Emmrich E., Wittbold P. (editors) Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series

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Atomistic to continuum coupling methods 2

Since we minimize the energy, this term plays no role, so we omit it. The atomistic

nergy hence becomes

(φ

, . . . , φ

) =

j6=i



− φ



−

f(x

)φ

We make the further approximation of nearest neighbour interaction. Note that this is

a strong assumption (see Section 5.1 for some discussion). On the other hand, it allows

us to pursue the analysis quite far. The atomistic energy hence becomes

(φ

, . . . , φ

) =

N−1



i+1

− φ



−

f(ih) φ

. (4.6)

Remark 4.2. Assume that V

= V

(r) attains its minimum at a unique value, that is

r = 1. When the material is submitted to no body forces (f ≡ 0) and no boundary con-

ditions, e

attains its minimum for conﬁgurations φ such that φ

i+1

− φ

= h. Hence,

the reference conﬁguration corresponds to atoms located on a periodic lattice of pa-

rameter h. This shows, at least in this simple case, that introducing the scaling (3.1) is

consistent with the assumption (H1).

We consider here that the microscopic equilibrium conﬁguration is a solution of the

variational problem

= inf



(φ

, . . . , φ

); φ ∈ X



, (4.7)

where the variational space is



φ ∈ R

N+1

; φ

= 0, φ

= a



. (4.8)

Note that we have imposed ﬁxed displacement boundary conditions. We look for a

global minimizer of the energy (see Section 5.2 for some alternative approaches).

In the continuum mechanics model, the solid deformation is described by the map

φ : Ω = (0, L) → R. Following the same arguments as in the proof of Theorem

3.1, one can show that, for sufﬁciently regular deformations φ and potentials V

, the

atomistic energy e

(φ(0), φ(h), . . . , φ(N h)) converges to the continuum mechanics

expression

(φ) =

(φ

′

(x)

) dx −

f(x) φ(x) d

x. (4.9)

We deﬁne the macroscopic conﬁguration as a solution of the minimization problem

= inf{E

(φ); φ ∈ X

} (4.10)

with



φ ∈ H

(0, L); φ(0) = 0, φ(L) = a



. (4.11)

We assume here that the energy E

(φ) is well-deﬁned as soon as φ ∈ H

(0, L).

206 F

rédéric Legoll

L = N

P h

Atomistic model Continuum model

Figure 2. Partition of Ω = (0, L) according to (4.12).

We expect that solving (4.9)–(4.10) gives a good approximation of the solution of

the atomistic problem when the equilibrium deformation is smooth. When non-regular

deformations are expected to appear, we approximate the solution of the atomistic

problem with the solution of a coupled model. In the spirit of (4.4), we introduce a

partition Ω = (0, L) = Ω

∪ Ω

of the domain, and we consider the coupled energy

(φ) =

Ω

[

(φ

′

(x)

) − f(x)φ(x)

]

i; i

h,ih+h∈Ω



i+1

− φ



−

i;i

h∈Ω

f(ih)φ

To simplify notations, we assume that Ω

and Ω

are connected subdomains of Ω. To

ﬁx ideas, we make the choice

Ω

= (0, P h], Ω

= (P h, L), (4.12)

for some P that depends on h such that P h is constant. Similar results are obtained

without making assumption (4.12) with more technical proofs (see [23, 24]).

The coupled energy hence becomes

(φ) =

[

(φ

′

(x)) − f(x)φ(x)

]

P −1



i+1

− φ



−

f(ih) φ

(4.13)

and is deﬁned on the variational space

(

φ; φ

|Ω

∈ H

(Ω

), φ

|Ω

= {φ

}

0≤i≤P

∈ R

P +1

= 0, φ(L) = a, φ

= φ(P h)

)

. (4.14)

The expression of the boundary conditions φ

= 0 and φ(L) = a reﬂects the fact

that 0 ∈ ∂Ω

and L ∈ ∂Ω

. The interface condition between the macroscopic and the

microscopic domains reads φ

= φ(P h). We hence enforce a continuity-like condition

on the deformation at the boundary.

Atomistic to continuum coupling methods 2

Remark 4.3. I

n a two-dimensional situation, writing down the interface condition is

less straightforward. Actually, there are several possibilities. One can ﬁrst decrease

the mesh size in Ω

down to the atomistic lattice size for mesh elements close to the

interface and enforce equality of displacements of atoms and mesh nodes. Since the

mesh size can only evolve smoothly (to keep good mesh qualities, hence accuracy, on

the domain Ω

), this strategy is expensive. Another possibility is to request that the

positions of the atoms along (or close to) the interface are obtained from interpolation

from the positions of the mesh nodes on the interface. This constraint can also be

implemented in the spirit of mortar ﬁnite element methods [21].

Hence, the coupled problem we consider is

= inf{E

(φ); φ ∈ X

}. (4.15)

The questions we address here are:

•

Is the deﬁnition (4.13) of the coupled energy always the most appropriate? We

require that the coupled energy is consistent with the atomistic energy, and that it

leads to an efﬁcient and accurate variational problem. The deﬁnition (4.13) is a

natural choice, but it is certainly not the only possible one.

•

How to (adaptively) deﬁne the partition Ω = Ω

∪ Ω

such that the solution of

the coupled problem (4.13)–(4.15) is a good approximation of the solution of the

atomistic problem (4.6)–(4.7)?

•

Can error bounds be obtained?

We study both the general case of a convex energy density and a speciﬁc example

of nonconvex energy, the Lennard–Jones case. See [23] and [24] for an analysis of a

very similar problem. In [23] and [24], we followed a proof strategy to obtain W

1,∞

estimates. In this contribution, we present a different proof strategy

and obtain H

estimates (see Theorem 4.15 below).

4.2 The convex case

We assume here that V

is a

convex potential. In this case, we show that we can

propose an a priori deﬁnition for the partition which is only based on properties of the

body forces f . Vaguely stated, the subdomain Ω

(in which the continuum mechanics

model is used) is the part of the domain Ω where the body force f and its derivative f

′

are small.

With thisdeﬁnition, we show that thesolution of the coupled problem (4.13)–(4.15)

is a good approximation of the solution of the atomistic problem (4.6)–(4.7): when

the atomic lattice parameter h goes to zero, the deformation and the strain given by

the coupled model converge to the deformation and the strain given by the atomistic

model. The main result of this section is Theorem 4.15 below.

This strategy is actually the one mentioned in Remark 1.3 of [23].

208 F

rédéric Legoll

We now proceed in details. We make the convexity assumption

(

H3) V

∈ C

(R), V

′

(1) = 0, ∃α, β ∈ R ∀x ∈ R : 0 < α ≤ V

′′

(x) ≤ β,

and the regularity assumption

(H4) f ∈ C(R).

Note that (H4) is required in order to give a sense to the atomistic energy (4.6). From

(H3), we deduce that, for all x and y,

(x −1)

≤ V

(x) −V

(1), (

4.16)

α(x −y)

≤

(

′

(x) −V

′

(y)

)

(x − y), (4.17)

′

(x) −V

′

(y)| ≤ β|x − y|, (4.18)

(x)| ≤ |V

(1)| +

|x −1|

, (

4.19)

(x) −V

(y)| ≤ β|x − y|

(

|x| + |y| + 2

)

. (4.20)

We ﬁrst consider the macroscopic problem (4.10), where the macroscopic energy is

(4.9), and the variational space is (4.11). In view of (4.19), E

(φ) is well-deﬁned for

φ ∈ X

Theorem 4.4. Under the assumptions (H3) and (H4), the macroscopic variational

problem (4.10) has a unique solution φ

Proof. We ﬁrst prove existence of a minimizer. On X

, the energy E

is lower-

bounded. Indeed, making use of (4.16), we obtain

(φ) ≥ V

(1) +

(φ

′

(x) −1)

x −

kfk

(0,

kφk

(0,L)

. (4.21)

Using a Poincaré inequality for the function ψ : x 7→ φ(x) − ax/L, which belongs to

(0, L), with the Poincaré constant C

Ω

, which only depends on Ω = (0, L), we ﬁnd

kφk

(0,L)

≤



kψk

(0,L)

+ C



≤ 2kψk

(0,L)

+ 2C

≤ 2C

Ω

kψ

′

(0,L)

+ 2C

≤ 4C

Ω

kφ

′

− 1k

(0,L)

where C is the H

(0, L)-norm of the function x 7→ a x/L. In the above bound, the

constant

C only depends on a and L. We now insert this estimate into (4.21) and

obtain

(φ) ≥ C +

8LC

Ω

kφk

(0,L)

−

kfk

(0,L)

kφk

(0,L)

(4.22)

Atomistic to continuum coupling methods 2

for some constant C.

Since the function t ∈ R

7→ C +

Ω

−

kfk

(0,

lower-bounded, we obtain that E

is lower-bounded on X

Let us consider a minimizing sequence of functions φ

. By deﬁnition, E

(φ

) is

bounded, hence, in view of (4.22), we obtain that kφ

(0,L)

is bounded. Since Ω =

(0, L) is bounded, we infer from Rellich’s theorem (compact embedding of H

(0, L)

into L

(0, L)) that there exists

φ ∈ X

uch that, possibly for a subsequence only, φ

converges to

φ w

eakly in H

(0, L) and strongly in L

(0, L).

The energy E

is continuous for the strong topology of H

(0, L). Indeed, for θ

and θ

in H

(0, L), we have, using (4.20),

(θ

) −E

(θ

)| ≤

(θ

′

(x)) −V

(θ

′

(x))| dx

|f(x)| |θ

(x) −θ

(x)| dx

≤ β

|θ

′

(x) −θ

′

(x)|

(

|θ

′

(x)| + |θ

′

(x)| + 2

)

+ kfk

(0,L)

kθ

− θ

(0,L)

≤ βkθ

− θ

(0,L)

k|θ

′

|+ |θ

′

| + 2k

(0,L)

+ kfk

(0,L)

kθ

− θ

(0,L)

This inequality yields the continuity of E

. Since E

is also convex, it is weakly

lower semi-continuous in H

(0, L) (see, e.g., [34, Corollaire III.8]). We hence infer

from φ

⇀

φ in H

(0,

L) that





≤ lim

infE

(φ

) = limE

(φ

) = I

Since

φ ∈ X

we also have I

≤ E





As a consequence, I

= E





and

a minimizer of (4.10).

Since V

is strictly convex, the energy E

is a strictly convex function on the con-

vex set X

, hence the minimizer of (4.10) is unique.

In order to write the weak formulation of (4.10), we introduce the forms

(φ

, ψ) =

′

(φ

′

(x)) ψ

′

(x) dx,

(ψ) =

f(x) ψ(x) dx,

deﬁned on H

(0, L) ×H

(0, L) and H

(0, L), respectively. Note that A

is nonlinear

with respect to φ.

Lemma 4.5. The macroscopic solution φ

deﬁned by Theorem 4.4 satisﬁes

∀ψ ∈ H

(0, L), A

(φ

, ψ) = B

(ψ). (4.23)

210 F

rédéric Legoll

In addition, the function ψ

: x 7→ φ

(x) − a

x/L is in H

(0, L) and satisﬁes the

estimate

kψ

(0,L)

≤

kfk

(0,

(4.24)

for some C that only depends on the domain Ω = (0, L).

Proof. Equation (4.23) is the weak formulation of the variational problem (4.10). It

implies that, in D

′

(0, L),

−V

′′

(φ

′

) φ

′′

= −

(

′

(φ

′

)

′

= f.

Since V

′′

(x) ≥ α > 0 and since f ∈ C

(0, L) ⊂ L

(0, L), we obtain that φ

′′

∈

(0, L), and

kψ

′′

(0,L)

= kφ

′′

(0,L)

≤

kfk

(0,

. (4.25)

Hence ψ

is in H

(0, L). Let us now upper-bound this function in the H

-norm. We

ﬁrst note that ψ

∈ H

(0, L). In view of (4.17) and (4.23), we have



′

−



(0,

≤



′

(φ

′

(x)) − V

′







′

(x) −



′

(φ

′

(x))



′

(x) −



= A

(φ

, ψ

)

= B

(ψ

)

≤ kfk

(0,L)

kψ

(0,L)

where we have used, in the second line, that



′

(x) −



x = 0 since ψ

(x) :=

(x) − ax/L vanishes for x = 0 and x = L. Using a Poincaré inequality for the

function ψ

, we obtain

kψ

(0,L)

≤

kfk

(0,

for a constant C that only depends on Ω = (0, L). This bound, combined with (4.25),

yields (4.24).

We now consider the microscopic problem (4.7), where the microscopic energy

(4.6) and the variational space is (4.8). The following theorem is the microscopic

counterpart of Theorem 4.4.

Theorem 4.6. Under the assumptions (H3) and (H4), the microscopic variational prob-

lem (4.7) has a unique solution φ

Atomistic to continuum coupling methods 2

We skip the proof of this theorem which follows the same lines as the proof of

heorem 4.4. In order to write the weak formulation of (4.7), we introduce the forms

(φ, ψ) = h

N−1

i=0

′



i+1

− φ



− ψ

(ψ)

= h

i=0

f(ih) ψ

deﬁned on R

N+1

× R

N+1

and R

N+1

, respectively, and the homogeneous microscopic

space



ψ ∈ R

N+1

; ψ

= 0, ψ

= 0



Lemma 4.7. The microscopic solution φ

deﬁned by Theorem 4.6 satisﬁes

∀ψ ∈ X

, A

(φ

, ψ) = B

(ψ). (4.26)

We now introduce a coupled problem based on a given partition of Ω = (0, L) =

Ω

∪Ω

. We recall that, in order to simplify the notations, we work with the choice

Ω

= (0, P h], Ω

= (P h, L),

for some P that depends on h such that P h is constant (see (4.12) and Figure 2).

The following theorem is the coupled counterpart of Theorem 4.4.

Theorem 4.8. Under the assumptions (H3) and (H4), the coupled variational problem

(4.15) has a unique solution φ

We skip the proof of this theorem, which again follows the same lines as the proof

of Theorem 4.4. In order to write the weak formulation of (4.15), we introduce the

forms

(φ, ψ) = h

P −1

i=0

′



i+1

− φ



− ψ

′

(φ

′

(x)) ψ

′

(x) dx,

(ψ) = h

i=0

f(ih) ψ

P h

f(x) ψ(x) dx,

deﬁned on X

× X

and X

, respectively, and the homogeneous coupled space



ψ ∈ X

; ψ

= 0, ψ(L) = 0



. (4.27)

Lemma 4.9. The coupled solution φ

deﬁned by Theorem 4.8 satisﬁes

∀ψ ∈ X

, A

(φ

, ψ) = B

(ψ). (4.28)

In addition, the function ψ

: x ∈ Ω

7→ φ

(x) −ax/L is in H

(Ω

) and satisﬁes the

estimate

kψ

(Ω

)

≤

kfk

(Ω

)

(4.29)

for some C that only depends on the domain Ω.

212 F

rédéric Legoll

Proof. T

he proof of (4.28) follows the same lines as the proof of (4.23) of Lemma 4.5.

The proof of (4.29) follows the same lines as the proof of (4.24): we make use of a

Poincaré inequality on Ω

, with a constant C

Ω

, which is next upper-bounded by the

Poincaré constant of Ω = (0, L).

We now aim at estimating the distance between φ

nd φ

. We introduce an aux-

iliary problem, which is the discretization of the coupled problem by a ﬁnite element

method, using a mesh on Ω

of mesh size h. Of course, the resulting problem is as

expensive to solve as the full microscopic problem (both problems involve as many

variables), but it helps us in the numerical analysis.

We hence introduce the space

= {θ ∈ X

; θ is piecewise afﬁne on Ω

on a mesh of size h},

and its homogeneous version

= X

∩ X

We introduce the discretized coupled problem

inf



(φ); φ ∈ X



. (4.30)

As for the coupled problem (4.15), the above problem has a unique solution φ

, which

satisﬁes the weak formulation (see Lemma 4.9)

∀ψ ∈ X

, A

(φ

, ψ) = B

(ψ). (4.31)

To estimate the difference between the solution φ

of the coupled problem and the

solution φ

of the atomistic problem, we ﬁrst estimate the distance between φ

and φ

solutions of the non-discretized and discretized coupled problem, respectively. Next,

we estimate the distance between φ

and φ

To measure distances in X

, we introduce the mixed semi-norm (recall Ω

(P h, L))

|ψ|

)

P h

′2

(x) dx + h

P −1

i=0



i+1

− ψ



e have the following estimates for the form A

Lemma 4.10. Assume that (H3) holds. For all u and g in X

, we have the strong

monotonicity

(u, u −g) −A

(g, u − g) ≥ α|u − g|

)

. (4.32)

For all u, g and θ in X

, we have the Lipschitz-continuity

(u, θ) −A

(g, θ)| ≤ 2β |u −g|

)

|θ|

)

. (4.33)

Proof. The bound (4.32) is a direct consequence of (4.17). The bound (4.33) follows

from (4.18).

Atomistic to continuum coupling methods 2

The above lemma shows that A

ulﬁlls the assumptions (of strong monotonicity

and Lipschitz continuity) of the theorem of Zarantonello [160, Theorem 25.B], which

is a generalization of the Lax-Milgram theorem to the nonlinear setting. The following

lemma provides an estimate of the distance between φ

and φ

, and is a generalization

of the Céa lemma.

Lemma 4.11. The coupled solution φ

, which is deﬁned by Theorem 4.8, and its dis-

cretized counterpart φ

, solution of (4.30), satisfy



− φ



)

≤

2β

h kφ

′′

(Ω

)

for some C that only depends on Ω.

Proof. Using (4.32), (4.28) and (4.31), we have



− φ



)

≤ A

(φ

, φ

− φ

) −A

(φ

, φ

− φ

)

= A

(φ

, φ

− φ

+ ψ) −A

(φ

, φ

− φ

+ ψ)

for any ψ ∈ X

. With (4.33), we therefore obtain for any ψ ∈ X



− φ



)

≤ A

(φ

, φ

− ψ) − A

(φ

, φ

− ψ)

≤ 2β



− φ



)

|φ

− ψ|

)

Hence,



− φ



)

≤

2β

ψ∈X

|φ

− ψ|

)

. (4.34)

We have shown in Lemma 4.9 that φ

∈ H

(Ω

). By a standard approximation result,

we thus have

inf

ψ∈X

|φ

− ψ|

)

≤ C h kφ

′′

(Ω

)

, (4.35)

where C only depends on Ω

and can be bounded by a constant that only depends on

Ω. Collecting (4.34) and (4.35), we obtain the claimed estimate.

We now estimate the distance between φ

nd φ

. We make use of the weak formu-

lations of thecorresponding variational problems (see Lemma 4.7 and equation (4.31)).

To compare the conﬁgurations φ

and φ

, that belong to different spaces, we introduce

the evaluation operator

E : X

→ X

ψ 7→ Eψ

deﬁned by (Eψ)

= ψ

for 0 ≤ i ≤ P , and (Eψ)

= ψ(ih) for ih ≥ P h. We also

introduce the interpolation operator

I : X

→ X

ψ 7→ Iψ

214 F

rédéric Legoll

deﬁned by (I

ψ)

= ψ

for 0 ≤ i ≤ P , and Iψ

|Ω

is the piecewise afﬁne interpolation

of ψ

|Ω

. We note that I(X

) ⊂ X

and

∀φ, ψ ∈ X

, A

(φ, ψ) = A

(Iφ, Iψ). (4.36)

Lemma 4.12. Assume that (H3) and (H4) hold. Then



− Iφ



)

≤

here

τ = sup

ψ∈X

, ψ6=0

(ψ) − B

(Eψ)|

|ψ|

)

. (

4.37)

Proof. Let ψ ∈ X

. Using (4.31) and (4.26), we obtain

(φ

, ψ) = B

(ψ)

= B

(Eψ) + B

(ψ) − B

(Eψ)

= A

(φ

, Eψ) + B

(ψ) −B

(Eψ)

= A

(Iφ

, ψ) + B

(ψ) −B

(Eψ),

where, in the last line, we have used (4.36) and the fact that IEψ = ψ for any ψ ∈ X

We hence obtain, for any ψ ∈ X

(φ

, ψ) − A

(Iφ

, ψ) ≤ τ|ψ|

)

where τ is deﬁned by (4.37). We now choose ψ = φ

− Iφ

and use (4.32):



− Iφ



)

≤ A

(φ

, φ

− Iφ

) −A

(Iφ

, φ

− Iφ

)

≤ τ



− Iφ



)

which yields the claimed bound.

Combining Lemma 4.11 and 4.12, we obtain the following result:

heorem 4.13. Assume that (H3) and (H4) hold. Then

|φ

− Iφ

)

≤

2β

h k φ

′′

(Ω

)

where τ is deﬁned by (4.37) and C only depends on Ω.

We now address the question of how to deﬁne the partition Ω = (0, L) = Ω

∪Ω

Our aim is to upper-bound τ that appears in the estimate of Theorem 4.13. We use the

following result, which can be proved by Taylor expansion: