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Sandia National Labs SAND 2010-8353J 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), (x; y), and (x; y), respectively. Symmetric and antisymmetric two-point functions satisfy, e.g., for the scalar two-point function (x; y), we have (x; y) = (y; x) and (x; y) = − (y; x), respectively. The dot (or inner) product of two vectors u; v 2 Rn is denoted by u · v 2 R; the dyad (or outer) product is denoted by u ⊗ w 2 Rn×k whenever w 2 Rk; given a second-order tensor (matrix) U 2 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 2 R3. The Frobenius product of two second-order tensors A 2 Rn×k and B 2 Rn×k, denoted by A: B, is given by the sum of the component-wise n×n product of the two tensors.
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