FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS

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1 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI ABSTRACT This paper is devoted to a new finite element consistency analysis of Cauchy Born approximations to atomistic models of crystalline materials in two and three space dimensions Through this approach new atomistic Cauchy Born models are introduced and analyzed These intermediate models can be seen as first level atomistic/quasicontinuum approximations in the sense that they involve only short-range interactions The analysis and the models developed herein are expected to be useful in the design of coupled atomistic/continuum methods in more than one dimension Taking full advantage of the symmetries of the atomistic lattice we show that the consistency error of the models considered both in energies and in dual W,p type norms is O(ε, where ε denotes the interatomic distance in the lattice INTRODUCTION Modern multiscale methods for the simulation of materials introduce several coupling mechanisms of the atomistic and the continuum descriptions aiming at the design of methods of atomistic accuracy with continuum cost To understand these mechanisms and the behavior of the coupled models is a challenge both from the modeling point of view and from the computational perspective It is known, for example, that ad-hoc coupling of models may lead to undesirable computational artifacts 6, 8 The development of the mathematical foundations of coupled multiscale models therefore seems necessary Although the mathematical theory of multiscale models is still quite limited at present, it is hoped that, ultimately, the availability of a comprehensive mathematical theory of multiscale models will enhance the development of efficient, accurate and robust numerical algorithms for multiscale models Indeed, the area of multiscale simulations in materials science is a very active field; see, for example, the review articles 9, 0 In particular, a problem that has received considerable attention from the engineering as well as from the mathematical point of view is the atomistic-to-continuum passage (cf 5, 9, 6,, 4, 3,, and the corresponding coupled methods for crystalline materials (cf 39, 7, 3, 0, 8, 4, 3, 5, 6, 5, 4, 36, 7, 6, 7, 30, 8, 9, 0, 35, 8, 37, 38, 43, 44,,, 34, 35, 4, 4 Much of the literature on atomistic/continuum coupling in crystals is concerned with the quasicontinuum method 39 and its variants In these methods, in regions of interest in the material (strong deformations, defects the atomistic model is kept, while in regions of smooth deformations the atomistic model is replaced with a continuum model discretized by finite elements Despite the increasing number of papers concerned with the numerical analysis of these methods, satisfactory analytical results are available in one space dimension only; in two and three space dimensions the precise formulation of efficient coupling methods is still in its infancy

2 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI This paper is devoted to a novel finite element consistency analysis of Cauchy Born approximations to atomistic models of crystalline materials in two and three space dimensions Through this approach new atomistic Cauchy Born models are introduced and analyzed These intermediate models can be seen as first level atomistic/quasicontinuum approximations in the sense that they involve only short-range interactions The analysis and the models developed herein are expected to be useful in the design of coupled atomistic/continuum methods in several space dimensions Specifically, we concentrate on the comparison of an atomistic model and its continuum Cauchy Born approximation, as in 9, 4; however, in contrast with the finite-difference -style analysis in 9, 4, here we develop a theoretical framework in the spirit of finite-element methods Taking full advantage of the symmetries we show that the consistency error of the models considered in dual W,p type norms is O(ε, ε being the interatomic distance As a consequence we provide an alternative finite element proof of the second order consistency result of the continuum Cauchy Born model derived by E & Ming 4 using finite difference techniques In addition we derive consistency results for the energies, by showing that all models considered have energies that are O(ε close to each other The first consistency results of the energies for the continuum Cauchy Born model approximating the atomistic model were derived in Blanc, LeBris & Lions 9 Our approach is based on the introduction and analysis of intermediate atomistic Cauchy Born models Since, most of the recent results concerning the construction and analysis of various quasicontinuum methods in one space dimension were based on the properties and the proper comparison of a similar intermediate atomistic Cauchy Born model involving only short-range interactions we expect that our approach can provide the appropriate analytical framework for the construction and analysis of various atomisitic/continuum methods in multiple space dimensions Notation Lattice, discrete domain, continuum domain We consider a simple d-dimensional lattice, which is generated by d linearly independent vectors of R d, d =, 3 For simplicity of the exposition we assume that the lattice L entire is generated by the unit coordinate vectors e,, e d of R d The extension of the consistency analysis developed in this paper to include any d linearly independent vectors of R d is straightforward, since the general case can be obtained by applying an affine map We note however that, unlike consistency, stability can be sensitive to the specific lattice structure (eg rectangular or triangular; see, 4, for example We will consider discrete periodic functions on L entire defined over a periodic domain L More precisely, let L := l = (l,, l d = n e + + n d e d : (n,, n d Z d N, N N d, N d The actual configuration of the atoms is thus a subset of R d, which we call discrete domain and denote by discr ; the corresponding continuum domain is denoted by ; ie, discr := x l = (x l,, x ld = ε l, l L,, := x x N, x N x Nd, x Nd o Here O o denotes the interior of the set O o Note that a possible confusion in the notation may arise when l i = i, i =,, d, since by x i we denote the coordinates of the continuum variable x; see below It will be clear however from the analysis which is the continuum variable Functions and spaces We consider atomistic deformations y l = y(x l, l L of the form y l = Fx l + v l, with v l = v(x l periodic with respect to L

3 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 3 Here F is a constant d d matrix with det F > 0 The corresponding spaces for y and v are denoted by X and V and are defined as follows: X := y : L R d, y l = Fx l + v l, v V, l L, V := u : L R d, u l = u(x l periodic with zero average with respect to L For functions y, v : L R d we define the inner product y, v ε := ε d y l v l For a positive real number s and p we denote by W s,p (, R d the usual Sobolev space of functions y : R d ; we shall use the same symbol for discrete functions defined on the lattice and for continuum functions It will be clear from the context which of the two is intended in a specific instance Further by W s,p # (, Rd we denote the corresponding Sobolev space of periodic functions with basic period By, we denote the standard L ( inner product The space corresponding to X in which the minimizers of the continuum problem are sought is X := y : R d, y(x = Fx + v(x, v V, where V := u : R d, u W k,p (, R d W,p # (, Rd, u dx = 0 Henceforth, for the sake of notational simplicity, we shall suppress the symbol R d in our notation for Sobolev spaces, and will simply write W s,p ( and W s,p # ( Difference quotients and derivatives We will use the notation D η y l := y l+η y l, l, l + η L, ε for the difference quotient (discrete derivative in the direction of the vector η For functions defined on the continuum domain, the following notation is used ζi φ(ζ := φ(ζ,, ζ d, ζ = (ζ,, ζ d, ζ i ζ φ(ζ := ζi φ(ζ, i ( α v(x := v(x, x α u i (x u(x := x α iα To avoid confusion we distinguish between derivatives with respect to arguments, denoted by ζi, which usually appear in composite functions, and derivatives with respect to the spatial variable x i, denoted by the symbol i Atomistic and Cauchy Born potential We consider the atomistic potential ( Φ a (y := ε d φ η (D η y l, where R is a given finite set of interaction vectors We allow the potential to vary with the type of bond, ie, φ η may vary with η On the other hand we assume standard conditions on the potential away from zero (cf 9: φ η are functions defined on R d \0, which are smooth for any ζ, ζ > ρ In fact we assume that there exist C ρ,k = C(ρ, k 0, such that D k ζ φ η(ζ C ρ,k, for ζ > ρ, and a multi-index k of length

4 4 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI k 3 Notice that we have not imposed any symmetry hypotheses on the potentials φ η In order to explore the consistency of the Cauchy Born approximation, we will consider sufficiently smooth diffeomorphisms y on The assumption that y is is natural since it excludes interpenetration In addition, it leads to the lower bound D η y l α(y, η > 0, 9, which is required in the course of bounding derivatives of φ η (D η y l It will be assumed throughout the paper that whenever the potential is applied to a smooth function y, y is a diffeomorphism on the domain For a given field of external forces f : L R d, where f l = f(x l, the atomistic problem reads as follows: (3 If such a minimizer exists, then where (4 find a local minimizer y a in X of : Φ a (y f, y ε DΦ a (y a, v ε = f, v ε, for all v V, DΦ a (y, v ε := ε d = ε d d i= ζi φ η (D η y l D η v l i ζ φ η (D η y l D η v l Throughout the rest of the paper we shall use the summation convention for repeated indices The corresponding Cauchy Born stored energy function is W (F = W CB (F := φ η (F η Then, the continuum Cauchy Born model is stated as follows: (5 find a local minimizer y CB in X of : Φ CB (y f, y, where the external forces f are appropriately related to the discrete external forces and Φ CB (y := W CB ( y(x dx If such a minimizer exists, then (6 DΦ CB (y CB, v = f, v, for all v V, where DΦ CB (y, v = Here the stress tensor S is defined, as usual, by S iα ( y(x vi (x dx = S iα ( y(x α v i (x dx, v V x α S := W (F F iα iα

5 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 5 A simple calculation yields the following relation between the stress tensor and the atomistic potential: W (F S iα = = φ η (F η F iα F iα (7 = F iα = φ η (F jβ η β = ζi φ η (F η η α ζi φ η (F jβ η β F jβ η β F iα Main results The question whether, and under what conditions, the continuum Cauchy Born model (5 approximates (3 is very delicate The analytical assessment of the quality of an approximating scheme in Numerical Analysis is based on the notions of consistency, stability and convergence Consistency essentially refers to the extent to which an exact smooth solution fails to satisfy the numerical scheme Given that the stability of the approximating scheme is satisfactory, usually the consistency error determines the order and therefore the quality of the approximation It is to be noted that the consistency error depends in an essential manner on the norm we use to measure it The choice of the norm is, in turn, dictated by the chosen method of stability/convergence analysis In this paper we focus on the consistency analysis of Cauchy Born approximations We then briefly discuss the consequences of our results in the convergence analysis To be more precise, let us assume that y is a smooth solution of the Cauchy Born problem (6 Then, our goal is to quantify the size of DΦ a (y, v ε f, v ε, for all v V such that v W,p ( = Here by v W,p ( we denote the W,p norm of the bilinear interpolant of v V In addition, y is identified with its bilinear interpolant Elements of V will be identified with elements of the finite element space V ε, see Section for the precise definitions Modulo a data approximation error in f, it suffices to estimate C V (y := sup DΦ a (y, v ε DΦ CB (y, v : v V with v W,p ( =, where in the last relation y is any smooth function We we shall refer to C V (y as the variational consistency error Similarly, one can define the energy consistency error C E (y := Φ a (y Φ CB (y Our aim is to show that both consistency errors are of second order in the lattice spacing ε; cf Sections 4 and 5 Our assumptions on the potentials and on y are standard; see the discussion following equation ( Related results for energy consistency and variational consistency in the case of a single potential φ η = φ were derived by Blanc, LeBris & Lions 9 and by E & Ming 4, respectively In 9 boundary effects were taken into account We note that our second order consistency results do not require symmetry of the potentials Thus, Theorem 4, where we show that C E (y = O(ε can be seen as an extension of Theorem in 9 where under similar assumptions on the potential it was proved that C E (y 0 as ε 0 The estimate C E (y = O(ε for symmetric potentials was proved in Theorem 3 of 9 allowing infinite-range interactions Regarding variational consistency, it was shown in 4 by asymptotic methods, still under the assumption of symmetric potentials, that DΦ a (y, v ε DΦ CB (y, v C(y, vε, with C(y, v depending on higher order derivatives of y and v Our main contribution in this paper is the analytical approach proposed, which is based on finite element analysis, and the new construction of an intermediate atomistic Cauchy Born model The analysis of 9 and 4 is based on finite difference techniques and on the direct comparison of the atomistic and

6 6 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI continuum Cauchy Born models The proposed atomistic Cauchy Born model, see Section 3, is not the standard model that one gets by replacing in the atomistic potential long-range interactions by short-range ones As was mentioned before, the available results concerning the construction and analysis of various quasicontinuum methods in one space dimension were heavily based on the use of intermediate atomistic Cauchy Born models The definition of these intermediate models is obvious in one dimension but, as our analysis shows, this is not so in multiple space dimensions The finite element approach taken in this paper is a natural choice for coupled methods, since, by construction, atomistic/continuum coupling methods assume finite element discretization of the continuum region We therefore expect that our approach can provide the appropriate analytical framework for the construction and analysis of various atomisitic/continuum methods in multiple space dimensions, which is at present lacking The paper is organized as follows Section is devoted to the introduction of the finite element notation and the construction of the atomistic Cauchy Born model starting from the continuum model and performing appropriate approximation steps In Section 3 we show that the atomistic model approximates the atomistic Cauchy Born model with the desired accuracy The use of the mesh symmetries is important in the analysis Section 4 provides our main results in the two-dimensional case regarding the order of accuracy of the energy and variational consistencies related to the comparison of the atomistic and Cauchy Born continuum models Section 5 is devoted to the extension of our results to three space dimensions Finally, in Section 6 we briefly discuss the applicability of our results in the convergence analysis, under suitable stability hypotheses CONSTRUCTION OF AN ATOMISTIC CAUCHY BORN MODEL In the sequel we provide a link between the continuum model and the atomistic model by introducing an intermediate model, which we call atomistic Cauchy Born model (A-CB To derive this model, we start from the continuum model and perform appropriate approximation steps, which finally yield the A-CB The final model has consistency error of the order O(ε compared to the continuum Cauchy Born model In the next section we show that the A-CB has O(ε consistency error compared to the original atomistic model Bilinear finite elements on the lattice Let V ε be the space of continuous piecewise bilinear periodic functions on the lattice L More precisely, let T := K : K = (x l, x l + (x l, x l +, x l = (x l, x l discr, V ε := v : R, v C(, v K Q (K and v l = v(x l periodic with respect to L, where Q (K denotes the set of all bilinear functions on K; ie, v K (x = α 0 + α x + α x + α 3 x x It is then well known that the elements of the linear space V ε can be expressed in terms of the nodal basis functions, Ψ l = Ψ l (x, l L, as v(x = v l Ψ l (x = v l Ψ l (x Ψ l (x, v l = v(x l, where we have used the fact that Ψ l (x can be written as the tensor product of the standard one-dimensional piecewise linear hat functions x Ψ l (x and x Ψ l (x with respect to the x and x variable, respectively Here Ψ l (x l = δ l l and Ψ l (x l = δ l l Preliminaries We shall make frequent use of the following version of the Bramble Hilbert lemma 3, Theorem 43 Its proof is completely straightforward and is therefore omitted

7 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 7 Lemma Let O be a bounded open set in R d and suppose that p and s 0 Suppose further that ζ is a linear functional on a linear subspace H of W s,p (O with the following property: C 0 > 0 v H : ζ(w C 0 w W s,p (O Then, for any w H and any set S Ker(ζ we have that ζ(w C 0 inf ϕ S w ϕ W s,p (O If, in addition, there exists a positive constant C, independent of diam(o, and a real number t > s such that inf ϕ S w ϕ W s,p (O C (diam(o t s w W t,p (O where is a seminorm on W t,p (O, then w W t,p (O, ζ(w C 0 C (diam(o t s w W t,p (O w H W t,p (O We note that when O = K, S = Q (K (with Q (K as defined above when d = and a completely analogous definition when d = 3; cf Section 5, s = 0,, p and with Q A w Q (K signifying the averaged Taylor polynomial of a smooth function w introduced in, Section 46, we have that w Q A w W s,p (K C ε s d αw L p (K, where C is a positive constant independent of ε Due to the fact that we work on Q (K the right-hand side of this estimate involves only part of the standard W,p (K seminorm, see, Section 46 for a proof and a related discussion Thus, for a linear functional ζ that vanishes on S = Q (K and such that ζ(w C 0 w L p (K for all w H = C(K, we have from Lemma with s = 0, t = and p that d ( ζ(w C 0 C ε αw L p (K α= α= w C(K W,p (K Next, we observe that in Q (K the following inverse inequality holds: for p and q, there exists a constant γ s,p,q such that ( ϕ W s,q (K γ s,p,q ε d/q d/p ϕ W s,p (K, ϕ Q (K, s = 0, For more general inverse inequalities involving variable degree derivatives we refer to, Section 45 We note that ( follows by a simple homogeneity argument on Q We will use the following consequence of (: if w V ε, and α K 0, then, for /p + /q =, (3 /q K α K α w L (K γ,p, K αk q w W,p (, w V ε, q, K T K T K T K α K α w L (K γ,, max K T α K w W, (, w V ε, q = Indeed, these inequalities follow by Hölder s inequality and by noting that, thanks to (, /p /p K α w p L (K K γ p,p, (ε d/p p w p W,p (K (4 K T K T = γ,p, w W,p (

8 8 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI Consistency analysis of the Cauchy Born model: d = Suppose that v V ε and let y be a sufficiently smooth function Our aim is to approximate DΦ CB (y, v = S iα ( y(x vi (x dx = S iα ( y(x α v i (x dx, v V ε x α It suffices to consider a (generic term of the above sum To simplify the notation we drop the vector indices and let g( y be the part that corresponds to the stress tensor In the sequel we will consider g( y(x v(x dx, v V ε Recall that, since v V ε, we have that v(x = v (l,l Ψ l (x Ψ l (x, l = (l, l L, v l = v(x l Thus v(x = = ( v (l,l ε xl,x l (x xl,x l +(x Ψ l (x ( v ε (l +,l v (l,l xl,x l +(x Ψ l (x Therefore, by defining G(x := G(x, x = we have that g( y(x v(x dx = ε = ε = ε = ε + ε xl + x l xl + x l xl + x l xl + =: A + A x l + x x N g( y(x dx x l G(x l +, x G(x l, x g( y(x, x dx, ( g( y(x v (l +,l v (l,l xl,x l +(x Ψ l (x dx v (l +,l v (l,l Ψ l (x dx Ψ l (x dx v (l +,l v (l,l ( G(x l +, x G(x l, x Ψ l (x dx v (l +,l v (l,l x l G(x l +, x G(x l, x dx v (l +,l v (l,l In the next lemma we show that A is second order accurate with respect to ε We refer to 40 for a similar result Lemma Let y be a smooth function; then xl + ( G(x l +, x G(x l, x Ψ l (x dx C(y K ε x l

9 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 9 Proof It suffices to consider the functional xl + ( ζ(g = G(x l +, x G(x l, x Ψ l (x dx, G C(K K e, x l where K e is the element that shares the edge x l, x l+ e with K, and to observe that ζ(ϕ = 0 for all ϕ P Indeed, if ϕ = x m then ζ(ϕ = (x l + m (x l m x l + ϕ = x m then ζ(ϕ = xl + x l x l ( Ψ l (x dx = 0, (x m (x m( Ψ l (x dx = 0 Finally, for ϕ = x (x x l, ζ(ϕ = x x l + ( l + x l (x x l Ψ l (x dx = 0 x l Next we observe that by Lemma we have, for any ϕ P, that ζ(g C max x l s x l + xl + x l G(s, x ϕ(s, x dx Cε G ϕ L (K K e G C(K K e, and if The proof is completed by using standard approximation properties of P (see, to bound inf ϕ P G ϕ L (K K e by C(y ε3 and noting that K = K e = ε The term A requires further simplification To this end, we note that A = xl + G(x l +, x G(x l, x dx v ε (l +,l v (l,l x l = xl + x l + g( y(x dx dx v ε (l +,l v (l,l x l x l = xl + x l + g( y(x dx dx v ε (l +,l v (l,l x l x l + xl + x l + g( y(x dx dx v ε (l +,l l L x l x + v (l,l + l = g( y(x dx D e v l + D e v l+e, K T K where l = (l, l We shall also require the following result Lemma 3 Let m K be the barycenter of K; then, there exists a constant C = C(y such that g( y(x dx K g( y(m K D e v l + D e v l+e C(y ε v K T K W,p (

10 0 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI Proof We denote by ζ the functional (5 ζ(w := K and observe that ζ(ϕ = 0 K w(x w(mk dx, for all ϕ Q (K Obviously ζ(w w L (K for every w C(K, and thus by Lemma and ( we have that d (6 ζ(w C ε αw L (K Therefore, using the fact that v V ε we deduce with w( = g( y( that g( y(x dx K g( y(m K D e v l + D e v l+e K T K C ε K T α= d K αg( y(x L (K v L (K The proof is thus complete in view of (3 We summarize what we have shown so far in the following proposition Proposition Suppose that y is a smooth function, and let m K be the barycenter of K; then, for any v V ε, the quantity A a,cb, v ε : = ε d Si ( y(m K D e v l + D e v l+e i K T + ε d Si ( y(m K D e v l + D e v l+e i K T (7 = ε d ζ φ η ( y(m K η η D e v l + D e v l+e K T + ε d ζ φ η ( y(m K η η D e v l + D e v l+e, K T is a second order approximation to DΦ CB (y, v in the sense that there exists a constant M = M(y, p, p, independent of v, such that DΦ CB (y, v A a,cb, v ε M ε v W,p ( 3 The atomistic Cauchy Born model We define the average difference quotient (discrete derivative as follows: D e v l := D e v l + D e v l+e, D e v l := D e v l + D e v l+e Thus we can define the discrete gradient matrix as v l α= iα = D e α v i l

11 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS We introduce the atomistic potential Φ a,cb (y := ε d = ε d φ η ( y l η W CB ( y l Notice that due to the definitions of D eα v l, α =,, as averages of discrete gradients, Φ a,cb is not the standard atomistic potential that one gets by replacing in the atomistic model long-range interactions by short-range ones In fact, in that case the corresponding atomistic potential would be Φ a,cb (y := ε d = ε d φ η ( y l η W CB ( y l, where yl iα = D e α y i l Now, for a given field of external forces f : L R d the atomistic Cauchy Born problem reads: find a local minimizer y a,cb in X of : Φ a,cb (y f, y ε If such a minimizer exists, then DΦ a,cb (y a,cb, v ε = f, v ε, for all v V The next lemma provides the link between A a,cb introduced earlier in Proposition and DΦ a,cb Lemma 4 Let y V ε ; then for, any v V ε, (8 A a,cb, v ε = DΦ a,cb (y, v ε Proof Obviously, (9 DΦ a,cb (y, v ε = ε d + ε d Si ( y l D e v l + D e v l+e i Si ( y l D e v l + D e v l+e i Hence, it suffices to observe that for y V ε and for K being the element with vertices x l, x l+e, x l+e +e, x l+e, see Fig, we have that y(m K = y l, for y V ε, and thus S iα ( y(m K = S iα ( y l

12 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI x l+η e x l+η x l+e x l x l+e x l+η e FIGURE Typical interaction vector η and the corresponding lattice points 3 COMPARISON OF ATOMISTIC CAUCHY BORN AND ATOMISTIC MODELS: d = To compare the atomistic and atomistic Cauchy Born models we start from (4 and note that: DΦ a (y, v ε = ε d ζ φ η (D η y l D η v l = ε d ζ φ η (D η y l D η e v l + D η e v l+η e + ε d ζ φ η (D η y l D η e v l + D η e v l+η e Due to the assumed periodicity of the lattice, we have that DΦ a (y, v ε = ε d ζ φ η (D η y l D η v l = ε d (3 ζφ η (D η y l + ζφ η (D η y l η e D η e v l ζφ η (D η y l + ζφ η (D η y l η e D η e v l +ε d Notice that when η, η are positive, D η e v l = D e v l + + D e v l+(η e, D η e v l = D e v l + + D e v l+(η e, while when, eg, η = σ < 0, (3 D η e v l = D e v l σ e D e v l e Therefore, we deduce that DΦ a (y, v ε = ε d Φ η,l, D e v l + ε d Φ η,l, D e v l,

13 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 3 where (33 Φ η,l, := η k=0 and Φ η,l, is defined analogously ζφ η (D η y l k e + ζφ η (D η y l k e η e, for η > 0 σ k= ζφ η (D η y l+k e + ζφ η (D η y l+k e η e, for η = σ < 0, In the sequel, we focus on the term Φ η,l, when η > 0 The other terms are treated in a similar manner As a first step we will compare Φ η,l, with where (34 Φ η,l, := η k=0 ζ φ η ( y(m k,η,lη, η > 0, m k,η,l is the midpoint of the side with endpoints x l k e, x l k e +η e, m,l We have the following result is the midpoint of the side with endpoints x l, x l+ e, and m,l is the midpoint of the side with endpoints x l, x l+ e Lemma 3 Let y be a smooth function; then, assuming that η > 0, we have that Φ η,l, Φ η,l, C(y ε Proof We begin by noting that (in the case of k = 0: ζφ η (D η y l + ζφ η (D η y l η e = ζφ η ( D η e y l + D η e y l+η e + D η e y l + D η e y l+η e + ζφ η ( D η e y l η e + D η e y l η e +η e + D η e y l η e + D η e y l η e +η e We shall use the following elementary bound: there exists a constant c = c (ψ such that, for any smooth function ψ and real numbers a, b, ψ(a + ( a + b (35 ψ(b ψ c a b We notice that D η e y l + D η e y l+η e + D η e y l + D η e y l+η e D η e y l η e + D η e y l η e +η e D η e y l η e + D η e y l η e +η e C ε In view of (35 it suffices to consider ζ φ η ( 4 D η e y l + 4 D η e y l+η e + 4 D η e y l + 4 D η e y l+η e + 4 D η e y l η e + 4 D η e y l η e +η e + 4 D η e y l η e + 4 D η e y l η e +η e = ζ φ η ( 4 D η e y l η e + 4 D η e y l η e +η e + D η e y l + 4 D η e y l+η e + 4 D η e y l η e

14 4 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI x l+η x l η e x l x l+e FIGURE Symmetry as a result of combining D η y l and D η y l η e Now, 4 D η e y l η e + 4 D η e y l η e +η e η y(m 0,η,l η y(x l + η y(x l+η e η y(m 0,η,l + 4 D η e y l η e η y(x l + 4 D η e y l η e +η e η y(x l+η e Similarly, C ε D η e y l + 4 D η e y l+η e + 4 D η e y l η e η y(m 0,η,l C ε Proceeding as above for any k, k η, we complete the proof Next we establish the following result, which is valid for all positive or negative η Lemma 3 Let y be a smooth function, and let m,l denote the midpoint of the side with endpoints x l, x l+ e (cf (34 above; then, η ζ φ η ( y(m,l η Φ η,l, C(y ε Proof Assume first that η > 0 It is a simple matter to observe that Φ η,l, is a sum of η terms involving function evaluations at points that are symmetrically placed with respect to m,l (if η is odd, one of these points is m,l Thus, Φ η,l, ζ φ η ( y(m,l η η C(y ε In the case where η = σ < 0 one can show that σ ζ φ η ( y(m k,η,lη Φ η,l, C(y ε k=

15 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 5 As before, the points m k,η,l are symmetrically placed with respect to m,l Hence σ σ k= ζ φ η ( y(m k,η,lη ζ φ η ( y(m,l η C(y ε, and the proof is complete We have therefore completed the proof of the following result Proposition 3 Let y be a smooth function, and let m,l, m,l be as in (34; then, for any v V ε, the quantity ζ φ η ( y(m,l η η De v l (36 A, v ε := ε d ζ φ η ( y(m,l η η De v l + ε d is a second order approximation to DΦ a (y, v ε in the sense that there exists a constant M = M(y, p, independent of v, such that DΦ a (y, v ε A, v ε M ε v W,p (, p 4 ENERGY AND VARIATIONAL CONSISTENCY: d = In this section we present our basic results in two space dimensions Similar results for the comparison of atomistic and atomistic Cauchy Born models, or atomistic Cauchy Born and continuum Cauchy Born models are proved in a similar fashion, however we omit their (entirely similar statements Theorem 4 (VARIATIONAL CONSISTENCY Let y be a smooth function; then, for any v V ε, the continuum Cauchy Born variation DΦ CB (y, v is a second order approximation to the atomistic variation DΦ a (y, v ε in the sense that there exists a constant M V = M V (y, p, p, independent of v, such that DΦ CB (y DΦ a (y, v ε, v M V ε v W,p ( Proof If x l is the bottom-left vertex of K, we denote by K e the element that shares the edge x l, x l+ e with K Similarly, we denote by K e the element that shares the edge x l, x l+ e with K Then, one can rewrite (7 as (4 A a,cb, v ε := ε d K T + ε d K T ζ φ η ( y(m K η η + ζ φ η ( y(m Ke η η De v l ζ φ η ( y(m K η η + ζ φ η ( y(m Ke η η De v l One should compare directly the above expression with (36 Since, obviously, ζ φ η ( y(m K η η + ζ φ η ( y(m η η Ke (4 ζ φ η ( y(m,l η η C(y ε, the proof is complete in view of Propositions and 3 Theorem 4 (ENERGY CONSISTENCY Let y be a smooth function; then, the continuum Cauchy Born energy Φ CB (y is a second order approximation to the atomistic energy Φ a (y in the sense that there exists a constant M E = M E (y, such that Φ CB (y Φ a (y M E ε

16 6 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI Proof The proof is similar to, though considerably simpler than, the proof of the variational consistency We start from the continuum Cauchy Born energy Φ CB (y = W CB ( y(x dx = W CB ( y(x dx K T K = K W CB ( y(m K + W CB ( y(x W CB ( y(m K dx =: I + I, K T K T K where as before m K is the barycenter of K Using similar arguments as in the proof of Lemma 3 we get I C(y ε As before we will compare I with the atomistic energy Φ a (y Since m K is the barycenter of K the key point here is to rearrange the terms in Φ a (y in order to create symmetries around the cell K In fact, using the assumed periodicity of the lattice, we have that Φ a (y = ε d φ η (D η y l = ε d 4 φ η (D η y l + φ η (D η y l (η e + φ η (D η y l (η e + φ η (D η y l (η e (η e Next we use similar splittings as in Lemma 3: D η y l = D η e y l + D η e y l+η e + D η e y l + D η e y l+η e, D η y l (η e = D η e y l (η e + D η e y l (η e +η e + D η e y l (η e + D η e y l (η e +η e, D η y l (η e = D η e y l (η e + D η e y l (η e +η e + D η e y l (η e + D η e y l (η e +η e, D η y l (η e (η e = D η e y l (η e (η e + D η e y l (η e (η e +η e Observe now that, for example, + D η e y l (η e (η e + D η e y l (η e (η e +η e D η e y l + D η e y l (η e = y l+η e y l (η e ε = η + y l+e y l ε y(m,l + y(m,l + O(ε = η y(m,l + O(ε, where m,l is the midpoint of the edge x l, x l+e Notice that the above relations do not depend on the signs of η, η Using similar groupings and the fact that y(m,l + y(m,l+e + y(m,l+η e + y(m,l (η e = η y(m K + O(ε, η 4 one can deduce, by adopting the arguments of Lemma 3 to our case, that Φ a (y = ε d φ η ( y(m K η + O(ε The proof is thus complete

17 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 7 5 ANALYSIS OF CAUCHY BORN APPROXIMATIONS: d = 3 The analysis presented in the previous sections can be extended to three space dimensions The arguments are similar, but there are certain steps that differ, so we shall focus our attention on these, and we shall only present the key points here In particular, we shall omit arguments that are similar to those in two dimensions, unless this is necessary Linking the continuum model to the atomistic model is based on a three-dimensional atomistic Cauchy Born model (A-CB As before, to derive this model we start from the continuum model and perform appropriate approximation steps The final model has consistency error of the order O(ε compared to the continuum Cauchy Born model and the original atomistic model Trilinear finite elements on the lattice Let V ε be the linear space of all periodic functions on the lattice L that are continuous and piecewise trilinear on More precisely, let T := K : K = (x l, x l + (x l, x l + (x l3, x l3 +, x l = (x l, x l, x l3 discr, V ε := v : R d, v C(, v K Q (K and v l = v(x l periodic with respect to L, where Q (K denotes the set of all trilinear functions on K As before the elements of V ε can be expressed in terms of the nodal basis functions Ψ l = Ψ l (x as v(x = v l Ψ l (x Ψ l (x Ψ l3 (x 3, v l = v(x l, where we have used the fact that Ψ l (x can be written as the tensor product of the standard one-dimensional piecewise linear hat functions Ψ li (x i Here Ψ li (x li = δ li li 5 Consistency analysis of the Cauchy Born model: d = 3 Let, as before, v V ε and let y be a sufficiently smooth function Our aim is to approximate DΦ CB (y, v = S iα ( y(x α v i (x dx, v V ε As before, we consider a generic term of the above sum of the form: g( y(x v(x dx, v V ε Recall that, since v V ε, Let us now consider v(x = = ( v (l,l,l 3 ε xl,x l (x xl,x l +(x Ψ l (x Ψ l3 (x 3 ( v ε (l +,l,l 3 v (l,l,l 3 xl,x l +(x Ψ l (x Ψ l3 (x 3 G(x := G(x, x, x 3 = x x N g( y(x, x, x 3 dx

18 8 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI Then, g( y(x v(x dx = ε = ε = ε = ε + ε ( g( y(x v (l +,l,l 3 v (l,l,l 3 xl,x l +(x Ψ l (x Ψ l3 (x 3 dx xl + xl3 + x l x l3 xl + xl3 + x l x l3 xl + xl3 + x l = : A + A x l3 xl + xl3 + x l x l + x l g( y(x dx v (l +,l,l 3 v (l,l,l 3 Ψ l (x Ψ l3 (x 3 dx dx 3 G(x l +, x, x 3 G(x l, x, x 3 Ψ l (x Ψ l3 (x 3 dx dx 3 v (l +,l,l 3 v (l,l,l 3 ( G(x l +, x, x 3 G(x l, x, x 3 v (l +,l,l 3 v (l,l,l 3 x l3 Ψ l (x Ψ l3 (x 3 4 dx dx 3 G(x l +, x, x 3 G(x l, x, x 3 4 dx dx 3 v (l +,l,l 3 v (l,l,l 3 As in two space dimensions (cf Lemma one can show that A is second order accurate with respect to ε The term A, on the other hand, requires further simplification To this end, notice that A = ε = ε = ε + ε + ε + ε = K T xl + xl l L ˆ K x l x l3 xl + xl3 + x l x l3 xl + xl3 + 4 x l x l3 xl + x l x l x l3 xl + xl3 + x l x l3 xl + x l3 + x l x l3 g( y(x dx 4 G(x l +, x, x 3 G(x l, x, x 3 4 dx v (l +,l,l 3 v (l,l,l 3 x l + g( y(x dx x l x l + g( y(x dx x l x l + x l g( y(x dx x l + x l g( y(x dx x l + dx v (l +,l,l 3 v (l,l,l 3 dx v (l +,l,l 3 v (l,l,l 3 dx v (l +,l, l 3 + v (l,l, l 3 + dx v (l +,l +,l 3 v (l,l +,l 3 dx v (l +,l +, l 3 + v (l,l +, l 3 + g( y(x dx x l D e v l + D e v l+e + D e v l+e3 + D e v l+e +e 3,

19 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 9 where l = (l, l, l 3, l = (l, l, l 3, and ˆl = (l, l, l 3 We define the average discrete derivatives as follows: D e v l = (5 D e v l + D e v l+e + D e v l+e3 + D e v l+e +e 4 3 As in Lemma 3, one can show, with m K signifying the barycenter of K, that there exists a constant C = C(y such that K T K g( y(x dx K g( y(m K D e v l C(y ε v W,p ( As in two space dimensions, one can define, for any v V ε, the quantity ζ φ η ( y(m K η η De v l (5 A a,cb, v ε : = ε d K T +ε d K T ζ φ η ( y(m K η η De v l ζ φ η ( y(m K η η 3 De3 v l +ε d K T We thus deduce that A a,cb, v ε is a second order approximation to DΦ CB (y, v in the sense that DΦ CB (y, v A a,cb, v ε M ε v W,p ( 5 The atomistic Cauchy Born model: d = 3 Using the above definition of the average discrete derivatives we define the discrete gradient matrix as v l iα := D e α v i l, and the atomistic potential Φ a,cb (y := ε d φ η ( y l η = ε d W CB ( y l Now, for a given field of external forces f : L R d the atomistic Cauchy Born problem reads as follows: If such a minimizer exists, then find a local minimizer y a,cb in X of : Φ a,cb (y a,cb f, y a ε DΦ a,cb (y a,cb, v ε = f, v ε, for all v V As in two space dimensions, one can link A a,cb to DΦ a,cb as follows: let y V ε ; then, for any v V ε, (53 A a,cb, v ε = DΦ a,cb (y, v ε

20 0 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI 53 Comparison of atomistic Cauchy Born and atomistic models: d = 3 To compare the atomistic and atomistic Cauchy Born models we start from (4: DΦ a (y, v ε = ε d ζ φ η (D η y l D η v l Then, one can split D η v l and D η v l as follows: and D η v l = D η e v l + D η e v l+η e + D η3 e 3 v l+η e +η e D η v l = D η e v l + D η3 e 3 v l+η e + D η e v l+η e +η 3 e 3 Using similar splittings but starting with D η e v l and D η3 e 3 v l, we end up with six alternative expressions for D η v l Thus, by replacing D η v l with its average and grouping terms of the same type of discrete derivative we deduce that DΦ a (y, v ε = ε d + ε d + ε d ζ φ η (D η y l 3 D η e v l + 6 D η e v l+η e + 6 D η e v l+η3 e D η e v l+η e +η 3 e 3 ζ φ η (D η y l 3 D η e v l + 6 D η e v l+η e + 6 D η e v l+η3 e D η e v l+η e +η 3 e 3 ζ φ η (D η y l 3 D η 3 e 3 v l + 6 D η 3 e 3 v l+η e + 6 D η 3 e 3 v l+η e + 3 D η 3 e 3 v l+η e +η e Hence, since the lattice is periodic, we deduce that (54 DΦ a (y, v ε 3 ζφ η (D η y l + 6 ζφ η (D η y l ηe + 6 ζφ η (D η y l η3e ζφ η (D η y l ηe η 3e 3 = ε d D ηe v l 3 ζφ η (D η y l + 6 ζφ η (D η y l ηe + 6 ζφ η (D η y l η3e ζφ η (D η y l ηe η 3e 3 + ε d + ε d D ηe v l 3 ζφ η (D η y l + 6 ζφ η (D η y l ηe + 6 ζφ η (D η y l ηe + 3 ζφ η (D η y l ηe η e D η3e 3 v l As before we split, for η, η, η 3 > 0, D η e v l = D e v l + + D e v l+(η e, D η e v l = D e v l + + D e v l+(η e, D η3 e 3 v l = D e3 v l + + D e3 v l+(η3 e 3 In case where η α < 0 we use splittings of the form (3 In the sequel we will assume that η, η, η 3 > 0 The general case can be treated by adopting the arguments presented in two space dimensions (cf (33 and

21 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS Lemma 3 Therefore, by using the notation (55 Φ η,l, := Φ η,l, := Φ η,l,3 := we finally deduce that η k=0 η k=0 η 3 k=0 3 ζφ η (D η y l k e + 6 ζφ η (D η y l k e η e + 6 ζφ η (D η y l k e η 3 e ζφ η (D η y l k e η e η 3 e 3, 3 ζφ η (D η y l k e + 6 ζφ η (D η y l k e η e + 6 ζφ η (D η y l k e η 3 e ζφ η (D η y l k e η e η 3 e 3, 3 ζφ η (D η y l k e3 + 6 ζφ η (D η y l k e3 η e DΦ a (y, v ε = ε d + ε d + 6 ζφ η (D η y l k e3 η e + 3 ζφ η (D η y l k e3 η e η e, Φ η,l, D e v l Φ η,l, D e v l + ε d Next, we shall focus on the comparison of Φ η,l, with the term where m k,η,l Φ η,l, = η k=0 ζ φ η ( y(m k,η,lη, Φ η,l,3 D e3 v l is the midpoint of the side with endpoints x l k e, x l k e +η e, and m,l is the midpoint of the side with endpoints x l, x l+ e Let k = 0 We group together the 3 -terms in Φ η,l, We use the splittings D η v l = D η e v l + D η3 e 3 v l+η e + D η e v l+η e +η 3 e 3 + D η e v l + D η3 e 3 v l+η e + D η e v l+η e +η 3 e 3, and D η v l η e η 3 e 3 = D η e v l η e η 3 e 3 + D η3 e 3 v l η3 e 3 + D η e v l + D η e v l η e η 3 e 3 + D η3 e 3 v l η e η 3 e 3 +η e + D η e v l η e +η e These choices were made in order to create the required symmetries In fact, by taking into account that D η v l and D η v l η e η 3 e 3 are grouped together in (55 and observing that D η e v l + D η e v l+η e +η 3 e 3 + D η e v l η e η 3 e 3 = 4η y(m 0,η,l + O(ε, D η e v l + D η e v l+η e +η 3 e 3 + D η e v l η e η 3 e 3 + D η e v l η e +η e = 4η y(m 0,η,l + O(ε, D η3 e 3 v l+η e + D η3 e 3 v l+η e + D η3 e 3 v l η3 e 3 + D η3 e 3 v l η e η 3 e 3 +η e = 4η 3 3 y(m 0,η,l + O(ε,

22 CHARALAMBOS MAKRIDAKIS AND ENDRE SÜLI yields a second order approximation to 4 6 ζφ η ( y(m 0,η,lη, upon employing similar arguments as in Lemma 3 The remaining 6 factor is due to the 6 -terms in Φ η,l, In fact, we use the splittings D η v l k e η e = D η e v l k e η e + D η3 e 3 v l k e + D η e v l k e +η 3 e 3 + D η e v l k e η e + D η3 e 3 v l η e + D η e v l η e +η 3 e 3 and D η v l k e η 3 e 3 = D η e v l k e η 3 e 3 + D η3 e 3 v l k e η 3 e 3 +η e + D η e v l k e +η e + D η e v l k e η 3 e 3 + D η3 e 3 v l η3 e 3 + D η e v l η3 e 3 +η 3 e 3 As before, D η e v l k e +η 3 e 3 + D η e v l k e η e + D η e v l k e +η e + D η e v l k e η 3 e 3 = 4η y(m 0,η,l + O(ε, D η e v l k e η e + D η e v l η e +η 3 e 3 + D η e v l k e η 3 e 3 + D η e v l η3 e 3 +η 3 e 3 D η3 e 3 v l k e + D η3 e 3 v l η e + D η3 e 3 v l k e η 3 e 3 +η e + D η3 e 3 v l η3 e 3 We thus deduce that Φ η,l, Φ η,l, C(y ε Finally, as in the case of two space dimensions, we have that η ζ φ η ( y(m,l η Φ η,l, C(y ε, = 4η y(m 0,η,l + O(ε, = 4η 3 3 y(m 0,η,l + O(ε with m,l signifying the midpoint of the side with endpoints x l, x l+ e By employing entirely similar arguments as in the case of two space dimensions, we then deduce that Theorem 4 holds in three space dimensions as well Analogously, Theorem 4 is also valid in three dimensions 6 REMARKS ON CONVERGENCE We briefly discuss here how the consistency results presented in this paper, combined with appropriate local stability properties of the Cauchy Born solution, can imply local convergence In one space dimension these results are based on a simple application of the inverse function theorem, see 36, 3, 33, see also 4 The stability in multiple space dimensions is very subtle (see, for example, 4 and is beyond the scope of this paper A key assumption in the inverse function theorem in the form presented in, Theorem is that G(w = DΦ a (w is differentiable with bounded inverse at a given point w X Next, for the sake of simplicity of the presentation, we shall assume that in an appropriate discrete space X ε, DΦ a (y CB evaluated on a sufficiently smooth solution y CB of (6, whose existence is assumed, is differentiable with bounded inverse Then, upon verifying two further assumptions on DΦ a and y CB one can apply the inverse function theorem to infer the local existence of a solution of the atomistic problem y a in a ball with center y CB Hence, in addition, one obtains an estimate of the form y a y CB Xε γ DΦ a (y CB f Yε

23 FINITE ELEMENT ANALYSIS OF CAUCHY BORN APPROXIMATIONS TO ATOMISTIC MODELS 3 Here γ depends on the norm of the inverse of the derivative of DΦ a (y CB and DΦ a (y CB f Yε = sup DΦ a (y CB f, v ε v Xε : v X ε with v Xε 0 Thus, to estimate the error we observe that DΦ a (y CB, v ε f, v ε = DΦ a (y CB, v ε DΦ CB (y CB, v + f, v f, v ε =: I + I Let us also suppose that X ε is a subspace of V ε equipped with W,p ( norm Then, the consistency result of Theorem 4 implies that In addition, I C(y CB ε v W,p ( C(y CB ε v Xε ( I = f, v f, v ε = fv QI (fv dx = ( fv QI (fv dx, K T K where Q I denotes the standard nodal interpolation operator on V ε Let us denote by ζ the functional ζ(w = (6 w QI (w dx, w C(K, K K and observe that ζ(ϕ = 0 for all ϕ Q (K Lemma and inequality ( then yield that (6 ζ(w C ε d αw L (K α= Further, since v V ε, we have that α(fv L (K 3 f W, (K v W, (K, and we deduce, using similar arguments as in the proof of Lemma 3, that We thus arrive at the error bound I C f W, ( ε v W,p ( C ε v Xε y a y CB Xε C(y CB, f ε, where y a y CB Xε is a discrete W,p ( norm corresponding to y a Q I y CB W,p ( Acknowledgements CM was partially supported by the FP7-REGPOT project ACMAC: Archimedes Center for Modeling, Analysis and Computations of the University of Crete ES was partially supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E03507/ REFERENCES A Abdulle, P Lin, and A V Shapeev Numerical methods for multilattices Technical report, arxiv:07346, 0 M Arndt and M Griebel Derivation of higher order gradient continuum models from atomistic models for crystalline solids Multiscale Model Simul, 4(:53 56 (electronic, M Arndt and M Luskin Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model Multiscale Model Simul, 7(:47 70, M Arndt and M Luskin Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel- Kontorova model Comput Methods Appl Mech Engrg, 97(49-50: , 008

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