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Extended quasicontinuum methodology for highly heterogeneous discrete systems

December 2023
International Journal for Numerical Methods in Engineering 125(6):e7415

DOI:10.1002/nme.7415

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CC BY-NC 4.0

Authors:

Benjamin Werner

Czech Technical University in Prague

Jan Zeman

Czech Technical University in Prague

Ondřej Rokoš

Eindhoven University of Technology

Lattice networks are indispensable to study heterogeneous materials such as concrete or rock as well as textiles and woven fabrics. Due to the discrete character of lattices, they quickly become computationally intensive. The QuasiContinuum (QC) Method resolves this challenge by interpolating the displacement of the underlying lattice with a coarser finite element mesh and sampling strategies to accelerate the assembly of the resulting system of governing equations. In lattices with complex heterogeneous microstructures with a high number of randomly shaped inclusions the QC leads to an almost fully-resolved system due to the many interfaces. In the present study the QC Method is expanded with enrichment strategies from the eXtended Finite Element Method (XFEM) to resolve material interfaces using nonconforming meshes. The goal of this contribution is to bridge this gap and improve the computational efficiency of the method. To this end, four different enrichment strategies are compared in terms of their accuracy and convergence behavior. These include the Heaviside, absolute value, modified absolute value and the corrected XFEM enrichment. It is shown that the Heaviside enrichment is the most accurate and straightforward to implement. A first-order interaction based summation rule is applied and adapted for the extended QC for elements intersected by a material interface to complement the Heaviside enrichment. The developed methodology is demonstrated by three numerical examples in comparison with the standard QC and the full solution. The extended QC is also able to predict the results with 5% error compared to the full solution, while employing almost one order of magnitude fewer degrees of freedom than the standard QC and even more compared to the fully-resolved system.

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Received: 6 September 2023 Revised: 1 December 2023 Accepted: 10 December 2023

DOI: 10.1002/nme.7415

RESEARCH ARTICLE

Extended quasicontinuum methodology for highly

heterogeneous discrete systems

Benjamin Werner1Jan Zeman1Ond

rej Rokoš2

1Faculty of Civil Engineering, Czech

Technical University in Prague, Prague,

Czech Republic

2Mechanics of Materials, Department of

Mechanical Engineering, Eindhoven

University of Technology, Eindhoven, The

Netherlands

Correspondence

Benjamin Werner, Faculty of Civil

Engineering, Czech Technical University

in Prague, Thákurova 7, 16629 Prague 6,

Czech Republic.

Email: benjamin.werner@cvut.cz

Funding information

Ministry of Education, Youth and Sports

of the Czech Republic, Grant/Award

Number:

CZ.02.2.69/0.0/0.0/18_053/0016980;

Czech Science Foundation, Grant/Award

Numbers: 19-26143X, 22-35755K; CTU

Global Postdoc Fellowship Program

Summary

Lattice networks are indispensable to study heterogeneous materials such as con-

crete or rock as well as textiles and woven fabrics. Due to the discrete character

of lattices, they quickly become computationally intensive. The QuasiContinuum

(QC) Method resolves this challenge by interpolating the displacement of the

underlying lattice with a coarser finite element mesh and sampling strategies to

accelerate the assembly of the resulting system of governing equations. In lattices

with complex heterogeneous microstructures with a high number of randomly

shapedinclusionstheQCleadstoanalmostfully-resolvedsystemduetothemany

interfaces. In the present study the QC Method is expanded with enrichment

strategies from the eXtended Finite Element Method (XFEM) to resolve mate-

rial interfaces using nonconforming meshes. The goal of this contribution is to

bridge this gap and improve the computational efficiency of the method. To this

end, four different enrichment strategies are compared in terms of their accuracy

and convergence behavior. These include the Heaviside, absolute value, modified

absolutevalueandthecorrectedXFEMenrichment.Itis shown that the Heaviside

enrichment is the most accurate and straightforward to implement. A first-order

interaction based summation rule is applied and adapted for the extended QC for

elements intersected by a material interface to complement the Heaviside enrich-

ment. The developed methodology is demonstrated by three numerical examples

in comparison with the standard QC and the full solution. The extended QC is

also able to predict the results with 5% error compared to the full solution, while

employing almost one order of magnitude fewer degrees of freedom than the

standard QC and even more compared to the fully-resolved system.

KEYWORDS

extended finite element method, lattice networks, micromechanics, multiscale, quasicontinuum

method

1INTRODUCTION

Numerical models of discrete lattice networks are beneficial to study the failure behavior of various heterogeneous

materials. These numerical models are simple to assemble and represent the underlying microstructure directly. They are

highly suitable as numerical models for cohesive-frictional materials such as concrete and rock.1,2 The lattice nodes rep-

resent center points of grains, connected by spring or beam elements and are arranged as regular or irregular lattices.1,3

This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any

medium, provided the original work is properly cited and is not used for commercial purposes.

Int J Numer Methods Eng. 2023;e7415. wileyonlinelibrary.com/journal/nme 1of26

https://doi.org/10.1002/nme.7415

2of26 WERNER  .

FIGURE 1 A schematic illustration of a QC model with an underlying X-braced lattice in grey, a finite element mesh in blue, repatoms

in black and sampling interactions in magenta.

A great advantage of these models is their numerical robustness in predicting crack growth and crack coalescence in

quasi-brittle materials4,5 due to their local representation of the microstructure. Continuum models, in contrast, are based

on homogenized material behavior and lead to an element size-dependency of strength and fracture toughness predic-

tions, stress-locking problems, and difficulties to capture multiple growing cracks.3,6 Over the past decades the approach

of using lattice networks to study numerically quasi-brittle materials such as concrete has been continuously improved by

inelastic damage behavior of the spring elements connecting the particles as well as friction between them.7,8 Discrete lat-

tice networks have also been applied to study fracture behavior of cortical bovine bone9or textiles and woven fabrics.10,11

The elements of the lattice are representing yarns and fibers of textiles or fabrics at the meso-scale level and thus directly

allow to incorporate bond failure in a straightforward manner. The fibrous material of paper has been represented by

lattice networks in numerical studies as well, and shown to reproduce the fracture processes accurately.12

Discrete lattices provide a simple way to investigate heterogeneous materials, but they are usually costly to simulate.

The QuasiContinuum (QC) method is a numerical multi-scale approach for lattice networks addressing that problem.

This numerical method was first introduced by Tadmor et al.13,14 and proposed for crystal lattices on the atomistic level

to investigate dislocations and nanoindentation. The methodology relies on two main steps. (i) Instead of considering all

degrees of freedom (DOF) of the full atomistic arrangement the system of equations in QC is reduced by interpolation.

Here, a subset of so-called representative atoms (repatoms) is selected for a triangulation or discretization (Figure 1)

with a linear interpolation to approximate the displacements of the full atomistic arrangement, in the spirit of the Finite

Element Method (FEM).*(ii) To avoid visiting every lattice site to determine the total potential energyof the system, and

to assemble the resulting system of governing equations, summation rules are used to further reduce the computational

costof the method. A certain set of sampling atoms or interactions is selected (Figure 1) with corresponding weight factors

to approximate the total potential energy of the entire system. The weight factors are thereby indicating the number of

sites the sampling atom is representing in a triangle, in analogy to the weights used in Gauss integration values. With

these two integral parts the QC method allows studying solids at the atomistic level or microscale at locations with highly

non-uniform deformations, while seamlessly transitioning to the continuum or macroscopic level at places with small

deformation gradients with limited computational efforts.16,17

The QC methodology has been applied to problems at different length scales, including simulations of nano cutting

of a copper crystal by Yang et al.18 Here, the QC has been extended with a material removal criterion to analyse the

ultra-precision process more accurately and to determine the influence of various parameters on the cutting force. Tran

et al.19 applied the method at the micro-scale to study the friction behavior of textured and rough aluminum surfaces. A

cylindrical indenter was moved along the surface and the friction coefficient was determined, among other parameters.

Besides investigations of atomistic arrangements, QC has been applied to lattices at the macroscopic level. Beex

et al.20 used the method to study fabric with elastic-plastic material behavior. Individual links of the lattice represent

yarn segments of the fabric while the lattice nodes serve as yarn-to-yarn contact points. To include dissipative material

behavior, the QC was reformulated in a virtual-power-based manner and extended to an interaction based summation

rule in Reference 20. Generalization to dissipative processes based on a variational formulation has been performed in

Reference 21. The QC method holds potential for irregular lattice arrangements, which is of great advantage for use in

concrete and rock and concrete analysis.22

Further extensions involve beam lattice structures using the corotational beam element formulation allowing for geo-

metrical nonlinearities. Phlipot and Kochmann23 presented two- and three-dimensional indentation tests and a mode

I sample with a pre-existing crack. They used multiple lattice unit cell geometries including triangular, hexagonal and

Body Centered Cubic (BCC) arrangements. Chen et al.24 studied elastic three-dimensional lattice structures with BCC

and Kelvin unit cells and presented multiple numerical examples including contact definition between indentor and

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WERNER  . 3of26

lattice structure. A more accurate refinement indicator for automatic adaptive meshing has also been developed. In

an earlier study, Chen et al.25 extended the approach with corotational beam elements to plastic material behavior.

Multiple beam elements with plastic hinges have been combined to represent one strut of the lattice to study indentation

in a three-dimensional BCC arrangement.

The interpolation step, when used with meshes, decreases the number of unknowns substantially, and is one of

the integral parts of the QC method. Automatic adaptive refinement of the discretization is a further key to reduce the

computational cost. During a numerical analysis the triangulation is refined at locations of high interest such as the

contact area in indentation scenarios,24,26 the surrounding of crack tips27,28 or along inhomogeneities like grain

boundaries29 or vacancies.30

In the XFEM such discontinuities can be incorporated using nonconforming meshes31–37 and are considered by

additional DOFs and enriched interpolation functions. Rokoš et al.38 used enrichments from XFEM to model crack propa-

gation in discrete lattice networks and expanded the idea of computational efficiency through automatic mesh adaptivity

in the QC method to the coarsening of the discretization. By including enriched interpolation functions the mesh is coars-

ened in the wake of the crack while the area around the tip of the propagating crack in dissipative lattice networks is fully

resolved. This leads to a significantly reduced number of DOFs while keeping high accuracy. To incorporate grain bound-

aries, vacancies and voids, material interfaces or other heterogeneities in numerical models they have to be fully resolved

by the discretization in the current state of the QC method. Including enriched interpolation functions for weak discon-

tinuities from the XFEM would allow to use a nonconforming triangulation. Consequently, the mesh does not have to be

refined at the interface, further decreasing the number of unknowns in the system of equations significantly.

In the energy-based QC method, the evolution of the entire system is governed by the total potential energy and its

accuracy depends on the applied summation rule. Therefore, a significant effort has been spent on defining suitable sum-

mation rules which introduce a minimal error and are at the same time computationally efficient. In the first studies

developing the QC method the atom closest to the quadrature point of the triangle was selected as the sampling atom.13,14

This resulted in a computationally efficient summation rule for large triangles but the selection procedure does not tran-

sition seamlessly to fully resolved regions experiencing high deformations. As a result, non-physical forces appear at

the interface between the fully and non-fully resolved domains and introduce errors in the analysis. The cluster based

summation rule of Knap and Ortiz26 resolves this problem by considering multiple sampling nodes located in a circle

around the repatoms. Eidel and Stukowski39 investigated the cluster summation rule in more detail and proposed strate-

gies to further reduce the error in the approximated potential energy. Yang et al.40 combined cluster and quadrature point

summation rule. A similar approach is the central summation rule by Beex et al.41 where the cluster is reduced to the

repatom itself and the remaining atoms are accounted for by the center atom. In an earlier study Beex et al.42 presented

an exact summation rule whereby every atom with a neighbour in an adjacent triangle is accounted for individually for

sampling and the center atom is representing the remaining atoms inside the triangle. This leads to small bands of sam-

pling atoms along the triangle edges, thus resulting in higher accuracy but requires more computational effort. Amelang

et al.43 as well as Amelang and Kochmann44 studied and systematically summarized existing summation rules. These

studies provide optimal summation rules for non-uniform meshes for one-, two- and three-dimensional lattices. The

selection procedure of sampling atoms is similar to the studies of Beex et al.41 and Yang et al.40 as a combination of a

cluster at the repatoms combined with a sampling atom at the quadrature point. For higher order interpolation functions

multiple quadrature points inside the triangle and additional sampling atoms on the edges are applied. Instead of using

sampling atoms to approximate the potential energy of the system, Beex et al.20 used the interactions between them. In

an X-braced lattice one link for each of the four directions is selected to represent all of them in a triangle. In addition,

links crossing triangle edges are accounted discretely in the summation rule which leads to an exact assembly of the

potential energy.

The main goal of the present study is incorporating weak discontinuities into the QC method by using enriched inter-

polation functions motivated by the XFEM to account for interfaces in heterogeneous discrete systems, to further reduce

the number of DOFs and speed up calculations. To quantify the error of the approach, the outcome is compared to the

standard QC with fully-resolved discontinuity boundaries as well as the underlying full direct numerical solution. Three

different numerical examples are investigated. The first two include a single stiff circular inclusion and a single stiff fiber

embedded in a soft matrix and studied under tensile loading. The third example represents a cross-section of a concrete

sample with multiple aggregates and fibers which are surrounded in a more compliant cement matrix, studied under ten-

sile loading using Periodic Boundary Conditions (PBC). For the extended as well as the standard QC the interaction based

first-order summation rule is applied and compared to the full summation. The main outcome of this study is the conver-

gence behavior of the standard QC and the extended QC with four different enrichment types (Heaviside, absolute value,

4of26 WERNER  .

FIGURE 2 Schematic illustration of the kinematic variables of an X-braced lattice.

modified absolute value, and corrected XFEM) using the full summation. The convergence of both QC approaches using

the interaction based first-order summation rule is demonstrated. Moreover, the accuracy and the numerical efficiency

of the extended and the standard QC are compared for all three numerical examples.

2METHODS

2.1 Full-lattice formulation

The lattice network considered is an X-braced arrangement in two dimensions where one atom is connected by inter-

actions to eight direct neighbours inside the domain Ω(Figure 2).†The location of each atom is described by a position

vector and the position vectors of all atoms are stored in a single column matrix r=[r1,…,rnato ]Tfor the entire system

with nato atoms. An equilibrium configuration rof the discrete system is determined by the minimization of the total

potential energy Πkat a time instant tkas

r(tk)=arg min



r∈2nato Πk(

r),k=1,…,nT,(1)

where 

rdenotes an arbitrary admissible position vector, and with the initial condition r(0)=r0and 0 =t0<t1<···<

tnT=Tbeing discretization of a considered time interval [0,T]. The potential energy of the lattice

Πk(

r)=

nint



=1k

(

r)(2)

is the sum of all elastic energies k

 of all interactions at time tk. The entire system contains nint interactions stored in an

index set Nint. Each interaction  ∈Nint is connecting the lattice sites and with the length

r =r−r2,(3)

determined by the Euclidean norm of the difference between the two position vectors rand r. The elastic energy of the

link between lattice sites and is defined as

k

(

r)=EA

2r

0

r −r

02(4)

determined from the Young’s modulus E, the link cross-section area A as well as the strain of the link, which is the

difference between the current

r and the initial length r

0of the interaction, divided by the initial length.

WERNER  . 5of26

After determining the potential energy of the whole lattice arrangement, the system of equations can be solved itera-

tivelyusing the Newton-Raphson method. Thepotentialenergy function is thereby approximatedby the Taylorexpansion,

resulting in the following iterative systems of equations

Ki

ri+1−

ri+fi=0(5)

where the interval force vector is defined as the gradient of Πk,

fi=f(

ri)= Πk(



r

r=

ri(6)

and the tangent stiffness matrix as the Hessian of Πk,

Ki=K(

ri)= 2Πk(



r

r

r=

ri,(7)

both determined at a time instance tk. Solving Equation (5) provides the location of the current increment of all position

vectors. The iteration continuous until the combination of the normalized position vector and the normalized force is

smaller than a predefined tolerance tol, that is,

(

ri+1−

ri)

rfree +fi

fDBC<tol.(8)

In Equation (8) the entire column matrix is split according to free and prediscribed DOFs r=[rfree,rDBC]with rDBC being

the position vectors of the atoms with fixed displacements and rfree atoms left to move freely. The global column matrix of

the internal forces f=[ffree,fDBC]is split equivalently and the reaction forces of the atoms with prescribe displacements,

fDBC, are used in the convergence criterion for normalization.

2.2 QC reduction by interpolation

To reduce the number of unknowns in Equation (5), atoms displacements are interpolated. The interpolation in the

present study contains two parts, the standard FEM interpolation by using a triangular mesh, and an enriched interpo-

lation motivated by the XFEM. Only the first part is applied in the standard QC Method, whereas a combination of both

results in the extended QC. The standard QC interpolation for the position vector of atom ,



r

qc =

∈Nrep

(r

0)

r

rep,∈Nato,(9)

is determined by the FE shape function (r

0)and the location of the repatoms 

r

rep with Nato and Nrep being index sets

containing all lattice atoms and repatoms.

For the extended QC, four different enrichment functions are applied to account for weak discontinuities within a

domain. This includes the Heaviside function,45 the absolute value enrichment,46 the modified absolute value enrich-

ment31 as well as the corrected XFEM.47 They will be established and explained in the following. Before we introduce

these four enrichment options, the general approximation is defined here. The position vector for each atom in the lattice,



r=

r

qc +n



j=1j(r

0)R(r

0)[•(r

0)−•(rj

0)]





j(r



g

j,∈Nato,

j∈N(10)

is determined by the superposition of the standard interpolation

r

qc from Equation (9) and the enrichment. The enriched

interpolation is calculated by the shape function j(r

0)with jbeing one of the enriched repatoms nin the index set N.

6of26 WERNER  .

-15 -10 -5 0 5 10 15

-15

-10

-5

(A)

-10

-5

10 20

-10 -10

-20 -20

(B)

FIGURE 3 (A) Example with a square inclusion and repatoms with the discontinuity Γdisc separating the two domains Ω1(compliant

matrix) and Ω2(stiff inclusion) together with the enriched elements in dark grey. (B) The signed distance function of the discontinuity.

The ramp function R(r

0)is only relevant for the corrected XFEM (explained later) and is shown in Figure 4D; for the other

three enrichment types it can be assumed equal to one for all atoms. The enriched degrees of freedom 

g

jadditionally

account for the interfaces inside elements. The enrichment function •(r

0)is one of the four options, explained in the

following, and applied in its shifted form to ensure the Kronecker-delta property (the shift is guaranteed by the second

term of the square bracket in Equation 10). That leads to a vanishing of the enrichment functions in all repatoms except

the ones belonging to an element with the interface and limits the additional DOFs to the minimum. In addition, the

enrichment functions are zero along the boundaries, which greatly simplifies the application of boundary conditions.

The enrichment functions are based on the signed distance function

(r

0)=r

0−c⋅sign(nT

c(r

0−c)) (11)

with the Euclidean norm of the difference between the location r

0of the atom in the initial configuration and the closest

location cto on the discontinuity Γdisc (red line in Figure 3A).32 The second term in Equation (11) determines the sign

for the current atom depending on the normal vector ncin c. This ensures different signs on both sides of Γdisc and is

shown in Figure 3B for a square inclusion as a cone like surface.

The Heaviside sign function

sign(r0)=









−0.5,(r0)<0,

0,(r0)=0,

+0.5,(r0)>0,

(12)

switches from +0.5to−0.5 through the enriched elements with zero at the interface and is the first enrichment type. It

is commonly used for strong discontinuities such as cracks but can be applied to weak discontinuities as well. In that

case the Heaviside sign function has to be zero along the interface.32 In analyses of continua using XFEM this condition

is enforced by using the Lagrange multipliers or Nitsche’s method.32,48,49 Due to the discrete character of the underlying

lattice network, this requirement can be easily applied by setting sign zero along the interface in Equation (12). Due to its

constant character, the Heaviside enrichment in its shifted form has the advantage to vanish outside elements crossed by

an interface. This avoids so called blending elements, characterized by only a subset of the elements nodes or repatoms

are enriched, and can lead to suboptimal convergence behavior.32 In Figure 4A, the enrichment function is shown in its

unshifted form as a surface along the discontinuity together with the triangulation of the square inclusion example from

WERNER  . 7of26

-1

10 20

-10 -10

-20 -20

-20 -15 -10 -5 0 5 10 15 20

-0.5

0.5

(A)

-1

10 20

-10 -10

-20 -20

-20 -15 -10 -5 0 5 10 15 20

-0.5

0.5

(B)

10 20

-10 -10

-20 -20

-20 -15 -10 -5 0 5 10 15 20

-0.5

0.5

(C)

10 20

-10 -10

-20 -20

-20 -15 -10 -5 0 5 10 15 20

-0.5

0.5

(D)

FIGURE 4 The four enrichment functions (A) Heaviside, (B) absolute value, (C) modified absolute value, and (D) corrected XFEM,

corresponding to a square inclusion and highlighted in red. In addition, cross-sections of individual enrichments are visualized along the

X1-axis for X2=0 in the insets. The insets include the repatoms as open circles, the atoms as grey dots as well as the atom on the interface

indicated as red dots.

Figure 3. Moreover, a cross section of the Heaviside function along the X1-axis for X2=0 is shown as well as the enriched

repatoms (indicated as orange circles).

The absolute value enrichment

abs(r0)=abs((r0)) (13)

describes the location of the discontinuity by using the absolute value of the signed distance function and is zero along

the discontinuity (Figure 4B). It is a common enrichment function in the context of XFEM to investigate continua to

account for weak discontinuities. A disadvantage of the absolute value enrichment are blending elements, which can lead

to suboptimal results.

The third type of enrichment is the modified absolute value function. The interface inside the enriched elements is

described by a ridge of the interpolation function (Figure 4C). The modified absolute value enrichment

mabs(r0)= n



j=1j(r0)⋅(rj

0)−

n



j=1j(r0)⋅(rj

0)

(14)

8of26 WERNER  .

-15 -10 -5 0 5 10 15

-15

-10

-5

(A)

-1

10 20

-10 -10

-20 -20

-20 -15 -10 -5 0 5 10 15 20

-0.5

0.5

(B)

FIGURE 5 (A) Example with a fiber as a discontinuity Γdisc embedded in a domain Ω, together with the enriched elements (dark grey)

and repatoms (orange circles). (B) Step enrichment function for a fiber with a cross-section of step visualized along the X1-axis for X2=0in

the inset. The inset includes the repatoms as open circles, the atoms as grey dots, as well as the atoms on the interface indicated as red dots.

is calculated by the standard FE shape function j(r0)at atom and the signed distance function at one of the enriched

repatoms n. It was first introduced by Moës et al.31 and addresses the suboptimal results of the absolute value function.

All repatoms outside the enriched elements are zero and therefore blending elements are avoided.

Finally, the fourth enrichment function is the corrected XFEM

cXFEM(r0)=abs((r0)) ⋅R(r0),(15)

which is calculated by the absolute value function multiplied with a ramp function R(r0).The product of the two functions

leads to the enrichment function visualized as a surface plot and its cross section in 2D along the X1-axis and for X2=0

in Figure 4D. The ramp function takes the value of one for all enriched repatoms and is zero for the remaining ones.

Identical to the Heaviside and absolute value enrichment the corrected XFEM is characterized by zero values along the

interface. Furthermore, equally to the modified absolute value enrichment the corrected XFEM function is zero for the

repatoms of all non-enriched elements and blending elements do not appear.

Besides weak discontinuities with an area domain such as the square inclusion with Ω2in Figure 3A, fibers are con-

sidered as weak discontinuities as well, and included by an enrichment function into the interpolation. The domain of

the weak discontinuity collapses to the line Γdisc (Figure 5A). Radtke et al.50–52 as well as Pike and Oskay53 have applied

the Heaviside step function

step(r0)=+0.5,(r0)=0,

0,(r0)>0,(16)

for such weak discontinuities, which is derived from the signed distance function and characterized by having the value

of +0.5alongΓdisc and zero otherwise (Figure 5B).

Independently of the enrichment strategy, the position vector of the nato atoms



r=𝚽

g(17)

can be expressed by the interpolation matrix 𝚽as well as the single column matrix of the generalized degrees of freedom



g=[



rrep,

g

1,…,

g

n]T(18)

WERNER  . 9of26

containing the standard DOFs at the repatoms 

rrep as well as the enriched ones 

gof the index set N.The

interpolation matrix 𝚽in Equation (17) is formed in two steps where the first part consists of the standard

FE interpolation

(ΦFE)(2−1)(2j−1)=j(r

0),for ∈Nato,

j∈Nrep,j=1,…,nrep

0,otherwise (19)

and the second part

(Φ)(2−1)(2j−1)=

j(r

0),for ∈Nato,j=1,…,n

0,otherwise (20)

results from the enrichment in Equation (10). Both are combined into the resulting interpolation matrix

𝚽=[𝚽FE,𝚽](21)

and applied in Equation (17).

2.3 QC reduction by summation rule

The second step to reduce the computational effort in the QC method is related to summation rules. Hereafter, the

potential energy is approximated by using a first-order interaction based summation rule, that is,

Πk(

r)≈

Πk(

r)= 

∈Ssam

int

k

(

r).(22)

Four interactions, one for each orientation (0◦,90

◦,±45◦), are selected inside an element. The potential energy

of the selected interaction k

, connecting atoms and , are multiplied with a weight factor  to represent

the potential energy of all interactions in that element. These interactions and their weight factors are stored

in an index set Ssam

int . By summing up the products of the energy k

 with the weight factors  in the set

Ssam

int , the potential energy 

Πk(

r)of the whole system is approximated. This summation rule was first suggested

by Beex et al.20 Here, the summation rule will be expanded for the extended QC with non-conforming meshes

with Heaviside enrichment and therefore to elements cut by a material interface. In the following the sum-

mation rule for the standard QC and the expansion for the extended QC is described in detail. It is referred

to as first-order summation rule, which is appropriate for a first-order interpolation but not suitable for higher

order polynomials.

For the standard QC the selection procedure of the interactions as well as the calculation of the weight factors is

outlined in Algorithm 1. For meshes with right-angled triangles‡and an X-braced lattice used herein, the number of

interactions and the weight factor scales with the area Aiof the i-th element. Therefore, the weight factors for elements

with edge size hlonger than the lattice spacing dand Ai>1

2d2are assigned with the element area (Algorithm 1,lines

3–6). For the the fully resolved elements the weight factors are set to 1 or 0.5 to represent themselves, depending if they

are located on the element edge or inside the element (Algorithm 1, lines 7–14). In Figure 6the selection procedure for the

sampling interactions of Algorithm 1is illustrated. The model has a stiff inclusion embedded in a soft matrix, indicated by

the red interface, and is fully resolved with triangles having the same size as the lattice spacing. With increasing distance

from the discontinuity the element size grows towards the predefined edge length. In Figure 6B, the sampling interactions

are shown enlarged. Close to the discontinuity all interactions are selected and added to the set Ssam

int with a weight factor

of  =1. The elements with bright blue sampling interactions in Figure 6B have an area of Ai=8 and each selected

sampling interaction is assigned a weight factor of  =Ai. The same is done for the elements with purple sampling

interactions between the fully resolved triangulation at the discontinuity with corresponding weight factors.

10 of 26 WERNER  .

Algorithm 1. First-order summation rule for standard QC

1: for i=1,…,nTdo Loop over all triangles

2: Calculate area Aiof current triangle based on lattice spacing d

3: if Ai>1

2d2then

4: Find center point of the longest edge of the triangle Pi

edge

5: Find four interactions (one for each orientation) inside triangle at Pi

edge

6: Add four interactions to Ssam

int set each with  =Ai

7: else

8: for j=1,…,nint do Loop over all interactions

9: Compute center point Pj

int of j-th interaction

10: Barycentric coordinates of center point Pj

int

11: if Pj

int inside the triangle then

12: Link is added to Ssam

int set with  =1

13: else if Pint is on triangle edge then

14: Link is added to Ssam

int set with  =0.5

15: end if

16: end for

17: end if

18: end for

19: for i=2,…,nsam do

20: Sort sampling links in Ssam

int

21: if Ssam

int (i−1)=Ssam

int (i)then Check and remove for double occurrence of

22: sampling links in ordered set Ssam

int

23: i−1

 =i−1

 +i

 Sum the weight factors

24: end if

25: end for

-15 -10 -5 0 5 10 15

X1 [mm]

-15

-10

-5

X2 [mm]

(A)

-6 -4 -2 0 2 4 6

X1 [mm]

-14

-12

-10

-8

-6

-4

X2 [mm]

(B)

FIGURE 6 (A) Triangulation of the standard QC along with the discontinuity (red) and the region of interest in a black frame, together

with (B) the magnification of the region of interest visualizing the sampling interactions for the triangles (bright blue:  =8, purple:

 =2or =4, green:  =1 and yellow:  =0.5).

WERNER  . 11 of 26

-15 -10 -5 0 5 10 15

X1 [mm]

-15

-10

-5

X2 [mm]

(A)

-6 -4 -2 0 2 4 6

X1 [mm]

-14

-12

-10

-8

-6

-4

X2 [mm]

(B)

FIGURE 7 (A) Triangulation of the extended QC along with the discontinuity (red) and the region of interest (highlighted as a black

frame) together with (B) the magnification of the region of interest showing the sampling interactions for the triangles (bright blue:  =8,

purple: 1 <

 <8, green:  =1 and yellow:  =0.5).

For the extended QC the selection procedure of the sampling interactions of triangles which are not cut by the dis-

continuity is identical to the standard QC algorithm for the coarse mesh. The four sampling interactions are located at

the center point of the longest edge of the element (Figure 7) and the weight factors are determined through the area of

the triangle (Algorithm 2, lines 26–30). For the triangles cut by the interface all interactions sharing one or two atoms

with the Γdisc are selected discretely. They are stored in the index set Sdisc

in which contains the interactions along the

interface. The weight factor is thereby  =1 if the link is inside the triangle or  =0.5ifitisonatriangleedge.

Furthermore, for each of the two domains of the cut triangle one sampling interaction in all four orientations (0◦,90

◦

and ±45◦) is selected. The links inside the inclusion are stored in the index set Sint

in and the ones in the domain out-

side the inclusion are stored Sint

out (Algorithm 2, lines 16–23). The three sets Sint

disc,Sint

in and Sint

out are all part of Sint

sam.After

leaving the for-loop the last four links in that domain are selected as sampling bonds (Algorithm 2, lines 24–25) and

appear therefore at corresponding locations in the triangle (purple in Figure 7B). The second for–loop (lines 31–34 in

Algorithm 2) checks for double occurrences of sampling links and is combining the weight factors. This ensures that

a sampling link sharing at least one atom with the interface and which is in addition shared between two triangles

(being on an edge) will be present only once in the set Sint

in with a weight factor of  =1 at the end of the procedure.

It was observed that diagonal sampling interactions with weight factors  =0.5 and connecting atoms in two differ-

ent triangles lead to errors in the energy approximation. Therefore, the last step in Algorithm 2is to increase the weight

factor to  =1 and compensate for this error. At the same time the weight factor of the sampling interaction in the

same direction in the neighbouring triangle (purple in Figure 7B) is reduced by 0.5. These weight factors account for

the exact number of interactions in every triangle as well as for the exact number of interactions in every domain of the

cut triangles.

3NUMERICAL EXAMPLES

The QC method in its standard and extended version is applied to investigate an example with a single stiff circular inclu-

sion (Section 3.1), a single fiber embedded in a soft matrix (Section 3.2) and a polished cross-section of a concrete sample

with multiple aggregates and fibers. The focus is thereby on the convergence behavior of the displacement, elastic energy,

homogenizedstressand stiffness of the standard QC and its extendedversion.The full and the first-order interaction based

summation rule are compared to each other for the three numerical examples. For all evaluations of the QC outcome the

full solution of the lattice network is used as a reference.

12 of 26 WERNER  .

Algorithm 2. First-order summation rule for extended QC

1: for i=1,…,nTdo Loop over all triangles

2: if distance function at least at one of the triangle atoms is 0 then

3: for j=1,…,nint do Loop over all interactions

4: Compute center point of interaction Pj

int

5: Obtain Barycentric coordinates of center point Pj

int

6: if Pj

int inside the triangle then

7: Interaction is added to a set of links of i-th triangle Si

int with  =1

8: else if Pj

int is on triangle edge then

9: Interaction is added to a set of links of i-th triangle Si

int with  =0.5

10: end if

11: if j-th interaction shares at least one atom with the interface then

12: Set flag = 0

13: else if j-th interaction is inside inclusion then

14: Set flag = −1

15: else if j-th interaction is outside inclusion then

16: Set flag = +1

17: end if

18: end for

19: for j=1,…,nintTri do Loop over all interactions of current triangle

20: if flag = 0 then

21: Add to Sdisc

int discretely with  =1 if interaction is inside the triangle

22: and  =0.5 if interaction is on triangle edge

23: else if flag = −1then

24: in

 =in

 +j



25: else if flag = +1then

26: out

 =out

 +j



27: end if

28: Add last interaction with flag = −1toSin

int and flag = +1toSout

int

29: with {Sdisc

int ,Sin

int,Sout

int }Ssam

int

30: end for

31: else Triangle is not cut by the interface

32: Find center point of the longest edge of the triangle Pi

edge

33: Find four interactions (one in each orientation) inside triangle

34: and connected to Pi

edge

35: Add four interactions to set Ssam

int each with  =Ai

36: end if

37: end for

38: for i=2,…,nsam do

39: Sort sampling links in Ssam

int

40: if Ssam

int (i−1)=Ssam

int (i)then Check and remove for double occurrence of

41: sampling links in ordered set Ssam

int

42: i−1

 =i−1

 +i

 Sum the weight factors

43: end if

44: end for

45: for i=1,…,nsam do

46: if i

 =0.5andi

int =±45◦then

47: Find sampling interactions of neighbouring triangle the i-th link is shared with

48: Decrease the weight factor  in that neighbouring triangle and direction by 0.5

49: Set i

 =1 for i-th interaction

50: end if

51: end for

WERNER  . 13 of 26

FIGURE 8 Numerical example with a single stiff circular inclusion with its center at Pcent =(−17,0)and a radius of R=40 mm (red)

and the applied boundary conditions including the prescribed displacement uDon Γ1and Γ3.

TABLE 1 Element size of the triangulation together with the number of DOFs for the standard and extended QC using heaviside

enrichment.

Element size [mm] 32 16 8 4 2

DOF standard QC 3150 3442 4818 10,658 34,674

DOF extended QC 194 634 2286 8662 33,678

3.1 Single stiff inclusion

The first numerical example is characterized by an inclusion with a radius of R=40 mm and its center point at Pcent =

(−17,0), embedded in a square domain with a edge length of 256 mm. The Young’s modulus of the inclusions is ten times

higher compared to the surrounding matrix. The underlying lattice has a spacing of d=1 mm and is subject to tensile

loading (Figure 8). The loading is applied with the displacement uDon the boundaries Γ1and Γ3. All four boundaries are

restricted in the X1direction while the nodes on Γ2and Γ4arefreetomoveintheX2direction.

3.1.1 Convergence behavior using the full summation rule

The example is simulated by using the full solution, the standard and extended QC Method. The extended QC includes

four different enrichment strategies introduced in Section 2.2. For both types of QC five different triangulations are used

and the coarsest one is shown in Figure 9. The standard QC discretization is conforming and fully resolved along the

discontinuity Γdisc (red in Figure 9) while the triangulation of the extended QC is characterized by a regular mesh. The

discretization in Figure 9is centrosymmetric and the uniform part of it has an edge length of 32 mm which leads to

approximately 3200 DOFs for the triangulation of the standard and 200 for the extended QC, while the full solution has

around 133,000 unknowns. Table 1gives an overview of all numbers of DOFs with the corresponding triangle size. The

full summation rule is applied for the standard and extended QC analyses and therefore the potential energy of the system

is determined by visiting every interaction of the lattice.

The convergence behavior of the extended and standard QC is evaluated by the relative error in elastic energy and

displacement magnitude. The relative error of the elastic energy

Π=ΠQC −Π

FS

ΠFS(23)

14 of 26 WERNER  .

-100 -50 0 50 100

X1 [mm]

-100

-50

100

X2 [mm]

(A)

-100 -50 0 50 100

X1 [mm]

-100

-50

100

X2 [mm]

(B)

FIGURE 9 Employed triangulation with an element size of 32 mm for the (A) standard and (B) extended QC along with the circular

inclusion in red.

(A) (B)

FIGURE 10 (A) The relative error in elastic energy, (B) the relative error in displacement of the extended QC with the Heaviside

(Heav), the absolute value (abs val), the modified absolute value (mod abs val) and the corrected XFEM (cXFEM) enrichment as well as the

standard QC of the numerical example with a single stiff inclusion.

is determined from the difference between the QC analyses and the full solution, with regard to the absolute value of the

elastic energy of the full solution ΠFS. Moreover, the relative error of all displacements

u=uQC −uFSL2

uFSL2(24)

is calculated, where u=r−r0. This leads to four curves for the extended QC with the Heaviside, absolute value, modified

absolute value and corrected XFEM enrichment and one curve from the standard QC (Figure 10).

The error of the extended QC decreases exponentially with an increasing number of DOFs and all four curves appear

as nearly straight lines in the double logarithmic diagrams. In addition, all four enrichment types result in a very similar

outcome, whereby the Heaviside enrichment leads to the smallest and the absolute value enrichment to the largest error

in elastic energy and displacement magnitude. The coarsest triangulation of the extended QC with 194 DOFs using the

WERNER  . 15 of 26

FIGURE 11 Field plots of the local error in displacement magnitude of the standard QC for the numerical example with a single stiff

inclusion for different sizes of the initial mesh. (A) Element size 32 mm. (B) Element size 8 mm. (C) Element size 2 mm.

FIGURE 12 Field plots of the local error in displacement magnitude of the extended QC with Heaviside enrichment for the numerical

example with a single stiff inclusion. (A) Element size 32 mm. (B) Element size 8 mm. (C) Element size 2 mm.

Heaviside enrichment has an error in elastic energy and displacement magnitude of approximately 2% for both quan-

tities. The same enrichment leads to the smallest error between 0.01% and 0.02% for both quantities using the finest

discretization with 33678 DOFs. The error for the standard QC ranges between 0.2% and 4 ×10−3% for the elastic energy

and between 0.65% and 8 ×10−3% for the displacement magnitude (Figure 10A,B).

In Figures 11,12 and 13 field plots of the absolute error of the displacement magnitude are visualising the distribution

of difference of both types compared to the full solution. Three out of the five discretizations are displayed, along with

the extended QC incorporating Heaviside (Figure 12) and absolute value enrichment (Figure 13). The absolute error of

the displacement magnitude

𝜺u

=uQC

−uFS

,=1,…,nato,(25)

is calculated for every atom as the difference between the QC prediction and the reference result of the full solution. For

the standard QC the error in displacement occurs throughout the whole domain of the compliant matrix, but is visibly

smaller inside the stiff inclusion (Figure 11). In the case of the extended QC the inclusion domain protrudes in the error

surface and especially the interface shows local maxima. The surrounding matrix domain is characterized by mostly

negative error values and indicates a too stiff prediction of the matrix behavior caused by coarse interpolation (Figures 12

and 13).

3.1.2 Convergence of the first-order summation rule

The first-order summation rule is characterized by selecting only one sampling interaction in all four orientations inside

a triangle or domain on both sides of the discontinuity. The selection procedure for the standard and extended QC is

explained in Section 2and visualized in Figures 6and 7. To quantify the influence of the first-order summation rule

relative to the full summation rule on the accuracy of the analyses results both approaches are compared. Since the

16 of 26 WERNER  .

FIGURE 13 Field plots of the local error in displacement magnitude of the extended QC with absolute value enrichment for the

numerical example with a single stiff inclusion. (A) Element size 32 mm. (B) Element size 8 mm. (C) Element size 2 mm.

(A) (B)

FIGURE 14 (A) The relative error in elastic energy, (B) the relative error in displacement of the extended QC using the Heaviside

enrichment (Heav) and the standard QC with full and first-order summation rule of the numerical example with a single stiff circular

inclusion.

absolute value, the modified absolute value as well as the corrected XFEM enrichment types need higher order sum-

mation rules the comparison is limited to the enriched interpolation with the Heaviside function. For all studied mesh

sizes the Heaviside enrichment led to the smallest error in elastic energy and displacement using the full summation rule

(Figure 10) and justifies this limitation. The full and first-order summation rule leads to very similar results of the relative

error of the elastic energy and displacement magnitude for the standard and extended QC (Figure 14). The difference of

relative error for the elastic energy and displacement magnitude between the summation rules is for all triangulations

smaller than 0.015% and decreases with element size. On the other hand, the number of sampling interactions and there-

fore the computational effort is substantially smaller. The full summation rule uses all 262,656 interactions while the

first-order summation rule reduces that number to 3.7% to 50.9% for the standard QC and 1% to 50.5% for the extended

QC, depending on the element size (Table 2).

3.2 Single fiber

The standard and extended QC is next applied to an example of a single fiber as weak discontinuity embedded in a

compliant matrix. The fibers Young’s modulus is 100 times higher than that of the matrix, and the fiber has its center at

Pcent =(−17,0)oriented under 45◦(Figure 15). It has a length of 80 mm and the entire model is 256 mm in width and

WERNER  . 17 of 26

TABLE 2 Number of sampling interactions for each element size of the triangulation for the standard and extended QC using the

heaviside enrichment.

Element size [mm] 32 16 8 4 2

Sampling interactions standard QC 9680 10,720 15,968 38,816 133,856

Sampling interactions extended QC 2628 4212 10,362 34,944 132,690

FIGURE 15 The numerical example with a single fiber with its center at Pcent =(−17,0)and a length of 80 mm (red) and the applied

boundary conditions including the prescribed displacement uDon Γ1and Γ3.

height. The boundary conditions are the same as in the previous numerical example with a prescribed displacement uD

on Γ1and Γ3in the X2direction and a restricted displacement in the X1direction on all four boundaries. The enrichment

for the extended QC is based on the step function in Equation (16).

The convergence behavior of the standard and extended QC for the relative error of the elastic energy and the dis-

placement magnitude according to Equations (23) and (24) is visualised in Figure 16 using the full solution as a reference.

The standard QC leads, for all triangulations and for both quantities, to a higher accuracy compared to the extended QC.

Nevertheless, the extended QC is predicting the elastic energy with a relative error smaller than 4.3×10−1% while the rel-

ative error for the displacement magnitude is below 1% (Figure 16), which is well acceptable. The first-order summation

rule has a marginal influence on the outcome for the two QC approaches and the largest difference is the relative error

of displacement magnitude 7 ×10−3% of the standard QC.

3.3 Multiple inclusions and fibers

The third example is representing a realistic and complex microstructure, where the inclusion geometry is taken from

a segmented concrete cross-section representing aggregates embedded in a cement matrix (Figure 17A). From the seg-

mented cross-section 31 inclusions are selected for the QC model and 42 fibers are added (Figure 17B). The standard

and extended QC is used to study the convergence behavior of the elastic energy, the displacement magnitude as well as

homogenized material parameters. The Young’s modulus of the matrix is 1 MPa, inclusions have 10 MPa and the fibers

have 100 MPa. The square domain size is 384 mm and the lattice spacing is d=1 mm.

For the standard QC the automatic refinement algorithm starts out with the same element size as in the previous two

examples, but due to the high number of interfaces and the requirement to resolve them fully leads to relatively fine mesh

throughout the whole domain (Figure 18). The four coarsest triangulation have therefore a similar number of DOFs and

only the finest triangulation with an element size of 2 mm has a substantially different amount of unknowns (Table 3).

18 of 26 WERNER  .

(A) (B)

FIGURE 16 (A) The relative error in elastic energy and (B) the relative error in displacement using the extended QC with the

Heaviside enrichment (Heav) and the standard QC with full and first-order summation rule of the fiber example.

(A) (B)

FIGURE 17 (A) The concrete cross-section together with the segmentation of the region of interest indicated by the frame. (B) The

schematic of the example with 31 inclusions and 42 fibers with periodic boundary conditions.

For the extended QC the inclusions and fibers have two separate level set functions and enrichments as described in

Section 2.2. For the inclusions the Heaviside sign function according to Equation (12) and for the fibers the Heaviside

step function in Equation (16) is used. The triangulations for the extended QC range between 800 and 82,000 DOFs,

the standard QC has between 50,000 and 99,000 DOFs (Table 3), while the full solution leads to approximately 296,000

unknowns.

Previous studies54,55 have shown that in densely packed inclusions the XFEM with one level set function representing

all inclusions can lead to artefacts. The enriched DOF is thereby unable to account for two different interfaces in neigh-

bouring elements. A multi level set approach, which lead to enriched DOF to account for only one interface, can avoid

this artefacts and further improve the convergence of effective elastic properties. In this study we limit ourselves to two

level set functions, one representing inclusions and one for fibers, although such extensions would be straightforward.

WERNER  . 19 of 26

-150 -100 -50 0 50 100 150

X1 [mm]

-150

-100

-50

100

150

X2 [mm]

(A)

-150 -100 -50 0 50 100 150

X1 [mm]

-150

-100

-50

100

150

X2 [mm]

(B)

FIGURE 18 Triangulation of (A) the standard and (B) extended QC with multiple inclusions and fibers.

TABLE 3 Element size of the triangulation together with the number of DOFs for the standard and extended QC.

Element size in mm 32 16 8 4 2

DOF standard QC 50,962 50,964 51,470 57,920 98,868

DOF extended QC 842 2398 7120 23,134 81,818

3.3.1 Periodic boundary conditions

For such complex microstructures homogenized material parameters are of interest and therefore the homogenized first

Piola-Kirchhoff stress tensor Pas well as the homogenized stiffness tensor Dare determined, and periodic boundary

conditions are applied to the Representative Volume Element (RVE) in Figure 17B. The periodicity conditions

rep −r−

rep =F(r+

0,rep −r−

0,rep)(26)

are used with r+

rep being the repatom locations on the boundaries ΓRand ΓTwith normal vectors N+and r−

rep the repatom

locations on ΓBand ΓLwith N−(Figure 17B). The indices indicate bottom, top, left and right part of the boundary. The

current location of the repatoms r+∕−

rep along the boundaries is calculated by the macroscopic deformation gradient Fand

the location of the repatoms in the initial configuration r+∕−

0,rep. Here, a deformation gradient

F=1.06 0

(27)

leading to a tensile loading in the X1direction is applied. For completeness, the details on implementing the boundary

conditions and extracting homogenized quantities are presented in Appendix A.

3.3.2 Convergence behavior

For the standard and extended QC, using the full and first-order summation rule, the convergence behavior of the elastic

energy and the displacement magnitude are determined according to Equations (23) and (24)(Figure19). The relative

20 of 26 WERNER  .

5 % relative error

(A)

5 % relative error

(B)

FIGURE 19 The relative error of (A) the elastic energy and (B) the displacement magnitude of the multi-inclusion and fiber example

using the extended QC with the Heaviside (Heav) and the absolute value (abs val) enrichment and the standard QC with full and first-order

summation rule.

TABLE 4 Number of sampling interactions for each element size of the triangulation for the standard and extended QC using heaviside

enrichment.

Element size [mm] 32 16 8 4 2

Sampling interactions standard QC 155,234 155,242 157,074 182,202 344,490

Sampling interactions extended QC 35,901 39,967 54,585 110,239 324,929

error of the elastic energy using the extended QC with Heaviside enrichment is between 36% and 0.3% and is there-

fore more than one order of magnitude higher compared to the single inclusion example (Figure 14B). The prediction

of the standard QC leads to relative errors between 0.4% and 0.1% of the elastic energy (Figure 19A). The relative error

of the displacement magnitude ranges between 5% and 0.1%, while the standard QC leads to errors smaller than 0.35%.

In addition, analyses using the absolute value enrichment with full summation rule are carried out. They are slightly

more accurate for the two coarser triangulations of the elastic energy, while the Heaviside enrichment is more accu-

rate for finer discretizations for the elastic energy and the displacement (Figure 19). The differences between the two

summation rules are similar to the two previous examples and are negligible. The largest deviation between the summa-

tion rules for both quantities is 0.08% and occurs in the elastic energy for the coarsest triangulation of the extended QC

(Figure 19A). The number of sampling interactions is reduced by the first-order summation rule from 590,592 to a range

between 26.3% and 58.3% for the standard QC and to 6.1% and 55% for the extended QC depending on the element size of

the mesh (Table 4).

In addition, the homogenized stress Pand stiffness tensor Dare determined. The relative error of both quantities

P=PQC −PFSF

PFSFand D=DQC −DFSF

DFSF(28)

is determined by the Frobenius norm of the tensor of the QC analyses and the full solution. The standard QC is predicting

both tensors with an error smaller than 0.4% for all triangulations (Figure 20). On the other hand, the extended QC leads

to a range of relative error between 37% and 0.3%, decreasing with a smaller element size. Nevertheless, the discretization

with an element size of 8 mm is predicting the elastic energy, the homogenized stress and homogenized stiffness below

the 5% threshold, which is acceptable for engineering practice. Furthermore, the error of the displacement magnitude

for the same numerical model is around 1% by using only 14% of the number of DOFs and 35% of sampling interactions

WERNER  . 21 of 26

5 % relative error

(A)

5 % relative error

(B)

FIGURE 20 The relative error of (A) the first Piola-Kirchhoff stress tensor Pand (B) the homogenized stiffness tensor Dof the

multiple inclusion and fiber example using the extended QC with the Heaviside (Heav) and the absolute value (abs val) enrichment and the

standard QC with full and first-order summation rule.

FIGURE 21 The elastic strain of (A) the full solution, together with (B) the standard QC and (C) extended QC for an element size of

8 mm.

compared to the coarsest standard QC model. Moreover, the numerical model of the extended QC with an element size

of 8 mm uses only 2.4% of DOFs and 9.2% of sample interactions compared to the full model.

To illustrate the results further, Figure 21 shows the distribution of local interaction strains



e =(

r −r

r

(29)

for the full solution as the reference along with the standard and extended QC for an element size of 8 mm. The results

show that the extended QC reproduces the overall strain distribution accurately, particularly in the low-strain regions of

stiff aggregates and fibers. Main discrepancies occur near the fiber ends characterized by strong strain concentrations. In

these regions, the standard QC method provides more detail, benefiting from the fully resolved interphases and fibers. In

contrast, the extended QC method provides more localized fields based on the adopted enrichment functions.

22 of 26 WERNER  .

4SUMMARY AND CONCLUSION

In the present study, the Quasicontinuum Method was extended by enrichment strategies to account for weak discon-

tinuities with nonconforming triangulations. The methodology was illustrated on the simplest case of two-dimensional

lattices with X-braced square unit cells and nonlinear elastic truss interactions. This still allows investigating complex

microstructures with densely packed inclusions and randomly shaped interfaces, that can be represented by the underly-

ing regular lattice, to represent them accurately while using a regular coarser mesh and reduce the computational burden.

The enrichment strategies include the Heaviside, absolute value, modified absolute value and corrected XFEM. Further-

more, the first-order summation rule has been applied for the standard QC and was expanded to account for different

domains inside one element cut by a material interface for the extended QC. Based on the comparative studies performed

for single and multiple fiber-inclusion composites we conclude that:

1. The Heaviside function has a slight advantage in accuracy of the example with a single stiff inclusion compared to

the other three enrichment strategies. In the example with multiple inclusions and fibers the Heaviside enrichment

leads to similar outcome as the absolute value. At the same time, the Heaviside function is the most straightforward

to implement and is able to account for randomly shaped inclusions in the form of the sign function and for fibers as

astepfunction.

2. The first-order interaction based summation rule has been developed for the extended QC and reduces the number of

sampling interactions substantially for both types of QC with less than 0.1% difference compared to the full summation

rule. The first-order summation rule complements the Heaviside enrichment of the extended QC, since the Heaviside

function requires only the selection of one set of four interactions at one location inside the element.

3. The extended QC with an element size of 8 mm is predicting the elastic energy as well as the homogenized stress and

stiffness tensor below the 5% threshold of relative error, acceptable in many engineering applications. The extended

QC model uses almost one order of magnitude fewer DOFs and one third of sampling interactions compared to the

standard QC. The extended QC uses only 2.4% DOFs and 9.2% of sampling interactions compared to the full model.

4. The extended QC with 8 mm element size predicts displacements with 1% relative error by using 14% of DOFs and one

third of sampling interactions in contrast to the standard QC. Compared to the full model only 2.4% DOFs and 9.2%

of sampling interactions are needed.

The extended QC developed in this manuscript can account for heterogeneous lattice networks of nonlinear elastic

truss interactions with geometrically complex microstructures. The methodology can potentially increase the computa-

tional efficiency of diverse applications of heterogeneous lattices. These include the analysis of concrete microstructures,

asshowninSection3.3,atomisticlattice simulations incorporating grain boundaries,16,29 or woven fabrics with embedded

deformable electronic components.20

The method can be easily combined with enrichments for strong discontinuities to predict fracture under mode I fail-

ure. Other extensions include modeling heterogeneous beam lattices, building on the works of Phillot and Kochmann 23

andChenetal.,

25 and incorporating mode II failure typical of cohesive-frictional materials, or utilizing embedded dis-

continuity techniques 56–58 to capture fracture processes and the effect of fiber reinforcements. Finally, higher order

interpolation or meshless approaches in combination with enrichment might improve the computational efficiency of

the extended QC for heterogeneous lattices even further. All topics lie within our interest and will be reported separately.

AUTHOR CONTRIBUTIONS

Benjamin Werner: Methodology, Software, Validation, Formal analysis, Investigation, Writing–Original Draft, Visu-

alization; Jan Zeman: Conceptualization, Methodology, Formal analysis, Writing–Review & Editing, Project adminis-

tration, Funding acquisition; Ond

rej Rokoš: Conceptualization, Methodology, Software, Formal analysis, Resources,

Writing–Review & Editing, Supervision, Funding acquisition.

ACKNOWLEDGMENTS

The work of Benjamin Werner received funding from projects No. CZ.02.2.69/0.0/0.0/ 18_053/0016980 awarded by the

Ministry of Education, Youth and Sports of the Czech Republic (from 02/2021 to 04/2022), No. 22-35755K awarded the

Czech Science Foundation (from 01/2023) and by the CTU Global PostFellowship Program (from 05/2022). The work of

Jan Zeman and Ond

rej Rokoš was supported by project No. 19-26143X awarded by the Czech Science Foundation.

WERNER  . 23 of 26

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

ENDNOTES

∗An alternative approach involves assuming a general coarse-scale mesh and enforcing the scale transition by the Largange multipliers; see,

for example, Reference 15 for a detailed study in this direction.

†Note that we have chosen the X-braced lattice only as an example. The same procedure can be applied to different unit cells by a suitable

redefinition of the interaction list.

‡Note that right-angle triangulation is used for convenience. Other triangulations or discretizations are possible, as usually applied in QC

formulations.

ORCID

Benjamin Werner https://orcid.org/0000-0003-2633-9064

Jan Zeman https://orcid.org/0000-0003-2503-8120

Ond

rej Rokoš https://orcid.org/0000-0003-2589-5333

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How to cite this article: Werner B, Zeman J, Rokoš O. Extended quasicontinuum methodology for highly

heterogeneous discrete systems. Int J Numer Methods Eng. 2023;e7415. doi: 10.1002/nme.7415

APPENDIX A. PERIODIC BOUNDARY CONDITIONS

For the implementation the periodicity conditions in Equation (26) are rewritten as

rep =rL

rep +r2

rep −r1

rep (A1)

rep =rB

rep +r4

rep −r1

rep (A2)

with r1

rep,r2

rep and r4

rep being the location of the corner repatoms and control points of the RVE (Figure 17B) on which the

displacement is prescribed by

rep =F⋅rk

0rep k=1,2,4.(A3)

The first Piola-Kirchhoff stress tensor

P=1

V0

k=1,2,4fk

DBC rk

0rep (A4)

is calculated through the initial volume V0, reaction forces at the three corner nodes fk

DBC and their location in the initial

configuration rk

0repof the RVE.

The fourth order tangent stiffness tensor

D=1

V0

i

jri

0rep

Kij

Mrj

0rep

LC (A5)

is determined through a rearranged stiffness matrix Kij

Mand the location of the corner repatoms ri

0repin the initial config-

uration. In addition, LC indicates the left conjugation and is defined as Tijkl =Tjikl for a fourth-order tensor.59 The stiffness

matrix Kij

Mfollows from the condensation of the system of equations

Kpp Kfp

Kpf Kff up

uf=fp

0(A6)

26 of 26 WERNER  .

withupbeing the prescribed displacement incrementsonthe three corner nodes and fpthecorrespondingreaction force

increments. On the other hand, ufcontains all remaining and therefore free DOFs and the stiffness matrix is partitioned

accordingly. The unknown displacement increments ufcan be solved for and substituted back leading to the expression

KMup=fpwith KM=Kpp −Kpf (Kff )−1Kfp (A7)

of the stiffness matrix. In a last step the reduced stiffness matrix KMis rewritten in a format of three 2 ×2 submatrices

satisfying



jKij

M⋅uj

p=fi

pwith i,j=1,2,4(A8)

and substituted in Equation (A5). The derivation of the first-order Piola-Kirchhoff stress tensor and the homogenized

fourth-order stiffness tensor is described in details by Kouznetsova et al.59

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Nov 2021
J Manuf Process

The development of numerical models applicable to ultra-precision machining presents serious challenges. Molecular dynamics methods can effectively capture nano-cutting operations at the micro-nanoscale but fail to accurately capture the realistically large-scale deformations involved in material removal, while it can be addressed by finite element methods but lacks the precision required for modeling nanoscale behaviors. The present study addresses this issue by developing a multi-scale numerical modeling method, the optimized Quasicontinuum (QC) method. Compared with the original QC method, the optimized QC method adds the material removal function, avoids unreasonable lattice excessive distortion and large-area, deep dislocation slip. The optimized QC method can more truly simulate the large deformation and material removal phenomenon of cutting, and greatly improve the calculation speed. The influence of cutting parameters such as cutting depth, tool rake angle and rounded tool cutting edge radius on cutting force is studied. The change of cutting depth has the most significant influence on cutting force, while the influence of tool rake angle and rounded tool cutting edge radius are relatively small. Therefore, the effect of cutting depth on cutting force is studied through the experiment of single crystal copper nano-cutting to verify the effectiveness of multi-scale method. The experimental results show that it is proportional to the simulation results.

A refinement indicator for adaptive quasicontinuum approaches for structural lattices

Article

Jan 2021

The quasicontinuum method is a concurrent multiscale approach in which lattice models are fully resolved in small regions of interest and coarse‐grained elsewhere. Since the method was originally proposed to accelerate atomistic lattice simulations, its refinement criteria – that drive refining coarse‐grained regions and/or increasing fully‐resolved regions – are generally associated with quantities relevant to the atomistic scale. In this contribution, a new refinement indicator is presented, based on the energies of dedicated cells at coarse‐grained domain surfaces. This indicator is incorporated in an adaptive scheme of a generalization of the quasicontinuum method able to consider periodic representative volume elements, like the ones employed in most computational homogenization approaches. However, this indicator can also be used for conventional quasicontinuum frameworks. Illustrative numerical examples of elastic indentation and scratch of different lattices demonstrate the capabilities of the refinement indicator and its impact on adaptive quasicontinuum simulations.

Generalized quasicontinuum modeling of metallic lattices with geometrical and material nonlinearity and variability

Article

Jul 2020
COMPUT METHOD APPL M

Extended quasicontinuum methodology for highly heterogeneous discrete systems

Abstract