

D(E) = dVRdE

(E). (15.28)

D(E) is thus the density of states contained between the hypersurfaces defined by E and E+dE,which defines the hypershell

Σ(E,∆E) = (q,p) ∶ E ≤H(q,p) ≤ E +∆E. (15.29)

The distribution function ρ of the microcanonical ensemble can now be defined as follows. Letus define ρ(E) such that it is constant within Σ(E,∆E) (later taking the limit ∆E → 0) andzero everywhere else. Note that we must have

∫Γρ(q,p;E,∆E) dq dp = 1, (15.30)

which results in

ρ(q,p;E,∆E) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1

VR(E +∆E) − VR(E)if (q,p) ∈ Σ(E,∆E),

0 else.

(15.31)

The phase average now becomes

⟨A⟩ = lim∆E→0

∫ΓA(q,p)ρ(q,p;E,∆E) dq dp = lim

∆E→0

∫Σ(E,∆E)A(q,p)dq dp

VR(E +∆E) − VR(E)

= lim∆E→0

1∆E [∫

VR(E+∆E)A(q,p)dq dp − ∫

VR(E)A(q,p)dq dp]

VR(E +∆E) − VR(E)∆E

= 1

D(E)∂

∂E∫VR(E)

A(q,p)dq dp

= 1

D(E)∂

∂E∫

ΓA(q,p)H (E −H(q,p)) dq dp,

(15.32)

where we used the Heaviside function in order to expand the integral to all of phase space inthe last line. Thus, we obtain

⟨A⟩ = 1

D(E) ∫ΓA(q,p) δ (E −H(q,p)) dq dp = ∫

ΓA(q,p)ρ(q,p,E)dq dp (15.33)

with the microcanonical distribution function

ρ(q,p,E) = 1

D(E)δ (E −H(q,p)) . (15.34)

Notice that the normalization constraint implies that

∫Γδ (E −H(q,p)) dq dp =D(E). (15.35)

Now that we have the probability density, we can look at macroscopic state variables. Forexample, we may define the internal energy as

U = ⟨H⟩, (15.36)

77


so that for an isolated system U = E. From thermodynamics, recall that

∂S

∂U= 1

Tand T = 1

∂S/∂E(15.37)

with entropy S and temperature T .

Boltzmann postulated the entropy as

S = kB ln Ω, (15.38)

which in our case can be restated as

S(E) = kB lnD(E), (15.39)

which in the thermodynamic limit of N →∞ can be shown to approach

S(E) = kB lnVR(E). (15.40)

Using (15.32) and the fact that E = const., let us compute

⟨xi∂H∂xj

⟩ = 1

D(E)∂

∂E∫VR(E)

xi∂H∂xj

dq dp

= 1

D(E)∂

∂E∫VR(E)

xi∂(H −E)∂xj

dq dp

= 1

D(E)∂

∂E[∫

VR(E)

∂xi(H −E)∂xj

dq dp − ∫VR(E)

δij(H −E)dq dp]

= 1

D(E)∂

∂E[∫

S(E)xi(H −E)nj dq dp − ∫

VR(E)δij(H −E)dq dp] .

(15.41)

Note that H = E on S(E), so that the first integral vanishes and we are left with

⟨xi∂H∂xj

⟩ = 1

D(E)∂

∂E∫VR(E)

δij(E −H)dq dp

=δij

D(E)∂

∂E∫

Γ(E −H)H(E −H)dq dp

=δij

D(E)[∫

ΓH(E −H)dq dp + ∫

Γ(E −H) δ(E −H)dq dp] .

(15.42)

Recall that the first integral is by definition VR(E) and notice that the second integral vanishes.Therefore,

⟨xi∂H∂xj

⟩ = δijVR(E)D(E)

= δijVR(E)V ′R(E)

= δij [∂

∂ElnVR(E)]

−1

= δijkB

∂S/∂E= δijkBT. (15.43)

Finally, let us take xi = pi (where i refers to one degree of freedom of one of the N atoms) sothat ∂H/∂xj = ∂H/∂pj = qj . Then,

δijkBT = ⟨pi∂H∂pj

⟩ = ⟨piqj⟩ = ⟨pipj

mj⟩ (15.44)

and considering i = j gives

⟨p2i

2mi⟩ = 1

2kBT. (15.45)

78


Summing over all 3N degrees of freedom thus gives the relation

3N

∑i=1

⟨p2i

2mi⟩ =

3N

∑i=1

⟨1

2miq

2i ⟩ = ⟨Kvibr.⟩ =

3N

2kBT ⇔ T = 2⟨Kvibr.⟩

3NkB, (15.46)

which links the average vibrational energy to the temperature of the ensemble.

Alternatively, using xi = qi gives

δijkBT = ⟨qi∂H∂qj

⟩ , (15.47)

so that summation over all 3 degrees of freedom of an atom α, again for i = j, yields

3kBT =3

∑i=1

⟨qi∂H∂qi

⟩ = −3

∑i=1

⟨qi∂V∂qi

⟩ = −⟨qα ⋅ fα⟩ (15.48)

and

−N

∑α=1

⟨qα ⋅ fα⟩ = −⟨W⟩ = 3NkBT, (15.49)

where W is often referred to as the virial of the system. Note that ⟨W⟩ = −2⟨Kvibr.⟩, which isreferred to as the virial theorem.

Without derivation, we just note that other continuum quantities can be derived similarly,including the virial stress tensor (in the current configuration, i.e., the Cauchy stresses)

σ = − 1

∣Ω∣

N

∑a=1

⟨pa ⊗ pa2ma

+ fa ⊗ qa⟩ with fa = −∂V

∂qa, (15.50)

which – in static equilibrium – agrees with the Cauchy-Born derivation of Section 14.

79


16 Atomistic Solution Algorithms

The equations of motion,

miqi = fi = −∂V∂qi

for i = 1, . . . ,N (16.1)

are to be solved numerically.

16.1 Zero-Temperature Molecular Statics

The simplest case, molecular statics at zero temperature, seeks the solution of

0 = fi = −∂V∂qi

for i = 1, . . . ,N. (16.2)

Owing to the symmetry of atomic lattices, many local energy minima exist (and oftentimesmultiple global maxima), which is why the solution algorithm can be numerically demanding.In addition, for large numbers N the assembly and storage of system-wide matrices becomescomputationally intractable, which is why matrix-free iterative solvers are popular. Typicalsolution algorithms include solvers of conjugate gradient and steepest descent type; the so-called Fast Inertial Relaxation Engine (FIRE) is a related popular scheme.

Special attention must be given to the application of boundary conditions. Essential BCsto individual atoms are generally to be avoided and one rather introduces padding regions toapply essential BCs over large numbers of atoms. Periodic BCs are a common way to simulateRVEs.

A general complication of particle methods such as atomistics (when compared to, e.g., FEM)is the necessity for frequent neighborhood searches and the associated updates of neighborlists of all N atoms.

16.2 Molecular Dynamics

Molecular Dynamics (MD) integrates the dynamic equations of motion over time, which isgenerally done in an explicit fashion to avoid the assembly and solution of the global systemfor computational reasons.

The most common scheme is called velocity-Verlet and can be summarized as follows. Usingtime increments ∆t = tα − tα−1 we integrate the equation of motion to obtain

q(tα +∆t) = qα+1 = qα + qα∆t + 1

2qα∆t2 +O(∆t3), (16.3)

and we know that

aαi = qαi = fαi /mi. (16.4)

Thus, for known initial conditions q0 = q1(0), . . . ,qN(0) and p0 = p1(0), . . . ,pN(0) we canintegrate the above equations of motion. Note that, in order to ensure velocity-time reversibleintegration, one usually uses an average acceleration scheme, which uses

vα+1 = vα + aα + aα+1

2∆t (16.5)

80


rather than a fully explicit scheme. The advantage is that we may conclude

q(tα −∆t) ≈ qα − qα∆t + 1

2qα∆t2 = qα − vα∆t + 1

2aα∆t2 (16.6)

so that

qα = qα−1 + vα∆t − 1

2aα∆t2

= qα−1 + vα−1∆t + aα−1 + aα

2∆t2 − 1

2aα∆t2

= qα−1 + vα−1∆t + ∆t2

2aα−1.

(16.7)

Note that one generally stores q(t) and v(t) = p(t)/m (rather than q and p). The generalalgorithm for zero-temperature MD is as follows:

(i) start with t0 = 0 and known q0 and v0 = p0/m(ii) fα = −∂V/∂q(qα)

(iii) aα = fα/m(iv) while t ≤ tend:

qα+1 = qα + vα∆t + fα∆t2/2mfα+1 = −∂V/∂q(qα+1)aα+1 = fα+1/mvα+1 = vα +∆t(aα + aα+1)/2tα = tα +∆t

Note that the force calculation fα = −∂V/∂q(qα) also involves neighborhood updates (i.e.,for each atom whose force is to be computed one finds, stores and frequently updates theneighboring atoms within the Verlet radius, which is typically chosen larger than the cut-offradius to reduce the frequency of neighborhood updates).

Without proof, we mention that the velocity-Verlet scheme is (approximately) energy con-serving ; i.e., it conserves the total Hamiltonian in an approximate sense (more specifically, itconserves a quantity known as shadow Hamiltonian, which converges to the exact Hamiltonianas ∆t→ 0; the error in the Hamiltonian scales of O(∆t2)). This is an important feature, whichmakes the velocity-Verlet scheme directly applicable to the canonical (NV E) ensemble. Forexample, for the simple harmonic oscillator discussed before, the shadow Hamiltonian can bederived exactly as

H∗(p, q) =H(p, q) − k2∆t2

8mq2, (16.8)

which obviously agrees with the exact Hamiltonian H in the limit ∆t→ 0.

We can easily compute the instantaneous temperature of the system in a postprocessing step,viz. (15.48) yields

T (tα) = 2

3NkB

N

∑i=1

mi

2∥δvαi ∥

2 (16.9)

where δvαi = vi(tα) − P (tα)/mi as discussed in Section 15.2. Note that the temperature T (t)fluctuates, and it makes more sense to compute the long-term average T as an approximationof the phase average.

81


16.3 Finite Temperature

Finite-temperature calculations usually require monitoring and/or controlling of the systemtemperature T . Irrespective of the ensemble being used, one can set the initial temperatureTini of an atomic ensemble as follows.

First, one initializes all particles with a random velocity and, in a subsequent step, ensuresthat the average momentum P = ∑imivi vanishes. This is accomplished, e.g., by adjusting allparticle velocities as

v0 ← v0 − P0

Mwith, as before, M =

N

∑i=1

mi. (16.10)

Next, using (16.9) compute the instantaneous temperature T 0 and rescale all atomic velocitiesby

v0 ←√

Tini

T 0v0 (16.11)

such that

T 0 = 2

3NkB

N

∑i=1

mi

2∥δv0

i ∥2 = 2

3NkB

N

∑i=1

mi

2

Tini

T 0∥v0

i ∥2 = Tini

T 0

2

3NkB

N

∑i=1

mi

2∥v0

i ∥2 = Tini. (16.12)

Maintaining a constant temperature, such as when using the microcanonical (NV T ) en-semble, is a whole different challenge that is commonly met by using special time-integrationschemes that modify atomic velocities at each time step.

Fixing the total kinetic energy can be accomplished, e.g., by Gauß’ principle of least constraint.To this end, notice that the classical equations of motion can be restated as minimizers of

C =N

∑i=1

mi

2(qi −

fimi

)2

because 0 = ∂

∂qC =mi (qi −

fimi

) =miqi − fi. (16.13)

This allows us to add constraints via Lagrange multipliers. E.g., the constraint

c =N

∑i=1

mi

2∥qi∥2 − 3

2NkBT = 0 (16.14)

can be differentiated to give

∂c

∂t=N

∑i=1

miqi ⋅ qi, (16.15)

which motivates the definition

C∗ = C − λN

∑i=1

miqi ⋅ qi. (16.16)

The resulting modified equations of motion are

∂

∂qC∗ =miqi − fi − λmiqi = 0 ⇔ miqi = fi + λmiqi. (16.17)

82


The Lagrange multiplier λ can be obtained from inserting qi = fi/mi + λqi into the constraintequation (16.15), which yields

N

∑i=1

mi (fi/mi + λqi) ⋅ qi = 0 ⇔ λ = − ∑Ni=1 fi ⋅ qi∑Na=1ma ∥qa∥2

. (16.18)

Overall, we thus arrived at

miqi = fi −∑Nb=1 fb ⋅ qb∑Na=1ma ∥qa∥2

qi. (16.19)

Note that this formulation does fix the average total kinetic energy but it does not allow any con-trol of the atomic fluctuations with respect to the mean motion (as required in the temperaturecalculation). This is commonly achieved by so-called thermostats.

16.4 Thermostats

For example, the Langevin thermostat modifies the equations of motion into

miqi = fi − γimiqi + gi(t), (16.20)

where γi is a damping/viscosity constant and gi denotes a random, time-varying noise (or force)that must is uncorrelated, i.e., it must satisfy

⟨gi⟩ = 0 ⇔ ⟨gik(t), gjl (t

′)⟩ = 2γimikBTδklδijδ(t − t′). (16.21)

This can be achieved in practice, while enforcing a temperature T , if the force is chosen tofollow a random, normal distribution with variance

σ2i = 2γimikBT /∆t (16.22)

at each time step of step size∆t. The damping term can be interpreted by Brownian motionat a significantly higher collision rate than that of atomic interactions.

A similar scheme is the Andersen thermostat which also introduces random forces as in theLangevin approach but does so not at every time step.

The probably most common thermostat is the Nose-Hoover thermostat, which starts witha fictitious Hamiltonian

H∗ = P 2

2M+N

∑i=1

∥pi∥2

2miQ2+ V (q) + 3NkBT lnQ. (16.23)

H∗ includes the 1-D motion of a fictitious atom having mass M , position Q and momentum P ;and it uses rescaled atomic momenta pi = Qmiqi. Applying Hamilton’s equations of motion toall particles i = 1, . . . ,N as well as to the fictitious atom results

dqidt

= ∂H∗

∂pi= pimiQ2

,dpidt

= −∂H∗

∂qi= fi,

dQ

dt= ∂H

∗

∂P= P

M,

dP

dt= −∂H

∗

∂Q= 1

Q(N

∑i=1

∥pi∥2

miQ2+ 3NkBT)

(16.24)

If we introduce a rescaled time t′ defined by dt = Q dt′ (and using the dot for derivatives withrespect to t′), the above equations reduce to

dqidt

= qiQ

= pimiQ

,dpidt

=˙piQ

= fi,dQ

dt= QQ

= P

M, (16.25)

83


and

QdP

dt= P =

N

∑i=1

∥pi∥2

miQ2+ 3NkBT =

N

∑i=1

∥pi∥2

mi+ 3NkBT. (16.26)

Multiplying the first equation by miQ and differentiating with respect to t′ leads to

miQqi = pi ⇒ miQqi +miQqi = ˙pi = Qfi. (16.27)

Next, division by Q and solving for the acceleration term produces

miqi = fi −miQ

Qqi = fi −mi

P

Mqi. (16.28)

Finally, by exploiting (16.26) we arrive at a system of evolution equations:

miqi = fi − γmiqi,

γ = 1

M(N

∑i=1

∥pi∥2

mi− 3NkBT) .

(16.29)

Thus, the Nose-Hoover thermostat is similar in nature to the Langevin/Andersen thermostats;here, the viscous drag coefficient γ is not constant but evolves with time and presents an intrinsicforce that aims to drive the system to the enforced temperature T . The only parameter to beadjusted is the fictitious mass M which my be interpreted as the inertia of the viscous coefficient.

We note that the same MD algorithm introduced above for zero-temperature conditions canbe applied at finite temperature, if we modify the atomic forces according to each thermostat.E.g., for the Langevin thermostat we would define atomic forces

f∗i (tα) = fi(tα) − γmiqi(tα) +G(tα). (16.30)

Analogously, the Nose-Hoover thermostat uses forces

f∗i (tα) = fi(tα) − γ(tα)qi(tα) (16.31)

and additionally updates the viscous coefficient γ at each time step in the average-accelerationfashion according to

γα+1 = γα + ∆t

2(γα + γα+1) . (16.32)

Finally, note that one can analogously impose an average stress tensor, the most frequentformulation of which is known as the Parrinello-Rahman approximation.

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17 Multiscale Modeling Techniques

Atomistics is a prime example for why multiscale techniques are required. Molecular dynamicsand statics are convenient methods but computer architectures limit the accessible length andtime scales. Length scales are limited by the total number of atoms that can be simulated. Hereare recent benchmark simulations that indicate the trends in feasible numbers of particles overthe years, limited by the size of computer memory (i.e., the need to store atomic positions andmomenta):

2003: 108 atoms (LANL Q) (Kadau et al., 2004)

2007: 1010 atoms (BlueGene/L) (Kadau et al., 2007)

2012: 1012 atoms (BlueGene/Q) (Habib et al., 2012)

2015: 1013 particles (Titan) (Koumoutsakos, 2015)

2018-20: 1015 atoms (prediction of exascale computing initiatives)

Even though the number of particles that can be simulated continues to increase due to in-creasingly large memory on supercomputers, the same cannot be said about time scales becausecommunication times and CPU speeds will not advance at the same pace as memory. Therefore,even though computer architectures promise increasing sample sizes, time scales will remain abottleneck.

This motivates techniques to bridge across scales, which is usually achieved in one of two ways:

hierarchical scale-bridging, also known as vertical scale-bridging, assumes a sepa-ration of scales so that a hierarchy of scales can be established. Information from thelower scales is passed on to the larger scales by the definition of effective measures.Computational homogenization and FE2, discussed before, are prime examples: RVE-information is passed on to the macroscale, based on the spatial separation between micro-and macroscales. Difficulties arise when a clear spatial separation of scales is not war-ranted (e.g., when macroscale features are on the same scale as microscale specifics) orwhen localization occurs on the microscale (e.g., when a crack develops in the RVE whichdevelops into a crack on the macroscale).

concurrent scale-bridging, also known as horizontal scale-bridging, does not as-sume a separation of scales but patches together multiple simulation techniques concur-rently. For example, using atomistics in one half of the simulation domain and continuummechanics and the other half. Challenges here arise from identifying compatible consti-tutive descriptions on both sides of the interface separating the two domains, and fromestablishing an interface that is compatible with both descriptions and allows informationto flow from one domain to another. Examples are the Coupled Atomistics-Discrete Dis-location (CADD) method, the Quasicontinuum (QC) method, Macroscopic-Atomistic-AbInitio-Dynamics (MAAD), and the Bridging Domain Method (BDM).

The idea behind concurrent techniques is to limit lower-scale resolution to where it is indeedneeded, while using an efficient large-scale description in the remaining simulation domain. Forexample, when running atomistic simulations at low temperature, interesting features occurin the vicinity of defects, interfaces and surfaces. This motivates to use atomistic techniquesnear those features while using a more efficient description in those regions deforming more

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homogeneously far away from defects, interfaces and surfaces. Ideally, such domain decomposi-tion should be performed adaptively, so that no knowledge is required a-priori about where theexpensive lower-scale resolution will be required during a simulation.

Examples of both hierarchical and concurrent scale-bridging techniques are shown in Fig. 6.

(Abraham et al., 1998)

Figure 6: Examples of hierarchical (left) and concurrent (right) scale-bridging techniques. Hi-erachical techniques pass information between scales by assuming a separation of scales. Con-current techniques as the shown MAAD scheme (adopted from Abraham et al. (1998)) coupledifferent techniques in the same simulation domain. Here, e.g., highly-accurate tight-binding(TB) is used at a crack tip, with molecular dynamics (MD) applied further awar from the tip,and finite elements (FE) used in the far field.

Going beyond atomistics, the state of the art in material modeling offers highly-accurate meth-ods for each individual scale, from density functional theory (DFT) and molecular dynamics(MD) at the lower scales all the way up to continuum theories and associated computationaltools for the macroscale and structural applications. Unfortunately, a wide gap exists due to alack of models applicable at the intermediate scales (sometimes referred to as mesoscales). Here,the continuum hypothesis fails because the discreteness of the atomic crystal becomes apparent,e.g., through the emergence of size effects. At the same time, atomistic techniques tend to incurprohibitively high computational expenses when reaching the scales of hundreds of nanometersor microns. Mastering this gap between atomistics and the continuum is the key to understand-ing a long list of diverse open problems. These include the mechanical response of nanoporous ornanostructured (i.e., nanocrystalline or nanotwinned) metals, the effective mechanical proper-ties of nanometer- and micron-sized structures, devices and engineered (meta)materials, furtherthe underlying mechanisms leading to inelasticity and material failure, or heat and mass transferin nanoscale materials systems. Overall, there is urgent need for techniques that bridge acrosslength and time scales in order to accurately describe, to thoroughly understand, and to reliablypredict the mechanics and physics of solids.

Over the decades, various methodologies have been developed to bridge across scales. On theone hand, hierarchical schemes are the method of choice when a clear separation of scales canbe assumed. In this case, homogenization techniques can extract the effective constitutive re-sponse at the macroscale from representative simulations at the lower scales. Examples includemultiple-level finite-element (FE) analysis or FEn Feyel and Chaboche (2000); Miehe et al.(1999); Schroder (2000), as well as the homogenization of atomistic ensembles to be used at the

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material-point level in macroscale FE simulations Chung (2004); Clayton and Chung (2006);Park et al. (2005); Shephard et al. (2004). By contrast, concurrent scale-coupling methods avoidthe aforementioned separation of scales and instead integrate different constitutive descriptionsinto a single-scale model by spatially separating domains treated e.g. by first-principles, molecu-lar dynamics, discrete defect mechanics, and continuum theories. Prominent examples compriseCoupled Atomistic/Discrete-Dislocation (CADD) models Curtin and Miller (2003); Nair et al.(2010); Shilkrot et al. (2004) and AtoDis Brinckmann et al. (2012), furthermore the BridgingDomain Method (BDM) Belytschko and Xiao (2003) and Bridging Scale Decomposition Liuet al. (2006), as well as Macroscopic, Atomistic, Ab Initio Dynamics (MAAD) Abraham et al.(1998); Broughton et al. (1999) which couples several scales. In such methods, a key chal-lenge arises from the necessity to pass information across interfaces between different modeldomains. To pick out one example, CADD is an elegant methodology which embeds a smallMD domain into a larger region treated by a Discrete Dislocation (DD) description. Here, thepassing of lattice defects across the interface between the two domains is a major challenge.In order to circumvent such difficulties, coarse-graining techniques apply the same lower-scaleconstitutive description to the entire model but scale up in space and/or in time. Classicalexamples include Coarse-Grained MD (CGMD) Rudd and Broughton (2005a,b) as well as thequasicontinuum (QC) method Tadmor et al. (1996). We note that, while upscaling in time andin space are equally important, the primary focus here will be on spatial coarse-graining, whichis the main achievement of the QC approximation. Up-scaling of atomistic simulations in thetime domain has been investigated in the context of MD simulations, see e.g. Kim et al. (2014);Voter (1997); Voter et al. (2002), and can be added to a spatial coarse-graining scheme suchas the QC method. A particular QC formulation (applicable to finite temperature) to study,e.g., long-term mass and heat transfer phenomena was introduced recently Ariza et al. (2012);Venturini et al. (2014); Wang et al. (2015). For alternative finite-temperature QC extensionssee e.g. Diestler et al. (2004); Dupuy et al. (2005); Hai and Tadmor (2003); Kulkarni et al.

L SENGTH CALES

TS

IME

CA

LE

S

ContinuumTheories

DiscreteDislocationDynamics

mm mmnmÅ

fs

ps

s

s

MolecularDynamics

QuantumMechanics

High-Res.TEM

SEM, TEM

DIC

ECCI

EBSD

Optical Microscopy

crystalplasticity

phasefield models

continuumdislocation

theory

straingradienttheories

MolecularDynamics

Monte-Carlo

DFT

CADD

Figure 7: Bridging across scales in crystalline solids: from the electronic structure all the wayup to the macroscale (including some of the prominent modeling and experimental techniques).

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(2008); Marian et al. (2010); Rudd and Broughton (2005a); Shenoy et al. (1999); Tadmor et al.(2013); Tang et al. (2006).

Spatial coarse-graining reduces the number of degrees of freedom by introducing geometricconstraints, thereby making the lower-scale accuracy efficiently available for larger-scale simu-lations Iyer and Gavini (2011); Izvekov and Voth (2005); Suryanarayana (2011); Tadmor et al.(1996). Coarse-graining strategies offer a number of advantages over domain-coupling methods:(i) the model is solely based on the lower-scale constitutive laws and hence comes with supe-rior accuracy (in contrast to coupling methods, there is no need for a separate and oftentimesempirical constitutive law in the continuum region); (ii) the transition from fully-resolved tocoarse-grained regions is seamless (no approximate hand-shake region is required between differ-ent domains in general); (iii) depending on the chosen formulation, model adaptation techniquescan efficiently reduce computational complexity by tying full resolution to those regions whereit is indeed required (such as in the vicinity of lattice defects or cracks and voids).

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18 The Quasicontinuum Method

The quasicontinuum method4 was introduced to bridge from atomistics to the continuum byapplying finite element interpolation schemes to atoms in a crystal lattice (Miller et al., 1998;Phillips et al., 1999; Tadmor et al., 1996). This is achieved by three integral components:geometric constraints (which interpolate lattice site positions from the positions of a reducedset of representative atoms), summation rules or sampling rules (which avoid the computationof thermodynamic quantities from the full atomistic ensemble), and model adaptation schemes(which localize atomistic resolution and thereby efficiently minimize the total number of degreesof freedom). To date, numerous QC flavors have been developed which mainly differ in the choiceof how the aforementioned three aspects are realized. Despite their many differences, all QC-based techniques share as a common basis the interpolation of atomic positions from a set ofrepresentative atoms as the primary coarse-graining tool.

The family of QC methods has been applied to a wide range of problems of scientific and tech-nological interest such as studies of the interactions of lattice defects in fcc and bcc metals.Typical problems include nanoindenation (Li et al., 2011; Lu et al., 2012; Tadmor et al., 1999),interactions of lattice defects (Hardikar et al., 2001; Phillips et al., 1999; Shimokawa et al., 2007)or of defects with nanosized cracks (Huai-Bao et al., 2011; Ringdalen Vatne et al., 2011; Tadmorand Hai, 2003) and nanovoids (Ariza et al., 2012; Ponga et al., 2015; Yu and Shen, 2010), orwith individual interfaces (Li et al., 2010; Yoshihiko et al., 2005). Beyond metals, the developedmodeling techniques can be extended, e.g., to ferroelectric materials and ionic crystals via elec-trostatic interactions (Dayal and Marshall, 2011) and multi-lattice approaches (Abdulle et al.,2012; Dobson et al., 2007a), further to non-equilibrium thermodynamics (Kulkarni et al., 2008;Marian et al., 2010; Rudd and Broughton, 2005a; Shenoy et al., 1999; Venturini et al., 2014),also to structural mechanics (Beex et al., 2011, 2014b) and two-dimensional materials (Parket al., 2010; Wang and Guo, 2013), and in principle to any material system with crystallineorder as long as a suitable position-dependent interatomic potential is available.

In this chapter, we will review the common theoretical basis of the family of QC methods,followed by the description of particular QC schemes and available techniques. We then proceedto highlight past and recent applications and extensions of the QC technique to study themechanics and physics of solids and structures, and we conclude by summarizing some openquestions and challenges to be addressed in the future or subject to ongoing research. Sincesuch a book chapter cannot give a full account of all related research, we focus on – in ourview – important QC advances and contributions and apologize if we have overlooked specificlines of research. For a list of related publications, the interested reader is also referred to theqcmethod.org website maintained by E. B. Tadmor and R. E. Miller (Tadmor and Miller, 2015).

18.1 Representative atoms and the quasicontinuum approximation

The rich family of QC methods is united by the underlying concepts of (i) significantly reducingthe number of degrees of freedom by suitable interpolation schemes, and (ii) approximating thethermodynamic quantities of interest by summation or sampling rules. Let us briefly outlinethose concepts before summarizing particular applications and extensions of the QC method.

In classical mechanics, an atomistic ensemble containing N atoms is uniquely described by howtheir positions q = q1, . . . ,qN and momenta p = p1, . . . ,pN with pi =miqi evolve with timet. Here and in the following, mi represents the mass of atom i, and dots denote material time

4This section dedicated to the quasicontinuum method is adopted from (Kochmann and Amelang, 2016).

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derivatives. Then, the ensemble’s total Hamiltonian H is given by

H(q,p) =N

∑i=1

∣pi∣2

2mi+ V (q), (18.1)

where the first term accounts for the kinetic energy and the second term represents the potentialenergy of the ensemble with V denoting a suitable atomic interaction potential which we assumeto depend on atomic positions only. The time evolution of the system is governed by Hamilton’sequations, which yield Newton’s equations of motion for all atoms i = 1, . . . ,N ,

miqi = fi(q) = −∂V

∂qi(q), (18.2)

where fi(q) represents the total (net) force acting on atom i. We note that V can, in principle,depend on both positions p and momenta q (the latter may be necessary, e.g., when usingthe quasiharmonic and other approximations for quasistatic finite-temperature formulationswithout time discretization). For conciseness, we here restrict ourselves to position-dependentpotentials only; the extension to include momenta is straight-forward. As one of the cornerstonesof the QC method, we limit our analysis to crystalline solids in which the ground state atomicpositions coincide with sites of a regular lattice described by a set of Bravais vectors that spanthe discrete periodic array of lattice sites. In most materials (including metals, ceramics andorganic materials), the interaction potential usually allows for an additive decomposition, i.e.

V (q) =N

∑i=1

Ei(q) (18.3)

with Ei being the energy of atom i. Some authors in the scientific literature have preferredto distinguish between external and internal forces by introducing external forces fi,ext on allatoms i = 1, . . . ,N so that

fi(q) = fi,ext(qi) −∂V

∂qi(q). (18.4)

Ideally (and in any reasonable physical system), such external forces are conservative and derivefrom an external potential Vext(q) and we may combine internal and external forces into asingle potential, which is tacitly assumed in the following. Examples of conservative externalforces include gravitation, long-range Coulombic interactions, or multi-body interactions such asduring contact. For computational convenience, the latter is oftentimes realized by introducingartificial potentials, see e.g. (Kelchner et al., 1998) for an effective external potential for sphericalindenters.

Due to limitations of computational resources, it is generally not feasible to apply the aboveframework to systems that are sufficiently large to simulate long-range elastic effects, even forshort-range interatomic potentials. However, except in the vicinity of flaws and lattice defectssuch as cracks and dislocations, respectively, the local environment of each atom in a crystallattice is almost identical up to rigid body motion. Therefore, the QC approximation replacesthe full atomistic ensemble by a reduced set of Nh ≪ N representative atoms (often referred toas repatoms for short). This process is shown schematically in Fig. 8.

Let us denote the repatom positions by x(t) = x1(t), . . . ,xNh(t). The approximate current

positions qhi and momenta phi of all atoms i = 1, . . . ,N are now obtained from interpolation, i.e.we have

qi ≈ qhi =Nh

∑a=1

Na(Xi)xa (18.5)

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Figure 8: Illustration of the QC methodology: identification of atoms of interest (e.g., high-centrosymmetry lattice sites near a dislocation core), reduction to the set of repatoms (coars-ening the full atomic lattice by choosing a small set of representative atomic sites), and theinterpolation of atomic positions (blue in the small inset) from repatom positions (red) with anexample repatom distribution around a dislocation and a grain boundary in two dimensions.

and consequently

pi ≈ phi =mi qhi =mi

Nh

∑a=1

Na(Xi) xa. (18.6)

Na(Xi) is the shape function of repatom a evaluated at the position Xi of lattice site i inthe undeformed (reference) configuration. As an essential feature, one usually requires thiscoarse-graining scheme to locally recover the exact atomic ensemble when all atoms are turnedinto repatoms. Therefore, shape functions should be chosen to satisfy the Kronecker propertyNa(Xb) = δab for all 1 ≤ a, b ≤ Nh with δij denoting Kronecker’s delta (δij = 1 if i = j and 0otherwise).

By borrowing concepts from the finite element method, the above geometric constraints withinthe original QC method (Knap and Ortiz, 2001; Shenoy et al., 1998; Tadmor et al., 1996) madeuse of an affine interpolation on a Delaunay-triangulated mesh. More recent versions haveexplored higher-order polynomial shape functions (Kwon et al., 2009; Park and Im, 2008) aswell as meshless interpolations using smoothed-particle approaches (Xiao and Yang, 2007; Yangand Xiao, 2005) or local maximum-entropy shape functions (Kochmann and Venturini, 2014).

The introduction of the geometric constraints in (18.5) has reduced the total number of in-dependent degrees of freedom from d ×N in d dimensions to d ×Nh. Therefore, the approxi-mate HamiltonianHh of the coarse-grained system, now involving approximate atomic positionsqh = qh1 , . . . ,qhN and momenta ph = ph1 , . . . ,phN, only depends on the positions and momentaof the repatoms through (18.5):

Hh(x, x) =N

∑i=1

∣phi ∣2

2mi+ V (qh). (18.7)

Instead of solving for the positions and momenta of all N lattice sites, the QC approximationallows us to update only the positions and momenta of the Nh repatoms, which requires tocompute forces on repatoms. These are obtained from the potential energy by differentiation,which yields the net force on repatom k:

Fk(x) = −∂V (qh)∂xk

=N

∑j=1

fhj (qh)Nk(Xj) (18.8)

with

fhj (qh) = −∂V (qh)∂qhj

= −N

∑i=1

∂Ei(qh)∂qhj

, (18.9)

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the total force acting on atom j. It is important to note that, in an infinite Bravais lattice (i.e.,in a defect-free single-crystal in the absence of external loading or free surfaces), the forces onall atoms vanish, i.e. we have fhi (qh) = 0, so that the net forces on all repatoms vanish as well(Fk = 0).

18.2 Summation rules and spurious force artifacts

Summation rules have become an integral ingredient of all QC methods (even though they aresometimes not referred to as such). Although the above introduction of repatoms has reducedthe total number of degrees of freedom significantly from d ×N to d ×Nh, the calculation ofrepatom forces still requires computing the forces between all N atoms and their on averageNb neighbors located within each atom’s radius of interaction. These O(N × Nb) operationsbecome computationally not feasible for realistically-sized systems. We note that, in principle,summations in (18.8) can be reduced to the support of the shape functions (i.e., to regions inwhich Nk(Xj) ≠ 0) but even this is prohibitively expensive in regions of dilute repatom concen-trations. Therefore, summation rules or sampling rules have been introduced to approximatethe thermodynamic quantities of interest of the full atomistic ensemble by those of a small setof carefully-chosen lattice sites (in the following referred to as sampling atoms), comparable toquadrature rules commonly found in the finite element method.

QC summation rules have either approximated the energy of the system (so-called energy-basedQC ) (Amelang et al., 2015; Eidel and Stukowski, 2009) or the forces experienced by the repatoms(force-based QC ) (Knap and Ortiz, 2001). Within each of those two categories, multiple versionshave been proposed. Here, we will first describe what is known as fully-nonlocal energy-basedand force-based QC formulations and then extract the original local/nonlocal QC method as aspecial case of energy-based QC.

18.2.1 Energy-based QC

In energy-based nonlocal QC formulations, the total Hamiltonian is approximated by a weightedsum over a carefully selected set of sampling atoms (which do not have to coincide with therepatoms introduced above). To this end, we replace the sum over index i in (18.3) or (18.8)by a weighted sum over Ns carefully-chosen sampling atoms, see e.g. (Amelang et al., 2015;Eidel and Stukowski, 2009; Espanol et al., 2013). Consequently, the total potential energy isapproximated by

V (qh) =N

∑i=1

Ei(qh) ≈ V (qh) =Ns

∑α=1

wαEα(qh), (18.10)

where wα is the weight of sampling atom α. Physically, wα denotes the number of lattice sitesrepresented by sampling atom α.

The force experienced by repatom k is obtained by differentiation in analogy to (18.8):

Fk(x) = −∂V (qh)∂xk

= −Ns

∑α=1

wαN

∑j=1

∂Eα(qh)∂qhj

Nk(Xj). (18.11)

Now, repatom force calculation has been reduced to O(Ns × Nb) operations (accounting forthe fact that the sum over j above effectively only involves Nb neighbors within the potential’scutoff radius). The selection of sampling atoms and the calculation of sampling atom weightsaims for a compromise between maximum accuracy (ideally Ns = N and wα = 1) and maximumefficiency (requiring Ns ≪ N and wα ≫ 1).

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In order to seamlessly bridge from atomistics to the continuum without differentiating be-tween atomistic and coarse-grained regions, the above summation rules can be interpreted asa fully nonlocal QC approximation which treats the entire simulation domain in the samefashion. Summation rules now differ by the choice of (i) sampling atom locations and (ii) sam-pling atom weights. Successful examples of summation rules introduced previously includenode-based cluster summation (Knap and Ortiz, 2001) with sampling atoms located in clustersaround repatoms, and quadrature-type summation (Gunzburger and Zhang, 2010; Yang et al.,2013) with sampling atoms chosen nearest to Gaussian quadrature points with or without therepatoms included as sampling atoms. Furthermore, element-based summation rules have beenintroduced in the nonlocal QC context (Gunzburger and Zhang, 2010; Yang et al., 2013) andhave demonstrated superior accuracy over traditional cluster-based summation schemes (Iyerand Gavini, 2011). Recently, a central summation rule was proposed in (Beex et al., 2014a)as an ad-hoc compromise between local and quadrature summation rules, which is similar inspirit and contained as a special case within the optimal summation rules recently introducedin (Amelang et al., 2015). Fig. 9 illustrates some of the most popular summation rules.

Many summation rules were introduced in an ad-hoc manner to mitigate particular QC deficien-cies (e.g., cluster rules were introduced to remove zero-energy modes in force-based QC (Knapand Ortiz, 2001)) or by borrowing schemes from related models (e.g., finite element quadraturerules (Gunzburger and Zhang, 2010)). The recent optimal sampling rules (Amelang et al., 2015)followed from optimization and ensure small or vanishing force artifacts in large elements whileseamlessly bridging to full atomistics. Their new second-order scheme also promises superioraccuracy when modeling free surfaces and associated size effects at the nanoscale (Amelang andKochmann, 2015).

18.2.2 Force artifacts

Summation rules on the energy level not only reduce the computational complexity but they alsogive rise to force artifacts, see e.g. (Amelang et al., 2015; Eidel and Stukowski, 2009) for reviewswithin the energy-based context. By rewriting (18.11) without neighborhood truncations as

Fk(x) = −Ns

∑α=1

wαN

∑j=1

∂Eα(qh)∂qhj

Nk(Xj) =N

∑j=1

⎛⎝−Ns

∑α=1

wα∂Eα(qh)∂qhj

⎞⎠Nk(Xj), (18.12)

and comparing with (18.8), we see that the term in parentheses in (18.12) is not equal to forcefhj and hence does not vanish in general even if fhj = 0 for all atoms (unless we choose all atomsto be sampling atoms, i.e. unless Ns = N and wα = 1). As a consequence, the QC representation

“optimal” rules

rinodal rule

cluster rule

local QC

quadrature rule

Figure 9: Illustration of popular summation rules with small open circles denoting lattice sites,solid circles representing repatoms, and large open circles are sampling atoms (crossed circlesdenote sampling atoms whose neighborhoods undergo affine deformations). Optimal summationrules (Amelang et al., 2015) are shown of first and second order (the gray sampling atoms onlyexist in the second-order rule).

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with energy-based summation rules shows what is known as residual forces in the undeformedground state which are non-physical.

In uniform QC meshes (i.e. uniform repatom spacings, uniform sampling atom distribution, anda regular mesh), these residual forces indeed disappear due to symmetry; specifically, the sumin parentheses in (18.12) cancels pairwise when carrying out the full sum. Residual forces henceappear only in spatially non-uniform meshes, see also (Iyer and Gavini, 2011) for a discussion.In general, one should differentiate between residual and spurious forces (Amelang et al., 2015;Eidel and Stukowski, 2009). Here, we use the term residual force to denote force artifactsarising in the undeformed configuration. In an infinite crystal, these can easily be identified bycomputing all forces in the undeformed ground state. However, force errors vary nonlinearlywith repatom positions. Therefore, we speak of spurious forces when referring to force artifactsin the deformed configuration. As shown e.g. in (Amelang et al., 2015), correcting for residualforces by subtracting those as dead loads does not correct for spurious forces and can lead toeven larger errors in simulations. Indeed, in many scenarios this dead-load correction is noteasily possible since the undeformed ground state may contain, e.g., free surfaces. In this case,the computed forces in the undeformed ground state contain both force artifacts and physicalforces. Thus, without running a fully-atomistic calculation for comparison, it is impossible todifferentiate between residual force artifacts and physical forces arising, e.g., due to surfacerelaxation (Amelang and Kochmann, 2015). Even though residual and spurious forces areconservative in the energy-based scheme, they are non-physical and can drive the coarse-grainedsystem into incorrect equilibrium states. Special correction schemes have been devised whicheffectively reduce or remove the impact of force artifacts, see e.g. the ghost force correctiontechniques in (Shenoy et al., 1998; Shimokawa et al., 2004). However, such schemes require aspecial algorithmic treatment at the interface between full resolution and coarse-grained regions,which may create computational difficulties and, most importantly, requires the notion of aninterface.

Based on mathematical analyses, further energy-based schemes have been developed, whichblend atomistics and coarse-grained descriptions by a special formulation of the potential energyor the repatom forces. Here, an interfacial domain between full resolution and coarse regions isintroduced, in which the thermodynamic quantities of interest are taken as weighted averagesof the exact atomistic and the approximate coarse-grained ones (Li et al., 2014; Luskin et al.,2013). Such a formulation offers clear advantages, including the avoidance of force artifacts.It also requires the definition of an interface region, which may be disadvantageous in a fully-nonlocal QC formulation with automatic model adaption since no notion of such interfacesexists strictly (Amelang et al., 2015).

18.2.3 Force-based QC

In order to avoid force artifacts, force-based summation rules were introduced in (Knap andOrtiz, 2001). These do not approximate the Hamiltonian but the repatom forces explicitly.Ergo, the summation rule is applied directly to the repatom forces in (18.8), viz.

Fk(x) =N

∑i=1

fhi (qh)Nk(Xi) ≈ Fk(x) =Ns

∑α=1

wα fhα(qh)Nk(Xα). (18.13)

As a consequence, this force-based formulation does not produce residual forces because fhα(qh) =0 in an undeformed infinite crystal. The same holds true for an infinite crystal that is affinelydeformed: for reasons of symmetry the repatom forces cancel, and spurious force artifacts areeffectively suppressed.

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Yet, this approximation gives rise to a new problem: the force-based method is non-conservative;hence, there is no potential from which repatom forces derive. This has a number of drawbacks,as discussed e.g. in (Eidel and Stukowski, 2009; Iyer and Gavini, 2011; Miller and Tadmor,2009). In quasistatic problems, the non-conservative framework may lead to slow numericalconvergence, cause numerical instability, or converge to non-physical equilibrium states, see e.g.the analyses in (Dobson et al., 2010a,b,c; Luskin and Ortner, 2009; Ortner, 2011). Furthermore,for dynamic or finite-temperature scenarios, a QC approximation using force-based summationrules cannot be used to simulate systems in the microcanonical ensemble (where the system’senergy is to be conserved; other ensembles may, of course, be used). Finally, repatom masses,required for dynamic simulations, are not uniquely defined because there is no effective kineticenergy potential when using force-based summation rules (see also Section 18.2.5 below). Incontrast, energy-based summation rules lead to conservative forces and to strictly-symmetricstiffness matrices with only the six admissible zero eigenvalues (Eidel and Stukowski, 2009).Moreover, the proper convergence of energy-based QC techniques was shown recently for har-monic lattices (Espanol et al., 2013).

18.2.4 Local/nonlocal QC

The original QC method of (Tadmor et al., 1996) uses affine interpolation within elements andmay be regarded as a special energy-based QC scheme, in which atomistic and coarse-grainedregions are spatially separated. In the atomistic region and its immediate vicinity, the fullatomistic description is applied and one solves the discrete (static) equations of motion foreach atom and approximations thereof near the interface (nonlocal QC ). In the coarse-grainedregion, a particular element-based summation rule is employed (local QC ): for each element, theenergy is approximated by assuming an affine deformation of all atomic neighborhoods withinthe element, so that one Cauchy Born-type sampling atom per element is sufficient and thetotal potential energy is the weighted sum over all such element energies. This concept hasbeen applied successfully to a myriad of examples, primarily in two dimensions (or in 2.5D byassuming periodicity along the third dimension). Since the element-based summation rule in thenonlocal coarse region produces no force artifacts (Amelang et al., 2015), force artifacts here onlyappear at the interface between atomistic and coarse-grained regions and have traditionally beencalled ghost forces (Miller et al., 1998; Shenoy et al., 1998). For a comprehensive comparison ofthis technique with other atomistic-to-continuum-coupling techniques see (Miller and Tadmor,2009). Fig. 10 schematically illustrates the concepts of local/nonlocal and fully-nonlocal QC.

18.2.5 Repatom masses

Besides repatom forces, energy-based summation rules provide consistent repatom masses, asmentioned above. When approximated by the QC interpolation scheme, the total kinetic energyof an atomic ensemble becomes

1

2

N

∑i=1

mi (qhi )2 = 1

2

N

∑i=1

mi

RRRRRRRRRRR

Nh

∑a=1

Na(Xi) xaRRRRRRRRRRR

2

= 1

2

Nh

∑a=1

Nh

∑c=1

xa ⋅ (N

∑i=1

miNa(Xi)Nc(Xi)) xc,

(18.14)

where the term in parentheses may be interpreted as the components Mhac of a consistent mass

matrix Mh. In avoidance of solving a global system during time stepping, one often resortsto mass lumping in order to diagonalize Mh. Applying the energy-based summation rule to

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Figure 10: Schematic view of the local/nonlocal (left) and fully-nonlocal (right) QC formula-tions. The shown nonlocal scheme uses a node-based cluster summation rule. The inset on theright illustrates the affine interpolation of atomic positions from repatoms, which is the samein both formulations.

the total Hamiltonian (and thus to the kinetic energy) yields a consistent approximation of thekinetic energy with mass matrix components

Mhac =

N

∑i=1

miNa(Xi)Nc(Xi) ≈ Mhac =

Ns

∑b=1

wbmbNa(Xb)Nc(Xb). (18.15)

In case of mass matrix lumping, the equations of motion of all repatoms now become

mhk xk = Fk(x), (18.16)

which can be solved by (explicit or implicit) finite difference schemes. The calculation of repatommasses is of importance not only for dynamic QC simulations, but it is also relevant in quasistaticfinite-temperature QC formulations.

18.2.6 Example: Embedded Atom Method

The QC approximation outlined above is sufficiently general to model the performance of mosttypes of crystalline solids, as long as their potential energy can be represented as a function ofatomic positions. Let us exemplify the general concept by a frequently-used family of interatomicpotentials for metals (which has also been used most frequently in conjunction with the QCmethod). The Embedded Atom Method (EAM) (Daw and Baskes, 1984) defines the interatomicpotential energy of a collection of N atoms by

Ei(q) =1

2∑j≠i

Φ(rij) +F(ρi), ρi =∑j≠if(rij). (18.17)

The pair potential Φ(rij) represents the energy due to electrostatic interactions between atomi and its neighbor j, whose distance rij is the norm of the distance vector rij = qi − qj . ρidenotes an effective electron density sensed by atom i due to its neighboring atoms. f(rij) isthe electron density at site i due to atom j as a function of their distance rij . F(ρi) accountsfor the energy release upon embedding atom i into the local electron density ρi. From (18.2),the exact force fk acting on atom k is obtained by differentiation, viz.

fk(q) = −N

∑i=1

∂Ei(q)∂qk

= −N

∑j∈nI(k)

[Φ′(rkj) + F ′(ρk) +F ′(ρj) f ′(rkj)]rkj

rkj. (18.18)

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Since most (non-ionic) potentials are short-range (in particular those for metals), one can ef-ficiently truncate the above summations to include only neighboring atoms within the radiusof interaction (nI(i) denotes the set of neighbors within the sphere of interaction of atom i).Introducing the QC approximation along with an energy-based summation rule now specifiesthe approximate repatom force as

Fk(x) = −Ns

∑α=1

wα ∑j∈nI(α)

[1

2Φ′(rhαj) +F ′(ρhα) f ′(rhαj)]

rhαj

rhαj[Nk(Xα) −Nk(Xj)] , (18.19)

which completes the coarse-grained description required for QC simulations. Note that, asdiscussed above, repatom forces (18.19) are prone to produce residual and spurious force artifactsin non-uniform QC meshes.

18.3 Adaptive refinement

One of the general strengths of the QC method is its suitability for adaptive model refinement.This is particularly useful when there is no a-priori knowledge about where within a simulationdomain atomistic resolution will be required during the course of a simulation. For example,when studying defect mechanisms near a crack tip or around a pre-existing lattice defect, it maybe sufficient to restrict full atomistic resolution to regions in the immediate vicinity of thosemicrostructural features and efficiently coarsen away from those. However, when investigatingmany-defect interactions or when the exact crack path during ductile failure is unknown, itis beneficial to make use of automatic model adaptation. By locally refining elements downto the full atomistic limit, full resolution can effectively be tied to evolving defects in an effi-cient manner. In the fully-nonlocal QC method (Amelang et al., 2015), this transition is trulyseamless as there is no conceptual differentiation between atomistic and coarse domains. Inthe local/nonlocal QC method as well as in blended QC formulations the transition is equallypossible but may require a special algorithmic treatment.

Mesh refinement requires two central ingredients: a criterion for refinement and a geometricrefinement algorithm. The former can be realized on the element level, e.g., by checking invari-ants of the deformation gradient within each element and comparing those to a threshold forrefinement (Knap and Ortiz, 2001). Alternatively, one can introduce criteria based on repatomsor sampling atoms, e.g., by checking the energy or the centrosymmetry of repatoms or sam-pling atoms and comparing those to refinement thresholds. The most common geometric toolis element bisection; however, a large variety of tools exist and these are generally tied to thespecifically-chosen interpolation scheme.

Unlike in the finite element method, mesh adaptation within the QC method is challengingsince element vertices, at least in the most common implementations, are restricted to sitesof the underlying Bravais lattice (this ensures that full atomistics is recovered upon ultimaterefinement). The resulting constrained mesh generation and adaptation has been discussedin detail e.g. in (Amelang, 2015). While mesh refinement is technically challenging but con-ceptually straight-forward, mesh coarsening presents a bigger challenge. Meshless formula-tions (Kochmann and Venturini, 2014; Xiao and Yang, 2007; Yang and Xiao, 2005) appearpromising but have not reached sufficient maturity for large-scale simulations.

18.4 Features and Extensions

The QC methods described above have been applied to a variety of scenarios which go beyondthe traditional problem of quasistatic equilibration at zero-temperature. In particular, QC

97


extensions have been proposed in order to describe phase transformations and deformationtwinning, finite temperature, atomic-level heat and mass transfer, or ionic interactions such asin ferroelectrics. In addition, the basic concept of the QC method has been applied to coarse-grain discrete mechanical systems beyond atomistics. Here, we give a brief (non-exhaustive)summary of such special QC features and extensions.

18.4.1 Multilattices

The original (local) QC approximation applies an affine interpolation to all lattice site positionswithin each element in the coarse domain, which results in affine neighborhood changes withinelements (the same applies to the optimal summation rules within fully-nonlocal QC). By usingCauchy-Born (CB) kinematics, one can account for relative shifts within the crystalline unitcells such as those occurring during domain switching in ferroelectrics or during solid-state phasetransitions and deformation twinning. The CB rule is thus used in coarse-grained regions torelate atomic motion to continuum deformation gradients. In order to avoid failure of the CBkinematics, such a multilattice QC formulation has been augmented with a phonon stabilityanalysis which detects instability and identifies the minimum required periodic cell size forsubsequent simulations. This augmented approach has been referred to as Cascading Cauchy-Born Kinematics (Dobson et al., 2007b; Sorkin et al., 2014) and has been applied, among others,to ferroelectrics (Tadmor et al., 2002) and shape-memory alloys (Dobson et al., 2007b).

18.4.2 Finite temperature and dynamics

Going from zero to finite temperature within the QC framework is a challenge that has resultedin various approximate descriptions but has not been resolved entirely. In contrast to full atom-istics, the physical behavior of continua is generally governed by thermodynamic state variablessuch as temperature and by thermodynamic quantities such as the free energy. Therefore,many finite-temperature QC frameworks have constructed an effective, temperature-dependentfree energy potential; for examples see (Dupuy et al., 2005; Hai and Tadmor, 2003; Kulkarniet al., 2008; Marian et al., 2010; Shenoy et al., 1999; Tadmor et al., 2013; Tang et al., 2006).Dynamic simulations of finite-temperature and non-equilibrium processes raise additional ques-tions. Here, we discuss a few representative finite-temperature QC formulations (often referredto as hotQC techniques); for reviews see also (Kulkarni, 2007; Tadmor et al., 2013; Venturiniet al., 2014; Venturini, 2011) and references therein, to name but a few.

Within the local/nonlocal QC framework, finite-temperature has often been enforced in differ-ent ways in the atomistic and the continuum regions, see e.g. (Tadmor et al., 2013). While inthe atomistic region a thermostat can be used to maintain a constant temperature, an effec-tive, temperature-dependent free energy is formulated in the coarse domain, e.g., based on thequasiharmonic approximation (Tadmor et al., 2013). For quasistatic equilibrium calculations,the dynamic atomistic ensemble is then embedded into the static continuum description of thecoarse-grained model. Alternatively, in dynamic hotQC simulations of this type, all repatoms(including those in both regions) evolve dynamically. The quasiharmonic approximation wasshown to be well-suited to describe, e.g., the volumetric lattice expansion at moderate temper-atures (where the phonon-based model serves as a leading-order approximation). Drawbacksinclude the inaccuracy observed at elevated temperature levels. Computational drawbacks interms of instability or technical difficulties may arise from the necessity to compute up to third-order derivatives of the interatomic potentials. These must hence be sufficiently smooth butoften derive from piece-wise polynomials or tabulated values in reality.

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Temperature and dynamics are intimately tied within any framework based on atomistics. Inprinciple, the zero-temperature QC schemes described in previous sections can easily be ex-panded into dynamic calculations by accounting for inertial effects and solving the dynamicequations of motion instead of their static counterparts. In non-uniform QC meshes, however,this inevitably leads to the reflection and refraction of waves at mesh interfaces and withinnon-uniform regions. While such numerical artifacts may be of limited concern within, e.g.,the finite element method, they become problematic in the present atomistic scenario, sinceatomic vibrations imply heat. For example, when high-frequency, short-wavelength signals can-not propagate into the coarse region, the fully-refined region will trap high-frequency (phonon)motion, which leads to artificial heating of the atomistic region and thus to non-physical simula-tion artifacts. Dynamic finite-temperature QC techniques hence face the challenge of removingsuch artifacts.

Any dynamic atomistic simulation must cope with the small time steps on the order of femtosec-onds, which are required for numerical stability of the explicit finite difference schemes. This, ofcourse, also applies to QC. In general, techniques developed for the acceleration of MD can alsobe applied to the above QC framework. One such example, hyperQC (Kim et al., 2014) borrowsconcepts of the original hyperdynamics method (Voter, 1997) and accelerates atomistic calcu-lations by energetically favoring rare events through the modification of the potential energylandscape. This approach was reported as promising in simple benchmark calculations (Kimet al., 2014).

As a further example, finite-temperature QC has been realized by the aid of Langevin ther-mostats (Marian et al., 2010; Venturini, 2011). Here, the QC approximation is applied withinthe framework of dissipative Lagrangian mechanics with a viscous term that expends the ther-mal energy introduced by a Langevin thermostat through a random force at the repatom level.In a nutshell, the repatoms can be pictured as an ensemble of nodes suspended in a viscousmedium which represents the neglected degrees of freedom. The effect of this medium is ap-proximated by frictional drag on the repatoms as well as random fluctuations from the thermalmotion of solvent particles. As a consequence, high-frequency modes (i.e., phonons) not trans-mitted across mesh interfaces are dampened out by the imposed thermostat in order to samplestable canonical ensemble trajectories. This method is anharmonic and was used to studynon-equilibrium, thermally-activated processes. Unfortunately, the reduced phonon spectra inthe coarse-grained domains result in an underestimation of thermal properties such as thermalexpansion. For a thorough discussion see (Venturini, 2011).

In contrast to the above finite-temperature QC formulations which interpret continuum ther-modynamic quantities as atomistic time averages, one may alternatively assume ergodicity andinstead take advantage of averages in phase space. This forms the basis of the maximum-entropyhotQC formulation (Kulkarni, 2007; Kulkarni et al., 2008; Venturini, 2011). By recourse tomean-field theory and statistical mechanics, this hotQC approach is based on the assumptionof maximizing the atomistic ensemble’s entropy. To this end, repatoms are equipped with anadditional degree of freedom which describes their mean vibrational frequency and which mustbe solved for in addition to the repatom positions at a given constant temperature of the system.This approach seamlessly bridges across scales and does not conceptually differentiate betweenatomistic and continuum domains. As described in the following section, this concept has alsobeen extended to account for heat and mass transfer (Ponga et al., 2015; Venturini et al., 2014).

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18.4.3 Heat and mass transfer

The aforementioned maximum-entropy framework has been applied to study not only finite-temperature equilibrium mechanics but has also been extended to describe non-equilibriumheat and mass transfer in coarse-grained crystals. As explained above, the motion of eachatom can be decomposed into high-frequency oscillations (i.e., heat) and its mean path (slow,long-term motion). The principle of maximum entropy along with variational mean-field theoryprovides the tools to apply this concept to the QC method by providing governing equationsfor the repatom positions as well as their vibrational frequencies (Kulkarni et al., 2008; Ven-turini et al., 2014; Venturini, 2011). In addition, the same framework is suitable to describemulti-species systems by equipping repatoms with additional degrees of freedom that repre-sent the local chemical decomposition. For an atomistic ensemble, this yields an effective totalHamiltonian (Ponga et al., 2015; Venturini et al., 2014; Venturini, 2011)

Heff(q,p,ω) =N

∑i=1

∣pi∣2

2mi+ ⟨Vi⟩(q,p,ω,x)

+3kBθi [3

2N + log ( h ωi

kBθi) − 1 +

n

∑k

xik logxik] ,(18.20)

where ⟨⋅⟩ denotes the phase-space average to be computed, e.g., by numerical quadrature; theassociated probability distribution and partition function are available in closed form (Ven-turini et al., 2014). N , kB and h are the total number of atoms, Boltzmann’s and Planck’sconstants, respectively. ωi represents the vibrational frequency of atom i, and θi is its absolutetemperature. In addition to obtaining Hamilton’s equations of motion for the mean positionsq(t) and momenta p(t) of all N atoms, minimization of the above Hamiltonian also yields thevibrational frequencies ωi as functions of temperature. In order to describe n different species,xik denotes the effective molar fraction of species k at atomic site i (i.e., for full atomisticsit is either 1 or 0, in the coarse-grained context it will assume any value xik ∈ [0; 1]). mi isan effective atomic mass. In addition to interpolating (mean) atomic positions and momentaas in the traditional QC method, one now equips repatoms with vibrational frequencies andmolar fractions as independent degrees of freedom to be interpolated: (q(t),p(t),ω,x). Whilegoverning equations for atomic positions follow directly, the modeling of heat and mass transferrequires (empirical) relations for atomic-level mass and heat exchange to complement the aboveframework (Venturini et al., 2014).

We note that the evaluation of the above Hamiltonian requires effective interatomic potentialsfor many-body ensembles with known average molar fractions xik. Possible choices includeempirical EAM potentials (Foiles et al., 1986; Johnson, 1989) and related lower-bound approxi-mations (Venturini et al., 2014) as well as DFT-informed EAM potentials (Ramasubramaniamet al., 2009; Ward et al., 2012).

18.4.4 Ionic crystals

Ionic interactions, such as those arising, e.g., in solid electrolytes or in complex oxide ferroelectriccrystals, present an additional challenge for the QC method because atomic interactions cease tobe short-range. The QC summation or sampling rules take advantage of short-range interatomicpotentials which admit the local evaluation of thermodynamic quantities of interest, ideally atthe element level. When long-range interactions gain importance, that concept no longer appliesand new effective summation schemes must be developed such as those reported in (Marshalland Dayal, 2014).

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Figure 11: Schematic view of a coarse-grained truss network undergoing indentation (left; shownare truss members, nodes, representative nodes, and example sampling truss members withinone element) and the simulation result of a truss lattice fracturing during three-point bending(right, color-coded by the axial stress within truss members which are modeled as elastic-plastic).

18.4.5 Beyond atomistics

The QC approximation is a powerful tool whose application is not necessarily restricted toatomistic lattices. In general, any periodic array of interacting nodes can be coarsened by acontinuum description which interpolates the full set of nodal positions from a small set ofrepresentative nodes. Therefore, since its original development for coarse-grained atomistics,the QC method has been extended and applied to various other fields.

As an example at even lower length scales, coarse-grained formulations of Density FunctionalTheory have been reported (Gavini et al., 2007; Suryanarayana et al., 2013), which consider-ably speed up electronic calculations. While the key concepts here appertain to the particularformulation of quantum mechanics based on electron densities, the coarse-graining of the set ofatomic nuclei may be achieved by recourse to the QC method.

As a further example at larger scales, the QC method can also be applied at the meso- andmacroscales in order to coarse-grain periodic structures such as truss or fiber networks. Here,interatomic potentials are replaced by the discrete interactions of truss members or fibers, andthe collective response of large networks is again approximated by the selection of representativenodes and suitable interpolation schemes. This methodology was applied to truss lattices (Beexet al., 2011, 2014b,c) and electronic textiles (Beex et al., 2013). As a complication, nodalinteractions in those examples are not necessarily elastic, so that internal variables must beintroduced to account for path dependence in the nodal interactions and dissipative potentialscan be introduced based on the virtual power theorem or variational constitutive updates.Fig. 11 shows a schematic of the coarse-graining of truss lattices as well as the simulated failureof a coarse-grained periodic truss structure loaded in three-point bending. Detailed results willbe shown in Section 19.3.

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19 Applications

It is a difficult task to summarize all applications of the QC method to date; for a thoroughoverview the interested reader is referred to the QC references found in (Tadmor and Miller,2015). Without claiming completeness, the following topics have attracted significant interestin the QC community (for references see (Tadmor and Miller, 2015)):

size effects and deformation mechanisms during nanoindentation,

nanoscale contact and friction, nanoscale scratching and cutting processes,

interactions between dislocations and grain boundaries (GBs), GB mechanisms,

phase transformations and deformation twinning, shape-memory alloys,

fracture and damage in fcc/bcc metals and the brittle-to-ductile transition,

void and cavity growth at zero and finite temperature,

deformation and failure mechanisms in single-, bi-, and polycrystals, in particular themechanics of nanocrystalline solids,

crystalline sheets and rods, including graphene and carbon nanotubes,

ferroelectrics and polarization switching,

electronic textile, fiber networks and truss structures.

In the following sections, we will highlight specific applications of the QC method applied tofcc metals and truss structures. The benchmark examples have been simulated by the authors’fully-nonlocal, massively-parallel 3D QC code.

19.1 Nanoindentation

Nanoindentation is one of the most classical examples that has been studied by the QC methodsince its inception due to the relatively simple problem setup and the rich inelastic deformationmechanisms that can be observed. Oftentimes, the indenter is conveniently modeled by anexternal potential (Kelchner et al., 1998), which avoids the handling of contact and the presenceof two materials.

As an illustrative example of the nonlocal QC method, Fig. 12 shows the results of a virtualindentation test into single-crystalline pure copper (modeled by an extended Finnis-Sinclair po-tential (Dai et al., 2006)) with a spherical indenter of radius 5 nm up to a maximum indentationdepth of 3 nm. For ease of visualization, this example is restricted to two dimensions. Of course,this scenario could easily be simulated by MD but we deliberately present simple scenarios firstfor purposes of detail visualization. The four panels in Figs. 12 (a) to (d) illustrate the distri-bution of repatoms and sampling atoms along with the QC mesh. Automatic mesh adaptationleads to local refinement underneath the indenter and around lattice defects. The latter arevisualized by plotting the centrosymmetry parameter (Kelchner et al., 1998) of sampling atoms(which agree with lattice sites in the fully-resolved regions).

Fig. 13 shows analogous results obtained from a pyramidal indenter penetrating into the sameCu single-crystal. As before, full atomistic resolution is restricted to those regions where it is

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Figure 12: Nanoindentation into a Cu single-crystal: (a)-(d) show the distribution of repatomsin blue (bottom left panels) as well as color-coded by centrosymmetry (top left panels), thesampling atoms of the first- and second-order summation rules of (Amelang et al., 2015) (shownin red and green, respectively, in the bottom right panels), and the mesh (top right panels).(e)-(h) are zooms of (a)-(d) illustrating the sampling atom centrosymmetry. (i) shows theforce-indentation curve (force in eV/A).

indeed required: underneath the indenter as well as in the vicinity of lattice defects. Figs. 13(c) and (d) show the spreading of dislocations into the crystal after emission from the indenter,resulting in full resolution in the wake of the dislocations. The remainder of the simulationdomain remains coarse-grained, thus allowing for efficient simulations having significantly fewerdegrees of freedom than full atomistic calculations. As before, the distribution of repatoms andsampling atoms is shown along with the QC mesh.

The curves of load vs. indentation depth for both cases of spherical and pyramidal indentersin two dimensions are summarized in Fig. 14. As can be expected from experiments, resultsdemonstrate broad hysteresis loops stemming from incipient plasticity underneath the inden-ters. Data also show pronounced size effects in case of the spherical indenter as well as cleargeometrical effects for the (self-similar) pyramidal indenters.

Figs. 15 and 16 show 3D nanoindentation simulations with spherical and pyramidal indenters,respectively. Both graphics show results for Cu single-crystals modeled by an EAM poten-tial (Dai et al., 2006). The spherical indenter has a radius of 40 nm and results are shown up toan indentation depth of 3 nm; the pyramidal indenter has an angle of 65.3 against the vertical

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Figure 13: Nanoindentation into a Cu single-crystal with a conical indenter, showing the distri-bution of repatoms, sampling atoms, the QC mesh, and the local centrosymmetry. Colors andpanel partitioning are identical to those of Fig. 12 (a) through (d).

axis and a maximum indentation depth of 5 nm. Via the centrosymmetry parameter, latticedefects have been identified and highlighted (for better visibility, only those atoms are shownwhich contribute to lattice defects as identified by a higher centrosymmetry parameter).

As shown in these examples, the QC method efficiently reduces computational complexity byassigning full atomistic resolution where it is indeed required (underneath the indenter andnear defects). While full resolution provides locally the same accuracy of molecular statics ordynamics, the QC approximation efficiently coarse-grains the remainder of the model domain,thereby allowing for simulations of significantly larger sample sizes. Similarly, by placing grainor twin boundaries as well as cavities and pre-existing defects in the crystal underneath theindenter, the QC method has been used to study defect interactions (again, the interestedreader is referred to (Tadmor and Miller, 2015) for a full list of references).

19.2 Surface effects

Surface effects play an important role in the mechanics of nanoscale structures and devices. Atthose small scales, the abundance of free surfaces and the associated high surface-to-volumeratios give rise to – both elastic and plastic – size effects. Using the QC approximation nearfree surfaces introduces errors because (i) the associated interpolation of atomic positions mayprevent surface relaxation and (ii) as a consequence of the chosen summation rule, atoms belowthe surface may be represented by those and the surface and vice-versa. The fully-nonlocalQC method combined with the second-order summation rules of (Amelang et al., 2015) reducesthe errors near free surfaces and therefore has been utilized extensively to model deformationmechanisms near free surfaces, see e.g. (Amelang and Kochmann, 2015).

As an example, consider a thin single-crystalline plate (thickness 12 nm) with a cylindrical

104


indentation depth [A]5 10 15 20 25 30

indentation depth [A]5 10 15 20 25 30

25

20

15

10

5

0

30

35

14

10

8

6

2

0

16

18

4

12

inden

tati

on f

orc

e [e

V/A

]

5 nm

40 nm

45º

57º

65º

Figure 14: Load vs. indentation depth for spherical indenters (of radii 5, 7.5, 10, 15, 20, 30and 40 nm) and pyramidal indenters (of different pyramidal angles) in 2D for a (100) single-crystalline Cu.

Figure 15: QC Simulation of 3D nanoindentation with a spherical indenter (radius 40 nm) intosingle-crystalline (100) Cu up to a penetration depth of about 3 nm, for details see (Amelanget al., 2015).

hole, which is being pulled uniaxially at remote locations far away from the hole. Near the hole,full atomistic resolution is required to capture defect mechanisms, but MD simulations are tooexpensive to model large plates. Common MD solutions focus on a small plate instead andeither apply periodic boundary conditions (which introduces artifacts due to void interactions)or apply the remote boundary conditions directly (which also introduces artifacts due to edgeeffects). Here, we use full atomistic resolution near the hole and efficiently coarsen the remainingsimulation domain. By using the second-order summation rule of (Amelang and Kochmann,2015), the coarsening has no noticeable impact on the representation of the free surface. Fig. 17illustrates the microstructural evolution for hole radii of 1.5 and 4.5 nm. Dislocations areemitted from the hole at a critical strain level (which increases as the hole radius decreases);the particular dislocation structures are constrained by the free surfaces at the top and bottom.

19.3 Truss networks

As discussed above, the QC method applies not only to atomic lattices but can also form thebasis for the coarse-graining of discrete structural lattices, as long as a thermodynamic potentialis available which depends on nodal degrees of freedom. For elastic loading, the potential followsfrom the strain energy stored within structural components. When individual truss members

105


Figure 16: QC simulation of 3D nanoindentation with a pyramidal indenter (angle of 65.3

against the vertical axis) into single-crystalline (100) Cu up to a penetration depth of about5 nm.

are loaded beyond the elastic limit, extensions of classical QC are required in order to introduceinternal variables and account for such effects as plastic flow, or truss member contact andfriction. For such scenarios, effective potentials can be defined, e.g., by using the virtual powertheorem (Beex et al., 2011, 2014b,c). Alternatively, variational constitutive updates (Ortiz andStainier, 1999) can be exploited to introduce effective incremental potentials, as will be shownhere for a truss network. For simplicity, we restrict our study to elastic-plastic bars which onlyundergo nonlinear stretching deformation and large rotations (but do not deform in bending).

Consider a truss structure consisting of two-node bars having undeformed lengths Lij anddeformed lengths rij = ∣rij ∣ = ∣qi − qj ∣ with nodal positions qi (i = 1, . . . ,N). The total strain ofeach bar is given by ε = (rij − Lij)/Lij . By using variational constitutive updates (Ortiz andStainier, 1999), we can introduce an effective potential for elastic-plastic bars. To this end, weupdate history variables εnp (the plastic strain) and enp (the accumulated plastic strain) for eachbar at each converged load step n. Assume the elastic-plastic effective energy density is givenby

W (εn+1,∆εp; εnp , e

np) =

E

2[εn+1 − (εnp +∆εp)]

2 +H (enp + ∣∆εp∣)2 +∆t τ0 ∣

∆εp

∆t∣ , (19.1)

where the first term represents the elastic energy at the new load step n + 1 (with Young’smodulus E), the second term represents linear plastic hardening with hardening modulus H,and the final term defines dissipation in a rate-independent manner (τ0 is the critical resolvedshear stress, and ∆t = tn+1 − tn > 0 is a constant time increment). The effective potential energyof each bar (with cross-sectional area A) is thus given by

V (qn+1i ,qn+1

j ; εnp , enp) = A ⋅ inf

∆εp

⎧⎪⎪⎨⎪⎪⎩W

⎛⎝rn+1ij −LijLij

,∆εp; εnp , e

np

⎞⎠

⎫⎪⎪⎬⎪⎪⎭(19.2)

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Figure 17: QC simulation of dislocations emitted from cylindrical holes in thin films of single-crystalline Cu (hole radii are 1.5 and 4.5 nm in the top and bottom graphics, respectively) atstrains of (a) 1.40%, (b) 1.63%, (c) 1.88%, (d) 2.13% and (e) 1.33%, (f) 1.50%, (g) 1.73%, (h)1.93%.

and the new internal variables are

εn+1p = εnp +∆εp, en+1

p = enp + ∣∆εp∣. (19.3)

Minimization can be carried out analytically for this simple case. Stationarity yields the kineticrule of plastic flow:

∂W

∂∆εp= −E (εn+1 − εnp −∆εp) + 2H (enp + ∣∆εp∣) + τ0 sign ∆εp = 0. (19.4)

Consider first the case of ∆εp > 0, which leads to

∆εp =E (εn+1 − εnp) − (τ0 + 2Henp)

E + 2H> 0. (19.5)

Analogously, for ∆εp < 0 we have

∆εp =E (εn+1 − εnp) + (τ0 + 2Henp)

E + 2H< 0. (19.6)

If none of the two inequalities is satisfied, the bar deforms elastically and ∆εp = 0. Once the newvalues of the internal variables have been determined, the stress in the bar connecting nodes iand j is

σn+1ij = E

⎛⎝rn+1ij −LijLij

− [εnp,ij +∆εp,ij]⎞⎠

(19.7)

and the corresponding force is fij = Aσij . Finally, a failure criterion can be included by defining acritical stress σcr or εcr and removing bars whose stress or strain reaches the maximum allowablevalue.

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The total Hamiltonian of the truss structure with nodal positions q = q1, . . . ,qN and momentap = p1, . . . ,pN becomes

H(q,p) =N

∑i=1

∣pi∣2

2mi+N

∑i=1∑

j∈C(i)V (qn+1

i ,qn+1j ; εnp,ij , e

np,ij) , (19.8)

where we assumed that the mass of each bar is lumped to its nodes, so mi represents the totalmass of node i. C(i) denotes the star of each node, i.e. the set of all adjacent nodes connectedto node i through a bar. As the structure of (19.8) is identical to that of atomistics, cf. (18.1),the QC method can be applied in the very same manner as described above for atomisticensembles, including summation/sampling rules as well as adaptive remeshing. We note thatthe sums in (19.8) only involve nearest-neighbor interactions, which is why no force artifactsare expected from node-based summation rules as well as from those of (Amelang et al., 2015),even in non-uniform meshes (as long as elements are sufficiently large).

As an example of this approach, we show simulation results for a three-point bending testof a periodic truss lattice with an initial notch. For small numbers of truss members, suchsimulations can efficiently be run with full resolution (i.e., modeling each truss member as a barelement). Advances in additive manufacturing over the past decade have continuously pushedour frontiers of fabricating micron- and nanometer-sized truss structures. This has resultedin periodic and hierarchical truss lattices with an unprecedented architectural design spaceof (meta)material properties. As a consequence, modeling techniques are required that canefficiently predict deformation and failure mechanisms in truss structures containing millionsor billions of individual truss members. The 3D three-point bending scenario of Fig. 18 inone such example, where the QC method reduces computational complexity by efficient coarse-graining of the bar network (bars are assumed to be monolithic and slender, undergoing thestretching-dominated elastic-plastic model described above with a failure strain of 10%). Loadsare applied by three cylindrical indenter potentials. To account for manufacturing imperfectionsand to induce local failure, we vary the elastic modulus of all truss members by a Gaussiandistribution (with a standard deviation of 50% of the nominal truss stiffness). Fig. 18 comparesthe simulated truss deformation and stress distribution before and after crack propagation. (a)and (b) demonstrate schematically the coarse-grained nodes, whereas (c) through (f) illustratethe advancing crack front in the fully-resolved region. Like in the atomistic examples in theprevious sections, the QC method allows us to efficiently model large periodic systems withlocally full resolution where needed.

Of course, this is only an instructive example. Truss members may be subject to flexuraldeformation, nodes may contribute to deformation mechanisms, and during truss compressionone commonly observes buckling and densification accompanied by truss member contact andfriction. Model extensions for such effects have been proposed recently and are subject toongoing research, see e.g. (Beex et al., 2011, 2014b,c).

19.4 Summary and open challenges

Although the QC method was established almost two decades ago, many challenges have re-mained and new ones have aroused. These include, among others:

Adaptive coarsening : adaptive model refinement may be geometrically challenging inpractice but it is conceptually sound, ultimately turning all lattice sites into repatomsand sampling atoms and thus recovering (locally) molecular dynamics or statics. Modelcoarsening, in contrast, is challenging conceptually and a key open problem in many QC

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Figure 18: Failure of a periodic 3D truss network under three-point bending; color-code indicatesstresses within truss members (in order to account for imperfections and stress concentrations;stresses scaled to maximum). (a) and (b) visualize the representative nodes in the coarse-grainedQC description (full resolution around the notch and coarsening away from the notch; the threeindenters are shown as blue balls). (c/e) and (d/f) show, respectively, the deformed structurewith its stress distribution before and after the crack has advanced.

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research codes due to the Lagrangian formulation. Overcoming this difficulty will allow toefficiently coarsen the model by removing full resolution (e.g., in the wake of a travelinglattice defect which leaves behind a perfect crystal that is now fully resolved). Withoutadaptive model coarsening, simulations will accumulate atomistic resolution and ulti-mately turn large portions of the model domain into full MD, thus producing prohibitivecomputational expenses.

Large-scale simulations: the number of existing massively-parallel, distributed-memoryQC implementations is small for a variety of reasons, especially in three dimensions. Thislimits the size of domains that can be simulated.

Dynamics is a perpetual problem of the QC method. Non-uniform meshes result in wavereflections and wave refraction, which affects phonon motion and thereby corrupts heatpropagation within the solid. For these reasons, dynamic QC simulations call for newapproaches. Several finite-temperature QC formulations have aimed at overcoming someof the associated problems, but those generally come with phenomenological or simplifyingassumptions.

Mass and heat transfer is intimately tied to the long-term dynamics of a (coarse-grained)atomistic ensemble. Recent progress (Venturini et al., 2014) has enabled the compu-tational treatment via statistical mechanics combined with mean-field theory, yet suchapproaches require constitutive relations for heat and mass transfer at the atomic scale,and they have not been explored widely.

Potentials: every atomistic simulation stands and falls by the accuracy and reliability ofits interatomic potentials. Unlike MD, QC may require conceptual extensions when itcomes to, e.g., long-range interactions or interaction potentials that not only depend onthe positions of the atomic nuclei.

Surfaces play a crucial role in small-scale structures and devices and most QC variantsdo not properly account for free surfaces (misrepresenting or not at all accounting forsurface relaxation). Improved surface representations are hence an open area of research,see e.g. (Amelang and Kochmann, 2015) for a recent discussion.

Structures: rather recently, the QC approximation has been applied to truss and fibernetworks where new challenges arise due to the complex interaction mechanisms (includ-ing, e.g., plasticity, failure, or contact). This branch of the QC family is still in its earlystages with many promising applications.

Imperfections: The QC method assumes perfect periodicity of the underlying (crystal ortruss) lattice. When geometric imperfections play a dominant role (such as in many nano-and microscale truss networks), new extensions may be required to account for randomor systematic variations within the arrangement of representative nodes.

Of course, this can only serve as a short excerpt of the long list of open challenges associatedwith the family of QC methods and as an open playground for those interested in scale-bridgingsimulations.

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20 Concurrent Multiscale Challenges at Finite Temperature

20.1 Finite-Temperature Quasicontinuum

The QC method introduced above was inherently at zero temperature (0K), so that an exten-sion is required when dealing with problems at finite temperature. There have been multipleapproaches to including temperature. Two prominent ones are briefly discussed here.

20.1.1 Maximum-Entropy HotQC

Consider a system of N particleswith positions and momenta, respectively,

q = (q1,q2, ...,qN) (20.1)

p = (p1,p2, ...,pN) with pi =miqi. (20.2)

Following statistical mechanics, we define the phase space average for any quantity A as

⟨A⟩ = 1

N !h3n ∫ΓA(q,p)ρ(q,p)dq dp (20.3)

Thermal vibrations motivate the separation of atomic degrees of freedom into mean motion anddeviations. The average positions and momenta are, respectively, defined as

qa = ⟨qa⟩, pa = ⟨pa⟩. (20.4)

At zero temperature, we have qa = qa and pa = pa without thermal fluctuations. At finitetemperature, one defines the standard deviations as

⟨∣qa − qa∣2⟩ = 3τ2a ⟨∣pa − pa∣2⟩ = 3σ2

a (20.5)

as well as the ratio

ωa =σaτa. (20.6)

Rearranging (20.5) to express in terms of ω yields a constraint for each atom a:

⟨∣pa − pa∣2⟩ + ω2a⟨∣qa − qa∣2⟩ = 6σ2

a. (20.7)

Next, one seeks more information about the newly introduced variables by recourse to theprinciple of maximum entropy, classicaly known in information theory. It states that theprobability distribution ρ that best represents the current state of knowledge is the one withthe largest entropy, where entropy may be defined as

S [ρ] = −kB⟨log ρ⟩. (20.8)

Notice that ρ satisfies the conditions of a probability density, since we require 0 ≤ ρ(p,q) and

∫Γ ρ(p,q)dq dp = 1. Notice that the latter can also be written as ⟨1⟩ = 1.

Altogether, one may formulate the maximization problem with the constraints

⟨1⟩ = 1

⟨∣pa − pa∣2⟩ + ω2a⟨∣qa − qa∣2⟩ = 6σ2

a,

111


by introducing Lagrange multipliers λ and βa (a = 1, . . . ,N). The result is the following func-tional to be maximized with respect to the probability density ρ(q,p) at a given temperature:

L [ρ] = S [ρ] − λ⟨1⟩ − ∫Γ∑a

βaρ [∣pa − pa∣2 + ω2a∣qa − qa∣2] dp dq, (20.9)

It turns out the this maximization can be carried out analytically, leading to a probabilitydensity similar in nature to the one of the canonical ensemble, viz.

ρ (q,p) = 1

Zexp(−∑

a

∣pa − pa∣2 + ω2a∣qa − qa∣2

2σ2a

) with Z = 1

N !h3N

N

∏i

⎡⎢⎢⎢⎢⎣(

2πσ2i

ωi)

3⎤⎥⎥⎥⎥⎦. (20.10)

Insertion into the total entropy and algebraic rearrangements in the limit of large N yields

S =N

∑a

Sa where Sa = 3kB logσ2a

hωa+ 4kB − kB logN (20.11)

as well as the standard deviations and

σa =√hωa exp [1

6(SakB

− 4 + logN)] . (20.12)

Note that we observe a spread in the atomic motion with increasing entropy, as can be expected.

In order to proceed to a finite-temperature QC version, we reinterpret the mean Hamiltonianas the internal energy

E(p,q) = ⟨H⟩, (20.13)

for which a Legendre transform provides the Helmholtz free energy as

F (q,p,ω,θ) = infS

E −∑a

θaSa , (20.14)

where ω = ω1, . . . , ωN. Note that the associate absolute temperature of atom a is

θa =∂E

∂Sa= hωa

2makBexp( Sa

3kB− 4

3+ logN

3) , (20.15)

from which we obtain the relation between entropy and temperature of an atom a as

Sa = 3kB log (2kBmaθahωa

) + 4kb − kB logN. (20.16)

Finally, inserting everything into the energy gives us a potential

F (q,p,ω, θ) =N

∑a=1

[1

2

∣pa∣2

ma+ 3kBθa log

hωa2kbmaθa

+ ⟨Va(q,p,q,p,ω,θ) + kBθa(logN − 1)⟩] ,

(20.17)

which now depends on the mean motion of atoms (q,p) as well as the thermal statisticalinformation encoded in ω = ω1, . . . , ωN and the temperature θ. Notice that this new energymay replace the original Hamiltonian, now containing kinetic and potential energy contributionsof the mean motion as well as contributions from the thermal fluctuations’ kinetic energy andan explicit temperature contribution in the final term.

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In conclusion, we have replaced the Hamiltonian of the system of N atoms by a new, effectiveHamiltonian whose associated “equations of motion” for each atom a at a given temperature θread

maqa = −∂

∂qaF (q,p,ω, θ), (20.18)

0 = ∂

∂ωaF (q,p,ω, θ). (20.19)

The second equation is new and required to solve for the statistical fluctuations ω. This com-pletes the finite-temperature QC or hotQC formulation at constant temperature; see, e.g.,(Kulkarni et al., 2008; Venturini et al., 2014; Venturini, 2011) for details. Available extensionsincluded heat flux and nonuniform temperature distributions (Venturini et al., 2014).

20.1.2 Quasiharmonic approximation

As an alternative approach, one may start with the classical probabilty density of the canonicalensemble at finite temperature and apply phase space averages to the total Hamiltonian of asystem including mean motion and statistical deviations. Phase averages thus become

A(q, T ) = 1

h3NZ∫

ΓA (q + δq,p) exp [−H(q + δq,p)

kBT]χ(δq)dδq dp, (20.20)

with fluctuations δq = δq1, . . . , δqN and a restrictive indicator function χ ensuring the ad-missibility of δq. Carrying out all phase-space integrals and rearranging terms results in aneffective kinetic energy.

T (p, T ) =N

∑a=1

∥pa∥2

2ma+ 3kBT

2

N

∑a=1

log2πkBTma

h2p

(20.21)

with an adjustable parameetr hp. The potential energy is represented by a Taylor expansion,i.e.,

V(q + δq) = V(q) +N

∑a=1

∂V∂qa

⋅ δqa +1

2

N

∑a,b=1

δqa ⋅Φab(q) δqb + h.o.t. (20.22)

Dropping the higher-order terms results in the so-called quasiharmonic approximation, andthe force-constant matrix is defined as

Φab(q) =∂2V

∂qa ∂qb(q). (20.23)

A lengthy calculation finally leads to (with hp hq = h)

V(q, T ) = V(q) −N

∑a=1

kBT

2log

⎡⎢⎢⎢⎣

h6Nq detΦa(q)(2πkBT )3N

⎤⎥⎥⎥⎦, (20.24)

where Φa(q) denotes the force-constant matrix associated with atom a. With the above methodestablished for atomistic particle ensembles, it has been extended and applied to the QC for-mulation. For details, see e.g. Tadmor et al. (2013).

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20.2 Irving-Kirkwood Theory

As an independent finite-temperature coarse-graining theory (that is not related to QC), let usmention the approach that goes back to Irving and Kirkwood (1950) and is hence named theIrving-Kirkwood theory. In a nutshell, the idea is to produce continuum fields A(x, t) fromdiscrete atomistic ensembles by defining

A(x, t) = ∫ΓA(q,p;x) f(q,p, t)dq dp = ⟨A,f⟩, (20.25)

where f(q,p, t) is some appropriate distribution function. The goal is construct functionsA(q,p;x) such that the continuum balance equations have the form

∂A∂t

(x, t) = ∂

∂t⟨A,f⟩. (20.26)

As an example, one can define a pointwise mass density as

ρ∗(x, t) =N

∑a=1

ma⟨δ(qa −x), f⟩. (20.27)

Likewise, the pointwise velocity field is defined as

v∗(x, t) = 1

ρ(x, t)

N

∑a=1

ma⟨vaδ(qa −x), f⟩. (20.28)

The analogous procedure may be carried out, e.g., for the energy density. Once these continuumfield quantities exist, the continuum governing equations apply such as conservation of mass,linear momentum and energy.

In practice, one does not use the pure (”sharp”) delta-function definitions given above butspatial averaging by introducing some weighting function w, going back to Hardy (1982), suchas to introduce the continuum mass density as

ρ(x, t) = ∫d

Rw(y −x)ρ∗(y, t)dy (20.29)

and the continuum velocity field as

v(x, t) = ∫d

Rw(y −x)v∗(y, t)dy, (20.30)

and the analogous procedure for all other field variables.

This admits the extraction of continuum-level field quantities from discrete atomistics ensembles,which has been exploited, e.g., in the Concurrent Atomistic-Continuum (CAC) methodto couple atomistics and continuum simulations in a concurrent fashion similar in spirit to theQC metho; see e.g. Yang et al. (2015).

114


21 Discrete Dislocation Dynamics

Although we will not go into details, let us briefly mention the popular multiscale techniqueknown as discrete dislocation dynamics (DDD). The key concept is the description ofdislocation motion within an elastic continuum, thus coupling the discrete nature of dislocationlines and the continuum formulation of an elastic medium.

Consider a dislocation line segment s that is moving with a velocity v, such that the motion ofthe dislocation is typical described by

m v + 1

M(T, p)v = Fs, (21.1)

where m is an effective mass density of the dislocation segment, M denotes the dislocationmobility which generally depends on both temperature T and pressure p, and Fs is the glide forcevector per unit length of the dislocation. The glide force includes various force contributionsacting onto the dislocation segment, e.g.,

Fs = FPeierls +FSelf +FInteraction +FObstacle +FImage +FOsmotic +FThermal +FExternal, (21.2)

which include (in the listed order) the Peierls stress to drive a dislocation in its glide plane, adislocation self-force, an interaction forces due to all other dislocation segments, an interactionforces with obstacles (such as second-phase particles or interfaces and surfaces), image forces,osmotic forces associated with dislocation climb, thermal forces arising from thermal fluctua-tions, and externally applied loads. The individual contributions are calculated based on theclassical continuum theory of dislocations (and extensions), based on the stress field produced adislocation segment (Hirth and Lothe, 1982). The above equation is solved by implicit/explicittime integration. In case of free surfaces, a special treatment is required to deal with the latter,which is usually accomplished by the superposition of the associated problems in an infinitebody with dislocations and in a finite-size body without dislocations but representative forcefields.

115


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Index

Angstrom, 656-12 potential, 65

acceleration, 9acoustic tensor, 47affine displacement BCs, 24Andersen thermostat, 83anharmonic, 66Arruda-Boyce, 39atomistics, 59average strain-driven BCs, 30average stress-driven BCs, 30averaging theorems, 17

balance laws, 7binding energy, 63Bloch wave, 55Bohr’s radius, 58bond angle, 67Born-Oppenheimer approximation, 58boundary conditions, 13Bravais lattice, 42Brownian motion, 83

CAC, 114canonical ensemble, 74Cauchy stress tensor, 11central limit theorem, 16centrosymmetry, 68computational homogenization, 22Concurrent Atomistic-Continuum, 114concurrent scale-bridging, 85configuration, 15constitutive relations, 6constraint potential, 30Coulomb potential, 57Coulombic interaction, 63current configuration, 9cut-off radius, 68

DDD, 115De Broglie, 56de Broglie wavelength, 61deformation gradient, 9deformation mapping, 9density functional theory, 60density of states, 77Dirichlet boundary, 13discrete dislocation dynamics, 115

displacement field, 9dissociation energy, 66distribution function, 75divergence theorem, 7

effective mass density, 23effective material, 21effective response, 22Einstein-Planck relation, 56elasticity tensor, 12electric permittivity, 57electron density, 60electron-volts, 65Embedded Atom Method, 67ensemble, 15, 74ensemble average, 15, 75equilibrium spacing, 64ergodicity, 75Eulerian, 9exchange-correlation functional, 61extended Cauchy-Born rule, 39extensive, 7

Fast Inertial Relaxation Engine, 80FE2, 36First Brillouin Zone, 55First Piola-Kirchhoff, 10first Piola-Kirchhoff stress tensor, 11flux, 6force constant, 66Fourier transform, 42

gradient elasticity, 39grand canonical ensemble, 74ground state, 58

Hamilton’s equations, 62harmonic potential, 66Hartree energy, 61heat equation, 8heterogeneous problem, 21hierarchical scale-bridging, 85Hill-Mandel condition, 24homogeneous, 8homogenized problem, 21horizontal scale-bridging, 85hotQC, 113

incremental tangent modulus tensor, 11

124


initial boundary value problem, 13initial conditions, 13intensive, 7interatomic potential, 63internal energy, 7, 77ionic crystal, 64Irreducible Brillouin Zone, 55Irving-Kirkwood, 114isothermal-isobaric ensemble, 74isotropy, 8

kinematic variables, 6Kohn-Sham, 61

Lagrangian, 9Langevin thermostat, 83Lennard-Jones, 64linear elasticity, 12linearized kinematics, 11local Cauchy-Born rule, 38local energy balance, 7

macroscale, 15macrostates, 74mappings, 6material points, 6microcanonical ensemble, 74, 76microscale, 15microstates, 74mixed stress/strain-driven BCs, 30Molecular Dynamics, 80molecular statics, 62Morse potential, 66multi-lattice, 41multi-lattices, 72

Navier’s equation, 13Neumann boundary, 13nonlocal Cauchy-Born rule, 39nonlocal model, 39Nose-Hoover thermostat, 83

observable, 74orbitals, 58order of a PDE, 14

pair potentials, 63Parrinello-Rahman approximation, 84perfectly bonded, 17periodic BCs, 24, 28perturbation stress tensor, 46phase average, 75phase space, 73

phonon modes, 66Pontryagin duality, 43principle of maximum entropy, 111probability density, 58, 75

quasiharmonic approximation, 66, 113quasistatics, 10

realization, 15reciprocal lattice, 42reference configuration, 9representative volume element, 16Reuss bound, 26right Cauchy-Green tensor, 10rigid body rotation, 12rigid body translation, 11ringing artifacts, 50

Sachs model, 26sample enlargement, 16Schrodinger’s equation, 57separation of scales, 21short-range, 65stationary states, 58statistically homogeneous, 15statistically inhomogeneous, 15statistically representative, 16Stilinger-Weber, 67strain energy density, 11strain tensor, 12stress, 10stretch, 9surface relaxation, 69

Taylor model, 25time average, 74time-independent Schrodinger equation, 57total Hamiltonian, 62traction, 10trajectory, 73translational invariance, 15two-body potentials, 63

uniform traction BCs, 24unit cell, 15

van der Waals, 68velocity, 9velocity-Verlet, 80Verlet radius, 81vertical scale-bridging, 85virial, 79virial stress tensor, 79

125


virial theorem, 79Voigt bound, 26volume change, 9

zero-temperature MD, 81

126

computational multiscale modeling · here and in the following, we use dots to denote rates, i.e.,...

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