WU HAO

(Department of Mathematics and Statistics,Auburn University,AL,36830,US)

Abstract:In this paper,we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term,therefore greatly improve the convergence rate derived in literature.

Key words:Peridynamics,exponential convergence,nonlocal boundary value problem,analytic function, finite-dimensional approximation

1 Introduction

Peridynamics(see[1]–[2])is motivated in aid of modeling the problems from continuum mechanics which involve spontaneous discontinuity forms in the motion of a material system.By replacing differentiation with integration,Peridynamic(PD)equations remain equally valid both on and offthe points where a discontinuity in either displacment or its spatial derivatives is located.Also,under the assumption that particles separated by a finite distance can interact with each other,the PD model is a multiscale material model(see[3]–[5])and naturally falls into the category of nonlocal models(see[6]–[7]).There have been many theoretical works on the mathematical foundation of PD models associated with boundary value problems(BVPs)(see[8]–[15]),where the well-posedness results of the corresponding PD systems were established.Meanwhile,to simulate PD models numerically,various discretization methods have been studied including finite difference, finite element,quadrature and particle based methods(see[10],[16]–[24]).

In[10],a functional analytical framework was built up to study the linear bond-based peridynamic equations associated with a particular kind of nonlocal boundary condition.Investigated were the finite-dimensional approximations to the solutions of the equations obtained by spectral method and finite element method;as a result,two corresponding general formulas of error estimates were derived.According to these formulas,however,one can only conclude that the optimal convergence is algebraic.This motivates us to think whether or not the convergence rate is improvable.

In this paper,we focus on the one-dimensional(1-D)stationary problems.Based on the theoretical framework developed in[10], firstly we show that analytic data functions produce analytic solutions with some appropriate restrictions on kernel function.Afterwards,we are able to prove that those finite-dimensional approximations will achieve exponential convergence under the analyticity assumption of the input data.

The paper is organized as follows.We brie fl y introduce the general PD equilibrium models at the begining of Section 2,after that,in the same section,we give the de finition of the PD operator.The nonlocal BVP is formulated in Section 3,followed by a discussion about analyticity of solutions.We devote Section 4 to the expositions of our main results on exponential convergence of finite-dimensional approximations.

2 The 1-D Stationary Linear Bond-based PD Equation

Suppose that the body of a material occupies a reference con figuration in a finite bar I.Let x denote the position of a particle in the reference con figuration,u(x)be its displacement as the motion of the body occurs and F,a functional of the displacement,be the pairwise force function per unit reference length due to interaction between particles.Then for a given point x,the force(per unit reference length)produced from interaction with other particles is computed by

i.e.,

De finition 2.1Assume u ∈ L2de fined on the interval(?δ,π + δ)satis fies either or

The PD operator?Lδis de fined by

where cδ＞ 0,and for a nonnegative functionin L1(Bδ(0)),the kernel function σ=σ(|y|)satis fies

Note that,for smooth enough functions,(2.1)implies u(0)=u(π)=0 while(2.2)gives ux(0)=ux(π)=0.Moreover,with Fourier sine and cosine series expansion,the following odd and even expansions are naturally derived:

Correspondingly,denoting the PD operator byfor functions satisfying(2.1)and byfor those obeying(2.2),we have the following representations of the operators:

where

Similar to[10],we mainly focus on the functions satisfying condition(2.1)with the Fourier sine expansion;nevertheless,all the paralell conclusions can be derived for the functions subject to(2.2).

3 Nonlocal BVPs and Solutions

Now we formulate the nonlocal BVP as follows:

where u satis fies(2.1).

3.1 Solution Spaces

De finition 3.1The spacewhich depends on the kernel function σ,consists of all functions u∈L2for whichThe-norm is de fined by

The corresponding inner product inis given by

In addition,given an exponent s,one can de fine the general spaceby

3.2The Relations Betweenand Sobolev Spaces

Several results are stated in the following for the later use,but the proofs are omitted and can be found in[10].

Lemma 3.1With the assumption of σ in(2.3),the spacesatis fies

with ηδ(k)satisfying

As to other assumptions of σ,we have

Lemma 3.2(i)Assume that σ is such that,for some constants γ1＞ 0 and α ∈ (0,2),

Moreover,

(ii)Assume that σ is such that,for some constants γ2＞ 0 and β ∈ (?∞,2),

Moreover,

3.3 Properties of Solutions

With Fourier expansions(2.4)and(2.5),the PD equation in(3.1)is converted to

from which we derive

Thus,

Lemma 3.3Under the assumption(2.3)on σ,(3.9)is the unique solution of nonlocal BVP(3.1)in

See Lemma 3.7 in[10]for the proof.

As for the regularity,the following result can be derived by applying(3.9);see Lemma 3.8 in[10].

Lemma 3.4If σ satis fies both(2.3)and(3.6)with β ∈ [0,2),then for?β),the unique solutionMoreover,

The above lemma shows that the solution has more regularity than the data function with an extra assumption of σ.

Furthermore,we can prove that,with more restrictions on σ,the analyticity of data functions will result in analyticity of solutions,i.e.,

Lemma 3.5If σ satis fies(2.3),(3.3)and(3.6)with 0≤β≤α∈(0,2),then for f(x)being analytic on the interval[0,π]and admitting an analytic continuation to the strip|Imz|＜ τ0in the complex plane C,the solution u(x)is analytic on[0,π].

Proof. Note that f(z)is analytic on the rectangular region R:[0,π]× (?τ0,τ0)and can be transformed into the formal Fourier expansion:

Making the change of variables ζ=eiz,i.e.,z=argζ? iln|ζ|yields

Therefore,f(z)being analytic on R corresponds with v(ζ)being analytic in the upper annulus e?τ0＜ |ζ|＜ eτ0.

Next,we consider the similar transformation of u(z):

Let ζ=eiz.Then

Combining(3.4)with(3.7)obtains

Thus,the series w(ζ)has the same maximal convergent annulus as v(ζ)because the relations of their coefficients satisfy

4 Exponential Convergence of Finite-dimensional Approximations

In[10],the authors seek approximations to the solutions of(3.1)in finite-dimensional subspaces of.The forcing term discussed there are within general Sobolev spaces with finite orders.In this section,we prove,by assuming f is analytic,the error of these approximations decays exponentially.

Similar to[10],we first consider the finite-dimensional subspace Vnspanned by the first n Fourier sine modes.We have what follows:

Theorem 4.1If σ satis fies(2.3)and(3.6)with 0≤β ＜2,then for f(x)being analytic on[0,π]and admitting an analytic continuation to the strip|Imz|＜ τ0in the complex plane C,and any τ with 0 ＜ τ＜ τ0,we have

Proof. From the proof of(3.5),we know that f(z)takes the form of

which is analytic on the rectangular region R:[0,π]× (?τ0,τ0).Making the change of variable ζ=eiz,we derive

which is found to be analytic in the upper annulusThe right-hand side is a Laurent expansion whose coefficients are given by

For convenience we choose r=eτ,so r ＞ 1.On the other hand,Lemma 3.4 implies that both

are in finitely differentiable on[0,π].Thus,for any γ ≥ 0,we obtain with(3.7)

Then the following recursion formula can be derived:

With

we conclude for any γ≥0,

as n is large enough.Therefore,

Substituting it into(4.1)yields

To sum up,we have shown

Replacing r with eτand taking squre root of both sides,we finally acquire

Before going to our next result,we introduce an important fact about the approximation of analytic function.

De fine the operator in(n≥1)by interpolation in the n+1 Gauss-Lobatto points.

Lemma 4.1Assume that u is analytic on I=[0,π].Then there are c,τ＞ 0 depending only on u such that the Gauss-Lobatto interpolant inu satis fies

Proof. This is the immediate conclusion of the combination of Theorem 1.9.3 in[25]and Lemma 3.2.6 in[26].Alternatively,a detailed demonstration of this lemma can be found in[27].

Since both endpoints 0 and π are sampling points of the Gauss-Lobatto interpolation operator,u(0)=inu(0)and u(π)=inu(π).

This time we focus on the space Vnmade up by continuous piecewise polynomials that of degree from 0 to n and yields to(2.1).

Theorem 4.2Ifσ satis fies(2.3),(3.3)and(3.6)with 0≤β≤α∈(0,2),then for f(x)being analytic on the interval[0,π]and admitting an analytic continuation to the strip|Imz|＜ τ0in the complex plane C,we have

where un∈ Vn,c,τ＞ 0 are some constants independent of n and δ.

Proof. As 0≤β≤α∈(0,2),Lemma 3.2 with interpolation theory of Sobolev space demonstrate the following relations of spaces:

Thus Vnis a finite-dimensional subspace of

find a un∈Vnsuch thatwhere

has a unique solution un∈Vnwhich is also the best approximation of the solution(3.9)in Vnwithnorm,i.e.,

Thus,with(3.5)and(3.8)we have

On the other hand,Lemma 3.5 indicates solution u is analytic on[0,π].By Lemma 4.1,there exists a polynomial pnof degree n such that

where c,τ＞ 0 are independent of n.It follows that

where c is some constant independent of n and δ.

After incorporating the derivation into(4.2),we obtain the desired result.

5 Conclusions

We demonstrate the exponential convergence(EC)of finite-dimensional approximations discussed in[10]provided that forcing term is analytic.We point out that some other approximation methods can also be used to acquire EC.The hp-version finite element method(see[28]),equipped with proper combination of mesh re finement and increasing polynomial degree,is shown to be superior over h-and p- finite element methods.In addition,the recently developed Fourier continuation(or termed Fourier extension,see[29]–[30])techniques have demonstrated highly accurate approximations for non-periodic functions.The analysis(see[31])indicates that EC can be obtained with this method on evenly spaced points until the parameter dependent accuracy threshold is reached.The applications of all these methods to linear peridynamic BVPs remain to be investigated in the future.