NONLOCAL PROBLEMS WITH LOCAL DIRICHLET AND NEUMANN BOUNDARY CONDITIONS BURAK AKSOYLU AND FATIH CELIKER

Transcription

1 NONLOCAL PROBLEMS WITH LOCAL DIRICHLET AND NEUMANN BOUNDARY CONDITIONS BURAK AKSOYLU AND FATIH CELIKER Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 480, USA. Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 480, USA. Abstract. We present novel governing operators in the theory of peridynamics PD) which will allow the extension of PD to applications that require local boundary conditions BC). Due to its nonlocal nature, the original PD governing operator uses nonlocal BC. The novel operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local Dirichlet or Neumann BC. Our construction is straightforward and easily accessible. The main ingredients are antiperiodic and periodic extensions of kernel functions together with even and odd parts of functions. We also present governing operators that enforce antiperiodic and periodic BC and the corresponding compatibility conditions for the right hand side function in a given operator equation. Keywords: Nonlocal operator, peridynamics, boundary condition, integral operator. Mathematics Subject Classification 000): 74B99, 47G Introduction We present novel governing operators in the theory of peridynamics PD), a nonlocal extension of continuum mechanics developed by Silling [11]. We consider problems in 1D and choose the domain := [ 1, 1]. By suppressing the dependence of u on the time variable t, the original bond based PD governing operator is given as L orig ux) := Ĉx x)ux)dx Ĉx x)ux )dx, x. 1.1) address: burak@wayne.edu, celiker@wayne.edu. Date: February 4, 017. Burak Aksoylu was supported in part by the European Commission Marie Curie Career Integration grant and the Scientific and Technological Research Council of Turkey TÜBİTAK) MFAG 115F473 grant. 1

2 B. AKSOYLU AND F. CELIKER Due to its nonlocal nature, the operator L orig uses nonlocal boundary conditions BC); see [11, p.01]. We define the operator that is closely related to L orig as Lux) := cux) Ĉx x)ux )dx, x, where c := Cx )dx. We will prove that the two operators agree in the bulk. As the main contribution, we present novel governing operators that agree with L in the bulk of, and, at the same time, enforce local Dirichlet or Neumann BC. Since PD is a nonlocal theory, one might expect only the appearance of nonlocal BC. Indeed, so far the concept of local BC does not apply to PD. Instead, external forces must be supplied through the loading force density [11]. On the other hand, we demonstrate that the anticipation that local BC are incompatible with nonlocal operators is not quite correct. Hence, our novel operators present an alternative to nonlocal BC and we hope that the ability to enforce local BC will provide a remedy for surface effects seen in PD; see [9, Chap. 4, 5, 7, and 1] and [8, 10]. For x, x [ 1, 1], it follows that x x [, ]. Hence, in 1.1), the kernel function Cx) needs to be extended from to the domain of Ĉx x), which is := [, ]. The default extension is the zero extension defined by 0, x [, 1), Ĉx) := Cx), x [ 1, 1], 0, x 1, ]. Furthermore, the kernel function Cx) is assumed to be even. Namely, C x) = Cx). An important first choice of Cx) is the canonical kernel function χ δ x) whose only role is the representation of the nonlocal neighborhood, called the horizon, by a characteristic function. More precisely, for x, χ δ x) := { 1, x δ, δ) 0, otherwise. The size of nonlocality is determined by δ and we assume δ < 1. constructed by χ δ x), a kernel function used in practice is in the form 1.) Since the horizon is Cx) = χ δ x)µx), 1.3) where µx) L ) is even.

3 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS 3 We define the periodic and antiperiodic extensions of Cx) from to, respectively, as follows Cx + ), x [, 1), Cx + ), x [, 1), Ĉ a x) := Cx), x [ 1, 1], Ĉ p x) := Cx), x [ 1, 1], 1.4) Cx ), x 1, ], Cx ), x 1, ]. Even for smooth µx), note that Ĉx), Ĉ a x), and Ĉpx) are not necessarily smooth; see Figure.1. Throughout the paper, we assume that ux) L ) C 1 ). 1.5) Even and odd parts of the function u are used in the novel governing operators. Here we provide their definitions. We denote the orthogonal projections that give the even and odd parts, respectively, of a function by P e, P o : L ) L ), whose definitions are P e ux) := ux) + u x), P o ux) := ux) u x). 1.6) Theorem 1.1. Main Theorem) Let c = Cx )dx. The following operators M D and M N defined by MD c ) ux) := [Ĉa x x)p e ux ) + Ĉpx x)p o ux ) ] dx, MN c ) ux) := [Ĉp x x)p e ux ) + Ĉax x)p o ux ) ] dx, agree with Lux) in the bulk, i.e., for x 1 + δ, 1 δ). Furthermore, the operators M D and M N enforce homogeneous Dirichlet and Neumann BC, respectively. More precisely, for u±1) = 0 and u ±1) = 0, we obtain M D u±1) = 0 and M N u ±1) = 0, respectively Related Work and Structure of the Paper. In [7], one of our major results was the finding that, in R, the PD governing operator is a function of the governing operator of local) classical elasticity. This result opened the path to the introduction of local boundary conditions into PD theory. Building on [7], we generalized the results in R to bounded domains, a critical feature for all practical applications. In [], we laid the theoretical foundations and in [1], we applied the foundations to prominent BC such as Dirichlet and Neumann, as well as presented numerical implementation of the corresponding wave propagation. We carried out numerical experiments by utilizing M D and M N as governing operators in [1]. In [3], we studied other related governing operators. In [4], we present the extension of the novel operators to D. In [6], we study the condition numbers of the novel governing operators. Therein, we prove that the modifications made to the operator L orig to obtain the novel operators are minor as far as the condition numbers are concerned.

4 4 B. AKSOYLU AND F. CELIKER The rest of the article is structured as follows. In Sec., we present the main observation that leads to the construction of the novel operators that enforce Dirichlet and Neumann BC. In Sec. 3, we give the proof of main theorem. In Sec. 4, we show how to obtain the operators that enforce antiperiodic and periodic BC by choosing suitable combination of kernel functions. In Sec. 5, when an equation using the governing operators is solved, we show that the right hand side function should satisfy the same the BC enforced by the governing operator. In Sec. 6, we provide the highlights of the extension from the 1D construction to D. We conclude in Sec. 7.. The Main Observation and the Construction Let us study the definition of Ĉax) given in 1.4) by explicitly writing the expression of the kernel in 1.3) as follows χ δ x + )µx + ), x [, 1), Ĉ a x) = χ δ x)µx), x [ 1, 1], χ δ x )µx ), x 1, ]. Let us closely look at the first expression in the above definition of Ĉax) Ĉ a x) x [, 1) = χ δ x + )µx + )..1) The expression in.1) is equivalent to { µx + ), x + δ, δ) and x [, 1), Ĉ a x) x [, 1) = 0, x + / δ, δ) and x [, 1). Due to the following set equivalence.) {x : x+ δ, δ) and x [, 1)} = {x : x [ δ, +δ) [, 1) = [, +δ)}, the expression.) reduces to Ĉ a x) x [, 1) = Similarly, for x 1, ], we have Ĉ a x) x 1,] = { µx + ), x [, + δ), 0, x [ + δ, 1). { 0, x 1, δ], µx ), x δ, ]. Combining.3) and.4), for x [, ], we obtain µx ), x [, + δ), µx), x δ, δ), Ĉ a x) = µx + ), x δ, ], 0, otherwise..3).4).5)

5 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS y y 0.5 y x x -0.5 b a) Plot of Cx). bp x). b) Plot of C ba x). c) Plot of C d) Plot of Cx, x0 ). e) Plot of Cp x, x0 ). f) Plot of Ca x, x0 ). z g) Contour plot of Cx, x0 ). z 1.6 z h) Contour plot of Cp x, x0 ). i) Contour plot of nonnegative part of Ca x, x0 ). Figure.1. The kernel function Cx) = χδ x)µx) with χδ x) given in 1.), δ = 0.4, and µx) = 0.5 x. The zero, periodic, and antiperiodic extensions b bp x), and C ba x), respectively. For plotting, we of Cx) are denoted by Cx), C 0 b x), C bp x0 x), and C ba x0 x) defined by employ bivariate versions of Cx b 0 x), Cp x, x0 ) := C bp x0 x), and Ca x, x0 ) := C ba x0 x), Cx, x0 ) := Cx respectively.. x

6 6 B. AKSOYLU AND F. CELIKER Similarly, we obtain the following expression for the periodic extension µx ), x [, + δ), µx), x δ, δ), Ĉ p x) = µx + ), x δ, ], 0, otherwise. Lemma.1. Let the kernel function Cx) be in the form Cx) = χ δ x)µx),.6) where µx) L ) is even. Let Ĉx), Ĉax), and Ĉpx) denote the zero, antiperiodic, and periodic extensions of Cx) to := [, ], respectively. Then, Ĉx) = Ĉax) = Ĉpx), x + δ, δ)..7) Furthermore, we have the following agreement in the bulk. For x 1 + δ, 1 δ), Ĉx x) = Ĉax x) = Ĉpx x), x [ 1, 1]..8) Proof. By the definition of the functions Ĉax) and Ĉpx) in.5) and.6), respectively, they differ from Ĉx) only on [, + δ) δ, ]. Also, see Figure.1. Hence, Ĉx), Ĉ a x), and Ĉpx) coincide on [ + δ, δ), i.e.,.7) holds. Since for x in the bulk, i.e., x 1 + δ, 1 δ) and x in the range of integration, i.e., x [ 1, 1], we have x x + δ, δ). From.7), we conclude.8). Remark.. The kernel Ĉx) : [, ] R is a univariate function. The operator L orig utilizes Ĉx x). In order to visualize.7), it is more useful to define bivariate versions of Ĉx x), Ĉax x), and Ĉpx x), respectively, as follows C, C a, C p : [ 1, 1] [ 1, 1] R. For brevity, with a slight abuse of notation, we represent the bivariate functions using the same name of the univariate function C ), i.e., Cx, x ) := Ĉx x), C a x, x ) := Ĉax x), and C p x, x ) := Ĉpx x). This notation is also used in Figure.1. Hence, analogous to.7), in the bulk, i.e., x 1 + δ, 1 δ), kernel functions coincide Cx, x ) = C a x, x ) = C p x, x )..9) We first prove that the operators L and L orig agree in the bulk. Lemma.3. Lux) = L orig ux), x 1 + δ, 1 δ). Proof. For x in the bulk, we have x δ, x + δ) = x δ, x + δ). Hence, Ĉx x)dx = χ δ x x) µx x)dx

7 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS 7 = µx x)dx x δ,x+δ) = µx x)dx x δ,x+δ) = µx )dx δ,δ) = χ δ x )µx )dx = Cx )dx. The result follows. 3. Dirichlet and Neumann BC, and Differentiation Under the Integral Sign Imposing Neumann also antiperiodic and periodic) BC requires differentiation. Thus, we present technical details regarding differentiation under the integral sign which are provided in Lemma 3.1. The proof of Lemma 3.1 which is omitted here, is by the Lebesgue Dominated Convergence Theorem. Similarly, the limit in the definition of the Dirichlet BC can be interchanged with the integral, again by the Lebesgue Dominated Convergence Theorem. Lemma 3.1. Suppose that the function k : x x R satisfies the following conditions. 1) The function kx, x ) is measurable with respect to x for each x x. ) For almost every x x, the derivative k x x, x ) exists for all x x. 3) There is an integrable function l : x R such that k x x, x ) lx ) for all x x. Then, d kx, x )dx = dx x x k x x, x )dx. We use Lemma 3.1 to check if the operator M N enforces homogeneous Neumann BC. First, we want to identify the integrand associated to M N. We start with writing P e and P o explicitly and utilizing a simple change of variable as follows. MN c ) ux) = = [Ĉp x x) ux ) + u x )) + Ĉax x) ux ) u x ))] dx K N x, x )ux )dx, 3.1)

8 8 B. AKSOYLU AND F. CELIKER where K N x, x ) := 1 {[Ĉa x x) Ĉax + x) ] + [ Ĉ p x x) + Ĉpx + x) ]}. Analogous to the construction given in [1], we assume that Cx) L ), and hence, Ĉx), Ĉax), Ĉpx) L ). 3.) We are now in a position to determine the necessary conditions to apply Lemma 3.1. First, we set x = x = and kx, x ) = K N x, x )ux ). Considering the jumps and the fact that the BC is enforced at the boundary, we assume that Ĉ a x) and Ĉpx) are piecewise continuously differentiable in and continuously differentiable functions up to. Hence, the first two conditions of Lemma 3.1 are satisfied. To satisfy the third condition, we define and assume that lx ) := ess sup K N x x x x, x ) ux ), ess sup K N x x x x, x ) L x ). 3.3) The integrability of lx ) is sufficient to satisfy the third condition. We could choose any L p x ) space. We choose the space L x ) in 3.3) in order to align with the construction given in [1]. Since ux ) L x ), we obtain lx ) L x ). We are now ready to prove our Main Theorem. Proof. Proof of Thm. 1.1) We exploit.9) in constructing the governing operators that enforce Neumann and Dirichlet BCs by rewriting the L operator in the following way. For x 1 + δ, 1 δ), we have ) L c ux) = Ĉx x)ux )dx = Ĉx x)p e + P o )ux )dx [Ĉx = x)p e ux ) + Ĉx x)p o ux ) ] dx = [Ĉp x x)p e ux ) + Ĉax x)p o ux ) ] dx = M N c ) ux).

9 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS 9 Similarly, for x 1 + δ, 1 δ) L c ) ux) = [Ĉx x)p e ux ) + Ĉx x)p o ux ) ] dx = [Ĉa x x)p e ux ) + Ĉpx x)p o ux ) ] dx = M D c ) ux). Next, we show that M N and M D enforce homogeneous Neumann and Dirichlet BC, respectively. The operator M N : First we remove the points at which the derivative of K N x, x ) does not exist from the set of integration. Note that such points form a set of measure zero, and hence, does not affect the value of the integral. We differentiate both sides of 3.1) and apply Lemma 3.1 to interchange the the differentiation with the integral. We can differentiate the integrand K N x, x ) in a piecewise fashion and obtain where d [ MN c ) u ] x) = dx K N x x, x )ux )dx, 3.4) K N x x, x ) = 1 {[Ĉ a x x) Ĉ ax + x) ] + [ Ĉ px x) + Ĉ px + x) ]}. We check the boundary values by plugging x = ±1 in 3.4). d [ MN c ) u ] K N ±1) = dx x ±1, x )ux )dx. 3.5) The functions Ĉ a and Ĉ p are -antiperiodic and -periodic because they are the derivatives of -antiperiodic and -periodic functions, respectively. Hence, Ĉ a±1 + x ) = Ĉ a 1 + x ) and Ĉ p 1 + x ) = Ĉ p±1 + x ). 3.6) Hence, the integrand in 3.5) vanishes, i.e., K N x ±1, x ) = 0. Therefore, we arrive at d dx M Nu±1) = cu ±1). 3.7) Since we assume that u satisfies homogeneous Neumann BC, i.e., u ±1) = 0, we conclude that the operator M N enforces homogeneous Neumann BC as well. The operator M D : In order to check if the operator M D enforces homogeneous Dirichlet BC, we start again with writing P e and P o explicitly and utilizing a simple change of variables

10 10 B. AKSOYLU AND F. CELIKER as follows. MD c ) ux) = = where Ĉ a x x) ux ) + u x )) dx K D x, x )ux )dx, Ĉ p x x) ux ) u x )) dx K D x, x ) := 1 {[Ĉa x x) + Ĉax + x) ] + [ Ĉ p x x) Ĉpx + x) ]}. 3.8) By the Lebesgue Dominated Convergence Theorem, the limit in the definition of the Dirichlet BC can be interchanged with the integral. Now, we check the boundary values by plugging x = ±1 in 3.8). MD c ) u±1) = K D ±1, x )ux )dx. 3.9) Since Ĉa and Ĉp are -antiperiodic and -periodic, respectively, we have Ĉ a 1 + x ) = Ĉa±1 + x ) and Ĉ p 1 + x ) = Ĉp±1 + x ). 3.10) Hence, the integrand in 3.9) vanishes, i.e., K D ±1, x ) = 0. Therefore, we arrive at M D u±1) = cu±1). 3.11) Since we assume that u satisfies homogeneous Dirichlet BC, i.e., u±1) = 0, we conclude that the operator M D enforces homogeneous Dirichlet BC as well. Remark 3.. We have defined M D and M N in a way that they are linear bounded operators. More precisely, M D, M N LX, X) where X = L ) C 1 ). For M D, the choice of X can be relaxed as L ) C 0 ). This choice is implied when we study M D. Boundedness of M D and M N follows from the choice of 1.5) and 3.). In addition, since M D and M N are both integral operators, their self-adjointness follows easily from the fact that the corresponding kernels are symmetric due to evenness of C), i.e., K D x, x ) = K D x, x) and K N x, x ) = K N x, x). 4. Other Possible Boundary Conditions The construction employed to satisfy local BC is based on the following decomposition of ux ) ux ) = P e ux ) + P o ux ), and the agreement of Ĉx x) with Ĉax x) and Ĉpx x) in the bulk; see.8). By replacing Ĉx x) with either Ĉax x) or Ĉpx x), we have the following 4 combinations for the integrand of L Ĉx x)ux ) = { Ĉ a x x), Ĉpx x) } P e ux ) + { Ĉ a x x), Ĉpx x) } P o ux ).

11 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS 11 Denoting the choice of Ĉax x) and Ĉpx x) by a and p, respectively, the combinations ap and pa gave rise to Dirichlet and Neumann BC, respectively; see Thm Namely, Dirichlet: ap Ĉx x)ux ) = Ĉax x)p e ux ) + Ĉpx x)p o ux ), Neumann: pa Ĉx x)ux ) = Ĉpx x)p e ux ) + Ĉax x)p o ux ). We show that the combinations aa and pp give rise to antiperiodic and periodic BC, respectively. Namely, Antiperiodic: aa Ĉx x)ux ) = Ĉax x)p e ux ) + Ĉax x)p o ux ) = Ĉax x)ux ), Periodic: pp Ĉx x)ux ) = Ĉpx x)p e ux ) + Ĉpx x)p o ux ) = Ĉpx x)ux ). Then, the operators M a and M p are defined by Ma c ) ux) := Ĉ a x x)ux )dx, Mp c ) ux) := Ĉ p x x)ux )dx. We recall the space of functions used to enforce antiperiodic and periodic BC, respectively [1]. { u L ) C 1 ) : lim x 1 ux) = lim x 1 ux), { u L ) C 1 ) : lim ux) = lim ux), x 1 x 1 lim x 1 u x) = lim u x) }, x 1 lim x 1 u x) = lim u x) }. x 1 Since Ĉa and Ĉ a are -antiperiodic and Ĉp are Ĉ p -periodic, similar to 3.10) and 3.6), we have Consequently, Ĉ a 1 x ) = Ĉa1 x ), Ĉ p 1 x ) = Ĉp1 x ) Ĉ a 1 x ) = Ĉ a1 x ), Ĉ p 1 x ) = Ĉ p1 x ). In addition, by applying Lemma 3.1, we obtain Ma c ) u 1) = M a c ) u1), 4.1) Mp c ) u 1) = M p c ) u1). 4.) d [ Ma c ) u ] 1) = d [ Ma u c ) u ] 1), 4.3) dx dx d [ Mp c ) u ] 1) = d [ Mp c ) u ] 1). 4.4) dx dx These imply that the operators M a and M p enforce antiperiodic and periodic BC, respectively.

12 1 B. AKSOYLU AND F. CELIKER 5. Compatibility Conditions When we solve an equation using the operators M BC where BC {D, N, a, p}, i.e., M BC u = f BC, we want to identify the conditions imposed on f BC. Since the operator M BC enforces the corresponding BC, we observe that the same BC is imposed on f BC. To see this, we start by assuming that u satisfies the corresponding BC. Then, we choose f BC from the same space to which u belongs, i.e., f BC L ) C 1 ). From 4.1) and 4.), respectively, we immediately see that f a 1) = f a 1), f p 1) = f p 1). From 4.3) and 4.4), respectively, we also get f a 1) = f a1), f p 1) = f p1). In addition, from 3.11) and 3.7), respectively, we obtain f D ±1) = 0, df N ±1) = 0. dx 6. The Extension to a D Problem In this section, we present the extension of the present work to D problems. The main idea of this extension relies on our 1D construction but it is nontrivial. Its proof requires significant amount of technical detail. Here, we provide only a small part of the results without proof. We choose the domain in D to be = [ 1, 1] [ 1, 1]. There are various combinations of BC one can enforce. Here, we report only pure Dirichlet and pure Neumann BC, the D analogues of the ones presented in Thm The proofs, a comprehensive discussion, and numerical results are provided in [4]. In D, the governing operator in 1.1) takes the form L orig ux, y) := Ĉx x, y y)ux, y)dxdx Ĉx x, y y)ux, y )dx dy. Similar to 1), we define the operator that is closely related to L orig as Lux, y) := cux, y) Ĉx x, y y)ux, y )dx dy, x, y), where c = Cx, y )dx dy. The kernel function Cx, y) is assumed to be even. Namely, C x, y) = Cx, y).

13 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS 13 Similar to the 1D case, we choose the kernel function Cx, y) to be the canonical kernel function χ δ x, y) whose definition is given as follows. For x, y), { 1, x, y) δ, δ) δ, δ) χ δ x, y) := 0, otherwise. The agreement of the operators L and L orig in the 1D bulk shown in Lemma.1 carries over to the D bulk whose definition is given by bulk = {x, y) : x, y) 1 + δ, 1 δ) 1 + δ, 1 δ)}. Inspired by the projections that give the even and odd parts of a univariate function given in 1.6), we define the following operators that act on a bivariate function. whose definitions are P e,x, P o,x, P e,y, P o,y : L ) L ), P e,x ux, y ) := ux, y ) + u x, y ), P o,x ux, y ) := ux, y ) u x, y ), 6.1) P e,y ux, y ) := ux, y ) + ux, y ), P o,y ux, y ) := ux, y ) ux, y ). 6.) Each operator is an orthogonal projection and possesses the following decomposition property P e,x + P o,x = I x, P e,y + P o,y = I y. One can easily check that all four orthogonal projections in 6.1) and 6.) commute with each other. We define the following new operators obtained from the products of these projections. P e,x P e,y ux, y ) := 1 4{ [ux, y ) + ux, y )] + [u x, y ) + u x, y )] }, P e,x P o,y ux, y ) := 1 4{ [ux, y ) ux, y )] + [u x, y ) u x, y )] }, P o,x P o,y ux, y ) := 1 4{ [ux, y ) ux, y )] [u x, y ) u x, y )] }, P o,x P e,y ux, y ) := 1 4{ [ux, y ) + ux, y )] [u x, y ) + u x, y )] }. These operators are also orthogonal projections and satisfy the following decomposition property P e,x P e,y + P e,x P o,y + P o,x P e,y + P o,x P o,y = I x,y. They will be used in the definition of the operators M D and M N. Theorem 6.1. Main Theorem in D) Let := [ 1, 1] [ 1, 1] and the kernel function be separable in the form Cx, y) = Xx)Y y), 6.3)

14 14 B. AKSOYLU AND F. CELIKER where X and Y are even functions. Then, the operators M D and M N defined by MD c ) ux, y) := [ Xa x x)p e,x + X p x x)p o,x ][Ŷa y y)p e,y + Ŷpy ] y)p o,y ux, y )dx dy, MN c ) ux, y) := [ Xp x x)p e,x + X a x x)p o,x ][Ŷp y y)p e,y + Ŷay ] y)p o,y ux, y )dx dy, agree with Lux, y) in the bulk, i.e., for x, y) 1+δ, 1 δ) 1+δ, 1 δ). Furthermore, the operators M D and M N enforce pure Dirichlet and pure Neumann BC, respectively. MD c ) ux, ±1) = MD c ) u±1, y) = 0, [ MN c ) u ] x, ±1) = [ MN c ) u ] ±1, y) = 0, n n where n denotes the outward unit normal vector. Remark 6.. Although we assume a separable kernel function Cx, y) = Xx)Y y) as in 6.3), note that we do not impose a separability assumption on the solution ux, y). Remark 6.3. In Thm. 6.1, the function u is scalar valued which corresponds to the solution of a nonlocal diffusion problem. In higher dimensional PD problems, the function u is vector valued. The extension of our construction to such problems is the subject of ongoing work. 7. Conclusion We presented novel governing operators M D and M N in the theory of PD constructed by the guiding principle that they agree with the original PD operator L orig in the bulk, and, at the same time, enforce local Dirichlet or Neumann BC. We also presented the operators M a and M p that enforce local antiperiodic and periodic BC. In [5], we give an overview of local BC in general nonlocal problems. We believe that our contribution is an important step towards extending the applicability of PD to problems that require local BC such as contact, shear, and traction. For future research, we plan to investigate if our approach of enforcing local BC can be used to eliminate surface effects. Finally, we presented the extension of the 1D governing operators to D on rectangular domains. The generalization to 3D box domains is straightforward. The construction of the operators for general geometries remains to be an open problem and constitutes the subject of ongoing work. References [1] Aksoylu, B., Beyer, H.R., Celiker, F.: Application and implementation of incorporating local boundary conditions into nonlocal problems. Submitted.

15 NONLOCAL PROBLEMS WITH LOCAL BOUNDARY CONDITIONS 15 [] Aksoylu, B., Beyer, H.R., Celiker, F.: Theoretical foundations of incorporating local boundary conditions into nonlocal problems. Rep. Math. Phys. 017). In press. [3] Aksoylu, B., Celiker, F.: Comparison of nonlocal operators utilizing perturbation analysis. In: B.K. et al. ed.) Numerical Mathematics and Advanced Applications ENUMATH 015, Lecture Notes in Computational Science and Engineering, vol. 11, pp Springer 016) [4] Aksoylu, B., Celiker, F., Kilicer, O.: Local boundary conditions in nonlocal problems in higher dimensions. Submitted. [5] Aksoylu, B., Celiker, F., Kilicer, O.: Nonlocal problems with local boundary conditions: An overview. Submitted. [6] Aksoylu, B., Kaya, A.: Conditioning analysis of nonlocal problems with local boundary conditions. Submitted. [7] Beyer, H.R., Aksoylu, B., Celiker, F.: On a class of nonlocal wave equations from applications. J. Math. Phy. 576), ). DOI / Eid: 0690 [8] Kilic, B.: Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials. Ph.D. thesis, Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 008) [9] Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer 014). DOI / [10] Mitchell, J.A., Silling, S.A., Littlewood, D.J.: A position-aware linear solid constitutive model for peridynamics. J. Mech. Mater. Struct. 105), ) [11] Silling, S.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, )