Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor)
Q. Du · M. D. Gunzburger · R. B. Lehoucq · K. Zhou A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws
DRAFT date 15 May 2011 Received: date / Accepted: date 2010-8353J
Q. Du and K. Zhou were supported in part by the U.S. Department of Energy Office of Science under grant DE-SC0005346. M. Gunzburger was supported in part by the U.S. Department of Energy Office of Science under grant DE-SC0004970. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under contract DE-AC04-94AL85000. SAND Q. Du Department of Mathematics Penn State University Tel.: +1-814-865-3674 State College, PA 16802, USA Labs Fax: +1-814-865-3735 E-mail: [email protected] M. D. Gunzburger Department of Scientific Computing Florida State University Tallahassee FL 32306-4120, USA Tel.: +1-850-644-7060 Fax: +1-850-644-0098 E-mail: [email protected]
National R. B. Lehoucq Sandia National Laboratories P.O. Box 5800, MS 1320 Albuquerque NM 87185-1320, USA Tel.: +1-505-845-8929 Fax: +1-505-845-7442 E-mail: [email protected] K. Zhou Sandia Department of Mathematics Penn State University State College, PA 16802, USA Tel.: +1-814-865-3674 Fax: +1-814-865-3735 E-mail: [email protected] 2 Q. Du et al.
Abstract A vector calculus for nonlocal operators is developed, including the definition of nonlocal generalized divergence, gradient, and curl operators and the derivation of their adjoint operators. Analogs of several theorems and identities of the vector calculus for differential operators are also presented. A subsequent incorporation of volume constraints gives rise to well-posed volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators. This is demonstrated via some examples. Relationships between the nonlocal operators and their differential counter- parts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. An appli- cation of the nonlocal calculus is to pose abstract balance laws with reduced regularity requirements. Keywords nonlocal · vector calculus · continuum mechanics · peridynamics
1 Introduction
We introduce a vector calculus for nonlocal operators that mimics the classi- cal vector calculus for differential operators. We define, within a Hilbert space setting, generalized nonlocal representations of the divergence, gradient, and curl operators and deduce the corresponding nonlocal adjoint operators. Non- local analogs of the Gauss theorem and the Green’s identities of the vector calculus for differential operators are also derived. The nonlocal vector cal- culus can be used to define nonlocal volume-constrained problems that are analogous to boundary-value problems for partial differential operators. In addition, we establish relationships between the nonlocal operators and their differential counterparts. A balance law postulates that the rate of change of an extensive quantity over any subregion of a body is given by the rate at which that quantity is produced in the subregion minus the flux out of the subregion through its boundary. Along with kinematics and constitutive relations, balance laws are a cornerstone of continuum mechanics. Difficulties arise, however, in, for example, the classical equations of continuum mechanics due to, e.g., shocks, corner singularities, and material failure, all of which are troublesome when defining an appropriate notion for the “flux through the boundary of the subregion.” The nonlocal vector calculus we develop has an important ap- plication to balance laws that are nonlocal in the sense that subregions not in direct contact may have a non-zero interaction. This is accomplished by defining the flux in terms of interactions between possibly disjoint regions of positive measure possibly sharing no common boundary. An important feature of the nonlocal balance laws is that the significant technical details associated with determining normal and tangential traces over the bound- aries of suitable regions is obviated when volume interactions induce flux. Our nonlocal calculus, then, is an alternative to standard approaches for cir- cumventing the technicalities associated with lack of sufficient regularity in local balance laws such as measure-theoretic generalizations of the Gauss- Green theorem (see, e.g., [4,18]) and the use of the fractional calculus (see, e.g., [1,24]). Nonlocal vector calculus, volume-constrained problems and balance laws 3
Preliminary attempts at a nonlocal calculus were the subject of [11,12], which included applications to image processing1 and steady-state diffusion, respectively. However the discussion was limited to scalar problems. In con- trast, this paper extends the ideas in [11,12] to vector and tensor fields and beyond the consideration of image processing and steady-state diffusion. For example, the ideas presented here enable an abstract formulation of the bal- ance laws of momentum and energy and for the peridynamic2 theory for solid mechanics that parallels the vector calculus formulation of the balance laws of elasticity. The nonlocal vector calculus presented in this paper, however, is sufficiently general that we envisage application to balance laws beyond those of elasticity, e.g., to the laws of fluid mechanics and electromagnetics. The paper is organized as follows. The nonlocal vector calculus is de- veloped in Sections 2 and 3. In Section 2, nonlocal generalizations of the differential divergence, gradient, and curl operators and of the corresponding adjoint operators are given as are nonlocal generalizations of several theorems and identities of the vector calculus for differential operators. In Section 3, we introduce the notion of nonlocal constraint operators that allow us to define
2010-8353J nonlocal volume-constrained problems that generalize well-known boundary- value problems for partial differential operators. Amended versions of several of the theorems and identities of the nonlocal vector calculus are derived that account for the constraint operators. Connections between the nonlo- cal operators and their differential counterparts are made in Sections 4 and 5. In Section 4, we connect the two classes of operators in a distributional
SAND sense whereas, in Section 5, the connections are made between weighted in- tegrals involving the nonlocal adjoint operators and weak representatives of their differential counterparts. The connections made in those two sections justify the use of the terminology “nonlocal divergence, gradient, and curl” operators to refer to the nonlocal operators we define. In Section 6, a brief
Labs review of the conventional notion of a balance law is provided after which an abstract nonlocal balance law is discussed; also, a brief discussion is given of the application of our nonlocal vector calculus to the peridynamic theory for continuum mechanics.
2 A nonlocal vector calculus
National We develop a nonlocal vector calculus that mimics the classical vector calcu- lus for differential operators. The nonlocal vector calculus involves two types of functions and two types of nonlocal operators. Point functions refer to functions defined at points whereas two-point functions refer to functions de- fined for pairs of points. Point operators map two-point functions to point functions whereas two-point operators map point functions to two-point func- tions so that the nomenclature for operators refer to their ranges. Point and Sandia 1 The authors of [11] refer to [25] where a discrete nonlocal divergence and gra- dient are introduced within the context of machine learning. 2 Peridynamics was introduced in [20,22]; [23] reviews the peridynamic balance laws of momentum and energy and provides many citations for the peridynamic theory and its applications. See Section 6.1 for a brief discussion. 4 Q. Du et al. two-point operators are both nonlocal. Point operators involve integrals of two-point functions whereas two-point operators explicitly involve pairs of point functions. We now make more precise the definitions given above. To this end, for a positive integer d, let Ω denote an open subset of Rd. Points in Rd are denoted by the vectors x, y, or z and the natural Cartesian basis is denoted by e1,..., ed. Let n and k denote positive integers. Functions from Ω or subsets of Ω into Rn×k or Rn or R are referred to as point functions or point mappings and are denoted by Roman letters, upper-case bold for tensors, lower-case bold for vectors, and plain-face for scalars, respectively, e.g., U(x), u(x), and u(x), respectively. Functions from Ω × Ω or subsets of Ω × Ω to Rn×k, or Rn, or R are referred to as two-point functions or two-point mappings and are denoted by Greek letters, upper-case bold for tensors, lower-case bold for vectors, and plain-face for scalars, respectively, e.g., Ψ(x, y), ψ(x, y), and ψ(x, y), respectively. Symmetric and antisymmetric two-point functions satisfy, e.g., for the scalar two-point function ψ(x, y), we have ψ(x, y) = ψ(y, x) and ψ(x, y) = −ψ(y, x), respectively. The dot (or inner) product of two vectors u, v ∈ Rn is denoted by u · v ∈ R; the dyad (or outer) product is denoted by u ⊗ w ∈ Rn×k whenever w ∈ Rk; given a second-order tensor (matrix) U ∈ Rk×n, the tensor-vector (or matrix-vector) product is denoted by U · v and is given by the vector whose components are the dot products of the corresponding rows of U with v.3 For n = 3, the cross product of two vectors u and v is denoted by u × v ∈ R3. The Frobenius product of two second-order tensors A ∈ Rn×k and B ∈ Rn×k, denoted by A: B, is given by the sum of the component-wise n×n product of the two tensors. The trace of B ∈ R , denoted by tr B , is given by the sum of the diagonal elements of B. Inner products in L2(Ω) and L2(Ω × Ω) are defined in the usual manner. For example, for vector functions, we have Z n (u, v)Ω = u · v dx for u(x), v(x) ∈ R Ω Z Z n (µ, ν)Ω×Ω = µ · ν dydx for µ(x, y), ν(x, y) ∈ R Ω Ω with analogous expressions involving the Frobenius product and the ordinary product for tensor and scalar functions, respectively.
2.1 Nonlocal point divergence, gradient, and curl operators
The nonlocal point divergence, gradient, and curl operators map two-point functions to point functions and are defined as follows. These operators along with their adjoints are the building blocks of our nonlocal calculus. Definition 1 [Nonlocal point operators] Given the vector two-point function ν : Ω × Ω → Rk and the antisymmetric vector two-point function 3 In matrix notation, the inner, outer, matrix-vector products are given by x·y = xT y, x ⊗ y = xyT , and U · v = UT v. Nonlocal vector calculus, volume-constrained problems and balance laws 5
k α: Ω × Ω → R , the nonlocal point divergence operator D ν : Ω → R is defined as Z D ν(x) := ν(x, y) + ν(y, x) · α(y, x) dy for x ∈ Ω. (1a) Ω
Given the scalar two-point function η : Ω × Ω → R and the antisymmet- ric vector two-point function β : Ω × Ω → Rn, the nonlocal point gradient k operator G η : Ω → R is defined as Z G η(x) := η(x, y) + η(y, x)β(y, x) dy for x ∈ Ω. (1b) Ω
Given the vector two-point function µ: Ω × Ω → R3 and the antisymmetric vector two-point function γ : Ω × Ω → R3, the nonlocal point curl operator 3 C µ : Ω → R is defined as
2010-8353J Z C µ (x) := γ(y, x) × µ(x, y) + µ(y, x) dy for x ∈ Ω. (1c) Ω The nonlocal point operators D, G, and C map vectors to scalars, scalars to vectors, and vectors to vectors, respectively, as is the case for the divergence, gradient, and curl differential operators. Relationships between the nonlocal
SAND point operators and differential operators are made in Sections 4 and 5 where we demonstrate circumstances under which the nonlocal point operators are identified with the differential operators in the sense of distributions and as weak representations, respectively. For the sake of simplicity, in much of the rest of the paper, we introduce
Labs the following notation:
α := α(x, y) α0 := α(y, x) ψ := ψ(x, y) ψ0 := ψ(y, x) u := u(x) u0 := u(y) u := u(x) u0 := u(y)
and similarly for other functions.4 National 2.2 Nonlocal integral theorems
Definition 1 leads to integral theorems for the nonlocal vector calculus that mimic basic integral theorems of the vector calculus for differential opera- tors;in particular, (2a) can be viewed as nonlocal Gauss’ theorem. The an- tisymmetry, with respect to x and y, of the integrands in the definitions of
Sandia the nonlocal point operators (1) plays a crucial role.
Theorem 1 [Nonlocal integral theorems] Let the functions ν, η, and µ and the operators D, G, and C be defined as in Definition 1. Then, we
4 R 0 For example, from (1a), we have D ν (x) = Ω (ν + ν ) · α dy. 6 Q. Du et al.
have the following analogs of the integral theorems of the vector calculus for differential operators: Z D ν dx = 0 (2a) Ω Z G η dx = 0 (2b) Ω Z C µ dx = 0. (2c) Ω Proof From (1a), we have that Z Z Z D ν dx = ν + ν0 · α dy dx. Ω Ω Ω Because α is antisymmetric, the integrand in the double integral is anti- symmetric, i.e., we have that (ν + ν0) · α = −(ν0 + ν) · α0, from which it easily follows that that double integral vanishes. Thus, (2a) follows. The nonlocal integral theorems (2b) and (2c) follow in exactly the same way from 5 (1b) and (1c), respectively. The nonlocal integral theorems (2a)–(2c) have the simple consequences listed in the following corollary that can be viewed as nonlocal integration by parts formulas.
Corollary 1 [Nonlocal integration by parts formulas] Adopt the func- tions and operators appearing in Definition 1. Then, given the point functions u(x): Ω → R, v(x): Ω → Rn, and w(x): Ω → R3, we have6 Z Z Z uD ν dx + (u0 − u)α · ν dydx = 0 (3a) Ω Ω Ω Z Z Z v ·G η dx + (v0 − v) · βη dydx = 0 (3b) Ω Ω Ω Z Z Z w ·C µ dx − γ × (w0 − w) · µ dydx = 0. (3c) Ω Ω Ω Proof Let ξ = uν; then, (ξ + ξ0) · α = (uν + u0ν0) · α = u(ν + ν0) · α + (u0 − u)ν0 · α so that, from (1a), we have Z D(ξ) = u(ν + ν0) · α + (u0 − u)ν0 · α dy ∀ x ∈ Ω. Ω 5 Alternately, (1b) and (1c) can be derived from (1a); one simply chooses ν = ηb and ν = b × µ, respectively, in (1a) where b is a constant vector; one also has to associate β and γ with α. 6 Whenever ν and α are scalar valued functions, the integration by parts formula (3a) appears in [13, Lemma 2.1]. Nonlocal vector calculus, volume-constrained problems and balance laws 7
Then, (2a) results in Z Z Z Z Z D(ξ) dx = u (ν + ν0) · αdy dx + (u0 − u)ν0 · α dydx = 0 Ω Ω Ω Ω Ω so that, by (1a) and the anti-symmetry of α, Z Z Z uD ν dx + (u − u0)ν0 · α0 dydx = 0. Ω Ω Ω A change of variables in the double integral then results in (3a). In a similar manner, (3b) and (3c) can be derived from (2a) along with (1b) and (1c), respectively. Alternately, (3b) and (3c) easily follow by setting, for an arbitrary constant vector b, ν = ηb and ν = b × µ, respectively, in (3a) and also associating β and γ with α.
2010-8353J 2.3 Nonlocal adjoint operators
The adjoints of the nonlocal point operators are two-point operators and are defined as follows.
Definition 2 Given a point operator L that maps two-point functions F to point functions defined over Ω, the adjoint operator L∗ is a two-point operator SAND that maps point functions G to two-point functions defined over Ω × Ω that satisfies G, L(F ) − L∗(G),F = 0. (4) Ω Ω×Ω Here, the operators F and G may denote pairs of vector-scalar, scalar-vector,
Labs or vector-vector functions. Corollary 1 can be used to immediately determine the adjoint operators corresponding to the point operators defined in Definition 1.
Corollary 2 [Nonlocal adjoint operators] Given the point function u(x): Ω → R, the adjoint of D is the two-point operator given by D∗ u(x, y) = −(u0 − u)α for x, y ∈ Ω, (5a) National
∗ k n where D u : Ω × Ω → R . Given the point function v(x): Ω → R , the adjoint of G is the two-point operator given by
G∗ v(x, y) = −(v0 − v) · β for x, y ∈ Ω, (5b)
∗ 3 where G v : Ω × Ω → R. Given the point function w(x): Ω → R , the Sandia adjoint of C is the two-point operator given by
C∗ w(x, y) = γ × (w0 − w) for x, y ∈ Ω, (5c)
∗ 3 where C w : Ω × Ω → R . 8 Q. Du et al.
Proof Let F = ν and G = u in (4). Then, (5a) immediately follows from (3a). Similarly, (5b) and (5c) are immediate consequences of (3b) and (3c), respectively, and (4). We can rewrite nonlocal integration by parts formulas in terms of the non- local adjoint operators given by Corollaries 1 and 2, respectively as follows:
Z Z Z uD ν dx − D∗ u · ν dy dx = 0 (6a) Ω Ω Ω Z Z Z v ·G η dx − G∗ vη dy dx = 0 (6b) Ω Ω Ω Z Z Z w ·C µ dx − C∗ w · µ dy dx = 0. (6c) Ω Ω Ω
2.4 Nonlocal Green’s identities
Nonlocal Green’s identities can be derived from (4) by setting F = ΘL∗(H), where L∗(H) may be a scalar or vector or second-order tensor function in which cases Θ is a scalar or second-order tensor or fourth-order tensor func- tion, respectively. This leads to the nonlocal Green’s first identity G, L(ΘL∗(H)) − L∗(G),ΘL∗(H) = 0. (7) Ω Ω×Ω Suppose now that Θ is a symmetric tensor. Then, reversing the roles of G and H and subtracting the result from (7) yields the nonlocal Green’s second identity G, L ΘL∗(H) − H, L ΘL∗(G) = 0. (8) Ω Ω Using (7) and (8) immediately yields the following two sets of results. Corollary 3 [Nonlocal (generalized) Green’s first identities] Given the scalar point functions u(x): Ω → R and v(x): Ω → R and the two-point second-order tensor function Θ(x, y): Ω × Ω → Rn×n, then Z Z Z uD Θ ·D∗(v) dx − D∗(u) · Θ ·D∗(v) dydx = 0. (9a) Ω Ω Ω Given the vector point functions u(x): Ω → Rn and v(x): Ω → Rn and the two-scalar point function θ(x, y): Ω × Ω → R, then Z Z Z v ·G θG∗(u) dx − θG∗(v)G∗(u) dydx = 0. (9b) Ω Ω Ω Given the vector point functions u(x): Ω → R3 and w(x): Ω → R3 and the two-point fourth-order tensor function Θ(x, y): Ω × Ω → R3×3×3×3, then Z Z Z ∗ ∗ ∗ w ·C Θ : C (u) dx − C (w): Θ : C (u) dydx = 0. (9c) Ω Ω Ω Nonlocal vector calculus, volume-constrained problems and balance laws 9
Corollary 4 [Nonlocal (generalized) Green’s second identities] We assume the same notation as in Corollary 3 and we also assume that the tensors Θ appearing in (9a) and (9c) are symmetric. Then, Z Z uD Θ ·D∗(v) dx − vD Θ ·D∗(u) dx = 0 (10a) Ω Ω
Z Z v ·G θG∗(u) dx − u ·G θG∗(v) dx = 0 (10b) Ω Ω Z Z ∗ ∗ w ·C Θ : C (u) dx − u ·C Θ : C (w) dx = 0. (10c) Ω Ω
2.5 Further observations and results about nonlocal operators
2010-8353J In this subsection, we collect several observations and results that can be deduced from the definitions and results of Sections 2.1–2.4.
2.5.1 A nonlocal divergence operator for tensor functions
A nonlocal divergence operator for tensor functions is defined by applying SAND (1a) to each of the rows of the tensor.
Definition 3 Given the tensor two-point function Ψ : Ω × Ω → Rn×k and the antisymmetric vector two-point function α: Ω × Ω → Rk the nonlocal n point divergence operator Dt(Ψ)(x): Ω → R for tensors is defined as Labs Z 0 Dt(Ψ)(x) := Ψ + Ψ · α dy for x ∈ Ω. (11a) Ω A nonlocal Gauss theorem for tensor functions as well as the correspond- ing integration by parts formula and Green’s identities can be derived and the ∗ adjoint operator Dt can defined in the same way as was done above for the nonlocal divergence operator D for vector functions, e.g., see Corollary 2. In n National particular, Definition 2 implies that, given the point function v(x): Ω → R ,
∗ 0 Dt (v)(x, y) = −(v − v) ⊗ α for x, y ∈ Ω, (11b) ∗ n×k ∗ where Dt (v): Ω × Ω → R , i.e., Dt maps a vector point function to a two-point second-order tensor function.7
7 Despite our notational convention that upper case Greek and Roman fonts Sandia denotes point and two-point tensor points, respectively, we could, in a slight abuse ∗ of notation, abbreviate Dt(Ψ) by D(Ψ). In an analogous fashion, Dt (v) may be abbreviated by D∗(v) when the argument is a vector point function. Note that this notational abuse is customary for the differential divergence and gradient operators for which ∇· is used to denote the divergence operator for both vectors and tensors and ∇ is used to denote the gradient operator on both scalars and vectors. 10 Q. Du et al.
2.5.2 Nonlocal vector identities
Compositions of the point operators defined in Sections 2.1 and 2.5.1 with the corresponding adjoint two-point operators derived in Section 2.3 lead to the following nonlocal vector identities. In this proposition, we set α = β = γ and, for the first, second, and fourth results, i.e., those involving nonlocal curl operators, we set n = k = 3.
Proposition 1 The nonlocal divergence, gradient, and curl operators and the corresponding adjoint operators satisfy
∗ 3 D C (u) = 0 for u: Ω → R (12a) ∗ C D (u) = 0 for u: Ω → R (12b) ∗ ∗ n G (u) = tr Dt (u) for u: Ω → R (12c) ∗ ∗ ∗ 3 Dt Dt (u) − G G (u) = C C (u) for u: Ω → R . (12d)
Proof We prove (12d); the proofs of (12a)–(12c) are immediate after direct substitution of the operators involved. Let u: Ω → R3. Then, by (1c), (5c), (11a), and (11b) and recalling that α is an antisymmetric function, we have
Z ∗ ∗ 0 0 0 Dt Dt (u) − G G (u) = − (u − u) ⊗ α + (u − u ) ⊗ α · α dy Ω Z + (u0 − u) · α + (u − u0) · α0 α dy Ω Z = −2 (u0 − u)(α · α) − (u0 − u) · αα dy Ω Z = −2 α × (u0 − u) × α dy, Ω where, for the last equality, we have used the vector identity a × (b × c) = b(c · a) − c(a · b). A simple computation show that the last expression is ∗ equal to C C (u) so that (12d) is proved.
Functions of the form C∗(u) do not entirely comprise the null space of the operator D. In fact, for any antisymmetric two-point function ν(x, y), we have D ν = 0. However, functions of the form C∗(u) are the only symmetric two-point functions belonging to the null space of D. Analogous statements can be made for the null space of the operator C and two-point functions of the form D∗(u). The four identities in (12) are analogous to the vector identities associated with the differential divergence, gradient and curl operator:
∇ · (∇ × u) = 0, ∇ × (∇u) = 0, ∇ · u = tr(∇u), − ∇ · (∇u) + ∇(∇ · u) = ∇ × (∇ × u), Nonlocal vector calculus, volume-constrained problems and balance laws 11
respectively. Because (∇·)∗ = −∇, ∇∗ = −∇·, and (∇×)∗ = (∇×), these identities can be written in the form
∇ · (∇×)∗u) = 0, ∇ × (∇·)∗u) = 0, ∇∗u = tr (∇·)∗u, ∇ · (∇·)∗u − ∇ (∇)∗u = ∇ × (∇×)∗u,
which more directly correspond to (12a)–(12d), respectively. In particular, the identities (12a), (12b), and (12d) suggest that D∗, G∗, and C∗ can also be viewed as nonlocal analogs of the differential divergence, gradient and curl operators that, when operating on point functions, result in two-point functions.8 Another set of results for the nonlocal operators that mimic obvious prop- erties of the corresponding differential operators are given in the following proposition whose proof is a straightforward consequence of (6) and the def- initions of the operators involved.
Proposition 2 Let b and b denote a constant scalar and vector, respectively. 2010-8353J Then, the nonlocal adjoint divergence, gradient, and curl operators satisfy
D∗(b) = 0, G∗(b) = 0, and C∗(b) = 0.
Moreover, these three relationships are equivalent to the three nonlocal inte- gral theorems (2a)–(2c). SAND These results do not hold for the point divergence, gradient, and curl operators, e.g., D(b), G(b), and C(b) do not necessarily vanish for constants b and constant vectors b. However, there exist special situation for which D(b) = 0 holds, and likewise for the other point operators. One such case Labs occurs whenever Ω is symmetric with respect to x = y.
2.5.3 Nonlocal curl operators in two and higher dimensions
The nonlocal point and two-point curl operators defined in (1c) and (5c), re- spectively, were defined in three dimensions. These definitions can be gener- alized to arbitrary dimensions by replacing the vector cross product with the ∗
National wedge product so that, e.g., instead of (5c) we would have C (w): Ω × Ω → Rr given by C∗(w)(x, y) := γ ∧ (w0 − w),
where γ(x, y): Ω × Ω → Rr with r a positive integer. This suggests a possi- ble exterior calculus-based formalism. However, such a formalism is beyond the scope of this paper so that only the special cases of r = 3 and lower
Sandia dimensions are considered and the vector cross product is retained. We can also define nonlocal point and two-point curl operators in two space dimensions, i.e., for Ω ⊂ R2; in fact, we have two types of each kind of
8 Note that G∗ C µ 6= 0 and C∗ G η 6= 0 because of the nonlocality of the operators. 12 Q. Du et al.
9 curl operator. First, we assume, without loss of generality, that µ · e3 = 0, w · e3 = 0, and γ · e3 = 0 in (1c) and (5c). Then, for a vector two-point 2 function µ(x, y): Ω × Ω → R , we can view C µ as the nonlocal scalar point function defined by Z 0 0 C µ (x) = (µ2 + µ2)γ1 − (µ1 + µ1)γ2 dy for x ∈ Ω Ω and, for a vector point function w: Ω → R2, we can view C∗(w) as the nonlocal scalar two-point function defined by ∗ 0 0 C (w)(x, y) = (w2 − w2)γ1 − (w1 − w1)γ2 for x, y ∈ Ω.
Next, assume, again without loss of generality, that µ = µe3, w = we3, and γ ·e3 = 0 in (1c) and (5c). Then, for a scalar two-point function µ(x, y): Ω × Ω → R, we can view C(µ) as the nonlocal vector point function defined by Z 0 C(µ)(x) := (µ + µ )(γ2e1 − γ1e2) dy for x ∈ Ω Ω and, for a scalar point function w(x): Ω → R, we can view C∗(w) as the nonlocal vector two-point function defined by ∗ 0 C (w)(x, y) = (w − w)(γ2e1 − γ1e2) for x, y ∈ Ω.
3 Constraint regions, constraint operators, and volume-constrained problems
In addition to the divergence, gradient, and curl operators and integrals over a region in Rn, the theorems and identities of the vector calculus for differential operators also involve operators acting on functions defined on the boundary of that region and integrals over that boundary surface.10 The
9 This is analogous to the two types of curl operators in two dimensions, one operating on vectors the other on scalars, given by
∂u1 ∂u2 ∂u ∂u curl u = − and curl u = − e1 + e2. ∂x2 ∂x1 ∂x2 ∂x1
10 n For example, given a region Ω ⊂ R having boundary ∂Ω, the divergence theorem for a vector-valued function u states that Z Z ∇ · u dx = u · n dx Ω ∂Ω and the Green’s (generalized) first identity for scalar functions u and v states that, for tensor-valued “constitutive” functions Θ, Z Z Z u∇ · (Θ∇v) dx + ∇u · (Θ∇v) dx = u(Θ∇v) · n dx. Ω Ω ∂Ω The nonlocal divergence theorem (2a) should correspond to the first of these rela- tions and the nonlocal Green’s first identity (9a) should correspond to the second. However, we see that neither (2a) or (9a) contain terms that correspond to the boundary integrals in the above two relations. Nonlocal vector calculus, volume-constrained problems and balance laws 13
nonlocal theorems and identities developed in Section 2 do not contain terms analogous to the boundary integrals found in the corresponding differential operator versions nor do they involve auxiliary boundary operators. However, by viewing boundary operators in the vector calculus for differential operators as constraint operators,11 it is a simple matter to rewrite the nonlocal vector theorems and identities so that they do include such terms. The reason it was not necessary to introduce constraint operators and “boundary” integrals in the theorems and identities of the nonlocal vector calculus is that, in the nonlocal case, “boundary” operators must operate on functions defined over measurable volumes, and not on lower-dimensional manifolds [12,20]. As a result, the actions of these operators are, in a real sense, indistinguishable from those of the nonlocal operators we have already defined, except for the resulting domains.
3.1 Constraint regions 2010-8353J We divide the region Ω into disjoint open subsets Ωs and Ωc, i.e., we have Ω = (Ωs∪Ωc)∪(Ωs∩Ωc), where ( · ) denotes the closure. We refer to Ωs as the 12 solution domain and Ωc as the constraint domain. Note that no relation is assumed between Ωs and Ωc so that, e.g., the four configurations of Figure 1 are possible. The bottom-left sketch in that figure depicts a constraint region that is a strip following the boundary of Ωs. Not surprisingly, this particular
SAND situation is most closely related to boundary surfaces, but, in the nonlocal case, constraint regions may take other guises, as depicted in Figure 1. This is another reason why we first introduced the nonlocal vector calculus without such regions or without constraint operators. Labs 3.2 Nonlocal point constraint operators
The constraint operators corresponding to each of the nonlocal point op- erators defined in Definition 1 are defined as follows, where the two-point functions α, β, Ψ, η, etc. are defined as in Definition 1.13 Note that each definition also involves redefining, from Ω to Ωs, the domains resulting from the actions of the nonlocal point operators D, G, and C. National Definition 4 [Nonlocal point constraint operators] Let Ω = (Ω ∪ s Ωc)∪(Ωs ∩Ωc). Corresponding to the point divergence operator D ν : Ωs →
11 In addition to trying to mimic more closely the theorems and identities of the vector calculus for differential operators, we introduce constraint regions and con- straint operators because they are needed to describe nonlocal volume-constrained problems and showing their well posedness; see Section 3.5 for examples of nonlocal Sandia volume-constrained problems. 12 Why we use the terminology “solution domain” and “constraint domain” will become clear when, in Section 3.5, we introduce examples of nonlocal volume- constrained problems that are analogous to differential boundary-value problems. 13 A constraint operator corresponding to the nonlocal divergence operator for tensors defined in (11a) can be defined in an entirely analogous manner. 14 Q. Du et al.
Fig. 1 Four of the possible configurations for Ω = (Ωs ∪ Ωc) ∪ (Ωs ∩ Ωc), where Ωs and Ωc denote the solution and constraint domains, respectively.
R defined as in (1a) but for x restricted to Ωs, we have the point constraint operator N (ν): Ωc → R defined as Z 0 N (ν)(x) := − (ν + ν ) · α dy for x ∈ Ωc. (13a) Ω
k Corresponding to the point gradient operator G η : Ωs → R defined as in (1b) but for x restricted to Ωs, we have the point constraint operator k S(η): Ωc → R defined as Z 0 S(η)(x) := − (η + η )β dy for x ∈ Ωc. (13b) Ω
3 Corresponding to the point curl operator C µ : Ωs → R defined as in (1c) but for x restricted to Ωs, we have the point constraint operator T (µ): Ωc → R3 defined as Z 0 T (µ)(x) := − γ × (µ + µ ) dy for x ∈ Ωc. (13c) Ω Note that the definitions of the point operators and the corresponding point constraint operators are defined using the same integral formulas but the former is defined for x ∈ Ωs and the latter for x ∈ Ωc.
3.3 Nonlocal integral theorems with constraint operators
It is now a trivial matter to rewrite the nonlocal integral theorems given in Theorem 1 and the nonlocal integration by parts formulas given in Corollary 1 in terms of the nonlocal point operators and point constraint operators. We simply take advantage of the fact that the definitions of the two types of operators have similar definitions.
Theorem 2 [Nonlocal integral theorems with constraint operators] Assuming the notations and definitions found in Definition 4, we have the Nonlocal vector calculus, volume-constrained problems and balance laws 15
following analogs of the integral theorems of the vector calculus for differential operators: Z Z D ν dx = N (ν) dx (14a) Ωs Ωc Z Z G η dx = S(η) dx (14b) Ωs Ωc Z Z C µ dx = T (µ) dx. (14c) Ωs Ωc Corollary 5 [Nonlocal integration by parts formulas with constraint operators] Adopt the hypotheses of Theorem 2. Then, Z Z Z Z uD(ν) dx + (u0 − u)ξ · ν dydx = uN (ν) dx (15a) Ωs Ω Ω Ωc
Z Z Z Z 2010-8353J v ·G(η) dx + (v0 − v) · βη dydx = v ·S(η) dx (15b) Ωs Ω Ω Ωc
Z Z Z w ·C(µ) dx − γ × (w0 − w) · µ dydx Ωs Ω Ω Z (15c) SAND = w ·T (µ) dx. Ωc
3.4 Nonlocal adjoint operators and Green’s identities with constraint operators Labs
With the definition of the point constrained operators and the redefinition of the point operators, the definition of adjoint operators given in (4) can be rewritten as G, L(F ) − L∗(G),F = G, B(F ) , (16) Ωs Ω×Ω Ωc where B is an operator that maps two-point functions F to point functions ∗ ∗ ∗ National defined over Ωc. As a result, the adjoint operators D , G , and C given in Corollary 2 remain unchanged. We can then rewrite nonlocal integration by parts formulas with constraint operators equations given by Corollary 5 as follows: Z Z Z Z uD ν dx − D∗ u · ν dydx = uN (ν) dx (17a) Ωs Ω Ω Ωc
Sandia Z Z Z Z v ·G η dx − G∗ vη dydx = v ·S(η) dx (17b) Ωs Ω Ω Ωc Z Z Z Z w ·C µ dx − C∗ w · µ dydx = w ·T (µ) dx. (17c) Ωs Ω Ω Ωc 16 Q. Du et al.
The abstract Green’s first and second identities (7) and (8) now take the form
G, L(ΘL∗(H)) − L∗(G),ΘL∗(H) = G, B(ΘL∗(H)) (18) Ωs Ω×Ω Ωc and G, L ΘL∗(H) − H, L ΘL∗(G) Ωs Ωs (19) = G, B ΘL∗(H) − H, B ΘL∗(G) , Ωc Ωc respectively. Likewise, the nonlocal Green’s first identities (9a)–(9c) now re- spectively take the form Z Z Z uD Θ ·D∗(v) dx − D∗(u) · Θ ·D∗(v)ν dydx Ωs Ω Ω Z (20a) = uN Θ ·D∗(v) dx Ωc Z Z Z v ·G θG∗(u) dx − θG∗(v)G∗(u) dydx Ωs Ω Ω Z (20b) = v ·S θG∗(u) dx Ωc Z Z Z w ·C Θ : C∗(u) dx − C∗(w): Θ : C∗(u) dydx Ωs Ω Ω Z (20c) = w ·T Θ : C∗(u) dx Ωc and the nonlocal Green’s second identities (10a)–(10c) now respectively take the form Z Z uD Θ ·D∗(v) dx − vD Θ ·D∗(u) dx Ωs Ωs Z Z (21a) = uN Θ ·D∗(v) dx − vN Θ ·D∗(u) dx Ωc Ωc Z Z v ·G θG∗(u) dx − u ·G θG∗(v) dx Ωs Ωs Z Z (21b) = v ·S θG∗(u) dx − u ·S θG∗(v) dx Ωc Ωc
Z Z w ·C Θ : C∗(u) dx − u ·C Θ : C∗(w) dx Ωs Ωs Z Z (21c) = w ·T Θ : C∗(u) dx − u ·T Θ : C∗(v) dx. Ωc Ωc Nonlocal vector calculus, volume-constrained problems and balance laws 17
3.5 Examples of nonlocal volume-constrained problems
The nonlocal point operators, the corresponding nonlocal adjoint operators, and the corresponding nonlocal constraint operators given in (1), (5), and (13), respectively, can be used to define nonlocal volume-constrained prob- lems. Here, we merely state problems involving scalar and vector “second- order” operators. 14 Let Ωc = Ωc1 ∪ Ωc2, where Ωc1 ∩ Ωc2 = ∅, and let Θ4, Θ2, and θ de- note fourth-order tensor, second-order tensor, and scalar two-point functions, respectively. The nonlocal volume-constrained problems
∗ D Θ2 ·D (u) = f in Ω u = g in Ωc1 (22a) ∗ N Θ2 ·D (u) = h in Ωc2
∗ Dt Θ4 : D (u) = f in Ω t 2010-8353J u = g in Ωc1 (22b) ∗ Nt Θ4 : Dt (u) = h in Ωc2 ∗ ∗ C Θ4 : C (u) + G θG (u) = f in Ω u = g in Ωc1 (22c) ∗ ∗ SAND T Θ4 : C (u) + S θG (u) = h in Ωc2 are analogous to the second-order differential boundary-value problems −∇ · K2 · ∇u = f in Ω u = g on ∂Ω1 (23a) Labs K2 · ∇u · n = h on ∂Ω2 −∇ · K4 : ∇u = f in Ω u = g on ∂Ω1 (23b) K4 : ∇u · n = h on ∂Ω2 ∇ × K4 : ∇ × u − ∇ k∇ · u = f in Ω National u = g on ∂Ω 1 (23c) n × K4 : ∇ × u = h1 o on ∂Ω2, k∇ · u = h2
respectively, where ∂Ω = ∂Ω1 ∪ ∂Ω2 denotes the boundary of Ω with ∂Ω1 ∩ ∂Ω2 = ∅ and where K4, K2, and k denote fourth-order tensor, second-order
Sandia tensor, and scalar point functions, respectively. A special form of the nonlocal volume-constrained problem (22a) corre- sponding to a scalar-valued solution is studied in [12] by appealing to a vari- ational formulation. Well-posedness results are provided there for the case in
14 Either Ωc1 or Ωc2 may be empty. 18 Q. Du et al.
which the natural energy space (used to define the variational problem) is equivalent to L2(Ω), the space of square integrable functions. A rigorous con- nection to (23a) is demonstrated as well for such a case. Although the notion of a nonlocal vector calculus is not introduced, see the recent book [2] where the authors and their collaborators collect their substantial recent work on nonlocal diffusion and its relationship to (23a); in that work, the nonlocal operator in (22a) represents the infinitesimal generator. The two papers [3, 10] also consider various properties of the nonlocal operator in (22a). As shown in [7], more generally, the natural energy space associated with the nonlocal operators used in (22a) may be a strict subspace of L2(Ω), for instance a fractional Sobolev space. For the latter cases, the nonlocal variational problems possess smoothing properties akin to that for elliptic partial differential equations but with possibly reduced order [7].
4 Identification of nonlocal operators with differential operators in a distributional sense
In this section, we identify, in a distributional sense, nonlocal operators with their differential counterparts. For the sake of brevity, we mostly consider the nonlocal point divergence operator D and its adjoint operator D∗. However, analogous results also hold for the nonlocal gradient and curl operators G and C, respectively, and their adjoints. Further connections between the non- local and differential operators are made in Section 5 where we demonstrate circumstances under which the nonlocal point operators are weak represen- tations of the divergence, gradient, and curl differential operators. We begin with the identification of the nonlocal point divergence operator D with the divergence differential operator ∇·. This identification is subject to the understanding that the former operates on two-point functions while the latter operates on point functions.
∞ Proposition 3 Let ν ∈ C0 (Ω ×Ω), i.e., ν belongs to the space of compactly supported infinitely differentiable functions over the domain Ω×Ω. Select the (antisymmetric) distribution
α(y, x) = −∇yδ(y − x), (24a) where ∇ and δ(y − x) denote the differential gradient and Dirac delta distri- bution, respectively. Then,
D ν (x) = ∇x · ν(x, x), (24b) where ∇· denotes the differential divergence operator. In particular, we have
D u(x) + u(y) (x) = ∇x · u(x). (24c) Nonlocal vector calculus, volume-constrained problems and balance laws 19
Proof We have ∇x · ν(x, x) = ∇x · ν(x, y) |y=x + ∇y · ν(x, y) |y=x = ∇y · ν(y, x) |y=x + ∇y · ν(x, y) |y=x = ∇y · ν(y, x) + ν(x, y) |y=x Z = ∇y · ν(y, x) + ν(x, y) δ(y − x) dy Ω Z = − ν(y, x) + ν(x, y) ∇yδ(y − x) dy Ω = D ν(x)
so that the relations (24b) and (24c) now follow. Next, we identify the nonlocal Gauss theorem (2a) with the classical Gauss theorem for functions that are compactly supported within Ω.
2010-8353J Corollary 6 The nonlocal Gauss theorem (2a) reduces to Z ν(x, x) · nx = 0. ∂Ω
Proof The result is immediate from (24b) and (2a).
SAND The following proposition shows that the average of the action of D∗ on a point function can be related, in a distributional sense, to the action of −∇ on that function.
∞ Proposition 4 Let u ∈ C0 (Ω) and select α as in (24b). Then,
Labs Z ∗ D (u) dy = −∇xu. Ω Proof Using (5a) we have that Z Z ∗ D (u) dy = u(y) − u(x) ∇yδ(y − x) dy Ω Ω Z National = − δ(y − x)∇y u(y) − u(x) dy Ω Z ∗ = − δ(y − x)∇yu(y) dy = −∇xu(x) = (∇x·) u(x). Ω Now consider the composition of the nonlocal divergence operator and its adjoint, i.e., D D∗, which, according to the next proposition, can be
Sandia identified, in the sense of distributions, with the Laplace operator ∆ = ∇·∇.
∞ 2 1 Proposition 5 Let u ∈ C0 (Ω) and select |α(x, y)| = 2 ∆yδ(y − x), where ∆ denotes the differential Laplace operator. Then,
∗ D D u(x) = −∆xu(x). 20 Q. Du et al.
Proof Using (1a) and (5a), we have Z D D∗u(x) = −2 u(y) − u(x)|α(x)|2 dy Ω Z = − u(y) − u(x) ∆yδ(y − x) dy Ω Z = − δ(y − x)∆y u(y) − u(x) dy Ω Z = − δ(y − x)∆yu(y) dy Ω ∗ = −∆xu(x) = −∇x · ∇xu(x) = (∇x·)(∇x·) u(x),
where the differential Green’s second identity is used for the third equality. An immediate application is to relate the average of the product of operators ∗ ∗ D (·) ·D (·) to the weak Laplacian ∇x · ∇x in a distributional sense. Corollary 7 Green’s first identity (9a) with Θ = I reduces to Z Z Z ∗ ∗ ∇xu · ∇xv dx = D (u) ·D (v) dydx. Ω Ω Ω
5 Relations between weighted nonlocal operators and weak representations of differential operators
The conventional differential calculus only involves operators mapping point functions to point functions; on the other hand, the nonlocal operators de- fined earlier map two-point functions to point functions or, in case of the nonlocal adjoint operators, point functions to two-point functions. To further demonstrate that the nonlocal operators correspond to nonlocal versions of the conventional divergence, gradient, and curl differential operators, we use the nonlocal operators D, G, and C to define, in Section 5.1, correspond- ing weighted operators that map point functions to point functions. We also show that the adjoint operators corresponding to the weighted operators are weighted integrals of the nonlocal adjoint operators D∗, G∗, and C∗. Then, in Section 5.2, the weighted operators are shown to be nonlocal versions of the corresponding differential operators.
5.1 Nonlocal weighted operators
The nonlocal point operators defined in (1) induce new operators, referred to as weighted operators.
Definition 5 Let ω(x, y): Ω × Ω → R denote a non-negative scalar-valued two-point function. Let the operators D, G, and C be defined as in (1). Then, Nonlocal vector calculus, volume-constrained problems and balance laws 21
given the point function u(x): Ω → Rn, the weighted nonlocal divergence operator Dω(u): Ω → R is defined by Dω(u)(x) := D ω(x, y)u(x) (x) for x ∈ Ω. (25a)
Given the point function u(x): Ω → R, the weighted nonlocal gradient oper- k ator Gω(u): Ω → R is defined by Gω(u)(x) := G ω(x, y)u(x) (x) for x ∈ Ω. (25b)
Given the point function w(x): Ω → R3, the weighted nonlocal curl operator 3 Cω(u): Ω → R is defined by Cω(w)(x) := C ω(x, y)w(x) (x) for x ∈ Ω. (25c) The following result shows that the adjoint operators corresponding to the weighted nonlocal operators are determined as weighted integrals of the corresponding nonlocal two-point adjoint operators (5). 2010-8353J
Proposition 6 Let ω(x, y): Ω × Ω → R denote a non-negative scalar two- point function and let D∗, G∗, and C∗ denote the nonlocal adjoint operators ∗ k given in (5). The adjoint operator Dω(u)(x): Ω → R corresponding to the weighted nonlocal divergence operator Dω is given by Z SAND ∗ ∗ Dω(u)(x) = D (u)(x, y) ω(x, y) dy for x ∈ Ω (26a) Ω
∗ for scalar point functions u(x): Ω → R. The adjoint operator Gω(v)(x): Ω → R corresponding to the weighted nonlocal gradient operator Gω is given by Labs Z ∗ ∗ Gω(v)(x) = G (v)(x, y) ω(x, y) dy for x ∈ Ω (26b) Ω
n ∗ for vector point functions v(x): Ω → R . The adjoint operator Cω(w)(x): Ω 3 → R corresponding to the weighted nonlocal curl operator Cω is given by Z ∗ ∗ Cω(w)(x) = C (w)(x, y) ω(x, y) dy for x ∈ Ω (26c)
National Ω
for vector point functions w(x): Ω → R3. Proof We have that Z D (u), u = u(x)D ω(x, y)u(x) dx ω Ω Ω Sandia Z Z = D∗(u)(x, y) · ω(x, y)u(x) dydx Ω Ω Z Z = ω(x, y)D∗(u)(x, y) dy · u(x) dx, Ω Ω 22 Q. Du et al. where (25a) is used for the first equality and (4) and (5a) for the second. ∗ But, by definition, the adjoint operator Dω(·) corresponding to the operator Dω(·) satisfies u, D (u) = D∗ (u), u . ω Ω ω Ω Comparing the last two results yields (25a). The conclusions (25b) and (25c) are derived in a similar fashion. In direct analogy to the results of Section 2.5.1, we extend 25a and 26a to tensors and vectors, respectively.
Definition 6 Let ω(x, y): Ω × Ω → R denote a non-negative scalar two- point function and let Dt be given as in (11b). Given the tensor point function n×k U(x): Ω → R , the weighted nonlocal divergence operator Dt,ω(U): Ω → Rn for tensors is defined by Dt,ω U (x) := Dt ω(x, y)U(x) (x) for x ∈ Ω. (27)
∗ n×k Corollary 8 The adjoint operator Dt,ω(u)(x): Ω → R corresponding to the operator Dt,ω is given by Z ∗ ∗ Dt,ω u (x) = Dt (u)(x, y) ω(x, y) dy for x ∈ Ω (28) Ω n for vector point functions u: Ω → R .
5.2 Relationships between weighted operators and differential operators
We have seen that the nonlocal operators satisfy many properties that mimic their differential counterparts. In particular, we established, in Section 4, that the nonlocal point operators can be identified with the corresponding differential operators in a distributional sense. In this section, we establish rigorous connections between the weighted operators and their differential counterparts by first introducing a cut-off or horizon parameter ε > 0 and then analyzing what occurs as ε → 0, that is, in the local limit. To this end, we define d Bε(x) := {y ∈ R : |y − x| < ε}. Let Ω = Rd and let φ denote a positive radial function; select y − x α(x, y) = for x 6= y |y − x| ( (29a) |y − x|φ(|y − x|) y ∈ B (x) ω(|x − y|) = ε 0 otherwise with the function φ satisfying a normalization condition Z |y − x|2φ(|y − x|) dy = d, (29b) Bε(x) Nonlocal vector calculus, volume-constrained problems and balance laws 23
where d denotes the space dimension. Note that α is an antisymmetric function whereas ω is a symmetric func- tion. We then have, for a scalar function u, that the components of Gω u and ∗ Dω u given by (25b) and (26a), respectively, are given by, for j = 1, . . . , d, Z dju(x) := u(x + z) + u(x) zj φ(|z|) dz (30a) Bε(0) Z ∗ dj u(x) := − u(x + z) − u(x) zj φ(|z|) dz, (30b) Bε(0)
where zj denotes the j-th component of z. The following result ensues.
Lemma 1 Assume that ω and α are defined as in (29a). Let u: Rd → R ∗ ∗ and let dju(x) and dj u(x) denote the j-th components of Gω(u) and Dω(u), respectively. Then, ∗ dju = −dj u. (31) 2010-8353J Proof From (30) we have Z ∗ dju(x) + dj u(x) = −2u(x) yjφ(|y|)dy = 0 Bε(0)
from which the result directly follows. SAND Based on this lemma and the definition of the weighted operators, we have the following results. Corollary 9 Under the same conditions as in Lemma 1, we have
Labs ∗ ∗ ∗ Dω = −Gω, Gω = −Dω, and Cω = Cω. (32) Thus, under the hypotheses of this corollary, the identities in (32) mimic their counterparts in the vector calculus for differential operators. Identities such as those in Proposition 1 do not generally hold for the weighted operators due to their nonlocal properties.15 Synergistic with the results of Section 4, we have the following. Corollary 10 Select the singular distribution National
zjφ(|z|) = −∂jδ(z),
where ∂ju denotes the weak derivative of u with respect to xj. Then, ∗ dj = ∂j and dj = −∂j. The following lemma demonstrates that the components of the weighted
Sandia ∗ gradient operator Gω or, by (32), Dω, converge, as ε → 0, to the corresponding spatial derivatives.
15 The identities in Lemma 1 and Corollary 9 also hold for more general choices of α and ω, namely, those such that α is anti-symmetric and ω is a radial function with support in Bε(x). 24 Q. Du et al.
Proposition 7 Let ω be defined as in (29a) and satisfy (29b). Then, for ∗ j = 1, . . . , d, the weighted operators dj and dj defined by (30) are bounded linear operators from H1(Rd) to L2(Rd). Moreover, if u ∈ H1(Rd), then as ε → 0
kd u − ∂ uk 2 d → 0 (33a) j j L (R ) ∗ kd u + ∂ uk 2 d → 0, (33b) j j L (R )
where ∂ju denotes the weak derivative of u with respect to xj. Moreover, if Z |z|1+sφ(|z|) dz < ∞ for some 0 ≤ s ≤ 1, (33c) Bε(0)
∗ then, for j = 1, . . . , d, the weighted operators dj and dj are bounded linear operators from Ht(Rd) to Ht−s(Rd) for any t ≥ 0.
Proof The conclusion (33a) is equivalent to kd u − ∂ uk 2 d → 0, where dj dj L (R ) dju and ∂dju denote the Fourier transforms of dju and ∂ju, respectively. Under the condition (33c) for s ∈ [0, 1], if u ∈ Ht(Rd) for t ≥ 0, then Z iy·ξ dju = (e + 1) ub(ξ) yj φ(|y|) dy Bε(0) Z = i sin(y · ξ) ub(ξ) yj φ(|y|) dy Bε(0) Z X = i sin(yjξj) cos( ykξk) Bε(0) k6=j X + cos(yjξj) sin( ykξk) ub(ξ) yj φ(|y|) dy k6=j Z X = iub(ξ) sin(yjξj) cos( yiξk) yjφ(|y|) dy, Bε(0) k6=j
where the second and fourth equalities hold because of the symmetry of the domain of integration and the antisymmetry of the integrand with re- spect to the integration variable, respectively. Because for any 0 ≤ s ≤ 1, P s | sin(yjξj) cos( k6=j yiξk)| ≤ |yjξj| , we have Z s 1+s |dju| ≤ |ξ ub(ξ)| |yj| φ(|y|) dy Bε(0)
t−s d so that dju ∈ H (R ). In particular, for t = s = 1, we obtain that, under the condition (29b), the weighted operators dj, j = 1, . . . , d, defined in (30a) are bounded linear operators from H1(Rd) to L2(Rd). Nonlocal vector calculus, volume-constrained problems and balance laws 25
If u ∈ H1(Rd), we have that
|dju − ∂dju| ≤ |dju| + |∂dju| Z X ≤ sin(yjξj) cos( ykξk) ub(ξ) yjφ(|y|) dy + |ξjub| Bε(0) k6=j Z 2 ≤ |ξjub| yj φ(|y|) dy + |ξjub| Bε(0) 2 d ≤ 2|ξjub| ∈ L (R ), where the third inequality holds because | sin(x)| ≤ |x| and | cos(x)| ≤ 1. By Taylor’s theorem, we have Z 3 3 X 2 dju = i (yjξj + yj ξj cos(θ1)/6)(1 + ( ykξk) cos(θ2)/2) ub yjφ(|y|) dy Bε(0) k6=j Z 2 = iξjub yj φ(|y|) dy + Iε(ξ) = iξjub + Iε(ξ), 2010-8353J Bε(0)
where, for some θ1 and θ2, i Z I (ξ) := y4 cos(θ )φ(|y|) dy ξ3u ε 6 j 1 j b Bε(0) i Z
SAND X + y2( y ξ )2 cos(θ )φ(|y|) dy ξ u 2 j i i 2 j b Bε(0) k6=j i Z X + y4( y ξ )2 cos(θ ) cos(θ )φ(|y|) dy ξ3 u. 12 j k k 1 2 j b Bε(0) k6=j Labs Because ub is bounded a.e., we see that, for any ξ, 2 3 2 X 2 4 X 2 3 |Iε(ξ)| ≤ ε |ξj ub| + ε |( ξk) ξjub| + ε |( ξk) ξj ub| → 0 as ε → 0. k6=j k6=j Hence, we have
d dju → ∂dju a.e. for ξ ∈ R as ε → 0.
National Then, by the dominated convergence theorem, we obtain
kd u − ∂ uk 2 d → 0 as ε → 0. j j L (R ) ∗ Lemma 1 implies the same results hold for the operator dj . The above lemma implies that if ω satisfies (33c) for s = 0, then the weighted operators are bounded operators from L2(Rd) to L2(Rd). More Sandia ∗ generally, for φ satisfying (33c) with positive s, the operators dj and dj actually map a subspace of L2(Rd), for instance the fractional Sobolev space Hs(Rd), to L2(Rd), or even map L2(Rd) to H−s(Rd). We refer to [7] for related work. A direct consequence of Lemma 7 is the following result. 26 Q. Du et al.
Proposition 8 Under the condition of Lemma 7, the weighted operators Dω, ∗ ∗ ∗ Gω, and Cω and their adjoint operators Dω, Gω, and Cω are bounded linear operators from Ht(Rd) to Ht−s(Rd) for 0 ≤ s ≤ 1, where d = 3 for the weighted curl operators. Moreover, if u ∈ H1(Rd) and u ∈ [H1(Rd)]d,
∗ Dω(u) → ∇ · u Dω(u) → −∇u ∗ Gω(u) → ∇u Gω(u) → −∇ · u (34) ∗ Cω(u) → ∇ × u Cω(u) → ∇ × u, where the convergence as ε → 0 is with respect to L2(Rd).
Proof We prove the convergence of the operator Dω(u) to the divergence differential operators. First, note that
d d X X Dω(u) = diui and ∇ · u = ∂iui i=1 i=1 so that
d X kD (u) − ∇ · uk 2 d ≤ kd u − ∂ u k 2 d → 0 as ε → 0. ω L (R ) i i i i L (R ) i=1
The remaining results can be proved in a similar fashion. For the purpose of discussing nonlocal equations, we also need to consider ∗ combinations of the weighted operators such as Dω(C1Dω(u)), where C1(x) is a tensor point function that, for example, describes the point property of a material. Note that the two-point property is involved in the definition of the weighted operators. In the next corollary, we illustrate that whenever the horizon ε goes to zero, the combinations of the weighted operators converge to their local counterparts.
1 d 1 d d d d d Corollary 11 Let u ∈ H (R ), u ∈ [H (R )] , C1 : R → R × R in ∞ d d d ∞ d L (R × R ), and c2 : R → R in L (R ). Then,
∗ Dω C1 ·Dω(u) → −∇ · (C1 · ∇u) ∗ Gω c2Gω(u) → −∇(c2∇ · u) (35) ∗ Cω C1 ·Cω(u) → ∇ × C1 · (∇ × u) , where the convergence as ε → 0 is with respect to H−1(Rd).16 Proof Using Proposition 8 and a similar method of proof as for Lemma 7, the results are obtained. 16 Similar results can be obtained for the nonlocal divergence operator on tensors; ∗ d in particular, we have that Dt,ω(C3 : Dt,ω(u)) → −∇ · (C3 : ∇u) with C3 : R → d d d d ∞ d d d d R × R × R × R in L (R × R × R × R ). Nonlocal vector calculus, volume-constrained problems and balance laws 27
Let L denote a linear operator that commutes with the differential and nonlocal operators. We then have the following result.
Lemma 2 Let L denote a linear operator that commutes with the differential and nonlocal operators. Then, if Lu ∈ H1(Rd), ∗ Dω(Lu) → ∇ · Lu Dω(Lu) → −∇Lu ∗ Gω(Lu) → ∇Lu Gω(Lu) → −∇ · Lu (36) ∗ Cω(Lu) → ∇ × Lu Cω(Lu) → ∇ × Lu,
where the convergence as ε → 0 is with respect to L2(Rd). If L is selected as a differential operator with constant coefficients, or its formal inverse, we can observe convergence in either stronger or weaker norms.
2010-8353J 6 Nonlocal balance laws
Sections 2 and 3 introduced a nonlocal vector calculus. Combined with the analytical theory in sections 4 and 5, a potentially powerful formalism for instantiating the nonlocal balance laws of continuum mechanics is now at hand. We start by discussing the notion of an abstract balance law as given
SAND by d P(Ω) = Q(∂Ω),Ω ⊂ R (37) postulating that the production of an extensive quantity over the open sub- region Ω is equal to the flux of the quantity through the boundary ∂Ω of that subregion. The additive quantities P and Q are expressed in terms of Labs linear functionals over Ω and ∂Ω, respectively. The purpose of this section is to demonstrate that the flux is given by the volume interaction between Ω and Rn \ Ω. Subsection 6.1 demonstrates that the flux induced by the peridynamic nonlocal continuum theory is given by a volume interaction and is elegantly phrased using the nonlocal calcu- lus. However, assuming sufficient regularity, the volume interaction may be relegated to that on the surface ∂Ω. In particular, suppose that Z National Q(∂Ω) = q˘ dHd−1, (38) ∂Ω whereq ˘ and Hd−1 are the flux density and a Hausdorff measure over the sur- face ∂Ω, respectively; see [5, Chap. II] for a discussion. Then, under general conditions, Z Z Z Sandia q˘ dHd−1 = q · n dHd−1 = divq dx (39) ∂Ω ∂Ω Ω for some vector function q, where n denotes the outward unit normal vector for Ω. The relations (38) and (39) are abstract versions of Cauchy’s postu- late and theorem, respectively. The relation (39) establishes the fundamental 28 Q. Du et al. result that the flux densityq ˘ at a point on any orientable surface ∂Ω is linear in the outward normal vector n. An important consequence is that the flux represents an interaction between the subregions external and internal to Ω. This is made explicit by defining the bi-additive form
d d I(Ω, R \ Ω) := Q(∂Ω) ∀ Ω ⊂ R . (40a)
The balance law (37) implies that P(Rd) = 0 so that the addivity of P and bi-additivity of I grants the important relation
d d d I(Ω, R \ Ω) + I(R \ Ω,Ω) = 0 ∀ Ω ⊂ R . (40b)
d d d d Because R \ Ω2 = R \ (Ω1 ∪ Ω2) ∪ Ω1 and R \ Ω1 = R \ (Ω1 ∪ Ω2) ∪ Ω2 for all disjoint subregions Ω1 and Ω2, we have
d d I(Ω1,Ω2) + I(Ω2,Ω1) = I(Ω1, R \ Ω1) + I(Ω2, R \ Ω2) d − I Ω1 ∪ Ω2, R \ (Ω1 ∪ Ω2) = 0. (40c)
Then, because the disjoint regions Ω1 and Ω2 are arbitrary,
d I(Ω,Ω) = 0 ∀ Ω ⊂ R . (40d)
We have established that the balance law (37) implies that the flux Q induces the alternating (or antisymmetric) bilinear form I. The relations (40b)–(40d) correspond to the physical understanding of a flux. The first relation ex- plains that the interaction of the subregion external to Ω upon Ω is equal and opposed to the interaction of Ω upon Rd \ Ω. The latter two relations exemplify the action-reaction principle and that there is no self-interaction among subregions, respectively. For example, whenever (37) denotes the bal- ance of linear momentum, the principle of action-reaction is Newton’s third law. Additivity of linear momentum implies Newton’s third law and need not be assumed. The above discussion explains why the balance law (37) can be equiva- lently replaced by
d d P(Ω) = I(Ω, R \ Ω),Ω ⊂ R (41) for the alternating bilinear form I. The crucial observation is that Cauchy’s theorem (39) explains that the flux Q has the algebraic structure of an al- ternating (or antisymmetric) bilinear form encapsulating an interaction. In particular, the unit outward normal provides an orientation leading to (40b). Interaction is now determined by subregions and not by contact along their respective boundaries. Introduce the bilinear form Z Z d J (Ω1,Ω2) := f(x, y) dy dx ∀ Ω1,Ω2 ⊂ R , (42a) Ω1 Ω2 Nonlocal vector calculus, volume-constrained problems and balance laws 29
where f : Rd × Rd → R denotes an interaction density.17 The bilinear form is alternating (or antisymmetric) if and only if
d f(x, y) = −f(y, x) ∀ x, y ∈ R . (42b) In stark contrast to I, the flux induced by J dispenses the need for deter- mining a unit normal to an orientable surface and instead considers volume interactions among regions. Such a relation and (42a) can also be interpreted more broadly. Indeed, by prescribing certain continuity assumptions on the bilinear form J with respect to its argument in a suitable topology, the exis- tence of the interaction density f follows from the Schwartz Kernel theorem [19]. We now derive the field equation for the balance law
d d Pe(Ω) = J (Ω, R \ Ω),Ω ⊂ R . (43a) A result due to Fuglede [9] as used in [4, p. 298] supplies a production density d ρe: R → R that satisfies 2010-8353J Z Z Z ρe(x) dx = f(x, y) dy dx. Ω Ω Rd\Ω The antisymmetry of f grants Z Z Z Z f(x, y) dy dx = f(x, y) dy dx (43b) SAND Ω Rd\Ω Ω Rd and the field equation Z ρe(x) = f(x, y) dy (43c) Rd Labs follows. In an analogous fashion, the field equation for the balance (41) ρ = divq ensues and is understood in the sense of distributions. Because the production of density in the volume Ω is unique, the relation Z divq(x) = f(x, y) dy (43d) d
National R results. Evidently the right-hand side of (43d) is identified with the distribu- tion divq and the flux through ∂Ω is given by Z Z Z q · n dHd−1 = f(x, y) dy dx ∂Ω Ω Rd\Ω
Sandia so that the relation (43b) is identified as the divergence theorem. We conclude that the balances (41) and (43a) are equivalent in the sense of distributions. An important distinction between the two interactions I and J is that, by (39), the former vanishes when the intersection of the closures of the
17 See also [23, p. 85] and [17, p. 29]. 30 Q. Du et al.
subregion has nonzero intersection, e.g., Ω1 ∩Ω2 = ∅, whereas the interaction J may not. The balance (43a) is defined to be nonlocal when the distribution divq is replaced by the right-hand side of (43d) because points y 6= x are involved with respect to the interaction density. On the other hand, when the distribution divq is such that div can be identified with a linear differential operator, say ∇·, and q is suitably defined, the balance (43a) is local.18 We remark that under suitable regularity conditions, a closed form ex- pression for an operator A may be derived such that Z ∇ · A f(x, y) = f(x, y) dy Rd is satisfied; see [16,17]. For a variational characterization of the previous relation within the peridynamic theory, see [14].
6.1 Peridynamic continuum theory
The relation (43d) explains that the role of the function q for the local balance is now assumed by the interaction density f = f(x, y) which is characterized as a two-point function given its dependence on x and y. To complete the description of a balance law for a physical system, we assume further that the interaction density results from a mapping of an underlying field given by a point function. This mapping represents the constitutive relation describing the specific system. We now demonstrate that the nonlocal calculus developed in the earlier sections is synergistic with the peridynamic nonlocal theory of continuum mechanics. Suppose that the scalar interaction density f = f(x, y) is replaced by the vector interaction density f = f(x, y). Then a generalization of the linearized bond-based peridynamic operator introduced by Silling [20] is given by Z 1 ∗ T 0 − Dt Dt u = (α ⊗ α) · (u − u) dy, (44) 2 Ω
∗ ∗ where u represents the displacement field, Dtu = Dt u and Dt u = Dt u are given by (11a) and (11b), respectively, and the integrand is an interaction 0 ∗ T density f = (α⊗α)·(u −u). A mechanical perspective indicates that Dt u describes the deformation of u and that a constitutive relation maps the deformation to the force density given by the integral operator. Because the integrand is antisymmetric with respect to the arguments x and y, the operator (44) induces an interaction – that of force between subregions. If we require that the integral operator in (44) only contain only rigid motion in its null space, e.g.,
∗ T T Dt (Dt (Ax + c)) = 0, A = −A , c := constant vector, (45)
18 See [6, pp. 149–150] for a discussion. Nonlocal vector calculus, volume-constrained problems and balance laws 31
then a necessary and sufficient condition for (45) to hold is that α(x, y) = + y − x ζ(|y − x|) where ζ : R → R. The integral operator is then written as Z 1 ∗ T y − x ⊗ y − x − Dt Dt u(x) = · u(y) − u(x) dy, (46) 2 Ω σ(|y − x|) where σ := ζ−2 and so results in the linearized peridynamic bond-based force density. When Ω ≡ Rd, the paper [7] provides analytical conditions on σ describing the amount of smoothing associated with the integral operator and the well posedness of the balance of linear momentum and associated equilibrium equation. That paper also demonstrates that as ε → 0, 1 − D (D∗u)T → −µ∇ · (∇u) − 2µ∇(∇ · u), 2 t t the Navier operator of linear elasticity with Poisson ratio one-quarter; see also the paper [8]. The subsequent paper [26] considers volume-constrained 2010-8353J problems on bounded domains in R and squares in R2. The state-based peridynamic theory [22] requires the consideration of both volumetric and shear deformations. Similar to the above discussion on the bond-based models, we may also formulate the state-based peridynamic model in terms of the nonlocal operators. Let α and ω be given by (29a). Then, the linear state-based peridynamic integral operator [21] is given by SAND ∗ T ∗ −Dt,ω η Dt,ωu + (λGωu) I , (47)
∗ ∗ ∗ where η and λ are materials constants, Gωu = Gω u , Dt,w and Dt,w are ∗ given by (26b), (27) and (28), respectively. The scalar Gωu measures the ∗ Labs volumetric change, or dilatation, in the material so that Gωu I is a diago- nal tensor representing volumetric stress. This allows us to readily apply the nonlocal calculus to study the well-posedness of both free-space and volume- constrained linear peridynamic state-based balance laws, and also suggests why, in the limit as ε → 0, the operator given by (47) leads to the linear Navier operator of elasticity for linear isotropic materials with general Pois- son ratios, e.g., not relegated to a value of one-quarter. This is the subject of forthcoming research. National
Acknowledgements
The third author thanks Petronela Radu of the University of Nebraska for pointing out the paper [4]. Sandia References
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