On atomistic-to-continuum couplings without ghost forces

On atomistic-to-continuum couplings without ghost forces Dimitrios Mitsoudis ACMAC Archimedes Center for Modeling, Analysis & Computation Department of Applied Mathematics, University of Crete & Institute of Applied & Computational Mathematics FORTH International Conference on Applied Mathematics, September 16 20, Heraklion, Crete joint work with Charalambos Makridakis and Phoebus Rosakis Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 1 / 39

Outline Description of the problem and notation Atomistic-to-continuum passage: Consistency of Cauchy-Born approximations Makridakis & Süli, ARMA, 2013 Atomistic/Continuum coupling: Design of methods that are free of ghost forces Makridakis, Mitsoudis, Rosakis (to appear in AMRX) Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 2 / 39

Motivation Problems in materials science, esp. crystalline solids that exhibit defects (e.g. vacancies, cracks). Defects cannot be (in general) described with a continuum model. Try to obtain accurate solutions by decomposing the reference lattice into an atomistic region that contains the neighbourhood of defects (exact atomistic model is used) and a continuum region where the deformations are smooth (continuum elasticity model is used e.g. coarse f.e. discretizations). Atomistic-to-continuum (A/C) couplings may be viewed as a class of computational multiscale schemes. Goal: Achieve a significant reduction in computational cost compared to the full atomistic description Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 3 / 39

Notation Z 2 is the 2D lattice, and let {e 1, e 2} be the standard basis of R 2. Scaled lattice: εz 2 = {x l = (x l1, x l2 ) = εl, l Z 2 } lattice distance ε = 1/k, k Z + Consider discrete periodic functions on Z 2 defined over a periodic domain L. Ω := (0, M 1] (0, M 2], M 1, M 2 Z + Ω discr := εz 2 Ω, L := Z 2 1 Ω ε Keep in mind a similar setting in 3D Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 4 / 39

Notation Functions and spaces: We consider atomistic deformations y l = y(x l ), l L, where y l = F x l + v l, with v l = v(x l ) periodic w.r.t. L. F is a const., 2 2 matrix with det F > 0. The corresponding spaces for y and v are denoted by X and V: X = {y : Ω discr R 2, y l = F x l + v l, v V}, V = {u : Ω discr R 2, u l = u(x l ) periodic with zero average w.r.t L}. For y, v : Ω discr R 2 we define y, v ε := ε 2 l L y l v l Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 5 / 39

Notation X = {y : Ω R 2, y(x) = F x + v(x), v V }, V = {u : Ω R 2, u W k,p (Ω, R 2 ) W 1,p (Ω, R 2 ), Ω udx = 0}., standard L 2 inner product. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 6 / 39

Discrete and continuous derivatives To avoid confusion D ηy l = y l+η y l, ε l, l + η L, φ(ζ1, ζ2) ζi φ(ζ) =, ζ i { } ζ = (ζ 1, ζ 2), ζ φ(ζ) = ζi φ(ζ), i v(x) = v(x), x i { u i (x) u(x) = x α }iα ζi derivatives w.r.t. arguments (usually appear in composite functions) i derivatives w.r.t. the spatial variable x i i. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 7 / 39

The atomistic problem Consider the atomistic energy Φ a (y) := ε 2 φ η(d ηy l ). l L η R R: finite set of interaction vectors, φ η( ) suff. smooth pair potential and may vary with η. (Atomistic problem) For a given field of external forces f : Ω discr R 2, find a local minimizer y a X of : Φ a (y a ) f, y a ε If such a minimizer exists, then DΦ a (y a ), v ε = f, v ε, for all v V, where DΦ a (y), v ε = ε 2 l L η R ζ φ η (D ηy l ) D ηv l. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 8 / 39

The continuum Cauchy-Born model The corresponding Cauchy-Born stored energy function is W (F ) = W CB(F ) = η R φ η (F η). (Continuum Cauchy-Born problem) Find a local minimizer y CB in X of : Φ CB (y CB ) f, y CB, where Φ CB (y CB ) = W CB( y CB ) and f discrete external forces Ω Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 9 / 39

If such a minimizer exists, then DΦ CB (y CB ), v = f, v, v V, where DΦ CB (y CB ), v = S iα ( y(x)) αv i (x) dx The stress tensor is defined as { W (F ) S := and is related to the atomistic potential through Ω F iα }iα, S iα = W (F ) F iα = η R ζi φ η (F η) η α. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 10 / 39

Remarks The interaction potential is non-convex: e.g. φ η(r) = V ( r ), V Lennard-Jones: Reverse point of view: The atomistic problem is the exact problem (discrete difference scheme) and we want to find a continuum approximation (a PDE) Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 11 / 39

Consistency analysis We need a sharp analysis on the approximation properties of the continuum model Variational Consistency error: { C V (y) := sup DΦ a (y), v ε DΦ CB (y), v : v V with v W 1,p (Ω) = 1 where y is any smooth function. Energy Consistency error C E(y) := Φ a (y) Φ CB (y). }, Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 12 / 39

One-dimensional example. The simplest model: Φ a (y) = ε l L 2 φ r (D r y l ) = ε φ 1 (D 1y l ) + φ 2 (D 2y l ). l L r=1 Then the corresponding Cauchy-Born stored energy function is W (F ) = W CB (F ) = 2 r=1 φ r (F r) = φ 1 (F ) + φ 2 (2F ). Then the atomistic Cauchy-Born model is defined as Φ a,cb (y) = ε l L W (D 1y l ) = ε l L φ 1 (D 1y l ) + φ 2 (2D 1y l ). The continuous energy is Φ CB (y) = W (y (x))dx. Ω Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 13 / 39

Atomistic Cauchy-Born model in multi-d Construct an intermediate model (atomistic Cauchy-Born model (A-CB)) which may serve as a link between the continuous and the atomistic model. To construct this model we start from the continuous model and perform appropriate approximate steps yielding finally the A-CB model. Bilinear Finite Elements on the Lattice: Let V ε,q be the space of bilinear periodic functions on the lattice L. Specifically, let T Q = {K Ω : K = (x l1, x l1 +1) (x l2, x l2 +1), x l = (x l1, x l2 ) Ω discr}, V ε,q = {v : Ω R 2, v C(Ω), v K Q 1 (K) and v l = v(x l ) periodic}, where Q 1 (K) is the set of bilinear functions on K: v K (x) = α 0 + α 1 x 1 + α 2 x 2 + α 3 x 1 x 2. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 14 / 39

The atomistic Cauchy-Born model Definition of the model. We define average discrete derivatives as follows: D e1 v l = 1 } {D e1 v l + D e1 v l+e2, 2 D e2 v l = 1 } {D e2 v l + D e2 v l+e1, 2 and the discrete gradient matrix as { v l } iα = D eα v i l. We introduce the atomistic Cauchy-Born energy Φ a,cb (y) = ε 2 φ η ( y l η) l L η R = ε 2 l L W CB( y l ). Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 15 / 39

Now, for a given field of external forces f : Ω discr R 2 (Atomistic Cauchy-Born (A-CB) problem) Find a local minimizer y a,cb in X of : Φ a,cb (y a,cb ) f, y a ε. If such a minimizer exists, then DΦ a,cb (y a,cb ), v ε = f, v ε, for all v V. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 16 / 39

Variational consistency Theorem (Variational consistency) Let y be a smooth function; then, for any v V ε,q, there exists a constant M V = M V (y, p), 1 p, independent of v, s.t. DΦ CB (y), v DΦ a (y), v ε MV ε 2 v W 1,p (Ω). In addition, there exists a constant M V = M V (y, p), 1 p, independent of v, s.t. DΦ a,cb (y), v ε DΦ a (y), v ε M V ε 2 v W 1,p (Ω). Makridakis & Süli, ARMA, 2013 Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 17 / 39

Energy consistency Theorem (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) s.t. Φ a (y) Φ CB (y) M E ε 2. In addition, there exists a constant M E = M E (y) s.t. Φ a (y) Φ a,cb (y) M E ε 2. Makridakis & Süli, ARMA, 2013 Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 18 / 39

Remarks Construct A-CB models based on triangular and tetrahedral meshes Extension to multibody potentials Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 19 / 39

Towards constructing ghost-force free methods in multi-d What is a ghost-force free coupling? The energy E is said to be free of ghost forces, if DE(y F ), v = 0, y F (x) = Fx, for all appropriate v : Ω L R 2 such that v l = 0 outside a compact set. 2nd componet of y qc!x Ad-hoc coupling of 4energies leads to ghost forces... 2 0!2!4!6 60 x 10!3 50 40 40 30 20 20 10 0 0 Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 20 / 39 D + 2 (y qc!x) 2 0.2 0.1 0!0.1!0.2 60

State of the art 1D & 2D: Energy based couplings free of ghost forces have been constructed recently. 3D: (later) 1D : Li & Luskin, 2011 and Shapeev, 2011 2D : Shapeev, 2011 Other works (mainly special cases) : Belytschko et. al., 2002, Shimokawa et. al., 2004, E, Lu & Yang, 2006, Ortner & Zhang, 2011, Shapeev, 2011 (3D) See, also, Luskin & Ortner, Acta Numerica, 2013 for a recent review. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 21 / 39

Towards constructing ghost-force free methods in multi-d Notation Let Ω, Ω a and Ω each be the interior of the closure of the union of lattice cells K T Q and connected, and suppose Ω = Ω a Ω, Γ = Ω a Ω. Here Γ is the interface between the atomistic and the A-CB regions. Fix η R and define the bond b l = {x R 2 : x = x l+tη, 0 < t < 1}. B η is the set consisting of all bonds b = b l for l L. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 22 / 39

The approach of Shapeev Work with each bond separately Represent long-range differences as line integrals over bonds: y η = D ηy l b l In 2D is then possible to transform the assembly of line integrals over all possible interactions into an area integral through a counting argument known as bond density lemma Lemma (Shapeev) Let S be a set consisting of unions of triangles T T T. Then for any fixed η R the following identity holds: χ S dτ = S. b B b η Limitations: Lemma valid only in 2D, piecewise linears over triangles. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 23 / 39

Towards constructing ghost-force free methods in multi-d A bond volume approach Represent long-range differences as volume integrals over bond volumes. Construction of an underlined globally continuous function representing the coupled modeling method. Work in two phases: first, use in the continuum region appropriate atomistic Cauchy-Born models. Subsequently, use in the continuum region f.e. s of arbitrary high order in a coarser mesh. Works in both 2 and 3D. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 24 / 39

2D: bond volumes and long range differences We construct methods based on bond volumes instead of bonds. For fixed η R, bond volume B l, η is the interior of a rectangle with diagonal b l, i.e., Lemma B l, η is the open quadrilateral with vertices x l, x l+η1 e 1, x l+η, x l+η2 e 2. Let v Q 1(B l, η ). Then ε 2 D ηv l = 1 v(x)η dx. η 1 η 2 B l, η x`+ B`, x` Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 25 / 39

2D: bond volumes and energies The method is designed with respect to bond volumes B l,η. In particular, we consider three cases determined by the location of each bond volume B l,η a. The closure of the bond volume is contained in the atomistic region: B l,η Ω a b. The bond volume is contained in the region Ω : B l,η Ω c. The bond volume intersects the interface: Either B l, η Γ or if B l, η Ω a and B l,η Γ. We denote by B Γ the set of these bond volumes. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 26 / 39

2D: bond volumes and energies For a fixed η, the contribution to the energy corresponding to a) is: EΩ a a,η{y} = ε 2 φ η(d ηy l ). l L B l, η Ω a The contribution to the energy from the atomistic CB region will be E a,cb Ω,η {y} = Ω φ η( y(x)η)dx, where y(x) = l L K l Ω χ Kl (x) y l Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 27 / 39

2D: energies on the interface For each bond volume intersecting the interface we denote y l,η to be a continuous piecewise polynomial function on an appropriate decomposition T (B l,η ) of B l,η satisfying ( ) only requirement: conforming gluing of T (B l,η ) with the neighboring bond volumes. x`+ B`, x` Corresponding energy: E Γ,η {y} = l L B l, η B Γ 1 χ η Ωa φ η( y l,η η) dx. 1η 2 B l, η Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 28 / 39

Total energy the total energy is defined through E bv {y} = η R E η{y} where E η{y} = E a Ω a,η{y} + E a,cb Ω,η {y} + E Γ,η{y}. x`+ x`+ B`, B`, x` x` Figure: Alternative decompositions T (B l,η ) of B l,η for two different bonds. Energy free of ghost forces Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 29 / 39

Energy free of ghost forces: Idea of the proof First we fix η and we consider decompositions consisting of bond volumes which cover R 2 : } SB m η := {B l,η : (i) B l,η B j,η =, if l j, (ii) R 2 = B l,η, m = 1,..., η 1 η 2. The number of different such coverings is η 1 η 2. Note that bond volumes corresponding to different m may overlap, but within a single SB m η its elements form a decomposition of non-overlapping bond volumes. S k Bη S k Bη Figure: Two different coverings S k B η and S k B η Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 30 / 39

Energy free of ghost forces: Idea of the proof II For a fixed η we have (?) 0 = DE η(y F ), v = { =φ h (F η) ε 2 l L B l, η Ω a D ηv l + v(x)η dx + Ω The main idea of the proof is to write the above as l L B l, η B Γ η 1 η 2 1 v [m] (x)η dx η 1 η 2 m=1 Ω 1 } χ η 1 η 2 Ωa v l,η η dx B l, η where v [m], m = 1,..., η 1 η 2 are appropriate conforming functions (in H 1 (Ω)) each one associated to a different covering SB m η consisting of bond volumes. For the first term above note ε 2 l L B l,η Ω a D ηv l = l L B l,η Ω a 1 v [m] (x)η dx η 1 η 2 B l,η Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 31 / 39

Construction in 3D We need to work with tetrahedra x l+e3 x l+e2+e3 x l+e1+e3 x l+e1+e2+e3 x l x l+e2 x l+e1 x l+e1+e2 Figure: A type-a decomposition of the cell K l into six tetrahedra. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 32 / 39

Construction in 3D: Atomistic CB model in tetrahedra T T = {T Ω : T is a tetrahedron whose vertices are lattice vertices of K l, x l Ω discr}, V ε,t := {v : Ω R 2, v C(Ω), v T P 1 (T ) and v l = v(x l ) periodic with respect to L}, Φ a,cb (y) := ε3 6 l L T K l (T ) η R φ η ( y η) = ε3 6 l L T K l (T ) W CB( y). For v V ε,t { } v T := D eα vl i, iα where D eα vl i on T are the difference quotients of v along the edges of T with directions eα. E.g. De3 vl i = De v i 3 l and D e2 vl i = De v i 2 l+e 1 +e 3. xl+e3 xl+e1+e3 xl+e1+e2+e3 xl Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 33 / 39

A key result x l+η x l B l,η Figure: A bond volume B l, η and its type-a decomposition into six tetrahedra. Lemma Let v be a piecewise linear and continuous function on a type-a decomposition of the bond volume B l,η into tetrahedra. Then ε 3 1 D ηv l = v(x)η dx. η 1 η 2 η 3 B l,η Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 34 / 39

Sketch of proof 1 η 1 η 2 η 3 1 v(x)η dx = v ν η ds B l, η η 1 η 2 η 3 B l, η 1 3 { } = ( η i )v ds + η i v ds, η 1 η 2 η 3 i=1 B l, η ( e i ) B l, η (e i ) B l,η (e i ): face of B l,η with outward unit normal e i. if τ is a triangle on a face with vertices z i, then η i v ds = τ 3 η i v(z j ), τ 3 j=1 Since τ is one of the two triangles of B l, η (e i ), τ η i = ε2 2 η 1 η 2 η 3. 1 η 1 η 2 η 3 B l, η (e i ) η i v ds = ε2 6 2 { v(zj ) + 2 v( z j ) }. Note that x l+η (resp. x l ) is a shared vertex at each B l,η (e i ), (resp. B l, η ( e i )), i = 1, 2, 3. 1 v(x) η dx = ε 2 ( ) v l+η v l, η 1 η 2 η 3 B l, η j=1 Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 35 / 39

Construction in 3 D N.B. The result is sensitive to the particular decomposition of the bond volume B l,η into tetrahedra. x` Decomposition into 5 tetrahedra, one of them without faces on the boundary Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 36 / 39

Construction in 3 D: Energy on interface E Γ,η {y} = l L B l,η B Γ 1 η 1η 2η 3 B l,η χ Ωa φ η( y l,η η) dx. B`, Figure: A possible decomposition T (B l, η ) of B l, η. Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 37 / 39

Construction in 3D The energy is free of ghost forces, in the sense that DE bv (y F ), v = 0, y F (x) = F x, Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 38 / 39

Comments The previous analysis may lead to energy-based methods that employ coarse mesh and high-order f.e. approximations of the Cauchy-Born energy in the continuum region (still remaining ghost-force free). There are several alternative ways to treat the interface and its discretization. Possibility of using discontinuous finite elements Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 39 / 39

Comments The previous analysis may lead to energy-based methods that employ coarse mesh and high-order f.e. approximations of the Cauchy-Born energy in the continuum region (still remaining ghost-force free). There are several alternative ways to treat the interface and its discretization. Possibility of using discontinuous finite elements Thank you for your attention Dimitrios Mitsoudis (ACMAC) A/C couplings ICAM 2013, Heraklion, Crete 39 / 39