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5-2018 Properties and Convergence of State-based Laplacians Kelsey Wells [email protected]
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This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. PROPERTIES AND CONVERGENCE OF STATE-BASED LAPLACIANS
by
Kelsey Wells
A DISSERTATION
Presented to the Faculty of
The Graduate College at the University of Nebraska
In Partial Fulfilment of Requirements
For the Degree of Doctor of Philosophy
Major: Mathematics
Under the Supervision of Professor Petronela Radu
Lincoln, Nebraska
May, 2018 PROPERTIES AND CONVERGENCE OF STATE-BASED LAPLACIANS
Kelsey Wells, Ph.D. University of Nebraska, 2018
Adviser: Petronela Radu
The classical Laplace operator is a vital tool in modeling many physical behaviors, such as elasticity, diffusion and fluid flow. Incorporated in the Laplace operator is the requirement of twice differentiability, which implies continuity that many physical processes lack. In this thesis we introduce a new nonlocal Laplace-type operator, that is capable of dealing with strong discontinuities. Motivated by the state-based peri- dynamic framework, this new nonlocal Laplacian exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow better representation of physical phenomena at different scales and in materials with different properties. We obtain explicit rates of conver- gence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero. We study mathematical properties of this state-based Laplacian, including connections with other nonlocal and local coun- terparts. Finally, we study the solutions of the state-based Laplacian, and use the structure of the solutions to further exhibit the connections between other nonlocal and local Laplacians. iii
ACKNOWLEDGMENTS
In an to attempt to keep this section from getting out of hand, I will only list a handful of the many people that have impacted me on this journey. First, I would like to thank my advisor Petronela Radu for being incredibly patient and always believing in me more than I believed in myself. I have learned so much from her, and have been awed by her dedication to her work and her family. I am grateful that I found an advisor who inspires and motivates me, and I am especially glad that she was willing to talk about all parts of life, and not just the mathematics. I would also like to acknowledge my committee members Mikil Foss, Adam Larios and Florin Bobaru for their input and conversations about peridynamics. To that end, much thanks to Dr. Stewart Silling from Sandia Laboratories and Dr. Pablo Seleson from Oak Ridge National Laboratory for enlightening discussions which helped improve this thesis. There have been various people along the way who have played a critical role in getting me to where I am today. Not the least of which were plenty of teachers and fellow students from elementary school to now. In particular, my high school calculus teacher John Coxson in whose class I decided to become a high school math teacher, and while that didn’t happen, it certainly started me on the path that got me here. Valerie Shaw, for being a cool female mathematician and teacher before I realized how important those types of role models would be. Ruth Duersch, for a friendship starting in high school, but never ceasing no matter the distance between us. She has always been one of my biggest supporters, and for that I am grateful. Ben Woodruff, my intro to proofs teacher at BYUI, who unknowingly planted the idea of going to graduate school in my head. Danae Romrell, for being the type of woman and mathematics teacher I want to be. I owe much of my undergraduate iv mathematics knowledge to Ben and Danae, and am thankful to them for that, and for being wonderful mathematics and life mentors. I would not have survived college as a math major, nor would I have enjoyed it so throughly without Lise Deal. I’m still annoyed (if only slightly) that she graduated a semester early, and didn’t come to grad school with me. There were countless other friends that supported me throughout my college career, but I will only mention a couple more. Caitlin (Trani) Kunz, who has my deepest gratitude for being my room- mate and friend for years, and for constantly inspiring me with her work ethic. Brad Christensen for convincing me to read Brandon Sanderson’s books, which became an invaluable coping mechanism for stress while studying for qualifying exams, and during my final and most stressful year of graduate school. I am especially grateful for the many “coincidences” that led me to the University of Nebraska-Lincoln. I have time and time again been reminded of how lucky I am to be a part of the mathematics department at UNL. I am forever grateful to the faculty and staff for treating graduate students as human beings, and not minions, and for the many professors who encouraged me and supported me throughout my years here. Thanks to Marilyn Johnson for helping me (and all of us) stay on top of the deadlines needing to be met to successfully make it through graduate school. There have been many friends in Lincoln over the last five years that deserve mention, and though I am unable to name you all, please know how grateful I am. First, to my roommates, Tonya Randall, Callie Griffith and Cassie Heap, for being my emotional support, for believing in me, and for being an escape from the world of mathematics. My deepest gratitude for Gayle and Robert Vance, for being a constant in my life the last five years, and for all the treats and notes left on my car when it was the last one in the parking lot. To all my other “church” friends, thanks for the smiles, love, and support. To the many graduate students I’ve met during my time here - v thank you. Special shout out to Laura White for the hundreds of homework/comp problems we worked on together and for suggesting that I learn about Petronela’s work. To my other academic sibling Cory Wright, it was a joy to travel with the two of them, and to write together this year. Other important friendships over the years include Corbin Groothuis, Seth Lindokken, Stephanie Prahl, Chris Evans, Karina Uhing, Lara Ismert, and many others whom I apologize for not mentioning by name. Finally, I want to express my immeasurable gratitude to Meggan Hass. I would have quit long ago if it were not for her; thank you for being the best graduate school best friend ever. Lastly, I would like to thank my family for their constant support of my academic endeavors. My grandma for telling me that I wasn’t allowed to quit, and my grandpa for always talking about how proud he is of me. My Nana and Gramps for never ceasing to send me cards to remind me they are thinking about me. To my siblings Brady, McKenzie, Zachary and Sydnee for always clearing their schedule when I come to town so that we can spend as much time together adventuring as possible, and for forgiving me for living so far away. To my sibling-in-laws Andie and Tyler for being wiling to do the same. The most gratitude, of course, goes to my parents Scott and Kanani for raising me to believe I could do anything I put my mind to and then supporting me in all of my aspirations, even when it has continually put me farther from home than they might want - thank you! vi
Table of Contents
Notation Summary ...... 1
1 Introduction 5 1.1 Bond-based peridynamics ...... 10 1.2 State-based peridynamics ...... 13
2 The state-based Laplacian 17 2.1 Notation and peridynamic states ...... 19 2.2 State-based equation of motion and its linearization ...... 20 2.3 A doubly nonlocal Laplacian operator ...... 24 2.4 Kernels of the state-based Laplacian ...... 25 2.5 Second formulation for the state-based Laplacian ...... 28
3 Convergence 31 3.1 Convergence of the bond-based Laplacian ...... 32 3.2 Convergence of the state-based Laplacian ...... 37
4 Convolutions and the Fourier Transform 58 4.1 Convolution form of the state-based Laplacian ...... 59 4.1.1 The state-based Laplacian on the space of distributions . . . . 60 4.2 Connections between the state-based, bond-based and classical Lapla- cians ...... 61 vii
4.2.1 Solutions of the state-based Laplacian ...... 68
5 Final Remarks and Future Directions 74 5.1 Physical Aspects ...... 74 5.2 Mathematical Aspects ...... 75
Bibliography 77 1
Notation Summary
1. Point and set notation
• Vectors in Rn are denoted by bold letters, i.e. x, y, r, p, q.
• Vector valued functions are denoted by bold letters, i.e. u, ν, α.
• Ω is a domain of Rn, possibly unbounded, unless otherwise specified.
• Ω0 ⊆ Ω.
• ∂Ω is the boundary of the region Ω.
•H x is the horizon of x, taken to be the ball some length δ centered at x. See Figure 1.1
•B the body in the reference configuration. See section (2.1).
• ξ, ζ are bonds, often written as ξ = p − x. See Section 1.1.
•B δ(x) is the ball of radius δ centered at 0.
• δ0 is the Dirac mass measure centered at the origin.
• δµ, δγ, δη are the length scales associated with the support of each kernel, µ, γ, η respectively.
•B γ is the support of γ, i.e. the ball of radius δγ centered at 0. Similarly
for Bµ and Bη.
• wn is the volume of the unit ball in n dimensions.
p n • L (Ω) = {f :Ω → R |f is Lebesgue measurable, ||f||Lp(Ω) < ∞}, where
1 Z p p ||f||Lp(Ω) = |f| dx , for 1 ≤ p < ∞. Ω 2
∞ n • L (Ω) = {f :Ω → R |f is Lebesgue measurable, ||f||L∞(Ω) < ∞}, where
||f||L∞(Ω) < ∞ = ess sup |f|. Ω
• For 1 ≤ p ≤ ∞,Lp = Lp(Rn).
• For 1 ≤ p ≤ ∞, the Sobolev space W k,p(Ω) is defined to be
k,p n 1 α p W (Ω) = {f :Ω → R |f ∈ Lloc(Ω),D f ∈ L (Ω) for all |α| ≤ k},
1 α where Lloc is all locally summable functions, α is a multiindex, and D f =
α1 αn ∂x1 f + ··· + ∂xn f. Then
1/p Z X α p |D f| dx 1 ≤ p < ∞ |α≤k Ω ||f||W k,p(Ω) = X α ess sup |D f| p = ∞. Ω |α≤k
2. Functions
• F is the force in Example 1 and the flux in Example 2.
• ν is the unit outward normal.
• ρ(x) is the mass density at x.
• u ∗ γ = (u1 ∗ γ, u2 ∗ γ, . . .), is the convolution of a vector and a scalar function. See equation (4.2).
•h u, ϕi = (hu1, ϕi ,..., hun, ϕi) is the duality product of a vector and a scalar function.
• f is a pairwise force vector. See equation (1.15). 3
• b is a prescribed body force density. See equation (1.15).
• dV indicates a volume integral, whereas dσ indicates a surface integral.
• We denote fˆ to be the Fourier transform of f, and fˇ of F −1(f) to be the inverse Fourier transform of f.
• µ, γ, η are symmetric radial functions, called kernels. Prototypical forms of these kernels are given in (2.8).
• Scaling factors, πµ, πγ, πη given in (3.3), (2.10), and (2.11) respectively, with prototypical forms given in (3.6), (2.13), and (2.14).
b • σ(δµ) is the scaling of the bond-based Laplacian, Lµ[u], given in (3.2).
s • σ(γ, η) or σ(δγ, δη) is the scaling of the state-based Laplacian, Lγη[u], given in (3.9).
3. Local operators
•∇ u is the classical gradient.
• ∆u is the classical Laplacian.
• div(v) is the classical divergence operator.
4. Nonlocal operators and nonlocal calculus
b •L µ[u] is the bond-based Laplacian with kernel µ. See equation (1.19).
•D α[v] is the nonlocal divergence operator. See equation (1.20).
•G α[u] is the nonlocal gradient operator. See equation (1.21).
• A hξi for ξ ∈ Hx, the image of a peridynamic state acting on the bond ξ. See section (2.1).
• D hξ, ζi is a double state, which acts on two bonds. See section (2.1). 4
• A • B is the dot product for states. See section (2.1).
• (A • D)j hξi is the left product of vector state A and double state D. See section (2.1).
• (D • B)i hξi is the right product of B and D. See section (2.1).
† • D is the adjoint of D
•∇ ψ(A) is the Frechet derivative of ψ, and ∇∇ψ = ∇(∇ψ) is the second Fr´echet derivative of ψ. See section (2.1).
• T is a vector state that describes the force. See equation (2.1)
s • K is the double state-kernel of the state-based Laplacian Lγη[u]. See Sec- tion 2.2. 5
Chapter 1
Introduction
The focus of this thesis is an introduction and analysis of a new nonlocal Laplace-type operator. The classical Laplacian is used in a variety of models for physical phenom- ena such as elasticity and diffusion, including heat and fluid flow. Mathematically, the classical Laplacian requires input functions that are twice differentiable. This requirement inhibits the ability to accurately represent discontinuities, which often appear in models for cracks, anomalous diffusion, or transport. To address some of these issues we introduce a new Laplacian, an operator that behaves like the Laplacian on smooth functions, but requires much less regularity than the classical Laplacian. We begin with a short overview of the derivation of classical elasticity and diffusion equations to illustrate how the Laplacian incorporates physical properties in these key models. The derivations can be found in traditional texts such as [18].
Example 1 (Elasticity).
In a deformation process, let u(x, t) be the displacement at (x, t) and F be the force acting on Ω0, a subregion of a larger domain Ω. The net force through ∂Ω0 is given by Z − F · νdσ. (1.1) ∂Ω0 6
Using the Green-Gaus theorem we have
Z Z − F · ~νdσ = − divFdx. (1.2) ∂Ω0 Ω0
For elastic bodies we can take F = F(∇u), where ∇u is the elastic force, so that the higher magnitude of the gradient, the larger the force. Assuming linear elasticity, we use Hooke’s Law to obtain F = −c∇u. (1.3)
Hence we get Z Z Z − divFdx = c div(∇u)dx = c ∆udx, (1.4) Ω0 Ω0 Ω0 where ∆ is the classical Laplace operator. The acceleration in a subregion Ω0 of Ω is given by Z utt(x, t)dx. (1.5) Ω0
Using Newton’s law we find
Z 0 ρ(x)utt(x, t) − c∆u dx = 0, for all Ω ⊆ Ω, (1.6) Ω0 where ρ(x) is the density at x. Thus
ρ(x)utt(x, t) − c∆u = 0. (1.7)
Equation (1.7) is the classical wave equation.
Example 2 (Diffusion).
If we instead let u(x, t) be the density of some physical quantity at (x, t), F(x, t) 7
be the flux, and assume Fick’s law for diffusion, we have
F = −c∇u, (1.8)
which again produces (1.4). If we assume that no creation or destruction of mass happens inside Ω, then the change in total mass, with respect to time, is given by the flux through ∂Ω0. Hence we have,
d Z Z Z u(x, t)dx = − F · νdσ = c ∆udx, (1.9) dt Ω0 ∂Ω0 Ω0
thus producing Z 0 ut(x, t) − c∆u dx = 0, for all Ω ⊆ Ω. (1.10) Ω0
Finally, we obtain
ut − c∆u = 0, (1.11) the classical diffusion equation. If we assume that the process has finished, i.e. taking t → ∞, or that changes in time are very small, we have that ut = 0. Thus with c = 1, we get Laplace’s equation − ∆u = 0. (1.12)
Remark 1.1. Note that the diffusion equation also governs other physical processes, where Fick’s law for diffusion is replaced by Fourier’s law for thermodynamics, or Ohm’s law in electrostatics.
These models, or variations of them, are used in classical continuum mechanics where material is assumed to be continuously distributed instead of consisting of discrete particles. As mentioned previously, models of elasticity and diffusion, as written, necessitate our function u to be twice differentiable in space. This becomes 8 a glaring shortcoming as we often encounter materials and situations which are not C2. We can loosen regularity requirements by considering the weak versions of theses models
hρutt, ϕi − c h∆u, ϕi = 0, (1.13) and
hut, ϕi − c h∆u, ϕi = 0, (1.14)
∞ where ϕ ∈ Cc (Ω). In these weak forms, we are not viewing the models at a point (x, t), but rather through a test function ϕ. We are in essence gathering information about what is happening around the point (x, t). Although these weak forms reduce the amount of regularity we require on u, we still need u ∈ W k,p(Ω) for some p and k. While weak forms often prove useful and satisfactory, there are still situations when the capabilities of these models fall short of encompassing all of the physi- cal relevance we may want or need. Modeling a material that develops a crack, or modeling at the discrete or atomistic level is impossible in a differential framework. Other methods have been developed to compensate for these problems. For example, we could use fractional mechanics theories which typically include a separate set of equations to predict where a crack may develop and how fast it may grow (see [6] as an example). While these other methods work well in some circumstances, it is not clear that they are sufficient in every situation and with every length scale. A relatively recent method of dealing with these types of discontinuities has been the incorporation of nonlocal operators into physical models. By nonlocal, we mean that the operator takes into account neighbors of every point. Nonlocal operators often arise as integral or fractional-derivative operators, allowing a reduction of the smoothness required on the functions of which the operators are to be applied. Models 9 which use these nonlocal operators have been termed nonlocal models and they allow the study of solutions and domains beset by discontinuities. Over the past decades nonlocal theories have been successfully employed in mod- eling various types of phenomena, including nonlocal diffusion [1], fractional kinetics and anomalous transport, [54], speech signal modeling [2], viscoelasticity [47], image processing [24, 35], fluid mechanics [32], and phase separation [23, 21]. In addition, the theory of peridynamics, introduced by Stewart Silling [43, 46], incorporates non- local models in dynamic fracture; see also [13]. Mathematically, fractional Laplacians have been studied by many authors, see [8], [10], and [53] as examples, and other fractional operators have have been studied in papers such as [48]. An introductory text for fractional calculus is available in [37]. Integro-differential Laplacians have been studied mathematically in [9],[36], [27], [19], and [49]. In [25] and [15] the authors develop a nonlocal (integral) vector calculus that mimics classical calculus. For this thesis the motivation behind considering a nonlocal framework comes from the theory of peridynamics, introduced by Silling in [43], which is a reformulation of classical continuum mechanics. The first applications were to dynamic fracture, see [3] and thermal diffusion see [4, 38]. In addition, the peridynamic theory has been used to model composite laminates [29], concrete [30], porous media flow [31], and tumor growth [33]. The main focus here is on a new diffusion-type operator, which we will label the state-based Laplacian. The operator is introduced in Section 2.3 and it is inspired by the most general formualtion of peridynamics, the state-based theory. The purpose of this chapter is to give the reader sufficient motivation for the nonlocal framework of peridynamics, and to introduce that framework. We will further highlight the relevance of peridynamics as a type of nonlocal model, and as the framework in which the state-based Laplacian arises naturally. 10
1.1 Bond-based peridynamics
The power and convenience of having a cohesive set of equations, together with the ability to track dynamic fracture growth are some of the most significant benefits of Silling’s reformulation of classical continuum mechanics in [43]. In this paper, Silling names his theory Peridynamics for the greek words peri meaning near, and dynamics meaning force. He presents a nonlocal, unified approach to modeling discrete, discon- tinuous and continuous material. In this theory, the spatial derivatives in classical theory have essentially been replaced by integrals, thus allowing for lower regularity of solutions. In addition, the authors of [42] illustrate the relationship between peridy- namics and classical molecular dynamics, particularly how peridynamics can be seen as an upscaling of molecular dynamics. In the nearly two-decades since the release of this new formulation, peridynamics has been shown to be extremely effective at tracking dynamic fracture in different materials (homogeneous or heterogeneous); see fiber-reinforced composites [28], composite laminates [29], orthotropic material [22], layered glass [5], concrete [30]. In the original formulation of peridynamics each point x interacts with all its
neighbors within a domain Hx, called the horizon of x, taken to be a ball of radius δ
centered at x. If p ∈ Hx then ζ = p − x is called a bond for the point x (see Figure 1.1). In this context consider the cumulative force that is acting on x through its neighbors inside the horizon, a force that is expressed through integral operators. By replacing differential operators with integral operators we allow low-regularity solu- tions to satisfy elasticity models. The bond-based peridynamics equation of elasticity, as introduced by Silling [43], is given by
Z ρ(x)u¨(x, t) = f(u(q, t) − u(x, t), q − x)dVx + b(x, t), (1.15)
Hx 11
• p • δ x
Horizon of x (Hx)
Figure 1.1: The interaction of the points p has on x, in the bond-based theory of peridynamics.
where ρ is material density, and b is a prescribed body force density field. The displacement vector field is given by u, and f gives the force vector that the particle q exerts on the particle x. The form of f embodies the constitutive information of the material. From Newton’s third law, we require that
f(−(u(q, t) − u(x, t)), −(q − x)) = −f(u(q, t) − u(x, t), q − x). (1.16)
For linear peridynamic material, the right had side of (1.15) becomes
Z (u(y) − u(x)) µ(y − x)dy + b(x, t), (1.17) Hx
where from (1.16) we must have µ(ξ) = µ(−ξ), ∀ξ. If µ is a function such that µ ≡ 0 when |ξ| > δ, we obtain the protagonist of the bond-based formulation, the nonlocal Laplacian, Z Lµ[u](x) = (u(y) − u(x)) µ(y − x)dy. (1.18) Bδ(x)
In the above formula, Bδ(0) is the ball of radius δ centered at x, while the kernel µ measures the strength of the bond y − x. Observe that this operator is well-defined
even for very rough functions u : Rn → Rk, n, k ≥ 1, as long as the integration for 12
b k each component of u is valid, (Lµ ∈ R ). The constant δ > 0 is the radius of the horizon, and describes the range of nonlocality. It can vary from very small values (peridynamics) to very large ones (δ = ∞ in nonlocal diffusion [1]). Of interest to us is the case of a finite horizon as well as the transition to infinitesimal values; in other words, we study the limiting behavior of nonlocal operators as δ goes to zero. In order to obtain convergence, we will consider the scaled nonlocal Laplacian with scaling σ(δ), which for clarity we label as the bond-based Laplacian with kernel µ:
Z b Lµ[u](x) = σ(δ) (u(y) − u(x)) µ(y − x)dy. (1.19) Bδ(0)
The convergence of the bond-based Laplacian to the classical Laplacian, along with the scaling σ(δ), will be discussed in Chapter 3. To highlight an important structural connection between the classical Laplacian and the bond-based Laplacian we present definitions of nonlocal gradient and diver- gence operators. These operators were formulated in [15] to provide the framework for a nonlocal vector calculus.
Definition 1.2 (Nonlocal operators). Let v :Ω × Ω → Rk be a vector two-point function, u : Rn → R, and α(x, y):Ω × Ω → Rk, be an antisymmetric vector two-point function. The nonlocal divergence operator Dα on v is defined as
Z Dα[v](x) := (v(x, y) + v(y, x)) · α(x, y)dy for x ∈ Ω, (1.20) Ω where Dα[v]:Ω → R. The nonlocal two-point gradient operator is defined as
Gα[u](x) := (u(y) − u(x)) α(x, y) for (x, y) ∈ Ω × Ω, (1.21) 13
k where Gα :Ω → R .
Taking µ = α2/2 we find
Lµ[u](x) = Dα[Gα[u]](x) for x ∈ Ω. (1.22)
Thus, the nonlocal bond-based Laplacian can be written in the familiar “diver- gence of the gradient” from that appears in classical models, i.e. equation (1.4). Bond-based peridynamics and the bond-based Laplacian have been well studied, and are continuing to be studied. However, in bond-based models particles interact through a central potential, thus “seeing” only neighbors in their horizon [46, Item 1 in list on pg. 153]. A consequence of this formulation gives a restriction on the Poisson ratio of 1/4 in 3D or 2D plane strain, and 1/3 in 2D plane stress, see [50]. Moreover, the bond-based systems lack the generality of stress tensors that are usually considered in continuum mechanics as they impose only a pairwise force interaction on particles [46, Item 2 in list on pg. 153]. For a more detailed discussion of these aspects and the motivation for a more general theory, see [46]. The state-based theory of peridynamics, introduced in the next section, overcomes these issues and generalizes the bond-based theory. The connection between the Laplace type operators that appear in each of these formulations is one of the goals of this work and is studied further in Chapter 4.
1.2 State-based peridynamics
To overcome the deficiencies of the bond-based model, Silling et al. in [46] intro- duced the theory of state-based peridynamics, in which the force between points are expressed through general operators called states. A discussion of these states, as 14
ε
• p • • q δ x
Horizon of p Horizon of x
Figure 1.2: The indirect interaction that the point q has on x through their common neighbor p.
relevant to this thesis, is given in Section (2.1). These state operators allow indirect force interactions of a neighbor with its neighbor’s neighbors. A given point x will be affected directly by its neighbors p, as well as indirectly, by the neighbors q of p through the point p (see Figure 1.2). Mathematically, the interactions of the point x will be expressed through double integrals over the product space Bδ(x) × Bε(p), for every point p in the horizon of x. Thus, the points affecting the behavior at x can be ε + δ distance away from x. This setting allows a very general approach to modeling that can incorporate a wide variety of physical behavior. The bond-based theory then, becomes a particular case of the state-based setting where points interact not only with their immediate neighbors (direct interactions), but also with neighbors of the neighbors (indirect interactions). This composition of interactions could also be extended to model a broader range of phenomena, such as seen in nonlinear elasticity, viscoelasticity, and viscoplasticity (for bond-based formulations of these models see [17], [52], and respec- tively [20]). 15
In particular the state-based equation of motion is given by
Z ρ(x)u¨(x, t) = {T[x, t] hq − xi − T[q, t] hx − qi} dVq + b(x, t), (1.23)
Hx
where ρ is material density, and b is a prescribed body force density field. Above the operator T is called a vector state which when computed at the point x is applied to a bond q − x whose resulting action is the force which q exerts on x. Thus the right hand side of (1.23) describes the cumulative effect of all action-reaction forces between x and its neighbors, and provides a very general framework for incorporating the material constitutive restrictions. The focus of this work is on the study of a newly introduced state-based Laplacian operator:
Z Z s Lγη[u](x) = (γ(p − x) + γ(q − x)) η(q − p)[u(q) − u(x)]dqdp
Rn Rn Z Z − (γ(x − p) + γ(q − p)) η(q − x)[u(q) − u(p)]dqdp,
Rn Rn that arises naturally in the state-based formulation of peridynamics (again, the inte- gration is performed on each component of u). As motivated by the physical consid- erations above, this operator captures effects from a wider and more diverse range of interactions by looking at cumulative effects modeled through two integral operators with two (possibly different) kernels, γ and η. By incorporating two kernels the op- erator gains an additional degree of flexibility that is important in applications, thus increasing the physical relevance of the model. The engineering and computational communities have provided us with many studies for state-based models ([51], [41], [34], and [26]; also, see the overview paper [44]), but the theoretical investigations of 16 these doubly nonlocal operators are still in their early stages. 17
Chapter 2
The state-based Laplacian
As mentioned perviously, the state-based Laplacian arises naturally from the state- based formulation of peridynamics. The introduction of this new operator is the focus of this chapter. We will start with a discussion of the importance of a new nonlocal Laplacian. Then, we will build up the tools needed to present the state-based peridynamics theory. From there, we show how the state-based Laplacian originates from the linearized state-based peridynamic equation of motion. We conclude this chapter with a conversation about the kernels, γ and η, of the state-based Laplacian, and end the with a rewriting of the Laplacian which is more convenient to work with. This new nonlocal Laplacian was inspired by three particular choices for ker- nels given by Silling in [45]; the examples concern elastic materials (in bond-based framework), linear fluids, and linear isotropic solids. At a mathematical level the state-based Laplacian is a double convolution-type operator, which generalizes the operator (1.19), while also providing a “decomposition” of the operator with respect to the kernels γ and η. The role of each kernel will be discussed from a physical, as well as a mathematical point of view. Additionally, by writing the state-based Laplacian in convolution form we obtain an operator that is well-defined on spaces of very irregular functions, even on the space of distributions; see Chapter 4. To summarize, the main contributions of this thesis are: 18
• At a physical level we introduce a mathematical operator that captures non- local effects in materials that are more general than the ones modeled with the single integral, bond-based operator. While this operator appears naturally in the (very) general state-based formulation, it allows us through its specific form involving two kernels to incorporate a variety of examples. Thus, we introduce a framework in which the nonlocal Laplacian can model very different materials, or even different behavior. In this more general context we have the ability to study transitional behavior from one class of phenomena to another, as well as the transition from one type of material to another.
• At a mathematical level the double convolution operator gives us a novel way to model physical behavior in the space of discontinuous functions or distribu- tions. The transition to “smooth” behavior can be studied through convergence results of the nonlocal operator to the classical Laplacian as the horizons of in- teraction shrink to zero. We obtain explicit rates of convergence and we discuss the importance of regularity for functions on which the nonlocal operator is applied.
Finally, we make note of a couple of distinctions between this operator and other operators. First, the structure of the state-based Laplacian resembles the nonlocal biharmonic introduced in [39], due to the presence of the double nonlocality. However, we show that the doubly nonlocal Laplacian approaches a second, and not a fourth- order differential operator. This aspect will be discussed in more detail in Section 4.1. Also, we take a scalar-valued kernel rather than a tensor-valued kernel, (see (2.5))
s hence, in the one dimensional case Lγη is a nonlocal version of the Navier operator
s from elasticity, but in higher dimensions Lγη is a nonlocal version of a Laplace type
s operator, and not the Navier operator. Indeed, Lγη is missing the nonlocal counterpart 19
of the ∇divu term (see [36]).
2.1 Notation and peridynamic states
In this section we introduce notation that is used in the state-based peridynamic formulation. We follow the notation given in [45]. Let B be the body in the reference configuration, and let x ∈ B. We let δ > 0 be the horizon of x, and for q ∈ B such that |q − x| ≤ δ, as before, call ξ = q − x a
bond. Let Hx be the set of all such bonds of x. A peridynamic state A is a mapping
from Hx. Denote by A hξi for ξ ∈ Hx, the image of state acting on the bond ξ. If A is a scalar state then the value of A hξi is a scalar. Similarly, A is a vector state
when A hξi is a vector. A double state, D , takes two bonds ξ, ζ ∈ Hx and D hξ, ζi is a second order tensor. The dot product of two vector states A and B is
Z A • B = A hξi · B hξi dVξ.
Hx
The left product of vector state A and double state D is the vector state defined by
Z (A • D)j hξi = Aj hξi Dij hζ, ξi dVζ, ∀ξ ∈ Hx.
Hx
Then the right product of B and D is the vector state defined by
Z (D • B)i hξi = Dij hξ, ζi Bj hζi dVζ, ∀ξ ∈ Hx.
Hx 20
† The adjoint of D is denoted by D and defined by
† Dij hξ, ζi = Dji hζ, ξi ,
† with D being self-adjoint if D = D . Let V be the set of all vector states, and ψ : V → R. If ψ is Fr´echet Differentiable at A ∈ V, then for any a ∈ V
ψ(A + a) = ψ(A) + ∇ψ(A) • a + o(||a||)
where the Fr´echet derivative ∇ψ(A) is a vector state. If S : V → V then
S(A + a) = S(A) + ∇S(A) • a + o(||a||)
where ∇S(A) is a double state. If ∇ψ is Fr´echet differentiable, then the second Fr´echet derivative of ψ is a double state defined by ∇∇ψ = ∇(∇ψ) on V.
2.2 State-based equation of motion and its linearization
We restate the state-based equation of motion from (1.23). The displacement from the equilibrium position of a point x in the body B at time t ≥ 0, denoted by u(x, t), is described by the equation
Z ρ(x)u¨(x, t) = {T[x, t] hq − xi − T[q, t] hx − qi} dVq + b(x, t), (2.1)
Hx where ρ is material density, and b is a prescribed body force density field. Again, the resulting action of T is the force which q exerts on x. Thus the right hand side of 21
(1.23) describes the cumulative effect of all action-reaction forces between x and its neighbors. We now describe the linearization of (2.1), as can be found in [45]. For further discussion on linearized state-based peridynamics, see [40]. Let y be the motion of the peridynamic body B. The position of a point x ∈ B at time t ≥ 0 is y(x, t). Let Y[x, t] be the deformation state, the vector state defined by
Y[x, t] hq − xi = y(q, t) − y(x, t), (q − x) ∈ Hx.
A material is simple if T only depends on Y. Hence, if we assume the material is simple, then T[x, t] = Tˆ (Y[x, t], x).
If the material is also elastic, then there exists a function Wˆ : V × R3 → R3 such that for any Y Tˆ (Y, x) = ∇Wˆ (Y, x).
Now, deriving the linearized state-based model will give a linear integro-differential equation expressed in terms of the displacement. We define T(Y0) = T0, and
0 ∇Tˆ (Y ) = K, (2.2)
which is a double state, where T0 is the initial force and Y0[x]hq−xi = y0(q)−y0(x). Thus, linearizing T gives
0 T(U) = T + K • U,
where U is the displacement. If the material is elastic, then
ˆ 0 0 K = ∇T(Y ) = ∇∇Wˆ (Y ), 22
which implies that K† = K, hence K is symmetric. Assume b0 is a time independent body force density field on B, which gives an equilibrated deformation y0. Then subject the body to another body force density field b, so that bˆ = b0 + b.
The change in the displacement field is denoted u hence
y = y0 + u.
Finally, assume T[x] hp − xi = 0 whenever |p − x| > δ. Linearizing the possibly nonlinear equation of motion (1.23) produces
Z 0 ρ(x)u¨(x, t) = T [x] + K[x] • U[x] hp − xi dVp (2.3) B Z 0 ˆ − T [p] + K[p] • U[p] hx − pi dVp + b(x, t). B
Since y0 is equilibrated
Z T0[x] hp − xi − T0[p] hx − pi + b0(x) = 0.
B
Thus
Z ρ(x)u¨(x, t) = {(K[x] • U[x]) hp − xi − (K[p] • U[p]) hx − pi} dVqdVp + b(x, t). B 23
Using the definition of dot product of two vector states we have
Z Z ρ(x)u¨(x, t) = K[x] hp − x, q − xi · U[x] hq − xi dVqdVp B B Z Z − K[p] hx − p, q − pi · U[p] hq − pi dVqdVp + b(x, t). B B
Finally, writing out the definition of a dot product we obtain
Z Z ρ(x)u¨(x, t) = K[x] hp − x, q − xi (u(q, t) − u(x, t))dVqdVp (2.4) B B Z Z − K[p] hx − p, q − pi (u(q, t) − u(p, t))dVqdVp + b(x, t). B B
The double state-kernel K, at a point x, scalar valued, essentially weighs the interac- tions between two bonds, p−x and q−x, whose output is denoted by K[x] hp − x, q − xi. For a simple material, equation (2.4) above represents a linearized state-based model for an elastic material if and only if K is symmetric, i.e. for any two bonds ξ and ζ which share the same application point, K[x]hξ, ζi = K[x]hζ, ξi. See the discussion in [45, Section 4.2, Proposition 4.1].
In [45] several choices of the state-kernel K are considered, each of them leading to a different physical model. For K[x]hξ, ζi given in terms of a Dirac mass supported at ζ = ξ, one recovers the peridynamic bond-based formulation, which we will discuss in further detail in Chapter 4. Below, we consider a particular scalar-valued, choice of the state kernel K which will give rise to a new Laplacian-type operator. The definition and properties of this new operator, together with the connections to nonlocal and local Laplacians are discussed. 24
2.3 A doubly nonlocal Laplacian operator
Motivated by the discussion in the previous section, we consider the state-kernel K given by
K[x]hξ, ζi := [γ(ξ) + γ(ζ)]η(ζ − ξ), (2.5) where γ and η are symmetric functions, i.e. γ(−ζ) = γ(ζ), and η(−ξ) = η(ξ). Taking ξ = p − x and ζ = q − x, (2.5) becomes
K[x]hp − x, q − xi = [γ(p − x) + γ(q − x)]η(q − p). (2.6)
We are now in position to formally introduce the new Laplace-type operator.
s Definition 2.1. We define the nonlocal state-based Laplace operator Lγη with kernels γ and η, to be the operator given by
Z Z s Lγη[u](x) =σ(γ, η) (γ(p − x) + γ(q − x)) η(q − p)[u(q) − u(x)]dqdp (2.7)
Rn Rn Z Z − σ(γ, η) (γ(x − p) + γ(q − p)) η(q − x)[u(q) − u(p)]dqdp,
Rn Rn where σ(γ, η) is a normalizing factor which is given by (3.9).
In section 3.2 the scaling σ(γ, η) will be determined for kernels γ, η with finite radii of interaction, δγ, δη such that
s |Lγη[u](x) − ∆u(x)| → 0 as δγ, δη → 0, for u sufficiently smooth, and for every point x in the domain. 25
2.4 Kernels of the state-based Laplacian
Note from (2.5) that while K is symmetric with respect to the bonds ξ and ζ, i.e. K[x]hξ, ζi = K[x]hζ, ξi, the kernels γ, η play different roles in describing the dy- namics. Indeed, the kernel elongations of the bonds ξ and ζ are measured by the kernel γ, while η accounts for the interdependence between ξ and ζ. Thus the choice
η(ζ − ξ) = δ0(ζ − ξ), where δ0 is the Dirac mass centered at the origin, will yield the bond-based model, [45]. With the same choice for η, and γ given by two derivatives of the Dirac mass, we obtain the classical Laplacian, [14]. These connections are strengthened as we show convergence of the operator to the classical Laplacian in Chapter 3, and are made explicit in Chapter 4 when we introduce the convolution form of the operator (2.7). As previously done for bond-based peridynamics models, we will consider bounded regions of interactions for both stretching and bond interdependence effects, as given by γ, respectively η. Our specific assumptions for the kernels are given below.
Assumption 1. Assume that γ and η are nonnegative radial and integrable (γ, η ∈ L1) functions, so with an abuse of notation we write γ(ξ) = γ(|ξ|) and η(ζ) = η(|ζ|).
Assume that γ is supported inside the ball of radius δγ, and η is supported inside the
ball of radius δη so that we have
γ(|ξ|) = 0 for |ξ|> δγ, and η(|ζ|) = 0 for |ζ|> δη.
Assumption 2. We consider specific rational forms for γ and η that allow us to ex-
s plicitly compute the scaling for the operator Lγη which gives the convergence to the 26
classical Laplacian. For δη, δγ > 0 and α, β < n, the choices
1 1 α , |ξ| ≤ δγ β , |ζ| ≤ δη γ(ξ) = |ξ| , η(ζ) = |ζ| , (2.8) 0, |ξ| > δγ 0, |ζ| > δη produce the state-kernel K
1 1 1 + , |ξ| < δ , and |ζ − ξ| < δ |ξ|α |ζ|α |ζ − ξ|β γ η K[x]hξ, ζi = (2.9) 0 otherwise.
Remark 2.2. The cases where α, β = n is not considered in this thesis because of the lack of integrability of the kernels this produces, indeed, we would need to use the principal value of the integrals. Moreover, α, β = n require smoother functions u to be taken in the operator. For a discussion on bond-based kernels see [12, 11].
From this point on, to avoid confusion, we will refer to the horizon length of the bond-based Laplacian with kernel µ as δµ. Thus, every horizon length scale is immediately identifiable with the kernel it is associated with.
Next we introduce two functions πγ, πη : (0, ∞) → [0, ∞) related to our kernels γ and respectively η, which will be needed for the proof of our convergence result to allow us to move the derivatives on the function u through integration by parts. They are selected such that they satisfy
∇yπγ(|y|) = yγ(y), πγ(δγ) = 0, (2.10) and
∇rπη(|r|) = rη(r), πη(δη) = 0. (2.11) 27
With the same abuse of notation for radial functions, we have that πγ, and πη are given explicitly by
|y| |r| Z Z πγ(y) = πγ(|y|) := λγ(λ)dλ, and πη(r) = πη(|r|) := ρη(ρ)dρ. (2.12)
δγ δη
Under Assumption 2 we obtain
|ξ|2−α − δ2−α γ , if α 6= 2 2 − α πγ(ξ) = (2.13) ln(|ξ|/δγ), if α = 2,
and |ζ|2−β − δ2−β η , if β 6= 2 2 − β πη(ζ) = (2.14) ln(|ζ|/δη), if β = 2.
For notational simplicity, when we write Bγ we mean the ball of radius δγ centered
at zero. Similarly, Bη is the ball of radius δη centered at zero.
Lemma 2.3. Under Assumption 1 with πγ and πη satisfying (2.10) and (2.11), we have 1 Z Z |y|2γ(y)dy = − π (y)dy, (2.15) n γ Bγ Bγ and 1 Z Z |r|2η(r)dr = − π (r)dr. (2.16) n η Bη Bη Proof. We prove the first equality, the second follows in a similar fashion. By taking the inner product of (2.10) with y we obtain
2 y · ∇yπγ(y)dy = |y| γ(y). 28
Integration with respect to y on Bγ yields
Z Z 2 y · ∇yπγ(y)dy = |y| γ(y)dy.
Bγ Bγ
By performing an integration by parts on the left side, where ν is the normal derivative in the y direction, we have
Z Z Z y · ∇yπγ(y)dy = πγ(y)y · ν dy − div(y) · πγ(y)dy
Bγ ∂Bγ Bγ Z Z = πγ(y)y · ν dy − n πγ(y)dy.
∂Bγ Bγ
For y ∈ ∂Bγ, πγ(y) = πγ(δγ) = 0 thus (2.15) holds.
In order to employ the functions πγ and πη in the proof of our convergence result, we will need a different formulation for the state-based Laplacian, which is obtained in the next section.
2.5 Second formulation for the state-based Laplacian
The new expression for the state-based Laplacian will more easily allow us to identify the domains of integration for the variables, and simplify the integrand. This more convenient form is given by Proposition 2.4.
Proposition 2.4. Under Assumption 1, the state-based Laplacian can be written in 29
the following form:
Z Z s Lγη[u](x) = 2σ(δγ, δη) γ(y)η(r)[u(x + y + r) − u(x)
Bγ Bη
− u(x + r) + u(x + y)]drdy. (2.17)
Proof. By rearranging (2.7) we obtain
Ls [u] Z Z γη (x) = γ(p − x)η(q − p)[u(q) − u(x)]dqdp 2σ(δγ, δη) Rn Rn Z Z − γ(x − p)η(q − x)[u(q) − u(p)]dqdp
Rn Rn Z Z + γ(q − x)η(q − p)[u(q) − u(x)]dqdp
Rn Rn Z Z − γ(q − p)η(q − x)[u(q) − u(p)]dqdp. (2.18)
Rn Rn
We perform a change of variables in each of the above integrals
• y := p − x and r := q − p, in the first integral,
• y := p − x and r := q − x, in the second integral,
• y := q − x and r := p − q, in the third integral,
• and y := q − p and r := q − x in the fourth integral.
The resulting form is 30
Ls [u] γη (x) 2σ(δγ, δη) Z Z = γ(y)η(r)[u(y + x + r) − u(x) − u(x + r) + u(x + y)]drdy
Rn Rn Z Z + γ(y)η(r)[u(y + x) − u(x) − u(r + x) + u(r + x − y)]drdy.
Rn Rn
A final change of variables in
Z Z γ(y)η(r)u(r + x − y)drdy,
Rn Rn and the fact that γ(y) = γ(−y), together with Assumption 1 give (2.17).
We will use this form of the state-based Laplacian to prove the results of Chapter 3. 31
Chapter 3
Convergence
In the previous chapter we introduced a new nonlocal state-based Laplacian. We would like to spend some time analyzing the properties of this new operator. In particular, we would like to understand the operator as the nonlocality vanishes, i.e. we are interested in the results of the state-based Laplacian when we shrink the horizons of the state-based theory to zero. We start by reproving a result from [19] in Section 3.1. This result illustrates the convergence and rate of the bond-based Laplacian to the classical Laplacian, each applied to C4 functions, as the length of the horizon δµ goes to zero, where δµ is the horizon of the bond-based Laplacian associated
2 with the kernel µ. The rate of convergence is shown to be δµ. In Section 3.2 we prove two convergence results for the state-based Laplacian to the classical Laplacian as the state-based horizon lengths, δγ and δµ, go to zero. The first result gives convergence
2 2 for analytical functions on a bounded interval obtaining a convergence rate of δγ + δη. The final and main result of the chapter is convergence for C4 functions on possibly
n 2 unbounded domains in R , with a convergence rate of δγ. 32
3.1 Convergence of the bond-based Laplacian
The convergence of the bond-based Laplacian with kernel µ to the classical Lapla-
cian as the horizon δµ shrinks to zero has been studied in several papers. It has been shown that the rate of convergence for the nonlocal Laplacian to the classical
2 Laplacian, whenever applied to a sufficiently smooth function u, is proportional to δµ (the proportionality constant depends on bounds for the fourth derivative of u); see [14],[19],[36] where the arguments are based on the work in [7]; see also [49] where the analysis for numerical error is performed. Furthermore, in [36] the authors have shown strong convergence in L2 of nonlocal L2 solutions to classical solutions with
1 H0 Sobolev regularity. In [19] the authors produce a new technique to prove that the bond-based Lapla- cian converges to the classical Laplacian. They show the bond-based Laplacian ap- plied to sufficiently smooth functions provides an approximation for the classical
2 Laplacian for δµ near zero. In particular they show that the rate of convergence is δµ, where δµ is the length scale for the horizon of the bond-based model. In this section we reprove this result using the method in [19] in order to keep the presentation self contained, and as preface to the next section (3.2), where we will utilize this technique to prove our main result.
Theorem 3.1 (Foss, Radu). [19] Let Ω ⊂ Rn, n ≥ 1, possibly unbounded, and let u ∈ C4(Ω) with
(4) M4 := sup |u (x)| < ∞. (3.1) x∈Ω
Assume that µ is a nonnegative radial function and is supported inside the ball of
b radius δµ. Then, with scaling factor σ(δµ) given by (3.2), Lµ converges to ∆u at a
2 rate of δµ. 33
The scaling σ(δµ) of the bond-based Laplacian is shown in the proof of the theorem to be 2 σ(δµ) = −Z , (3.2) πµ(y)dy
Bµ
where πµ is the function associated with µ taking a similar form as πγ and πη in (2.10) and (2.11). That is
∇yπµ(|y|) = yµ(y), πµ(δµ) = 0, (3.3)
with the same abuse of notation for radial functions, πµ is given explicitly by
|y| Z πµ(y) = πµ(|y|) := λµ(λ)dλ. (3.4)
δµ
In particular, for the specific kernels
1 α , |ξ| ≤ δµ µ(ξ) = |ξ| , (3.5) 0, |ξ| > δµ
we have |ξ|2−α − δ2−α µ , if α 6= 2 2 − α πµ(ξ) = (3.6) ln(|ξ|/δµ), if α = 2,
and κ−n−2 2(2 − κ + n)δµ σ(δµ) = , (3.7) ωn−1
where wn−1 is the volume of the unit ball in n − 1 dimensions. We now prove Theorem 3.1. 34
Proof. From a change of variables of (1.19) we have
Lb [u] Z µ (x) = µ(y) u(x + y) − u(x)dy. σ(δµ) Bµ
Applying the fundamental theorem of calculus we obtain
1 Lb [u] Z Z µ (x) = µ(y) ∇u(x + sy)ydsdy, σ(δµ) Bµ 0
where ∇u is the Jacobian matrix for u. Using πµ as defined in (3.3) we have
1 b Z Z Lµ[u] (x) = ∇yπµ(y)∇u(x + sy)dsdy. σ(δµ) Bµ 0
Integrating by parts gives
1 1 b Z Z Z Z Lµ[u] y (x) = πµ(y)∇u(x + sy) dsdy − πµ(y)divy (∇u(x + sy)) dsdy. σ(δµ) δµ 0 ∂Bµ 0 Bµ
For y ∈ ∂Bµ, we have that πµ(y) = πµ(δµ) = 0, thus the first term vanishes and we obtain
1 b Z Z Lµ[u] (x) = − πµ(y)s∆u(x + sy)dsdy. σ(δµ) 0 Bµ 35
Adding and subtracting ∆u(x) we have
1 b Z Z Lµ[u] (x) = − πµ(y)s ∆u(x + sy) − ∆u(x) dsdy σ(δµ) 0 Bµ 1 Z Z − πµ(y)s∆u(x)dsdy.
0 Bµ
In order to make the coefficient of the classical Laplacian one in the second term, we take σ(δµ) as given in (3.2). With this choice of scaling we can write
1 Z Z b Lµ[u](x) − ∆u(x) = −σ(δµ) πµ(y)s ∆u(x + sy) − ∆u(x) dsdy.
0 Bµ
Integrating by parts with respect to s on the right hand side produces
Z 2 s=1 b 1 − s Lµ[u](x) − ∆u(x) = σ(δµ) πµ(y) ∆u(x + sy) − ∆u(x) dy 2 s=0 Bµ 1 Z Z 1 − s2 − σ(δ ) π (y) ∆∇u(x + sy)ydsdy. µ µ 2 Bµ 0
Evaluating the first integral at s = 0 and s = 1 we obtain
1 Z Z 1 − s2 Lb [u](x) − ∆u(x) = −σ(δ ) π (y) ∆∇u(x + sy)ydsdy. µ µ µ 2 Bµ 0 36
Once again, we integrate by parts with respect to s to get
Z 2 3 s=1 b 1 s s Lµ[u](x) − ∆u(x) = σ(δµ) πµ(y) − + ∆∇u(x + sy)y dy 3 2 6 s=0 Bµ 1 Z Z 1 s2 s3 − σ(δ ) π (y) − + ∆∇2u(x + sy)y · ydsdy. µ µ 3 2 6 Bµ 0
Evaluating at s = 0 and s = 1 we find that the first integral vanishes, leaving
1 Z Z 1 s2 s3 Lb [u](x) − ∆u(x) = −σ(δ ) π (y) − + ∆∇2u(x + sy)y · ydsdy. µ µ µ 3 2 6 Bµ 0
With M4 as defined in (3.1), we estimate
1 Z Z 1 s2 s3 |Lb [u](x) − ∆u(x)| ≤ M σ(δ ) π (y) − + |y|2dsdy µ 4 µ µ 3 2 6 Bµ 0 M σ(δ ) Z = 4 µ π (y)|y|2dy 8 µ Bµ
δµ M σ(δ )nω Z = 4 µ n−1 π (λ)λn+1dλ 8 µ 0 2 M4σ(δµ)δ Z ≤ µ π (y)dy, 8 µ Bµ
where in the last two steps we used the coarea formula and the fact that
n n−1 λ ≤ λ δµ.
Finally, taking σ(δµ) to be as defined in (3.2) we obtain 37
M |Lb [u](x) − ∆u(x)| ≤ 4 δ2. (3.8) µ 4 µ
We see that the bond-based Laplacian applied to u approaches the classical Lapla- cian applied to u at a rate that is quadratically dependent on the length scale of
the horizon, δµ. One might hope that a similar rate, one that depends explicitly on the length scales of both horizons, would hold in the convergence of the state-based Laplacian. We will show that the state-based Laplacian has a rate of convergence
2 2 δγ + δη only under strict conditions. Then we will loosen restrictions to show that we can still obtain convergence, but at a rate that is explicitly dependent only on the
2 length scale associated with γ, i.e. δγ, but with the condition that δη ≤ δγ.
3.2 Convergence of the state-based Laplacian
The main result of this section shows that the state-based Laplacian applied to suffi- ciently smooth functions provides an approximation for the classical Laplacian applied
to the same function, for δγ and δη close to zero. In fact, we exhibit a rate of conver- gence for the error of this approximation that is quadratic with respect to the kernel horizons. The scaling of the state-based Laplacian needed for this approximation will be shown to satisfy 1 σ(δγ, δη) = − Z Z , (3.9) 2 η(r)dr πγ(y)dy
Bη Bγ 38
s where η is the kernel in Lγη, and πγ is the function associated with γ given by (2.10). From Lemma (2.3) we find that this scaling is equivalent to
n σ(δγ, δη) = Z Z . (3.10) 2 η(r)dr y2γ(y)dy
Bη Bγ
In particular, for the specific kernels of (2.8), if α 6= 2, we have
β−n α−n−2 (n − β)(n − α + 2)nδη δγ σ(δγ, δη) = 2 , (3.11) 2wn−1
where wn−1 is the volume of the ball in n − 1 dimensions. We begin by showing that in one dimension the rate of the difference between the nonlocal Laplacian and the
2 2 classical Laplacian, when applied to analytic functions is of order δη + δγ.
Theorem 3.2. Let Ω ⊂ R be a bounded interval, and let u be analytic in Ω, with
M := sup |u(k)(x)| < ∞, k ≥ 4. (3.12) x∈Ω
For γ and η satisfying Assumption 1 above, and the scaling σ(δγ, δη) given by (3.10) with n = 1, we have
s 2 2 kLγη[u] − ∆ukL∞(Ω) < C(δη + δγ), (3.13)
as δγ, δη → 0, where the constant C depends on M given by (3.12).
Proof. From (2.17) we have
Ls [u] Z Z γη = γ(y)η(r)[u(x + y + r) − u(x) − [u(x + r) − u(x + y)]] drdy. 2σ(δγ, δη) Bγ Bη
Using the analytic expansion for u around x in the first term, and around x + y in 39
the third term we obtain
Ls [u] Z Z (y + r)2 γη = γ(y)η(r) u0(x)(y + r) + u00(x) (3.14) 2σ(δγ, δη) 2 Bγ Bη
∞ # (y + r)3 (y + r)4 X (y + r)n +u000(x) + u(4)(x) + u(n)(x) drdy 3! 4! n! n=5
Z Z (r − y)2 (r − y)3 − γ(y)η(r) u0(x + y)(r − y) + u00(x + y) + u000(x + y) 2 3! Bγ Bη
∞ # (r − y)4 X (r − y)n +u(4)(x + y) + u(n)(x + y) drdy. 4! n! n=5 Since γ(y) and η(r) are symmetric, each of the terms that is an odd power in y or r in the first integral on the right hand side of (3.14) is antisymmetric, with respect to y, or respectively r; hence, they vanish after integration. Similarly, in the second integral the terms containing odd powers of r are antisymmetric and therefore they also disappear (note that the same does not hold for y due to the presence of y in u(x + y)). We obtain:
Ls [u] Z Z y2 + r2 y4 + 6y2r2 + r4 γη = γ(y)η(r) u00(x) + u(4)(x) 2σ(δγ, δη) 2 4! Bγ Bη
∞ n # X X 2ny2n−2ir2i + u(2n)(x) drdy 2i (2n)! n=3 i=0 40
Z Z r2 + y2 − γ(y)η(r) −u0(x + y)y + u00(x + y) 2 Bγ Bη
3r2y + y3 r4 + 6r2y2 + y4 −u000(x + y) + u(4)(x + y) 3! 4!
∞ n−1 X X 2n − 1y2n−1−2ir2i − u(2n−1)(x + y) 2i (2n − 1)! n=3 i=0
∞ n # X X 2ny2n−2ir2i + u(2n)(x + y) drdy. 2i (2n)! n=3 i=0 Gathering the even derivative terms we have
Ls [u] Z Z y2 + r2 γη = γ(y)η(r) (u00(x) − u00(x + y)) 2σ(δγ, δη) 2 Bγ Bη
y4 + 6y2r2 + r4 + u(4)(x) − u(4)(x + y) 4!
∞ n # X X 2ny2n−2ir2i + u(2n)(x) − u(2n)(x + y) drdy 2i (2n)! n=3 i=0
Z Z 3r2y + y3 + γ(y)η(r) u0(x + y)y + u000(x + y) 3! Bγ Bη
∞ n−1 # X X 2n − 1y2n−1−2ir2i + u(2n−1)(x + y) drdy. 2i (2n − 1)! n=3 i=0 Employing analytic expansions in each of the even derivative terms near x gives 41
s Z Z " ∞ j ! 2 2 Lγη[u] X y y + r = − γ(y)η(r) u000(x)y + u(2+j)(x) 2σ(δ , δ ) j! 2 γ η j=2 Bγ Bη
∞ ! X yj y4 + 6y2r2 + r4 + u(5)(x)y + u(4+j)(x) j! 4! j=2
∞ ∞ ! n # X X yj X 2ny2n−2ir2i + u(2n+j)(x) drdy j! 2i (2n)! n=3 j=1 i=0
Z Z 3r2y + y3 + γ(y)η(r) u0(x + y)y + u000(x + y) 3! Bγ Bη
∞ n−1 # X X 2n − 1y2n−1−2ir2i + u(2n−1)(x + y) drdy. 2i (2n − 1)! n=3 i=0 As before, each of the odd power terms (in y) in the first integral are antisymmetric and vanish. Simplifying produces
s Lγη[u]
2σ(δγ, δη) " ∞ ∞ ! n # Z Z X X y2j X 2ny2n−2ir2i = − γ(y)η(r) u(2n+2j)(x) drdy (2j)! 2i (2n)! n=1 j=1 i=0 Bγ Bη
Z Z 3r2y + y3 + γ(y)η(r) u0(x + y)y + u000(x + y) 3! Bγ Bη
∞ n−1 # X X 2n − 1y2n−1−2ir2i + u(2n−1)(x + y) drdy. 2i (2n − 1)! n=3 i=0 42
Next, we perform an analytic expansion around x for each of the odd derivatives in the second integral to obtain
s Lγη[u]
2σ(δγ, δη) " ∞ ∞ ! n # Z Z X X y2j X 2ny2n−2ir2i = − γ(y)η(r) u(2n+2j)(x) drdy (2j)! 2i (2n)! n=1 j=1 i=0 Bγ Bη
" ∞ ! Z Z X yj + γ(y)η(r) u0(x) + u00(x)y + u(1+j)(x) y j! j=2 Bγ Bη
∞ ! X yj (3r2y + y3) + u000(x) + u(4)(x)y + u(3+j)(x) j! 3! j=2
∞ ∞ ! n−1 # X X yj X 2n − 1y2n−1−2ir2i + u(2n−1+j)(x) drdy. j! 2i (2n − 1)! n=3 j=0 i=0
Once again, the odd power terms (in y) in the second integral are antisymmetric and vanish. Simplifying and moving 2σ(δγ, δη) to the right side of the equation we obtain
s Lγη[u](x) " ∞ ∞ ! Z Z X X y2j = −2σ(δ , δ ) γ(y)η(r) u(2n+2j)(x) γ η (2j)! n=1 j=1 Bγ Bη n # X 2ny2n−2ir2i · drdy 2i (2n)! i=0
Z Z 2 00 + 2σ(δγ, δη) γ(y)η(r)y drdy u (x) Bγ Bη 43
∞ Z Z X y2j+1 +2σ(δ , δ ) γ(y)η(r) u(2+2j)(x) ydrdy γ η (2j + 1)! j=1 Bγ Bη
" ∞ ∞ ! Z Z X X y2j+1 + 2σ(δ , δ ) γ(y)η(r) u(2n+2j)(x) γ η (2j + 1)! n=2 j=0 Bγ Bη n−1 # X 2n − 1y2n−1−2ir2i · drdy. 2i (2n − 1)! i=0 The scaling given by (3.10) normalizes the coefficient of u00, so the error of the
s 00 approximation |Lγη[u](x) − u (x)| is given by the remaining terms:
s 00 |Lγη[u](x) − u (x)| " ∞ ∞ ! n # Z Z X X |y|2j X 2n|y|2n−2i|r|2i ≤ 2Mσ(δ , δ ) γ(y)η(r) drdy γ η (2j)! 2i (2n)! n=1 j=1 i=0 Bγ Bη
∞ Z Z X |y|2j+1 +2Mσ(δ , δ ) γ(y)η(r) |y| drdy γ η (2j + 1)! j=1 Bγ Bη
" ∞ ∞ ! Z Z X X |y|2j+1 + 2Mσ(δ , δ ) γ(y)η(r) γ η (2j + 1)! n=2 j=0 Bγ Bη n−1 # X 2n − 1|y|2n−1−2i|r|2i · drdy, 2i (2n − 1)! i=0 where M is defined in (3.12). Since |y| < δγ, and |r| < δη, we get
s 00 |Lγη[u](x) − u (x)| " ∞ ∞ 2j−2 ! n 2n−2i 2i # Z Z X X δγ X 2n δγ δη ≤ 2Mσ(δ , δ ) γ(y)η(r)y2drdy γ η (2j)! 2i (2n)! n=1 j=1 i=0 Bγ Bη 44
∞ 2j Z Z X δγ +2Mσ(δ , δ ) γ(y)η(r)y2drdy γ η (2j + 1)! j=1 Bγ Bη
" ∞ ∞ 2j ! n−1 2n−2−2i 2i # X X δγ X 2n − 1 δγ δη + 2Mσ(δ , δ ) γ η (2j + 1)! 2i (2n − 1)! n=2 j=0 i=0 Z Z γ(y)η(r)y2drdy.
Bγ Bη
Using σ(δγ, δη) as given in (3.10) we have
" ∞ ∞ 2j−2 ! n 2n−2i 2i # X X δγ X 2n δγ δη |Ls [u](x) − u00(x)| ≤ M γη (2j)! 2i (2n)! n=1 j=1 i=0
∞ 2j " ∞ ∞ 2j ! n−1 2n−2−2i 2i # X δγ X X δγ X 2n − 1 δγ δη +M + M . (2j + 1)! (2j + 1)! 2i (2n − 1)! j=1 n=2 j=0 i=0
Separating the n = 1 terms in the first set of summations, the j = 1 in the second summation, and the n = 2 terms in the third set of summations, we obtain
s 00 |Lγη[u](x) − u (x)| 2 2 ∞ 2j−2 " ∞ ∞ 2j−2 ! n 2n−2i 2i # δγ + δη X δγ X X δγ X 2n δγ δη ≤ M + M 2 (2j)! (2j)! 2i (2n)! j=1 n=2 j=1 i=0
2 ∞ 2j−2 δγ X δγ +M + M 6 (2j)! j=2
2 2 ∞ 2j " ∞ ∞ 2j ! n−1 2n−2−2i 2i # δγ + 3δη X δγ X X δγ X 2n − 1 δγ δη +M + M . 6 (2j + 1)! (2j + 1)! 2i (2n − 1)! j=0 n=3 j=0 i=0
Since all of the above series are convergent for δγ, δη < 1 we note that each term is 45
2 2 s 00 of order δγ or δη. Thus as δγ and δη shrink to zero, Lγη[u] converges to u at a rate of
2 2 δγ + δη.
Next we will present a much more general convergence result that holds in any dimension, under less regularity for u, however we add an additional restriction on the support of the kernels. The ideas follow the method developed in [19] and illustrated in the proof of Theorem 3.1 to show convergence of the bond-based Laplacian to the classical Laplacian.
Theorem 3.3. Let Ω ⊂ Rn, n ≥ 1, possibly unbounded, and let u ∈ C4(Ω) with
(4) M4 := sup |u (x)| < ∞. (3.15) x∈Ω
Let γ and η satisfy Assumption 1, with the restriction that δη ≤ δγ. If
−α −α c1|y| ≤ γ(y) ≤ c2|y| for 0 ≤ α < n, and 0 < c1 ≤ c2,
s then Lγη[u] with scaling factor σ(δγ, δη) given by (3.9) converges to ∆u at a rate of
2 δγ.
Proof. From (2.17) we have
Ls [u] Z Z γη (x) = γ(y)η(r) u(x + y + r) − u(x) 2σ(δγ, δη) Bγ Bη − [u(x + r) − u(x + y)] drdy.
Applying the fundamental theorem of calculus we obtain 46
1 Ls [u] Z Z Z γη (x) = γ(y)η(r) [∇u(x + s(y + r))](y + r)dsdrdy 2σ(δγ, δη) Bγ Bη 0
1 Z Z Z − γ(y)η(r) [∇u(x + y + s(r − y))](r − y)dsdrdy,
Bγ Bη 0 where ∇u is the Jacobian matrix for u. Expanding and collecting similar terms produces
Ls [u] γη (x) 2σ(δγ, δη) 1 Z Z Z = γ(y)η(r) (∇u(x + s(y + r)) + ∇u(x + y + s(r − y))) ydsdrdy
Bγ Bη 0
1 Z Z Z + γ(y)η(r) (∇u(x + s(y + r)) − ∇u(x + y + s(r − y)))rdsdrdy
Bγ Bη 0
Using πγ and πη as defined in (2.10) and (2.11) we obtain
s Lγη[u] =: I1 + I2 (3.16) 2σ(δγ, δη) where
1 Z Z Z I1 := η(r) ∇u(x + s(y + r))
Bγ Bη 0 + ∇u(x + y + s(r − y)) ∇yπγ(y)dsdrdy, 47
1 Z Z Z I2 := γ(y) ∇u(x + s(y + r))
Bγ Bη 0 − ∇u(x + y + s(r − y)) ∇rπη(r)dsdrdy.
Note that I1 and I2 are vector valued quantities. Integration by parts in I1 yields
1 Z Z Z I1 = − η(r)πγ(y)divy[∇u(x + s(y + r))]dydsdr
Bη 0 Bγ
1 Z Z Z − η(r)πγ(y)divy[∇u(x + y + s(r − y))]dydsdr
Bη 0 Bγ
1 Z Z Z y + η(r)πγ(y)∇u(x + s(y + r)) dydsdr δγ Bη 0 B∂δγ (0)
1 Z Z Z y + η(r)πγ(y)∇u(x + y + s(r − y)) dydsdr. δγ Bη 0 B∂δγ (0)
For y ∈ ∂Bγ, we have πγ(y) = πγ(δγ) = 0 thus the last two terms vanish, so I1 becomes
1 Z Z Z I1 = − η(r)s∆u(x + s(y + r))πγ(y)dydsdr
Bη 0 Bγ 48
1 Z Z Z − η(r)(1 − s)∆u(x + y + s(r − y))πγ(y)dydsdr,
Bη 0 Bγ which, after adding and subtracting ∆u(x) we can write as
1 Z Z Z I1 = − η(r)s[∆u(x + s(y + r)) − ∆u(x)]πγ(y)dydsdr
Bη 0 Bγ
1 Z Z Z − η(r)(1 − s)[∆u(x + y + s(r − y)) − ∆u(x)]πγ(y)dydsdr
Bη 0 Bγ
1 Z Z Z − ∆u(x) η(r)πγ(y)dydsdr.
Bη 0 Bγ
We use the same approach for I2; we first integrate by parts, using the fact that
πη(r) = πη(δη) = 0 for r ∈ ∂Bη, and then add and subtract ∆u(x) to obtain
1 Z Z Z I2 = − γ(y)s[∆u(x + s(y + r)) − ∆u(x)]πη(r)drdsdy
Bγ 0 Bη
1 Z Z Z + γ(y)s[∆u(x + y + s(r − y)) − ∆u(x)]πη(r)drdsdy.
Bγ 0 Bη
In order to make the coefficient of the Laplacian in the third integral of I1 equal to
1, we take σ(δγ, δη) as given by (3.9). With this choice of scaling we write
s Lγη[u](x) − ∆u(x) =: 2σ(δγ, δη)(J1 + J2 + J3 + J4) , 49 where
1 Z Z Z J1 = − η(r)πγ(y)s[∆u(x + s(y + r)) − ∆u(x)]dydsdr,
Bη 0 Bγ
1 Z Z Z J2 = − η(r)πγ(y)(1 − s)[∆u(x + y + s(r − y)) − ∆u(x)]dydsdr,
Bη 0 Bγ
1 Z Z Z J3 = − γ(y)πη(r)s[∆u(x + s(y + r)) − ∆u(x)]drdsdy,
Bγ 0 Bη and 1 Z Z Z J4 = γ(y)πη(r)s[∆u(x + y + s(r − y)) − ∆u(x)]drdsdy.
Bγ 0 Bη
Again, J1,J2,J3, and J4 are vector valued. We now look to bound each integral; we begin with J1. Integrating by parts with respect to s, and using antisymmetry of the integrands we obtain
Z Z 2 s=1 (1 − s ) J1 = (∆u(x + s(y + r)) − ∆u(x))η(r)πγ(y) dydr 2 s=0 Bη Bγ
1 Z Z Z (1 − s2) − ∆∇u(x + s(y + r))(y + r)η(r)π (y)dsdydr. 2 γ Bη Bγ 0
After evaluating at s = 0 and s = 1 in the first term gives,
1 Z Z Z (1 − s2) J = − ∆∇u(x + s(y + r))(y + r)η(r)π (y)dsdydr. 1 2 γ Bη Bγ 0
Integrating by parts with respect to s again we obtain 50
Z Z 3 s=1 1 s s J1 = − + ∆∇u(x + s(y + r))(y + r)η(r)πγ(y) dydr 3 2 6 s=0 Bη Bγ 1 Z Z Z 1 s s3 − − + ∆ ∇2u(x + s(y + r))(y + r) (y + r)η(r)π (y)dsdydr, 3 2 6 γ Bη Bγ 0 where ∇2 is the Hessian tensor. Evaluating the first integral at s = 1 yields a factor of zero, while evaluating at s = 0 produces an antisymmetric function which vanishes after integration. Hence, we have
1 Z Z Z 1 s s3 J = − − + ∆[∇2u(x + s(y + r))(y + r)](y + r)η(r)π (y)dsdydr. 1 3 2 6 γ Bη Bγ 0
Taking M4 as defined in (3.15) we estimate the magnitude of J1 as follows
1 Z Z Z 1 s s3 |J | ≤ M − + |y|2 + |r|2 + 2|yr| η(r)|π (y)|dsdydr 1 4 3 2 6 γ Bη Bγ 0
M Z Z ≤ 4 |y|2 + |r|2 η(r)|π (y)|dydr. 4 γ Bη Bγ 51
Using the coarea formula we obtain
M Z Z δγ 4 n+1 |J1| ≤ nωn−1 η(r)dr λ |πγ(λ)|dλ 4 0 Bη Z δη Z n+1 +nωn−1 ρ η(ρ)dρ |πγ(y)|dy 0 Bγ M Z Z ≤ 4 δ2 + δ2 η(r)|π (y)|dydr, 4 γ η γ Bη Bγ where in the last equality we use the coarea formula and the fact that
n n−1 n n−1 λ ≤ λ δγ and ρ ≤ ρ δη. (3.17)
Multiplying by 2σ(δγ, δη) given by (3.9) gives the bound
M |2σ(δ , δ )J | ≤ 4 δ2 + δ2 . γ η 1 4 γ η
To find a bound on J2, we first integrate by parts with respect to s to obtain
Z Z 2 s=1 2s − s − 1 J2 = − (∆u(x + y + s(r − y)) − ∆u(x)) η(r)πγ(y) dydr 2 s=0 Bη Bγ
1 Z Z Z 2s − s2 − 1 + (∆∇u(x + y + s(r − y))(r − y)) η(r)π (y)dydr, 2 γ Bη Bγ 0 after which evaluation at s = 0 and s = 1 gives 52
1 Z Z J = − (∆u(x + y) − ∆u(x)) η(r)π (y)dydr 2 2 γ Bη Bγ
1 Z Z Z 2s − s2 − 1 + ∆∇u(x + y + s(r − y)) 2 Bη Bγ 0 − ∆∇u(x) (r − y)η(r)πγ(y)dsdydr.
In the last line we have added the last term which is zero by the antisymmetry of the integrand. Using the fundamental theorem of calculus for the first integral and and integrating by parts with respect to s in the second integral we have
1 1 Z Z Z J = − (∆∇u(x + sy)y) η(r)π (y)dsdydr 2 2 γ Bη Bγ 0
Z Z s2 − s 1 − s3 + + ∆∇u(x + y + s(r − y)) 2 6 Bη Bγ s=1
− ∆∇u(x) (r − y)η(r)πγ(y) dydr s=0
1 Z Z Z s2 − s 1 − s3 − + ∆[∇2u(x + y + s(r − y))(r − y)](r − y) 2 6 Bη Bγ 0
· η(r)πγ(y)dsdydr.
Without changing the value of the integral we can insert again an antisymmetric integrand in the first integral. We also evaluate the second integral at s = 0 and s = 1, to produce 53
1 1 Z Z Z J = − (∆∇u(x + sy) − ∆∇u(x)) yη(r)π (y)dsdydr 2 2 γ Bη Bγ 0
1 Z Z − (∆∇u(x + y) − ∆∇u(x)) (r − y)η(r)π (y)dydr 6 γ Bη Bγ
1 Z Z Z s2 − s 1 − s3 − + ∆[∇2u(x + y + s(r − y))(r − y)](r − y) 2 6 Bη Bγ 0
· η(r)πγ(y)dsdydr.
Now, integrating by parts with respect to s in the first integral and applying the fundamental theorem of calculus in the second integral produces
Z Z s=1 1 J2 = (1 − s) (∆∇u(x + sy) − ∆∇u(x)) yη(r)πγ(y) dydr 2 s=0 Bη Bγ
1 1 Z Z Z − (1 − s)∆[∇2u(x + sy)y]yη(r)π (y)dydr 2 γ Bη Bγ 0
1 1 Z Z Z − ∆[∇2u(x + sy)y](r − y)η(r)π (y)dydr 6 γ Bη Bγ 0
1 Z Z Z s2 − s 1 − s3 − + ∆[∇2u(x + y + s(r − y))(r − y)](r − y) 2 6 Bη Bγ 0
· η(r)πγ(y)dsdydr. 54
Evaluating the first integral at s = 0 and s = 1 gives
1 1 Z Z Z J = − (1 − s)∆[∇2u(x + sy)y]yη(r)π (y)dydr 2 2 γ Bη Bγ 0
1 1 Z Z Z − ∆[∇2u(x + sy)y](r − y)η(r)π (y)dydr 6 γ Bη Bγ 0
1 Z Z Z s2 − s 1 − s3 − + ∆[∇2u(x + y + s(r − y))(r − y)](r − y) 2 6 Bη Bγ 0
· η(r)πγ(y)dsdydr.
By bounding the fourth order derivatives as we did with J1, and using |y| < δγ and
|r| < δη we obtain
7M Z Z |J | ≤ 4 δ2 + δ2 η(r)|π (y)|dydr, 2 12 γ η γ Bη Bγ
hence, 7M |2σ(δ , δ )J | ≤ 4 δ2 + δ2 . γ η 2 12 γ η
Using approaches similar to the ones employed to bound J1 and J2, we find the
following bounds for J3, respectively J4:
M Z Z |J | ≤ 4 |y|2 + |r|2 |π (r)|γ(y)drdy, 3 4 η Bγ Bη 55 and,
5M Z Z |J | ≤ 4 |y|2 + |r|2 |π (r)|γ(y)dydr. 4 6 η Bη Bγ
Using Lemma 2.3 we have that
M Z Z M Z Z |J | ≤ 4 |r|2η(r)|π (y)|dydr + 4 |r|4η(r)γ(y)dydr. 3 4 γ 4n Bη Bγ Bη Bγ
Then using (3.17) we find
M Z Z M Z Z |J | ≤ 4 δ2 η(r)|π (y)|dydr + 4 δ4 η(r)γ(y)dydr, 3 4 η γ 4n η Bη Bγ Bη Bγ thus,
Z γ(y)dy
M4 2 M4 4 Bγ |2σ(δγ, δη)J3| ≤ δη + δη . (3.18) 4 4n Z
πγ(y)dy
Bγ
−α −α Using the assumption that c1|y| ≤ γ(|y|) ≤ c2|y| where 0 ≤ α < n, and 0 < c1 ≤ c2, we get
4 M4 2 M4c2n(n + 2 − α) δη |2σ(δγ, δη)J3| ≤ δη + 2 . 4 4c1n(n − α) δγ
Hence, under the assumption δη ≤ δγ, we have
M4c1n(n − α) + M4c2n(n + 2 − α) 2 |2σ(δγ, δη)J3| ≤ δη. 4c1n(n − α) 56
Similarly, we find the bound on J4 to be
5M4c1n(n − α) + 5M4c2n(n + 2 − α) 2 |2σ(δγ, δη)J4| ≤ δη. 6c1n(n − α)
Putting all of these together we find
s |Lγη(u) − ∆u(x)| ≤ |2σ(δγ, δη)J1| + |2σ(δγ, δη)J2| + |2σ(δγ, δη)J3| + |2σ(δγ, δη)J4|
2 2 2 ≤ C δγ + δη ≤ Cδγ,
where the value of the constant C changes from line to line, and it depends on
M4, n, α, c1 and c2. This estimate shows that our nonlocal state-based Laplacian with
2 the scaling of (3.11) converges to the classical Laplacian at a rate of δγ. Note that the quadratic rate is independent of the dimension, but the constant C does depend on the dimension.
Remark 3.4. In Theorem 3.3 we can relax the growth restrictions on γ by assuming
instead that there exists a C1 > 0 such that
Z C Z γ(y)dy ≤ 1 π (y)dy . (3.19) 2 γ δγ Bγ Bγ
Since δη ≤ δγ, (3.18) combined with (3.19) implies
M (n + C ) |σ(δ , δ )J | ≤ 4 1 δ2. γ η 3 8n η
Similarly, 5M (n + C ) |σ(δ , δ )J | ≤ 4 1 δ2, γ η 4 6n η
and the rate of convergence in the theorem holds. 57
Furthermore, we can replace the condition δη ≤ δγ by the assumption that there
exists a C2 > 0 such that
Z C Z γ(y)dy ≤ 2 π (y)dy , 2 γ δη Bγ Bγ
which becomes a condition that links the growth of γ with the growth of η. We then obtain from (3.18) that
M (n + C ) |σ(δ , δ )J | ≤ 4 2 δ2, γ η 3 8n η and 5M (n + C ) |σ(δ , δ )J | ≤ 4 2 δ2. γ η 4 6n η
2 2 The resulting rate of convergence will be δγ + δη. 58
Chapter 4
Convolutions and the Fourier Transform
In the previous chapter we focused on showing that the state-based Laplacian applied to u converges to the classical Laplacian applied u, whenever u is sufficiently smooth. We will continue the course of making connections between the state-based Laplacian and the classical Laplacian. In addition, similarities between the bond-based and the state-based Laplacians are studied. From the geometrical description of the state- based peridynamics formulation given in Figure 4.1, one can imagine that if we shrunk
δη to zero, we should return the bond-based formulation of peridynamics. In [45], Silling highlights this idea by presenting an example, in the state-based formulation, using the Dirac mass centered at zero to show that this produces the bond-based formulation. The focus of this chapter will be on emphasizing these connections. We begin by showing that the state-based Laplacian (2.7) can be written as con-
s volutions of the kernels of Lγη with u, and discuss the properties emphasized by the convolution structure. As the bond-based Laplacian (1.19) can also be written as a
b convolution of the kernel of Lµ and u we again see how the state-based, bond-based and classical Laplacians are connected. We illustrate these connections by taking the kernels to be Dirac mass measures, or combinations of derivatives of the Dirac mass measure. We conclude by studying the solutions to the Cauchy problem using the Fourier transform, and show improved regularity of the solutions. 59
δη
• y • • r δγ x
Horizon of p Horizon of x
Figure 4.1: The indirect interaction that the point x has on y through their common neighbor r. Relabeled here, from Figure 1.2, for convenience.
4.1 Convolution form of the state-based Laplacian
Assuming that γ and η are L1 integrable, the state-based Laplacian defined in (2.7) can be expressed in terms of double and single convolutions as follows:
Ls [u] γη = (γ ∗ η ∗ u) − (η ∗ u) kγk + (γ ∗ u) kηk − u kγk kηk , (4.1) 2σ(γ, η) L1 L1 L1 L1 where we mean L1 = L1(Rn). The convolution above is performed component wise, so each component of a vector is convolved with the scalar kernels:
u ∗ γ = (u1 ∗ γ, u2 ∗ γ, . . .). (4.2)
The expression (4.1) is similar to the convolution form of the bond-based Laplacian from (1.19) as expressed by
L [u] µ = µ ∗ u − u kµk . (4.3) σ(µ) L1
For a physical interpretation of the operator Lµ when µ is a probability measure, in the context of nonlocal diffusion, see [1]. 60
Also, note that although the double integral form of the operator in its (2.7) or (4.1) form implies similarity to the biharmonic operator
2 2 Bµ[u] = Lµ[u] = u ∗ µ ∗ µ − 2u ∗ µ kukL1 + u kµkL1 ,
s introduced in [39], the convolution formulation of Lγη clearly shows that no choice of kernels γ and η will yield the nonlocal biharmonic. Indeed, in order to eliminate the single convolution term one would have to choose a kernel that would also elim- inate the double convolution term (single convolution is associated with bond-based Laplacian, while double convolution is associated with the state-based Laplacian). Finally, the doubly nonlocal state-based Laplacian was shown in Chapter 3 to con- verge to a second-order differential operator, while the nonlocal biharmonic provides and approximation to the classical biharmonic operator ∆2.
4.1.1 The state-based Laplacian on the space of distributions
s The convolution formulation (4.1) shows that the operator Lγη can be conveniently defined for functions u of different smoothness levels depending on choices of γ and η. In particular, γ and η in C∞ will allow choosing u less smooth and vice-versa. The support for each of the kernels γ and η could be taken to be unbounded, but for applications linked to peridynamics, the finite horizon is the relevant choice. In fact, both the state-based and bond-based Laplacians can be defined in the weak sense on the space of distributions. Recall the definition of a convolution between an L1
0 n ∞ n function and a distribution: Let f ∈ D (R ), and ϕ ∈ Cc (R ) withϕ ˜(x) = (−x), then for g ∈ L1(Rn), hf ∗ g, ϕi = hf, g ∗ ϕ˜i . (4.4) 61
Hence, if γ, η ∈ L1(Rn), then u can be taken in D0(Rn), the space of distributions. In addition, γ and η can be chosen to be Dirac masses, or derivatives of Dirac masses, as shown in the next section.
4.2 Connections between the state-based, bond-based and
classical Laplacians
From the geometrical view of the state-based peridynamic formulation in Figure 4.1, we note that if we “shrink” δη to zero, we should get back the bond-based peridynamic formulation. This provides the visual and physical motivation for what we prove in this section. We will show that if η is taken to be the Dirac mass, in the sense of
b distributions, we will indeed get back the bond-based Laplacian with kernel γ, Lγ. Similarly, one can imagine shrinking the horizon of the bond-based Laplacian to a point and recovering the classical Laplacian. In fact, it has been shown in [25] that taking µ to be a combination of derivatives of the Dirac mass measure in the bond- based Laplacian will give back the classical Laplacian. Taking these two together, by letting µ = γ in the bond-based Laplacian, we recover the classical Laplacian from the state-based Laplacian. We start by proving the following Lemma, which demonstrates how derivatives of the Dirac mass apply to functions in L1.
∞ n Lemma 4.1. Let ϕ ∈ Cc (R ), and δ0 the Dirac mass measure centered at the origin. For a function v ∈ L1(Rn), in the sense of distributions
* (k) + * (k) + ∂ k ∂ (k) δ0 ∗ v, ϕ = (−1) (k) v, ϕ . (4.5) ∂xj ∂xj 62
1 ∞ Proof. Let v ∈ L (R), and ϕ ∈ Cc (R). Takingϕ ˜(x) = ϕ(−x), we have
* + * + ∂(k) ∂(k) δ ∗ v, ϕ = δ , v ∗ ϕ˜ , (4.6) (k) 0 (k) 0 ¯ ∂xj ∂xj
∞ where v ∗ ϕ˜ ∈ Cc (R). When δ0 is the Dirac mass measure centered at zero we obtain
* (k) + * (k) + ∂ k ∂ (k) δ0, v ∗ ϕ˜ = (−1) δ0, (k) v ∗ ϕ˜ ∂xj ∂xj (k) ! k ∂ = (−1) v ∗ ϕ˜ . (4.7) (k) ∂xj x=0
Then using the definition of the convolution andϕ ˜ we get
(k) ! Z (k) ! ∂ ∂ v ∗ ϕ˜ = v (y)ϕ ˜(x − y)dy (k) (k) ∂xj x=0 R ∂xj x=0 Z (k) ! ∂ = v (y)ϕ(y − x)dy (k) R ∂xj x=0 ! Z ∂(k) = (k) v (y)ϕ(y)dy. (4.8) R ∂xj
Finally, using (4.6), (4.7) and (4.8) we have
* (k) + * (k) + ∂ k ∂ (k) δ0 ∗ v, ϕ = (−1) (k) v, ϕ . ∂xj ∂xj
We now show that if we let η converge to the Dirac mass measure in state-based Laplacian formula, we recover the bond-based Laplacian with kernel γ.
1 n Proposition 4.2 (State-based to bond-based). Let u ∈ L (R ). If ||ηi||L1(Rn) = 1 with ηi → δ0 as i → ∞, in the sense of distributions, where δ0 is the Dirac mass 63
measure centered at the origin, then Ls [u] → L [u], as i → ∞, in the sense of γηi γ distributions.
1 n ∞ n Proof. Let u ∈ L (R ), and ϕ ∈ Cc (R ) such that ||ηi||L1 = 1, and hηi, ϕi → hδ0, ϕi in the sense of distributions. We have
hηi ∗ u, ϕi = hηi, u ∗ ϕ˜i ,
∞ n whereϕ ˜(x) = ϕ(−x). Since, u ∗ ϕ˜ ∈ Cc (R ),
hηi, u ∗ ϕ˜i → hδ0, u ∗ ϕ˜i . (4.9)
Then by Lemma 4.1 and (4.9)
hηi ∗ u, ϕi → hu, ϕi , as i → ∞.
Using the formulation in (4.1), we have
Ls [u] γηi , ϕ = hu ∗ ηi ∗ γ, ϕi − hu ∗ ηi, ϕi + h||ηi||L1 u ∗ γ, ϕi − hu||γ||L1 , ϕi . 2σ(γ, ηi)
By the definition of ηi we get
2 1 2σ(γ, ηi) = − Z Z = −Z . (4.10) 2 ηi(r)dr πγ(y)dy πγ(y)dy
Rn Rn Rn 64
As i → ∞ we obtain
Z − π (y)dy Ls [u], ϕ → hu ∗ γ, ϕi − hu, ϕi + hu ∗ γ, ϕi − hu, ϕi γ γηi Rn = 2 hu ∗ γ, ϕi − hu, ϕi
∞ n for all ϕ ∈ Cc (R ). Thus
2 Ls [u], ϕ → − hu ∗ γ, ϕi − hu, ϕi (4.11) γηi Z πγ(y)dy
Rn
∞ n for all ϕ ∈ Cc (R ). Then we can rewrite the right hand side of (4.11) to give
2 2 −Z hu ∗ γ, ϕi − hu, ϕi = − Z hLγ[u], ϕi . (4.12) πγ(y)dy σ(γ) πγ(y)dy
Rn Rn
Using σ(γ) as defined in (3.2), we obtain
Ls [u], ϕ → hL [u], ϕi , (4.13) γηi γ
∞ n for all ϕ ∈ Cc (R ).
It has been shown in [25] that if µ is taken to be a combination of derivatives of the Dirac mass, then then the bond-based Laplacian becomes the classical Laplacian. We restate and prove this result here for completeness and to study in full detail the connections between the state-based, bond-based, and classical Laplacians.
1 n 2 Proposition 4.3 (Bond-based to classical). [25] Let u ∈ L (R ), and |y| µi ∈
1 n L (R ), supp(µi) ⊆ Bδ(0). If ||µi||L1(Rn) = 1 with µi → ∆δ0 + δ0 as i → ∞ in the 65
sense of distributions, where δ0 is the Dirac mass measure centered at the origin,
then Lµi [u] → ∆u, as i → ∞, in the sense of distributions.
1 n ∞ n Proof. Let u ∈ L (R ), ||µi||L1 = 1, ϕ ∈ Cc (R ), and hµi, ϕi → h∆δ0, ϕi + hδ0, ϕi in the sense of distributions. We have
hµi ∗ u, ϕi = hµi, u ∗ ϕ˜i , (4.14)
∞ n whereϕ ˜(x) = ϕ(−x). Since, u ∗ ϕ˜ ∈ Cc (R ), we get
hµi, u ∗ ϕ˜i → h∆δ0, u ∗ ϕ˜i + hδ0, u ∗ ϕ˜i . (4.15)
Then by Lemma 4.1,
hµi ∗ u, ϕi → h∆u, ϕi + hu, ϕi , as i → ∞. (4.16)
Using the formulation in (4.3), we have
Lµi [u] , ϕ = hu ∗ µi, ϕi − hu||µi||L1 , ϕi . (4.17) σ(µi)
As i → ∞ we obtain
L [u] µi , ϕ → h∆u, ϕi + hu, ϕi − hu, ϕi σ(µi) = h∆u, ϕi , (4.18) 66
∞ n for all ϕ ∈ Cc (R ). Lemma 2.3 with γ replaced by µi gives
−2 2n σ(µi) = Z = Z . (4.19) 2 πµi (y)dy |y| µi(y)dy
Rn Rn
∞ n 2 Now, we take ϕ ∈ Cc (R ) such that ϕ(y) = |y| for y in Bδ(0). Then
Z Z 2 |y| µi(y)dy = ϕ(y)µi(y)dy.
Rn Rn
Then, in the sense of distributions
2 2 2 2 µi, |y| n = µi, |y| → ∆δ0, |y| + δ0, |y| (4.20) R Bδ(0) as i → ∞. Thus, we obtain
2 2 2 ∆δ0, |y| + δ0, y = δ0, ∆|y|
2 = δ0, div(∇|y| )
= hδ0, div(2y)i
= hδ0, 2ni
= 2n. (4.21)
Hence, in the sense of distributions, from (4.19) and (4.21) we have that
σ(µi) → 1. (4.22) 67
From Propositions 4.2 and 4.3 we immediately get the following corollary. In the state-based Laplacian, if η is taken to be the Dirac mass measure, and γ is taken to be the combination of two derivatives of the Dirac mass and the Dirac mass, we return the classical Laplacian.
1 n Corollary 4.4 (State-based to classical). Let u ∈ L (R ) and supp(µi) ⊆ Bδ(0). If
||ηi||L1(Rn) = 1, ||γi||L1(Rn) = 1 with ηi → δ0, γi → ∆δ0 + δ0 as i → ∞, in the sense 2 1 n of distributions, and |y| µi ∈ L (R ), where δ0 is the Dirac mass measure centered at the origin, then Ls [u] → ∆u, as i → ∞, in the sense of distributions. γiηi
We end this section with the observation that taking γ to be only the Dirac mass, results in the state-based Laplacian being zero. Thus, in some sense, how “fast” we shrink γ is important.
1 n Proposition 4.5. Let u ∈ L (R ). If ||γi||L1 = 1 with γi → δ0 as i → ∞, in the
sense of distributions, where δ0 is the Dirac mass measure centered at the origin, then Ls [u] → 0, as i → ∞, in the sense of distributions. γiη
1 n ∞ n Proof. Let u ∈ L (R ), and ϕ ∈ Cc (R ) such that ||γi||L1 = 1, and hγi, ϕi → hδ0, ϕi in the sense of distributions. We have
hγi ∗ u, ϕi = hγi, u ∗ ϕ˜i , (4.23)
∞ n whereϕ ˜(x) = ϕ(−x). Since, u ∗ ϕ˜ ∈ Cc (R ) we get
hγi, u ∗ ϕ˜i → hδ0, u ∗ ϕ˜i . (4.24)
Then by Lemma 4.1
hγi ∗ u, ϕi → hu, ϕi as i → ∞. (4.25) 68
The formulation in (4.1) gives
Ls [u] γiη , ϕ = hu ∗ η ∗ γi, ϕi − hu ∗ η, ϕi + h||η||L1 u ∗ γi, ϕi − hu||η||L1 , ϕi . 2σ(γi, η)
As n → ∞, we obtain
Ls [u] γiη , ϕ → hu ∗ η, ϕi − hu ∗ η, ϕi + h||η||L1 u, ϕi − hu||η||L1 , ϕi 2σ(γi, η) = 0,
∞ n for all ϕ ∈ Cc (R ).
4.2.1 Solutions of the state-based Laplacian
In the previous section we studied in detail the convolution form (4.1) of the state- based Laplacian (2.7). The ability to write the state-based Laplacian in a combination of convolutions hints at using the Fourier transform to study the solutions of the op- erator. We would be remiss not to take this opportunity, as properties of convolutions can provide a rich analysis in the study of regularity and integrability properties of functions. In the previous section we showed how closely the state-based and bond- based Laplacians are related, and one might wonder if and how their solutions are related. We indeed find that the state-based solutions can be written as a convolution of the bond-based solution and another function. We begin with the solutions to the bond-based Laplacian (1.19). We will usev ˆ to denote the Fourier transform of a function, v.
Proposition 4.6 (Solution of the Cauchy problem for the bond-based Laplacian). 69
b The solution uµ of the problem
b n Lµ[u](x) = f(x), x ∈ R , (4.26)
is given by ˆ b f uˆµ = . (4.27) σ(µ) µˆ − kµkL1
b Proof. If Lµ[u] = f, then using the convolution structure given in (4.3), we have
σ(µ) µ ∗ u − u kµkL1 = f. (4.28)
Applying the Fourier transform we obtain
ˆ σ(µ) µˆuˆ − uˆ kµkL1 = f. (4.29)
Thus we find that ˆ b f uˆµ = . (4.30) σ(µ) µˆ − kµkL1
Proposition 4.7 (Solution of the Cauchy problem for the state-based Laplacian).
s The solution uγη of the problem
s n Lγη[u](x) = f(x), x ∈ R , (4.31) is given by ˆ s f uˆγη = . (4.32) 2σ(γ, η)(ˆη + ||η||L1 )(ˆγ − ||γ||L1 ) 70
s Proof. If Lγη[u] = f then using the convolution structure given in (4.1), we have
2σ(γ, η) (γ ∗ η ∗ u) − (η ∗ u) kγkL1 + (γ ∗ u) kηkL1 − u kγkL1 kηkL1 = f. (4.33)
Applying the Fourier transform we obtain
ˆ 2σ(γ, η) uˆηˆγˆ − uˆηˆ||γ||L1 + uˆγˆ||η||L1 − uˆ||γ||L1 ||η||L1 = f. (4.34)
Then we find that
fˆ uˆ = 2σ(γ, η) ηˆγˆ − ηˆ||γ||L1 +γ ˆ||η||L1 − ||γ||L1 ||η||L1 fˆ = . (4.35) 2σ(γ, η)(ˆη + ||η||L1 )(ˆγ − ||γ||L1 )
Proposition 4.8 (Structure of the solutions of the state-based Laplacian). The so- lution of the state-based Laplacian Cauchy problem stated in (4.31), can be written as 1 s ˆb uˆγη = 2||η||L1 uγ · (4.36) (ˆη + ||η||L1 )
b b where uγ is the solution to the bond-based Cauchy problem Lγu = f. Thus, the state
s solution uγη can be written as
s γ uγη = 2||η||L1 ub ∗ N (4.37)
where Nˆ = 1 . (ˆη+||η||L1 )
In Proposition 4.2, we showed that taking η to be δ0 in the state-based Laplacian produced the bond-based Laplacian, and thus we would hope that doing so would 71
produce solutions to the bond-based Laplacian. Indeed, if η = δ0 in the sense of ˆ 1 distributions, then N = 2 . Thus we have that
γ γ γ 2ub ∗ N = ub ∗ δ0 = ub , in the sense of distributions.
Similarly, in Proposition 4.3, we showed that taking µ to be ∆δ0 + δ0 in the bond- based Laplacian produced the classical Laplacian. Recall that the solution to the classical Cauchy problem ∆u = f, can be written as
! −fˆ uc = F −1 . |ξ|2
If µ = ∆δ0 + δ0, in the sense of distributions, then
fˆ fˆ −fˆ uˆb = = = . 2 2 σ(µ) µˆ − kµkL1 −|ξ| + 1 − 1 |ξ|
b c Hence uµ becomes u . Putting these last two statements together, in the state-based Laplacian, if we take η = δ0 and γ = ∆δ0 + δ0 in the sense of distributions, we have that
s c uγη = u .
If we have convergence to the Dirac masses instead of equality, then we get comparable convergence results. Our final two propositions involve using the convolution form of the solutions to the state-based Cauchy problem to improve the integrability and regularity of the solutions. 72
Proposition 4.9 (Improved integrability of solutions to the state-based Cauchy prob- lem). Taking N to be such that Nˆ = 1 , the integrability of the solution to the (ˆη+||η||L1 ) state-based Laplacian is given by
s b 1 1 1 ||u || r ≤ c||u || p ||N|| q for = + − 1, γη L γ L L r p q
where c = 2||η||L1 .
b k n m n s k+m n In addition, if uγ ∈ C (R ) and N ∈ C (R ) then uγη ∈ C (R ), for k, m ≤ ∞.
Proof. Apply Young’s inequality for convolutions:
1 1 1 ||f ∗ g|| r ≤ ||f|| p ||g|| q for = + − 1, L L L r p q
to (4.37).
We give the following corollary as examples of this proposition.
Corollary 4.10. The following are all examples of Proposition 4.9
b 1 n s 1 n 1. If uγ,N ∈ L (R ), then uγη ∈ L (R ).
b 2 n s ∞ n 2. If uγ,N ∈ L (R ), then uγη ∈ L (R ).
b 1 n ∞ n b ∞ n 1 n s 3. If uγ ∈ L (R ), N ∈ L (R ), or uγ ∈ L (R ), N ∈ L (R ), then uγη ∈ L∞(Rn).
b 1 n 2 n b 2 n 1 n s 2 n 4. If uγ ∈ L (R ), N ∈ L (R ), or uγ ∈ L (R ), N ∈ L (R ), then uγη ∈ L (R ).
b 1 n q n b q n 1 n 5. If uγ ∈ L (R ), N ∈ L (R ) or uγ ∈ L (R ), N ∈ L (R ), for q ≥ 1, then
s q n uγη ∈ L (R ). 73
b 2 n q n s p n 6. If uγ ∈ L (R ), and N ∈ L (R ) for 2/3 ≤ q ≤ 2, then uγη ∈ L (R ), where 2q p = 2−q .
2 n b q n s p n 7. If N ∈ L (R ), and uγ ∈ L (R ) for 2/3 ≤ q ≤ 2, then uγη ∈ L (R ), where 2q p = 2−q .
The following proposition gives an increase in regularity of solutions to the state- based problem, by increasing the differentiability of N.
Proposition 4.11 (Improved regularity of solutions to the state-based Cauchy prob-
ˆ 1 k,q n b p n lem). Take N to be such that N = . If N ∈ W (R ) and uγ ∈ L (R ), (ˆη+||η||L1 ) then 1 1 1 us ∈ W k,r( n) where = + − 1. γη R r p q
k,q n b p n j q n Proof. If N ∈ W (R ) and uγ ∈ L (R ), then ∂ N ∈ L (R ) for all j ≤ k. Then
j s b j ∂ uγη = 2uγ ∗ ∂ N, for all j ≤ k. Hence, for j ≤ k, by Young’s inequality we obtain
j s b j 1 1 1 ||∂ u || r ≤ 2||η|| 1 ||u || p ||∂ N|| q ≤ ∞, for = + − 1. γη L L γ L L r p q
s k,r n Thus, uγη ∈ W (R ).
Examples similar to Corollary 4.10 can be extrapolated for Proposition 4.11. 74
Chapter 5
Final Remarks and Future Directions
To summarize, the main ideas of this thesis revolve around the introduction of a new nonlocal Laplace-type operator, which is intimately connected to the state-based theory of peridynamics. The newly introduced state-based Laplacian offers an approx- imation of the classical Laplace operator for functions that are sufficiently smooth, however, it can be applied even to discontinuous functions or distributions.
5.1 Physical Aspects
As a first observation, the operator provides a lot more flexibility in modeling diffusion- type phenomena. Indeed, this flexibility is achieved through two (possibly different) kernels, each with its own rate of growth, and its own horizon. Since ∆ = div(∇), we do not have much control on either one of the differential operators, div or ∇, possibly just incorporate some variable coefficients. The behavior, however, is prescribed in a restrictive way, not allowing adjustment for the rates of the change. In contrast γ and η have the freedom to be taken from a large variety of spaces, including distributions, or even non-integrable kernels (which would give rise to fractional derivatives). In addition, our operator can be applied to vector-valued functions, as it acts on each component. 75
The state-based Laplacian is intimately connected to the theory of elasticity as given by state-based peridynamics. However, note that the operator was not designed to approximate the Navier operator (except for in the one dimensional case) from the system of classical elasticity [36], as we do not recover the term ∇div u in the limit as the horizon goes to zero; we only get ∆u in the limit. This is a result of taking scalar valued kernels rather than tensor valued. Thus, one direction of future research is to develop a double-kernel Navier-type operator. Of interest to the dynamic fracture community would be an operator that incor- porates fracture in a time-dependent state-based model. In peridynamics, fracture appears as a result of the modeling and it is not introduced through ad hoc systems of equations. In fact, every bond gets broken when the energy sustained surpasses a critical level. This bond-breaking condition could be given in implicit or explicit ways. The mathematical analysis prefers the second option, thus it would be desirable to incorporate fracture in the state-based model in an explicit way as done in [16] for a bond-based model.
5.2 Mathematical Aspects
The error bounds between the nonlocal and the local Laplacians
s (Lγη − ∆)(u) are the main focus of this work. The quadratic dependence on the horizon, for the error, agrees with results obtained from the bond-based framework. Moreover, our proof also shows that the quadratic rate of this convergence with respect to the horizons is optimal for functions u ∈ C4. The C4 level of regularity required for the functions that are considered for these error bound is the same as in the bond-based 76 formulation, which also needs four derivatives on the input function. Also, our bounds are obtained in L∞ for functions u that live in a much smaller space (C4). An interesting aspect worthy of future exploration is that in the proof of Theorem
3.3 it is not clear if δη ≤ δγ is a necessary condition for convergence. Our conjecture is that for δη > δγ the state Laplacian will not converge to the classical Laplacian, so we are working on producing a counterexample to this convergence result. A nu- merical convergence study for different choices of the two kernels, would be helpful in highlighting this relationship. This work will provide insight to the applied commu- nity regarding the optimal use of Laplacians (local vs. nonlocal) depending on the physical model. Recall that in Section 1.1 we showed that the bond-based Laplacian (1.19) can be written as the nonlocal divergence of the nonlocal gradient (see (1.22)). This structure is nice as it aligns with the classical definition of the Laplacian (see (1.4)). We have yet to find state-based versions of the divergence and the gradient that align mathematically to give us state-based Laplacian, or to be physically relevant. As this structure arises in the bond-based Laplacian, it would be ideal if we could extend that idea to the state-based Laplacian. A deeper study of the structure of this operator would be valuable in gaining a better understanding of the operator, and further illustrate its connections to the bond-based can classical Laplacians.
s In Chapter 4 we studied the Cauchy problem Lγη[u] = f on unbounded domains, but we have yet to produce promising results on the bounded domain. This, of course, is of high interest as we would like to prove well-posedness of state-based models. 77
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