Archive for Rational Mechanics and Analysis manuscript No.
(will be inserted by the editor)
Sandia National Labs SAND 2010-8353J
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
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.
Q. Du
Department of Mathematics
Penn State University
Tel.: +1-814-865-3674
State College, PA 16802, USA
Fax: +1-814-865-3735
E-mail: qdu@math.psu.edu
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: gunzburg@fsu.edu
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: rblehou@sandia.gov
K. Zhou
Department of Mathematics
Penn State University
State College, PA 16802, USA
Tel.: +1-814-865-3674
Fax: +1-814-865-3735
E-mail: zhou@math.psu.edu
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 counterparts are established, first in a distributional sense and then in a weak sense
by considering weighted integrals of the nonlocal adjoint operators. An application 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 classical 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. Nonlocal analogs of the Gauss theorem and the Green’s identities of the vector
calculus for differential operators are also derived. The nonlocal vector calculus 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 application 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 boundaries of suitable regions is obviated when volume interactions induce flux.
Our nonlocal calculus, then, is an alternative to standard approaches for circumventing the technicalities associated with lack of sufficient regularity in
local balance laws such as measure-theoretic generalizations of the GaussGreen theorem (see, e.g., [4, 18]) and the use of the fractional calculus (see,
e.g., [1, 24]).
Sandia National Labs SAND 2010-8353J
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 contrast, 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 balance 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 developed 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
nonlocal volume-constrained problems that generalize well-known boundaryvalue 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 nonlocal 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
sense whereas, in Section 5, the connections are made between weighted integrals 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
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
We develop a nonlocal vector calculus that mimics the classical vector calculus 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 defined for pairs of points. Point operators map two-point functions to point
functions whereas two-point operators map point functions to two-point functions so that the nomenclature for operators refer to their ranges. Point and
1
The authors of [11] refer to [25] where a discrete nonlocal divergence and gradient 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
product of the two tensors. The trace of B ∈ Rn×n , 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



u · v dx
for u(x), v(x) ∈ Rn
 (u, v)Ω =
ΩZ Z


µ · ν dydx for µ(x, y), ν(x, y) ∈ Rn
 (µ, ν)Ω×Ω =
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 =
x y, x ⊗ y = xyT , and U · v = UT v.
T
Nonlocal vector calculus, volume-constrained problems and balance laws
5
α : Ω × Ω → Rk , 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)
Sandia National Labs SAND 2010-8353J
Ω
Given the scalar two-point function η : Ω × Ω → R and the antisymmetric vector two-point
function β : Ω × Ω → Rn , the nonlocal point gradient
operator G η : Ω → Rk 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
function γ : Ω × Ω → R3 , the nonlocal point curl operator
two-point
3
C µ : Ω → R is defined as
C µ (x) :=
Z
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
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
the following notation:
α := α(x, y)
u := u(x)
α0 := α(y, x)
0
u := u(y)
ψ := ψ(x, y)
u := u(x)
ψ 0 := ψ(y, x)
u0 := u(y)
and similarly for other functions.4
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 operators;in particular, (2a) can be viewed as nonlocal Gauss’ theorem. The antisymmetry, with respect to x and y, of the integrands in the definitions of
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
For example, from (1a), we have D ν (x) = Ω (ν + ν 0 ) · 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)
ZΩ
C µ dx = 0.
(2c)
Ω
Proof From (1a), we have that
Z
Z Z
D ν dx =
Ω
Ω
ν + ν 0 · α dy dx.
Ω
Because α is antisymmetric, the integrand in the double integral is antisymmetric, i.e., we have that (ν + ν 0 ) · α = −(ν 0 + 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
(1b) and (1c), respectively.5
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 functions 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
Z
Ω
Ω
Z Z
(v0 − v) · β η dydx = 0
(3b)
γ × (w0 − w) · µ dydx = 0.
(3c)
v · G η dx +
Ω
Z
w · C µ dx −
Ω
Ω
Z Z
Ω
Ω
Ω
Proof Let ξ = uν; then,
(ξ + ξ 0 ) · α = (uν + u0 ν 0 ) · α = u(ν + ν 0 ) · α + (u0 − u)ν 0 · α
so that, from (1a), we have
Z D(ξ) =
u(ν + ν 0 ) · α + (u0 − u)ν 0 · 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
0
D(ξ) dx =
u
(ν + ν ) · αdy dx +
(u0 − u)ν 0 · α dydx = 0
Sandia National Labs SAND 2010-8353J
Ω
Ω
Ω
Ω
Ω
so that, by (1a) and the anti-symmetry of α,
Z
Z Z
uD ν dx +
(u − u0 )ν 0 · 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.
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
that maps point functions G to two-point functions defined over Ω × Ω that
satisfies
G, L(F ) Ω − L∗ (G), F Ω×Ω = 0.
(4)
Here, the operators F and G may denote pairs of vector-scalar, scalar-vector,
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)
where D∗ u : Ω × Ω → Rk . Given the point function v(x) : Ω → Rn , the
adjoint of G is the two-point operator given by
G ∗ v (x, y) = −(v0 − v) · β for x, y ∈ Ω,
(5b)
where G ∗ v : Ω × Ω → R. Given the point function w(x) : Ω → R3 , the
adjoint of C is the two-point operator given by
C ∗ w (x, y) = γ × (w0 − w) for x, y ∈ Ω,
(5c)
where C ∗ w : Ω × Ω → R3 .
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 nonlocal adjoint operators given by Corollaries 1 and 2, respectively as follows:
Z
Z Z
D∗ u · ν dy dx = 0
uD ν dx −
Ω
Z
Ω
Z Z
G ∗ v η dy dx = 0
(6b)
C ∗ w · µ dy dx = 0.
(6c)
v · G η dx −
Ω
Z
Ω
w · C µ dx −
Ω
Z Z
Ω
(6a)
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 function, respectively. This leads to the nonlocal Green’s first identity
G, L(ΘL∗ (H)) Ω − L∗ (G), ΘL∗ (H) Ω×Ω = 0.
(7)
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
= 0.
(8)
− H, L ΘL∗ (G)
G, L ΘL∗ (H)
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)
Sandia National Labs SAND 2010-8353J
Ω
Z
Ω
∗
v · G θG (u) dx −
Ω
Z
Z
u · G θG ∗ (v) dx = 0
(10b)
Ω
∗
w · C Θ : C (u) dx −
Ω
Z
u · C Θ : C ∗ (w) dx = 0.
(10c)
Ω
2.5 Further observations and results about nonlocal operators
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
(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
point divergence operator Dt (Ψ )(x) : Ω → Rn for tensors is defined as
Z
Dt (Ψ )(x) :=
Ψ + Ψ 0 · α dy for x ∈ Ω.
(11a)
Ω
A nonlocal Gauss theorem for tensor functions as well as the corresponding 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
particular, Definition 2 implies that, given the point function v(x) : Ω → Rn ,
Dt∗ (v)(x, y) = −(v0 − v) ⊗ α for x, y ∈ Ω,
(11b)
where Dt∗ (v) : Ω × Ω → Rn×k , 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
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
D C ∗ (u) = 0
C D∗ (u) = 0
∗
Dt Dt∗ (u)
Dt∗ (u)
∗
G (u) = tr
− G G ∗ (u) = C C (u)
for u : Ω → R3
(12a)
for u : Ω → R
(12b)
n
(12c)
3
(12d)
for u : Ω → R
for u : Ω → R .
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
Dt Dt∗ (u)
∗
− G G (u) = −
Z (u0 − u) ⊗ α + (u − u0 ) ⊗ α0 · α dy
Ω
Z +
(u0 − u) · α + (u − u0 ) · α0 α dy
Z Ω
= −2
(u0 − u)(α · α) − (u0 − u) · α α dy
ZΩ = −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
Sandia National Labs SAND 2010-8353J
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 properties of the corresponding differential operators are given in the following
proposition whose proof is a straightforward consequence of (6) and the definitions of the operators involved.
Proposition 2 Let b and b denote a constant scalar and vector, respectively.
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 integral theorems (2a)–(2c).
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
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), respectively, were defined in three dimensions. These definitions can be generalized to arbitrary dimensions by replacing the vector cross product with the
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 possible 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
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
Note that G ∗ C µ 6= 0 and C ∗ G η 6= 0 because of the nonlocality of the
operators.
8
12
Q. Du et al.
curl operator.9 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
function µ(x, y) : Ω × Ω → R2 , we can view C µ as the nonlocal scalar
point function defined by
Z
C µ (x) =
(µ2 + µ02 )γ1 − (µ1 + µ01 )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
C ∗ (w)(x, y) = (w20 − w2 )γ1 − (w10 − 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
C(µ)(x) :=
(µ + µ0 )(γ2 e1 − γ1 e2 ) 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
C ∗ (w)(x, y) = (w0 − w)(γ2 e1 − γ1 e2 )
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
curl u =
∂u2
∂u1
−
∂x2
∂x1
and
curl u = −
∂u
∂u
e1 +
e2 .
∂x2
∂x1
10
For example, given a region Ω ⊂ Rn 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 relations 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.
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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
We divide the region Ω into disjoint open subsets Ωs and Ωc , i.e., we have
Ω = (Ωs ∪Ωc )∪(Ω s ∩Ω c ), where ( · ) denotes the closure. We refer to Ωs as the
solution domain12 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
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.
3.2 Nonlocal point constraint operators
The constraint operators corresponding to each of the nonlocal point operators 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.
Definition 4 [Nonlocal point constraint operators] Let Ω =
(Ωs ∪
Ωc )∪(Ω s ∩Ω c ). 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 constraint operators because they are needed to describe nonlocal volume-constrained
problems and showing their well posedness; see Section 3.5 for examples of nonlocal
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 volumeconstrained 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 ) ∪ (Ω s ∩ Ω c ), 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
N (ν)(x) := − (ν + ν 0 ) · α dy
for x ∈ Ωc .
(13a)
Ω
Corresponding to the point gradient operator G η : Ωs → Rk defined as
in (1b) but for x restricted to Ωs , we have the point constraint operator
S(η) : Ωc → Rk defined as
Z
S(η)(x) := − (η + η 0 )β dy
for x ∈ Ωc .
(13b)
Ω
Corresponding to the point curl operator C µ : Ωs → R3 defined as in (1c)
but for x restricted to Ωs , we have the point constraint operator T (µ) : Ωc →
R3 defined as
Z
T (µ)(x) := −
γ × (µ + µ0 ) 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
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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
0
uD(ν) dx +
(u − u)ξ · ν dydx =
uN (ν) dx
(15a)
Ωs
Ω
Z
Ω
Ωc
Z Z
0
v · G(η) dx +
Ωs
(v − v) · β η dydx =
Ω
Z
Z
v · S(η) dx
Ω
(15b)
Ωc
Z Z
w · C(µ) dx −
Ωs
Ω
γ × (w0 − w) · µ dydx
Ω
Z
w · T (µ) dx.
=
(15c)
Ωc
3.4 Nonlocal adjoint operators and Green’s identities with constraint
operators
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 ) Ω − L∗ (G), F Ω×Ω = G, B(F ) Ω ,
(16)
s
c
where B is an operator that maps two-point functions F to point functions
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
Z
Ω
v · G η dx −
Ωs
Z
Ωs
w · C µ dx −
Ω
Ωc
Z Z
Ω
Z Z
Ω
Ω
G ∗ v η dydx =
Ω
C ∗ w · µ dydx =
Z
v · S(η) dx
(17b)
w · T (µ) dx.
(17c)
Ωc
Z
Ωc
16
Q. Du et al.
The abstract Green’s first and second identities (7) and (8) now take the
form
(18)
G, L(ΘL∗ (H)) Ω − L∗ (G), ΘL∗ (H) Ω×Ω = G, B(ΘL∗ (H)) Ω
c
s
and
G, L ΘL∗ (H)
− H, L ΘL∗ (G)
Ωs
Ωs
∗
= G, B ΘL (H)
− H, B ΘL∗ (G)
Ωc
(19)
,
Ωc
respectively. Likewise, the nonlocal Green’s first identities (9a)–(9c) now respectively take the form
Z
Z Z
uD Θ · D∗ (v) dx −
D∗ (u) · Θ · D∗ (v)ν dydx
Ωs
Z Ω Ω
(20a)
=
uN Θ · D∗ (v) dx
Ωc
Z
Z Z
v · G θG ∗ (u) dx −
θG ∗ (v)G ∗ (u) dydx
Ωs
Ω Ω
Z
v · S θG ∗ (u) dx
=
(20b)
Ωc
Z Z
Z
w · C Θ : C ∗ (u) dx −
C ∗ (w) : Θ : C ∗ (u) dydx
Ωs
Z Ω Ω
=
w · T Θ : C ∗ (u) dx
(20c)
Ωc
and the nonlocal Green’s second identities (10a)–(10c) now respectively take
the form
Z
Z
vD Θ · D∗ (u) dx
uD Θ · D∗ (v) dx −
Ωs
Ωs
Z
Z
(21a)
=
uN Θ · D∗ (v) dx −
vN Θ · D∗ (u) dx
Ωc
Ωc
Z
Z
u · G θG ∗ (v) dx
v · G θG ∗ (u) dx −
Ωs
Ωs
Z
Z
∗
=
v · S θG (u) dx −
u · S θG ∗ (v) dx
Ωc
(21b)
Ωc
Z
Z
w · C Θ : C ∗ (u) dx −
u · C Θ : C ∗ (w) dx
Ωs
Ωs
Z
Z
=
w · T Θ : C ∗ (u) dx −
u · T Θ : C ∗ (v) dx.
Ωc
Ωc
(21c)
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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 problems. Here, we merely state problems involving scalar and vector “secondorder” operators.
Let Ωc = Ωc1 ∪ Ωc2 , where14 Ωc1 ∩ Ωc2 = ∅, and let Θ 4 , Θ 2 , and θ denote fourth-order tensor, second-order tensor, and scalar two-point functions,
respectively. The nonlocal volume-constrained problems

∗
in Ω

 D Θ 2 · D (u) = f
u=g
in Ωc1
(22a)


∗
N Θ 2 · D (u) = h
in Ωc2

∗
D
Θ
:
D
(u)
=f
in Ω

t
4
t

u=g
in Ωc1


∗
Nt Θ 4 : Dt (u) = h
in Ωc2

∗
∗
in Ω

 C Θ 4 : C (u) + G θG (u) = f
u=g
in Ωc1


∗
∗
T Θ 4 : C (u) + S θG (u) = h
in Ωc2
(22b)
(22c)
are analogous to the second-order differential boundary-value problems

in Ω

 −∇ · K2 · ∇u = f
u=g
on ∂Ω1
(23a)


K2 · ∇u · n = h
on ∂Ω2

in Ω

 −∇ · K4 : ∇u = f
u=g
on ∂Ω1


K4 : ∇u · n = h
on ∂Ω2

in Ω

 ∇ × K4 : ∇ × u − ∇ k∇ · u = f


u=g
on ∂Ω1
o


n × K4 : ∇ × u = h1

on ∂Ω2 ,

k∇ · u = h2
(23b)
(23c)
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
tensor, and scalar point functions, respectively.
A special form of the nonlocal volume-constrained problem (22a) corresponding to a scalar-valued solution is studied in [12] by appealing to a variational 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 connection 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 nonlocal and differential operators are made in Section 5 where we demonstrate
circumstances under which the nonlocal point operators are weak representations 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) = −∇y δ(y − x),
(24a)
where ∇ and δ(y − x) denote the differential gradient and Dirac delta distribution, 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)
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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) ∇y δ(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.
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).
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,
Z
D∗ (u) dy = −∇x u.
Ω
Proof Using (5a) we have that
Z
Z
D∗ (u) dy =
u(y) − u(x) ∇y δ(y − x) dy
Ω
Ω
Z
=−
δ(y − x)∇y u(y) − u(x) dy
ZΩ
=−
δ(y − x)∇y u(y) dy = −∇x u(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
identified, in the sense of distributions, with the Laplace operator ∆ = ∇ · ∇.
Proposition 5 Let u ∈ C0∞ (Ω) and select |α(x, y)|2 = 12 ∆y δ(y − x), where
∆ denotes the differential Laplace operator. Then,
D D∗ u(x) = −∆x u(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
Z Ω
=−
u(y) − u(x) ∆y δ(y − x) dy
ZΩ
=−
δ(y − x)∆y u(y) − u(x) dy
ZΩ
=−
δ(y − x)∆y u(y) dy
Ω
= −∆x u(x) = −∇x · ∇x u(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
∇x u · ∇x v 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 defined 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, corresponding 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,
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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 operator Gω (u) : Ω → Rk 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
Cω (u) : Ω → R3 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).
Proposition 6 Let ω(x, y) : Ω × Ω → R denote a non-negative scalar twopoint function and let D∗ , G ∗ , and C ∗ denote the nonlocal adjoint operators
given in (5). The adjoint operator Dω∗ (u)(x) : Ω → Rk corresponding to the
weighted nonlocal divergence operator Dω is given by
Z
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
Z
∗
Gω (v)(x) =
G ∗ (v)(x, y) ω(x, y) dy for x ∈ Ω
(26b)
Ω
for vector point functions v(x) : Ω → Rn . The adjoint operator Cω∗ (w)(x) : Ω
→ R3 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)
Ω
for vector point functions w(x) : Ω → R3 .
Proof We have that
Dω (u), u Ω =
Z
u(x)D ω(x, y)u(x) dx
ZΩ Z
=
D∗ (u)(x, y) · ω(x, y)u(x) dydx
ZΩ ΩZ
=
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, Dω (u) Ω = Dω∗ (u), u 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 twopoint function and let Dt be given as in (11b). Given the tensor point function
U(x) : Ω → Rn×k , 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)
∗
Corollary 8 The adjoint operator Dt,ω
(u)(x) : Ω → Rn×k corresponding to
the operator Dt,ω is given by
Z
∗
Dt,ω
Dt∗ (u)(x, y) ω(x, y) dy for x ∈ Ω
(28)
u (x) =
Ω
for vector point functions u : Ω → Rn .
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
Bε (x) := {y ∈ Rd : |y − x| < ε}.
Let Ω = Rd and let φ denote a positive radial function; select

y−x

for x 6= y
 α(x, y) =


|y − x|
(
|y − x|φ(|y − x|) y ∈ Bε (x)



 ω(|x − y|) = 0
otherwise
with the function φ satisfying a normalization condition
Z
|y − x|2 φ(|y − x|) dy = d,
Bε (x)
(29a)
(29b)
Nonlocal vector calculus, volume-constrained problems and balance laws
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Sandia National Labs SAND 2010-8353J
where d denotes the space dimension.
Note that α is an antisymmetric function whereas ω is a symmetric function. 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
dj u(x) :=
u(x + z) + u(x) zj φ(|z|) dz
(30a)
Bε (0)
d∗j u(x) := −
Z
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 dj u(x) and d∗j u(x) denote the j-th components of Gω (u) and Dω∗ (u),
respectively. Then,
dj u = −d∗j u.
(31)
Proof From (30) we have
dj u(x) + d∗j u(x) = −2u(x)
Z
yj φ(|y|)dy = 0
Bε (0)
from which the result directly follows.
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
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
zj φ(|z|) = −∂j δ(z),
where ∂j u denotes the weak derivative of u with respect to xj . Then,
dj = ∂j
and
d∗j = −∂j .
The following lemma demonstrates that the components of the weighted
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 d∗j defined by (30) are bounded
linear operators from H 1 (Rd ) to L2 (Rd ). Moreover, if u ∈ H 1 (Rd ), then as
ε→0
kdj u − ∂j ukL2 (Rd ) → 0
(33a)
kd∗j u
(33b)
+ ∂j ukL2 (Rd ) → 0,
where ∂j u denotes the weak derivative of u with respect to xj . Moreover, if
Z
|z|1+s φ(|z|) dz < ∞
for some 0 ≤ s ≤ 1,
(33c)
Bε (0)
then, for j = 1, . . . , d, the weighted operators dj and d∗j are bounded linear
operators from H t (Rd ) to H t−s (Rd ) for any t ≥ 0.
d
Proof The conclusion (33a) is equivalent to kdd
j u − ∂j ukL2 (Rd ) → 0, where
d
dd
j u and ∂j u denote the Fourier transforms of dj u and ∂j u, respectively.
Under the condition (33c) for s ∈ [0, 1], if u ∈ H t (Rd ) for t ≥ 0, then
Z
dd
ju =
Bε (0)
(eiy·ξ + 1) u
b(ξ) yj φ(|y|) dy
Z
=i
Bε (0)
Z
sin(y · ξ) u
b(ξ) yj φ(|y|) dy
=i
X
sin(yj ξj ) cos(
yk ξk )
Bε (0)
k6=j
X
+ cos(yj ξj ) sin(
yk ξk ) u
b(ξ) yj φ(|y|) dy
k6=j
Z
= iu
b(ξ)
X
sin(yj ξj ) cos(
yi ξk ) yj φ(|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 respect to the integration
variable, respectively. Because for any 0 ≤ s ≤ 1,
P
| sin(yj ξj ) cos( k6=j yi ξk )| ≤ |yj ξj |s , we have
s
|dd
b(ξ)|
j u| ≤ |ξ u
Z
|yj |1+s φ(|y|) dy
Bε (0)
t−s
so that dd
(Rd ). In particular, for t = s = 1, we obtain that, under
ju ∈ H
the condition (29b), the weighted operators dj , j = 1, . . . , d, defined in (30a)
are bounded linear operators from H 1 (Rd ) to L2 (Rd ).
Nonlocal vector calculus, volume-constrained problems and balance laws
25
If u ∈ H 1 (Rd ), we have that
d
d
d
|dd
j u − ∂j u| ≤ |dj u| + |∂j u|
Z
X
sin(yj ξj ) cos(
≤
yk ξk ) u
b(ξ) yj φ(|y|) dy + |ξj u
b|
Bε (0)
k6=j
Sandia National Labs SAND 2010-8353J
Z
≤ |ξj u
b|
Bε (0)
yj2 φ(|y|) dy + |ξj u
b|
≤ 2|ξj u
b| ∈ L2 (Rd ),
where the third inequality holds because | sin(x)| ≤ |x| and | cos(x)| ≤ 1. By
Taylor’s theorem, we have
Z
X
dd
(yj ξj + yj3 ξj3 cos(θ1 )/6)(1 + (
yk ξk )2 cos(θ2 )/2) u
b yj φ(|y|) dy
ju = i
Bε (0)
Z
= iξj u
b
Bε (0)
k6=j
yj2 φ(|y|) dy + Iε (ξ) = iξj u
b + Iε (ξ),
where, for some θ1 and θ2 ,
Z
i
y 4 cos(θ1 )φ(|y|) dy ξj3 u
b
Iε (ξ) :=
6 Bε (0) j
Z
X
i
yj2 (
yi ξi )2 cos(θ2 )φ(|y|) dy ξj u
b
+
2 Bε (0)
k6=j
Z
X
i
+
yj4 (
yk ξk )2 cos(θ1 ) cos(θ2 )φ(|y|) dy ξj3 u
b.
12 Bε (0)
k6=j
Because u
b is bounded a.e., we see that, for any ξ,
X
X
|Iε (ξ)| ≤ ε2 |ξj3 u
b| + ε2 |(
ξk )2 ξj u
b| + ε4 |(
ξk )2 ξj3 u
b| → 0
k6=j
as ε → 0.
k6=j
Hence, we have
d
d
dd
j u → ∂j u a.e. for ξ ∈ R
as
ε → 0.
Then, by the dominated convergence theorem, we obtain
kdj u − ∂j ukL2 (Rd ) → 0
as ε → 0.
Lemma 1 implies the same results hold for the operator d∗j .
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
generally, for φ satisfying (33c) with positive s, the operators dj and d∗j
actually map a subspace of L2 (Rd ), for instance the fractional Sobolev space
H s (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 H t (Rd ) to H t−s (Rd ) for 0 ≤ s ≤ 1, where d = 3 for the
weighted curl operators. Moreover, if u ∈ H 1 (Rd ) and u ∈ [H 1 (Rd )]d ,
Dω (u) → ∇ · u
Dω∗ (u) → −∇u
Gω (u) → ∇u
Gω∗ (u) → −∇ · u
Cω (u) → ∇ × u
Cω∗ (u)
(34)
→ ∇ × 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
Dω (u) =
d
X
di ui
i=1
and
∇·u=
d
X
∂i ui
i=1
so that
kDω (u) − ∇ · ukL2 (Rd ) ≤
d
X
kdi ui − ∂i ui kL2 (Rd ) → 0
as ε → 0.
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 (C1 Dω∗ (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.
Corollary 11 Let u ∈ H 1 (Rd ), u ∈ [H 1 (Rd )]d , C1 : Rd → Rd × Rd in
L∞ (Rd × Rd ), and c2 : Rd → R in L∞ (Rd ). Then,
Dω C1 · Dω∗ (u) → −∇ · (C1 · ∇u)
(35)
Gω c2 Gω∗ (u) → −∇(c2 ∇ · u)
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;
∗
in particular, we have that Dt,ω (C3 : Dt,ω
(u)) → −∇ · (C3 : ∇u) with C3 : Rd →
d
d
d
d
∞
d
d
d
R × R × R × R in L (R × R × R × Rd ).
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.
Sandia National Labs SAND 2010-8353J
Lemma 2 Let L denote a linear operator that commutes with the differential
and nonlocal operators. Then, if Lu ∈ H 1 (Rd ),
Dω (Lu) → ∇ · Lu
Dω∗ (Lu) → −∇Lu
Gω (Lu) → ∇Lu
Gω∗ (Lu) → −∇ · Lu
Cω (Lu) → ∇ × Lu
Cω∗ (Lu)
(36)
→ ∇ × 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.
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
by
P(Ω) = Q(∂Ω), Ω ⊂ Rd
(37)
postulating that the production of an extensive quantity over the open subregion 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
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 calculus. However, assuming sufficient regularity, the volume interaction may be
relegated to that on the surface ∂Ω. In particular, suppose that
Z
Q(∂Ω) =
q̆ dHd−1 ,
(38)
∂Ω
where q̆ and Hd−1 are the flux density and a Hausdorff measure over the surface ∂Ω, respectively; see [5, Chap. II] for a discussion. Then, under general
conditions,
Z
Z
Z
q̆ 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 postulate and theorem, respectively. The relation (39) establishes the fundamental
28
Q. Du et al.
result that the flux density q̆ 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
I(Ω, Rd \ Ω) := Q(∂Ω)
∀ Ω ⊂ Rd .
(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
I(Ω, Rd \ Ω) + I(Rd \ Ω, Ω) = 0
∀ Ω ⊂ Rd .
(40b)
Because Rd \ Ω2 = Rd \ (Ω1 ∪ Ω2 ) ∪ Ω1 and Rd \ Ω1 = Rd \ (Ω1 ∪ Ω2 ) ∪ Ω2
for all disjoint subregions Ω1 and Ω2 , we have
I(Ω1 , Ω2 ) + I(Ω2 , Ω1 ) = I(Ω1 , Rd \ Ω1 ) + I(Ω2 , Rd \ Ω2 )
− I Ω1 ∪ Ω2 , Rd \ (Ω1 ∪ Ω2 ) = 0. (40c)
Then, because the disjoint regions Ω1 and Ω2 are arbitrary,
I(Ω, Ω) = 0
∀ Ω ⊂ Rd .
(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 explains 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 balance 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 equivalently replaced by
P(Ω) = I(Ω, Rd \ Ω),
Ω ⊂ Rd
(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 alternating (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
J (Ω1 , Ω2 ) :=
f (x, y) dy dx
∀ Ω1 , Ω2 ⊂ Rd ,
(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
∀ x, y ∈ Rd .
Sandia National Labs SAND 2010-8353J
f (x, y) = −f (y, x)
(42b)
In stark contrast to I, the flux induced by J dispenses the need for determining 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 existence of the interaction density f follows from the Schwartz Kernel theorem
[19].
We now derive the field equation for the balance law
e
P(Ω)
= J (Ω, Rd \ Ω),
Ω ⊂ Rd .
(43a)
A result due to Fuglede [9] as used in [4, p. 298] supplies a production density
ρe: Rd → R that satisfies
Z
Z Z
ρe(x) dx =
f (x, y) dy dx.
Ω
Rd \Ω
Ω
The antisymmetry of f grants
Z Z
Z Z
f (x, y) dy dx =
Ω
Rd \Ω
Ω
f (x, y) dy dx
(43b)
Rd
and the field equation
Z
ρe(x) =
f (x, y) dy
(43c)
Rd
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)
Rd
results. Evidently the right-hand side of (43d) is identified with the distribution 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
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., Ω 1 ∩Ω 2 = ∅, 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 expression for an operator A may be derived such that
∇ · A f (x, y) =
Z
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
T 1
=
(α ⊗ α) · (u0 − u) dy,
(44)
− Dt Dt∗ u
2
Ω
where u represents the displacement field, Dt u = Dt u and Dt∗ u = Dt∗ u
are given by (11a) and (11b), respectively, and the integrand is an interaction
T
density f = (α⊗α)·(u0 −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.,
Dt (Dt∗ (Ax + c))T = 0,
18
A = −AT ,
See [6, pp. 149–150] for a discussion.
c := constant vector,
(45)
Nonlocal vector calculus, volume-constrained problems and balance laws
31
Sandia National Labs SAND 2010-8353J
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
T y−x ⊗ y−x
1
· u(y) − u(x) dy,
(46)
− Dt Dt∗ u(x)
=
2
σ(|y − x|)
Ω
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
− Dt (Dt∗ u)T → −µ∇ · (∇u) − 2µ∇(∇ · u),
2
the Navier operator of linear elasticity with Poisson ratio one-quarter; see
also the paper [8]. The subsequent paper [26] considers volume-constrained
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
T
∗
−Dt,ω η Dt,ω
u + (λGω∗ u) I ,
(47)
∗
where η and λ are materials constants, Gω∗ u = Gω∗ u , Dt,w and Dt,w
are
given by (26b), (27) and (28), respectively. The scalar Gω∗ u measures the
volumetric change, or dilatation, in the material so that Gω∗ u I is a diagonal tensor representing volumetric stress. This allows us to readily apply the
nonlocal calculus to study the well-posedness of both free-space and volumeconstrained 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 Poisson ratios, e.g., not relegated to a value of one-quarter. This is the subject
of forthcoming research.
Acknowledgements
The third author thanks Petronela Radu of the University of Nebraska for
pointing out the paper [4].
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