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Particle- and crack-size dependency of lithium-ion battery materials LiFePO4

  • Received: 09 November 2015 Accepted: 31 January 2016 Published: 02 February 2016
  • Lithium-ion batteries have become a widely-used commodity for satisfying the world’s mobile power needs. However, the mechanical degradation of lithium-ion batteries initiated by micro cracks is considered to be a bottleneck for advancing the current technology. This study utilizes a finite element method-based virtual crack closure technique to obtain particle- and crack-size-dependent estimates of mixed-mode energy release rates and stress intensity factors. Interfacial cracks in orthotropic bi-materials are considered in the current study, whereas the crack extension along the interface is assumed. The results show that energy release rate, stress intensity factor, and the propensity of crack extension are particle- and crack-size- dependent. In particular, our results show that for smaller plate-like LiFePO4 particles (100 nm × 45 nm), a crack has lesser tendency to extend if crack-to-particle size is less than 0.2, and for 200 nm × 90 nm particles, similar results are obtained for crack-to-particle sizes of less than 0.15. However, for larger particles (500 nm × 225 nm), it requires an almost flawless particle to have no crack extension. Therefore, the current study provides insight into the fracture mechanics of LiFePO4 and the associated crack-to-particle size dependency to prevent crack extensions.

    Citation: Michael A. Stamps, Jeffrey W. Eischen, Hsiao-Ying Shadow Huang. Particle- and crack-size dependency of lithium-ion battery materials LiFePO4[J]. AIMS Materials Science, 2016, 3(1): 190-203. doi: 10.3934/matersci.2016.1.190

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  • Lithium-ion batteries have become a widely-used commodity for satisfying the world’s mobile power needs. However, the mechanical degradation of lithium-ion batteries initiated by micro cracks is considered to be a bottleneck for advancing the current technology. This study utilizes a finite element method-based virtual crack closure technique to obtain particle- and crack-size-dependent estimates of mixed-mode energy release rates and stress intensity factors. Interfacial cracks in orthotropic bi-materials are considered in the current study, whereas the crack extension along the interface is assumed. The results show that energy release rate, stress intensity factor, and the propensity of crack extension are particle- and crack-size- dependent. In particular, our results show that for smaller plate-like LiFePO4 particles (100 nm × 45 nm), a crack has lesser tendency to extend if crack-to-particle size is less than 0.2, and for 200 nm × 90 nm particles, similar results are obtained for crack-to-particle sizes of less than 0.15. However, for larger particles (500 nm × 225 nm), it requires an almost flawless particle to have no crack extension. Therefore, the current study provides insight into the fracture mechanics of LiFePO4 and the associated crack-to-particle size dependency to prevent crack extensions.


    We consider the following hypersingular integral equation on a circle:

    $ c+2πc=   f(t)sin2ts2dt=g(s),s(c,c+2π). $ (1.1)

    Equation (1.1), considered the natural integral equation for harmonic problems, aligns with the form derived by Yu [28,29]. While natural integral equations themselves hold limited direct significance for solving boundary value problems (BVPs), they become instrumental in unbounded domain BVPs [8]. By introducing artificial boundaries such as circles or ellipses and applying corresponding natural integral equations on these boundaries, domain decomposition methods and coupled algorithms can be effectively constructed [29].

    Integrals of this kind possess several definitions and we present our definition as follows:

    $ c+2πc=   f(t)sin2ts2dt=limϵ0{(sϵc+c+2πs+ϵ)f(t)sin2ts2dt8f(s)ϵ} $ (1.2)

    and

    $ c+2πc=   f(t)sin2ts2dt=limϵ0{(sϵc+c+2πs+ϵ)f(t)sin2ts2dt4f(s)cotϵ2}. $ (1.3)

    The equivalent of (1.2) and (1.3) can be similarly obtained in [32].

    The composite middle rectangle rule remains applicable when the singular point coincides with the midpoint of each subinterval, leading to a superconvergence phenomenon. This allows us to derive the error expansion of the error functional. We leverage these asymptotic expansion results to develop an algorithmic scheme for solving hypersingular integral equations.

    The hypersigular integral defined on the integral

    $ ba=f(t)(ts)2dt=limε0{sεaf(t)(ts)2dt+bs+εf(t)(ts)2dt2f(s)ε},s(a,b), $ (1.4)

    has been paid much attention recently.

    Hypersingular integrals frequently arise in boundary element methods and various physical problems, including fracture mechanics, elasticity, acoustics, and electromagnetic scattering [31]. Numerous methods have been devised to handle these integrals, such as Gaussian quadrature [9,10], Newton-Cotes rules [3,12,16,17,19,25,26,27,28], transformation methods [5,7,11], and others [2,4,11,24,30]. These studies rigorously analyze trapezoidal-type quadrature formulas for weakly singular, singular (Cauchy principal value), and hypersingular integrals, providing full asymptotic expansions for error analysis [33,34,35,36,37]. Reference [38] explored product integration using a variant of the generalized Euler-Maclaurin summation formula, while [39] applied this formula to study the convergence of weakly singular Fredholm and Volterra integral equations. Hypersingular integral equations in potential theory was studied in [40], and modal analysis of a submerged elastic disk was studied by [43]. Superconvergence results for the hypersingular integral equation was presented in [41]. Singular integral operators [45], finite-part integrals of highly oscillatory functions [42], and hypersingular integrals [44] have been paid much attention in recent years.

    The extrapolation method[13,21,22] is based on the error function as

    $ T(h)-a_0 = a_1h^2+a_2h^4+a_3h^6+\cdots, $

    where $ T(0) = a_0 $ and $ a_j $ are constants independent of $ h $. In [14], a generalized trapezoidal rule for numerical computation of hypersingular integrals on intervals was introduced, with asymptotic error expansion proven. In [15], the composite trapezoidal rule for Hadamard finite-part integrals with the hypersingular kernel $1/\sin^2(x-s)$ was discussed, obtaining the main part of the asymptotic error expansion. In [18], two extrapolation algorithms were presented for hypersingular integrals on intervals, proving their convergence rates, which match those of classical middle rectangle rule approximations. In [20], an extrapolation algorithm for supersingular integrals was introduced.

    Before presenting our main results, we first let

    $ \phi_{ik}(t) = \left\{ 1(ki)!11=Mik(τ,t)(τt)2dτ,|t|<1,1(ki)!11Mik(τ,t)(τt)2dτ,|t|>1, \right. $ (1.5)

    where $ \tau\in[-1, 1] $ and

    $ Mik(τ,t)=τi(τt)ki=Fi(τ)(τt)ki. $ (1.6)

    Set $ J : = (-\infty)\bigcup(-1, 1)\bigcup(1, \infty) $, $ W : C(J)\rightarrow C(-1, 1) $ and let

    $ Tik(τ):=ϕik(τ)+j=1ϕik(2j+τ)+j=1ϕik(2j+τ), $ (1.7)

    be the linear operator.

    Compared with Gaussian quadrature and Newton-Cotes rules, the extrapolation method is more effective. In the following, the asymptotic error expansion of the middle rectangle rule for Hadamard finite-part integrals on a circle is given, based on the asymptotic error expansion, and an algorithm to solve the hypersingular integrals equation is presented. The asymptotic error expansion is

    $ En(f,s)=i=0hi2i+1f(i+1)(s)ai(τ), $ (1.8)

    where $ a_i(\tau) $ are functions independent of $ h $ defined as

    $ ai(τ)={ik=i1(1)k(k+1)!Tki(τ),i1,12Tki(τ),i=0. $ (1.9)

    This paper provides the asymptotic error expansion of the middle rectangle rule for Hadamard finite-part integrals on circles. Based on this expansion, an algorithm for solving hypersingular integral equations is proposed. To avoid computing $a_i(\tau)$, an extrapolation algorithm is suggested for a given $\tau$. A series of $s_j$ values approximate the singular point $s$ with mesh refinement. Using extrapolation techniques, we achieve higher-order accuracy and obtain a posteriori error estimates. Additionally, a collocation scheme is constructed to solve hypersingular integral equations via extrapolation methods, with proven convergence rates.

    Let $ c = t_0 < t_1 < \cdots < t_{n-1} < t_{n} = c+2\pi $ be a uniform partition of the interval $ [c, c+2\pi] $ with mesh size $ h = 2\pi/n $ and set

    $ ti=t0+(i1)h,i=1,2,,n, $ (2.1)
    $ ˆti=ti+h2,i=1,2,,n. $ (2.2)

    We define $ f_C(t) $, the constant interpolation for $ f(t) $, as

    $ fC(t)=f(ˆti),  i=1,,n, $ (2.3)

    and a linear transformation

    $ t=ˆti(τ):=(τ+1)(ti+1ti)/2+ti,i=1,,n1,  τ[1,1], $ (2.4)

    from the reference element $ [-1, 1] $ to the subinterval $ [t_{i}, t_{i+1}] $.

    Replacing $ f(t) $ in (1.1) with $ f_{C}(t) $ produces the composite rectangle rule:

    $ In(f,s):=c+2πc=   fC(t)sin2ts2dx=ni=1ωi(s)f(ti)=I(f;s)En(f,s), $ (2.5)

    where $ \omega_i(s) $ denotes the Cotes coefficients given by

    $ ωi(s)=2cot(ti1s2)2cot(tis2) $ (2.6)

    and $ E_{n}(f, s) $ is the error functional.

    Let

    $ γ(τ)=γ(h,s)=min1in|sti|h=1|τ|2 $ (2.7)

    and

    $ In,i(s)={titi1tˆtisin2ts2dt,im,tmtm1=      tˆtmsin2ts2dt,i=m. $ (2.8)

    We also define

    $ Fi(τ)=τi $ (2.9)

    and

    $ ϕi,i+1(t)={1211  Fi(τ)τtdτ,|t|<1,1211Fi(τ)τtdτ,|t|>1. $ (2.10)

    If $ F_{i}(\tau) $ is replaced by the Legendre polynomial, it is known that $ \phi_{i, i+1} $ defines the Legendre function of the second kind [1]. Let

    $ ϕii(t)={1211= Fi(τ)(τt)2dτ,|t|<1,1211Fi(τ)(τt)2dτ,|t|>1 $ (2.11)

    and

    $ ϕik(t)=1211Fi(τ)(τt)ki2(ki)!dτ,  k>i+1. $ (2.12)

    Define $ K_s(t) $ as

    $ Ks(t)={(ts)2sin2ts2ts,4,t=s. $ (2.13)

    Theorem 2.1. Assume $ f(t) \in C^{\infty}[c, c+2\pi] $. For the middle rectangle rule $ I_n(f; s) $ defined in (2.5) and $ a_i(\tau) $ defined in (1.9), we have

    $ En(f,s)=i=0hi2i+1f(i+1)(s)ai(τ) $ (2.14)

    where $ s = t_{m-1}+(1+\tau)h/2, \ \ m = 1, 2, \cdots, n $.

    Then the error function can be written as

    $ En(f,s)=f(s)2a0(τ)+hf(s)22a1(τ)++hi2i+1f(i+1)(s)ai(τ)+ $ (2.15)

    From the error functional, we know that it is not convergence when the first term is $ a_{0}(\tau) $. By the first part of $ a_{0}(\tau) = 0 $, we have the convergence rate $ O(h) $ which is called the superconvergence phenomenon.

    Define

    $ Mjik(t,s)=(ttj1/2)i(ts)ki=Fji(t)(ts)ki,  ki, $ (3.1)

    where

    $ Fji(t)=(ttj1/2)i. $ (3.2)

    By (2.4), we have

    $ Mjik(t,s)=hk+22k+2τi(τcj)ki=hk+22k+2Mik(τ,cj)=hk+22k+2Fi(τ)(τcj)ki, $ (3.3)

    where

    $ Mik(τ,cj)=τi(τcj)ki=Fi(τ)(τcj)ki, $ (3.4)
    $ cj=2(stj1)/h1 $ (3.5)

    and $ F_{i}(\tau) $ is defined as in(2.9).

    Lemma 3.1. Let $ K_s(t) $ be defined as in (2.13). For $ t \in (t_{j-1}, t_{j}) $, by linear transformation (2.6), we have

    $ Ks(t)=Kcj(τ),  τ(1,1) $ (3.6)

    where

    $ Kcj(τ)=4+4l=1(τcj)2(τcj2ln)2+4l=1(τcj)2(τcj+2ln)2 $ (3.7)

    and $ c_{j} $ is defined as in (3.5).

    Proof. By the identity in [1],

    $ π2sin2πt=l=l=1(t+l)2, $ (3.8)

    and then we get

    $ 1sin2ts2=4(ts)2+4l=11(ts2lπ)2+4l=11(ts+2lπ)2 $ (3.9)

    and

    $ (ts)2sin2ts2=4+4l=1(ts)2(ts2lπ)2+4l=1(ts)2(ts+2lπ)2. $ (3.10)

    Then

    $ Ks(t)=(ts)2sin2ts2=4+4l=1(τcj)2(τcj4lπ/h)2+4l=1(τcj)2(τcj+4lπ/h)2=4+4l=1(τcj)2(τcj2ln)2+4l=1(τcj)2(τcj+2ln)2=Kcj(τ), $

    which completes the proof.

    Lemma 3.2. (See [18, Lemma 3.2]) Let $ P_{n}(t), n = 0, 1, \cdots $, be the Legendre function [1] defined as

    $ Pn(t)=12n[n2]r=0(1)r(2n2r)!r!(nr)!(n2r)!tn2r. $ (3.11)

    For the polynomial $ t^n, n = 0, 1, \cdots $, it can be expanded by the Legendre function as

    $ tn=n!2n[n2]k=0(2n4k+1)Pn2k(t)k!(32)nk, $ (3.12)

    where $ (a)_{n} = (a)(a+1)\cdots(a+n-1) $.

    Lemma 3.3. Let $ \phi_{i, i+1}(t) $ and $ \phi_{ii}(t) $ be defined as in (2.10) and (2.11), respectively. Then

    $ ϕi,i+1(t)={i1+1j=1ω2j1Q2j1(t),i=2i1,i1j=0ω2jQ2j(t),i=2i11 $ (3.13)

    and

    $ ϕii(t)={Q0(t)+i1j=1ajQ2j(t),i=2i1,Q1(t)+i1j=1bjQ2j1(t),i=2i11, $ (3.14)

    where

    $ ωj=2j+1211Fi(τ)Pj(τ)dτ $ (3.15)

    and

    $ a_{j} = -(4j+1)\sum\limits_{k = 1}^{j}\omega_{2k-1}, $
    $ b_{j} = -(4j-1)\sum\limits_{k = 1}^{j}\omega_{2k-2}. $

    Proof. For $ i = 2i_{1} $,

    $ F_{i}(\tau) = \tau^{i} $

    leads to

    $ Fi(τ)=i1+1j=1ω2j1P2j1(τ), $ (3.16)

    where $ \omega_{2j-1} $ is defined as in (3.15) and $ P_{j}(\tau) $ are Legendre polynomials. The first part of (3.13) follows immediately from the definition of $ \phi_{i, i+1}(\tau) $. Since

    $ \sum\limits_{j = 1}^{i_1+1}\omega_{2j-1} = \sum\limits_{j = 1}^{i_1+1}\omega_{2j-1}P_{2j-1}(1) = F_{i}(1) = 1, $

    then we have

    $ \phi_{i, i+1}(t) = \sum\limits_{j = 1}^{i_1}\frac{a_{j}}{4j+1}[Q_{2j+1}(t)-Q_{2j-1}(t)] $

    with $ a_{j} = -(4j+1)\sum_{k = 1}^{j}\omega_{2k-1} $, which leads to the first part of (3.14) by using the recurrence relation [1].

    $ Pl+1(t)Pl1(t)=(2l+1)Pl(t),  l=1,2,, $ (3.17)

    which completed the proof.

    Lemma 3.4. Assume $ s \in (t_{m-1}, t_{m}) $, for some $ m $, $ c_{j} $ defined as in (3.5). Then, we have

    $ ψik(cj)={2khk+1tmtm1=      Mjik(t,s)sin2ts2dt,j=m,2khk+1tjtj1Mjik(t,s)sin2ts2dt,jm. $ (3.18)

    Proof. By the equation of (1.2),

    $ tmtm1=      Mjik(t,s)sin2ts2dt $ (3.19)
    $ =tmtm1=      Fmi(t)Ks(t)(ts)2dt=limε0{sεtm1Fmi(t)Ks(t)(ts)2dt+tms+εFmi(t)Ks(t)(ts)2dt2Fmi(s)Ks(s)ε}=hk+12k+1limε0{(cm2εh1+1cm+2εh)Fi(τ)Ks(τ)(τcm)2dτhFi(cm)Ks(cm)ε}=hk+12k+111= Fi(τ)Ks(τ)(τcm)2dτ=hk+12kψii(cm). $ (3.20)

    The second identity can be obtained similarly.

    Lemma 3.5. Under the assumption of Lemma 3.4, then we have

    $ ψik(cj)={2khk+1tmtm1=      Mmik(t,s)sin2ts2dt,j=m,2khk+1tjtj1Mjik(t,s)sin2ts2dt,jm. $ (3.21)

    Proof. We have:

    $ tmtm1=      Mmik(t,s)sin2ts2dt=tmtm1       Fmi(t)Ks(t)tsdt=limε0{sεtm1Fmi(t)Ks(t)tsdt+tms+εFmi(t)Ks(t)tsdt}=hk+12k+1limε0{(cmε1+1cm+ε)Fi(τ)Ks(τ)τcmdτ}=hk+12k+111 Fi(τ)Ks(τ)τcmdτ=hk+12kψi,i+1(cm). $ (3.22)

    The second part of this Lemma 3.4 can be obtained similarly.

    Lemma 3.6. For $ k > i+1 $, it holds that

    $ ψik(cj)=2khk+1tjtj1=   Mjik(t,s)(ki)!sin2ts2dt. $ (3.23)

    Proof. The proof of (3.23) can be obtained similarly to Lemma 3.4 or Lemma 3.5.

    Lemma 3.7. Suppose $ f(t) \in C^{\infty}[c, c+2\pi] $. If $ s\neq t_j $, for any $ j = 1, 2, \cdots, n $, then it holds that

    $ f(t)fC(t)=i=0k=i(1)i+1f(k)(s)i!Mjik(t,s)(ki)!. $ (3.24)

    Proof. By taking the Taylor expansion for $ f(t_{j-1/2}) $, we have

    $ f(t_{j-1/2}) = f(t)+\sum\limits_{i = 0}^{\infty}\frac{f^{(i)}(t)}{i!}(t_{j-1/2}-t)^{i} $

    and thus,

    $ f(t)fC(t)=i=1(1)i+1f(i)(t)i!(ttj1/2)i $ (3.25)

    and

    $ f(i+1)(t)=k=if(k)(s)(ts)ki(ki)!. $ (3.26)

    Combining (3.25) and (3.26) leads to (3.24).

    Define

    $ Hj(t)=f(t)fC(t)i=0k=i(1)i+1f(k+1)(s)h(i+1)!Mjik(t,s)(ki)!,  t(tj1,tj). $ (3.27)

    Proof.

    $ (tm1c+c+2πtm  )f(t)fC(t)sin2ts2dt=nj=1,jmtjtj1f(t)fC(t)sin2ts2dt=i=0k=i(1)i+1f(k+1)(s)h(i+1)!(ki)!nj=1,jmtjtj1Mjik(t,s)sin2ts2dt. $ (3.28)

    By (3.27) of $ \mathcal{H}_j(t) $, we have

    $ tmtm1=      f(t)fC(t)sin2ts2dt=tmtm1=      Hm(t)sin2ts2dt+i=0k=i(1)i+1f(k+1)(s)h(i+1)!(ki)!tmtm1=      Mmik(t,s)sin2ts2dt. $ (3.29)

    Putting (3.28) and (3.29) together yields

    $ c+2πc=     f(t)fC(t)sin2ts2dt=i=1k=i(1)i+1hkf(k+1)(s)(i+1)!(ki)!nj=1ψik(τ)=i=0hi2i+1f(i+1)(s)ai(τ) $ (3.30)

    where

    $ ai(τ)={ik=i1(1)k+1(k+1)!Tki(τ),i>0,12Tki(τ),i=0. $ (3.31)

    The proof is complete.

    For $ i = 1 $,

    $ a0(τ)=ni=1titi1=   tˆtisin2ts2dt=2hk=1ni=1{sin[k(tis)]+sin[k(ti1s)]}4k=11kni=1{cos[k(tis)]cos[k(ti1s)]}=4hk=1ni=1sin[k(tis)]=4hj=1nsin[nj(t1s)]=8πj=1sin[j(1+τ)π]=4πcot(1+τ)π2=4πtanτπ2, $ (3.32)

    where

    $ \sum\limits_{i = 1}^n \sin \left \lbrack k (t_i-s)\right \rbrack = \left \lbrace nsin[k(t1s)],k=nj,0,otherwise\right. $

    has been used. When $ \tau = 0 $, we get $ a_{0}(\tau) = 0 $ [6]. This is not the same as the case with a singular point located on the interval. The reason is that on the circle each subinterval is equal so we just consider the case of $ s $ located at the middle of the interval.

    At last, we suggest the modified composite middle rectangle rule, denoting by $ \tilde{I}_{n}(f, s) $, defined by

    $ ˜In(f,s)=In(f,s)4πf(s)tanτπ2. $ (3.33)

    Following asymptotic expansion,

    $ En(f,s)=i=0hi2i+1f(i+1)(s)ai(τ). $ (4.1)

    For the given $ s $ and positive integer $ n_0 $, we have

    $ m_0: = \frac{n_0(s-c)}{2\pi}. $

    First, we partition $ [c, c+2\pi] $ into $ n_0 $ equal subintervals to get a mesh denoted by $ \Pi_1 $ with mesh size $ h_1 = 2\pi/n_0 $. Then $ \Pi_1 $ is refined to get mesh $ \Pi_2 $ with mesh size $ h_2 = h_1/2 $. In this way, a series of meshes $ \{\Pi_j\}(j = 1, 2, \cdots) $ is obtained in which $ \Pi_j $ is refined from $ \Pi_{j-1} $ with mesh size denoted by $ h_j $. The extrapolation scheme is presented in Table 1.

    Table 1.  Extrapolation scheme of $T_i^{(j)}$.
    $T(h_{1})=T_1^{(1)}$
    $T(h_{2})=T_1^{(2)}$ $T_2^{(1)}$
    $T(h_{3})=T_1^{(3)}$ $T_2^{(2)}$ $T_3^{(1)}$
    $T(h_{4})=T_1^{(4)}$ $T_2^{(3)}$ $T_3^{(2)}$ $T_4^{(1)}$
    $T(h_{5})=T_1^{(5)}$ $T_2^{(4)}$ $T_3^{(3)}$ $T_4^{(2)}$ $T_5^{(1)}$
    $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$

     | Show Table
    DownLoad: CSV

    Define

    $ sj=s+τ+12hj,  j=1,2,, $ (4.2)

    and

    $ T(hj)=I2j1n0(f,sj). $ (4.3)

    The extrapolation algorithm takes the form of:

    $ Step one: Compute  T(j)1=T(hj),  j=1,,m. $
    $ Step two:Compute  T(j)i=T(j+1)i1+T(j+1)i1T(j)i12i11,  i=2,,m,    j=1,,mi. $

    Theorem 4.1. Under the same condition of Theorem 2.1, for a given $ \tau $ and (4.2), we have

    $ |I(f,s)T(j)i|Chi $ (4.4)

    and

    $ \left|\frac{T_{i-1}^{(j+1)}-T_{i-1}^{(j)}}{2^{i-1}-1}\right|\leq Ch^{i-1}. $

    Proof. By Eq (4.1), we get

    $ I(f,s)T(hj)=I(f,s)I(f,sj)+I(f,sj)T(hj)=I(f,s)I(f,sj)+i=0hij2iai(τ)f(i+1)(sj). $ (4.5)
    $ I(f;sj)=I(f;s)+I(f;s)τ+12hj+I(f;s)(τ+12hj)2++I(l2)(f;s)(τ+12hj)l2+, $ (4.6)

    and

    $ f(i+1)(sj)=f(i+1)(s)+f(i+2)(s)τ+12hj+f(i+3)(s)2!(τ+12hj)2++f(l)(s)(li1)!(τ+12hj)li1+. $ (4.7)

    Putting (4.5)–(4.7) together, we have

    $ I(f,s)T(hj)=i=0bi(s,τ)hij, $ (4.8)

    where

    $ bi(s,τ)=f(i+1)(s)ik=1ak(τ)2k(τ+12)ik1(ik)!(τ+1)i2ii!I(i)(f,s), $ (4.9)

    where $ b_i(s, \tau) $ is a constant for a given $ \tau $. By (4.8), we also have

    $ I(f,s)T(hj+1)=i=0bi(s,τ)hij+1. $ (4.10)

    By (4.8) and (4.10), with $ h_j = 2h_{j+1} $, we have

    $ I(f,s)=2T(hj+1)T(hj)+i=2bi(s,τ)(12i11)hij+O(hl1j)=T(j)2+i=2bi(s,τ)(12i11)hij, $ (4.11)

    which implies

    $ I(f,s)T(j)2=i=2bi(s,τ)(12i11)hij $ (4.12)

    and

    $ T(j)2=2T(hj+1)T(hj). $ (4.13)

    The accuracy $ O(h^{3}) $ can be obtained after performing the extrapolation process.

    In order to directly use the error expansion of (4.1), we presented the following the new partition as $ c = t_{00} < t_{01} < \cdots < t_{0, n-1} < t_{0n} = c+2\pi $ with $ t_{0i} = t_{i}+\frac{h}{2}, i = 0, 1, \cdots, n-1 $ and

    $ f0C(t)=f(t0i). $ (4.14)

    Then we have the approximate formula:

    $ ˜In(f;s):=c+2πc=    f0C(t)sin2ts2dt $ (4.15)
    $ =nj=0˜ωj(s)f(tj)=c+2πc=   f(t)sin2ts2dt˜En(f,s), $ (4.16)

    and get

    $ ˜ωi(s)={2cot(t0s2)2cot(t01s2),i=0,2cot(t0is2)2cot(t0,i+1s2),0<i<n,2cot(t0ns2)2cot(tn+1s2),i=n, $ (4.17)

    and then we find the following theorem similarly as with Theorem 2.1.

    Theorem 4.2. Assume $ f(t) \in C^{\infty}[c, c+2\pi] $. For the rectangle rule $ \tilde{I}_n(f; s) $ defined in (4.15) and $ a_{2i+1}(\tau) $ defined in (1.9), independent of $ h $ and $ s $, such that

    $ ˜En(f,s)=c+2πc=     f(t)f0C(t)sin2ts2dt=i=0h2i22i+1f(i+1)(s)a2i+1(τ) $ (4.18)

    where $ s = t_{m-1}+(1+\tau)h/2, \ \ m = 1, 2, \cdots, n $.

    Similarly as extrapolation scheme $ T(h_j) $, new extrapolation scheme is presented in Table 2.

    $ ˜T(hj)=˜I2j1n0(f,sj). $ (4.19)
    Table 2.  Extrapolation scheme of $\tilde{T}_i^{(j)}$.
    $\tilde{T}(h_{1})=\tilde{T}_1^{(1)}$
    $\tilde{T}(h_{2})=\tilde{T}_1^{(2)}$ $\tilde{T}_2^{(1)}$
    $\tilde{T}(h_{3})=\tilde{T}_1^{(3)}$ $\tilde{T}_2^{(2)}$ $\tilde{T}_3^{(1)}$
    $\tilde{T}(h_{4})=\tilde{T}_1^{(4)}$ $\tilde{T}_2^{(3)}$ $\tilde{T}_3^{(2)}$ $\tilde{T}_4^{(1)}$
    $\tilde{T}(h_{5})=\tilde{T}_1^{(5)}$ $\tilde{T}_2^{(4)}$ $\tilde{T}_3^{(3)}$ $\tilde{T}_4^{(2)}$ $\tilde{T}_5^{(1)}$
    $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$

     | Show Table
    DownLoad: CSV

    Theorem 4.3. Under the asymptotic expansion of Theorem 4.2, we have

    $ |I(f,s)˜T(j)i|Ch2i $ (4.20)

    and the posteriori asymptotic error estimate is given by

    $ \left|\frac{\tilde{T}_{i-1}^{(j+1)}-\tilde{T}_{i-1}^{(j)}}{4^{i-1}-1}\right|\leq Ch^{2i-2}. $

    The proof of Theorem 4.3 can be similarly obtained as with Theorem 4.1, so we omitted it here.

    Consider

    $ 14π2π0=   f(t)sin2ts2dt=g(s),  s(0,2π), $ (5.1)

    with

    $ 2π0g(t)dt=0. $ (5.2)

    As in [29], under the compatibility condition (5.2), there exists a unique solution up to an additive constant for (5.1). With periodical condition

    $ 2π0f(t)dt=0, $ (5.3)

    there is a unique solution of (5.1).

    Applying $ \tilde{I}_{n}(f, s) $ to approximate the hypersingular integral in (5.1) and using $ \hat{t}_k = t_{k-1}+h/2(k = 1, 2, \cdots, n) $ of each subinterval to be a collocation point, we get

    $ 12πnm=1(cotˆtktm2cotˆtktm12)fm=g(ˆtk),  k=1,2,,n, $ (5.4)

    denoted by

    $ AnFan=Gen, $ (5.5)

    where

    $ An=(akm)n×n,akm=12π(cotˆtkxm2cotˆtkxm12),k,m=1,2,,n,Fan=(f1,f2,,fn)T,Gen=(g(ˆt1),g(ˆt2),,g(ˆtn))T, $ (5.6)

    and $ f_k = f(\hat{t}_k)(k = 1, 2, \cdots, n) $. Obviously, $ \mathcal{A}_n $ is a circulant matrix and also a symmetric Toeplitz matrix. Since,

    $ nm=1akm=12πnm=1(cotˆtktm2cotˆtktm12)=0, $ (5.7)

    we see that $ \mathcal{A}_n $ is singular.

    The regularizing factor $ \gamma_{0n} $ (see Reference [23]) in (5.4) is introduced, which leads to

    $ {γ0n+12πnm=1(cotˆtktm2cotˆtktm12)fm=g(ˆtk),k=1,2,,n,nm=1fm=0, $ (5.8)

    where $ \gamma_{0n} $ has the form

    $ γ0n=12πnk=1g(ˆtk)h. $ (5.9)

    System (5.8) is denoted as

    $ An+1Fan+1=Gen+1, $ (5.10)

    where

    $ An+1=(0eTnenAn),Fan+1=(γ0nFan),Gen+1=(0Gen), $ (5.11)

    and $ e_n = (\underbrace{1, 1, \cdots, 1}_{n}) $.

    Linear system (5.8) can be rewritten as

    $ {γ0n+12πnm=1fm+1fmhcotˆtktm2h=g(ˆtk),  k=1,2,,n,12πnm=1fm+1fmhh=0, $ (5.12)

    where $ f_1 = f_{n+1} $ has been used. Let $ v_m = -(f_{m+1}-f_m)/h $, and we get

    $ {γ0n+12πnm=1cotˆtktm2vmh=g(ˆtk),k=1,2,,n,12πnm=1vmh=0. $ (5.13)

    The following lemma has been proved in [23].

    Lemma 5.1. ([23, Theorem 6.2.1, §6.2, Chapter 6]) For Eq (5.13), its solution is

    $ vm=h2πnk=1cottmˆtk2f(ˆtk). $ (5.14)

    Lemma 5.2. ([6, Lemma 4.2]) Let $ \mathcal{B}_{n+1} = (b_{ik})_{(n+1)\times (n+1)} $ be the inverse matrix of $ \mathcal{A}_{n+1} $, defined in (5.10). Then,

    (1) $ \mathcal{B}_{n+1} $ is expressed as

    $ Bn+1=(b00B1 B2Bn), $ (5.15)

    where

    $ B1=(b01,b02,,b0n),B2=(b10,b20,,bn0)T, $ (5.16)
    $ bi0=b0i=1n,1in, $ (5.17)
    $ bik=h22π[n1m=icotˆtktm21nn1m=1mcotˆtktm2], $ (5.18)
    $ 1in1,1kn,bnk=h22nπn1m=1mcotˆtktm2,1kn. $ (5.19)

    (2) $ \mathcal{B}_n $ is a Toeplitz and also circulant matrix.

    (3) for $ i = 1, 2, \cdots, n $, we have

    $ nk=1|bik|C. $ (5.20)

    Circulant matrices have unique advantages in matrix operations, such as matrix multiplication. It can greatly save the calculation time and improve the calculation efficiency. The circular matrix only needs to store the elements of the first row (or the first column) to completely determine the entire matrix, because the remaining row (or column) elements are obtained from the cyclic shift of the first row (or column) elements. Circulant matrices have many good mathematical properties, such as the fact that their eigenvalues and eigenvectors can be calculated analytically, which makes it easier to analyze and deal with circulant matrices than general matrices in solving linear equations and eigenvalue problems.

    We present the main theorem of this section.

    Theorem 5.1. The solution of linear system (5.8) or (5.10)is

    $ f(ˆti)fi=i=0h2i22i+1f(i+1)(s)a2i+1(τ). $ (5.21)

    Proof. Let $ {\bf F}_{n+1}^e = (0, f(\hat{t}_1), f(\hat{t}_2), \cdots, f(\hat{t}_n))^T $ be the exact vector. Then, from (5.10), we get

    $ Fen+1Fan+1=Bn+1(An+1Gen+1Gen+1), $ (5.22)

    which implies

    $ f(ˆti)fi=bi0nm=1g(ˆtm)+nk=1bik˜En(f,ˆtk)=12πnm=1g(ˆtm)h+nk=1bik˜En(f,ˆtk), $ (5.23)

    where $ \{b_{ik}\} $ are the entries of $ \mathcal{B}_{n+1} $ and $ \tilde{E}_{n}(f, s) $ is defined in (4.15).

    For the first part of $ \frac1{2\pi} \sum_{m = 1}^n g(\hat{t}_m) h $ of the rectangle rule to compute the definite integral $ \frac1{2\pi} \int_{0}^{2\pi} g(t) dt $.

    Then we have

    $ f(ˆti)fi=nk=1bik˜En(f,ˆtk)=nk=1biki=0h2i22i+1f(i+1)(ˆtk)a2i+1(τ)=i=0h2i22i+1a2i+1(τ)nk=1bikf(i+1)(ˆtk). $ (5.24)

    By (5.24) and (5.20), we obtain

    $ b_{ik} = \frac{c}{n}+O(h^3) $

    and

    $ \sum\limits_{k = 1}^nb_{ik}f^{(i+1)}(\hat{t}_k) = C+O(h^2). $

    Then we have

    $ f(ˆti)fi=i=0h2i22i+1a2i+1(τ), $

    where (5.19) and (5.20) has been used.

    Example 6.1. We consider (1.1) with density function $ f(t) = 1+2\cos(t)+ 2\cos(2t), c = -\pi, c+2\pi = \pi $.

    Numerical results are presented in Table 3 for $ s = t_{[n/4]}+(1+\tau)h/2 $. The right half of Table 3 shows that the accuracy of $ \tilde{I}_{n}(s, f) $ is always $ O(h^2) $. However, from the left half, the accuracy is $ O(h^2) $ if $ s $ is located at the superconvergence point $ (\tau = 0) $, and if $ \tau\ne 0 $, the middle rectangle rule is divergent.

    Table 3.  Errors of $I_{n}(s, f)$ and $\tilde{I}_{n}(s, f)$ with $s = t_{[n/4]}+(1+\tau)h/2$.
    $I_{n}(f, s)$ $\hat{I}_{n}(f, s)$
    $n$ $\tau=0$ $\tau=2/3$ $\tau=-2/3$ $\tau=-2/3$ $\tau=2/3$
    $32$ 5.3695e-004 -1.2446e+000 1.0651e+000 1.8350e-003 -7.6074e-004
    $64$ 1.2851e-004 -1.1295e+000 1.0421e+000 4.5684e-004 -1.9980e-004
    $128$ 3.1416e-005 -1.0741e+000 1.0310e+000 1.1397e-004 -5.1140e-005
    $512$ 7.7655e-006 -1.0470e+000 1.0256e+000 2.8464e-005 -1.2933e-005
    $1024$ 1.9303e-006 -1.0336e+000 1.0229e+000 7.1123e-006 -3.2516e-006

     | Show Table
    DownLoad: CSV

    Example 6.2. We still consider $ f (t) = 1+2\cos(t)+ 2\cos(2t), c = -\pi, c+2\pi = \pi $. For $ s = -\pi/2 $, and $ g(s) = 5.026548245744136e+001 $, we use $ s = t_{[n/4]}+(\tau+1) h/2 $ with $ \tau = 0 $ to compute $ s = -\pi/2 $.

    For $ s = 0 $ and $ g(s) = -7.539822368615504e+01 $, we also use $ s = t_{[n/2]}+(\tau+1) h/2 $ with $ \tau = 0 $ to compute $ s = 0 $.

    In Tables 4 and 6, we present the error of the middle rectangle rule as $ h^2 $, $ h^{4} $, $ h^{6} $, and $ h^{8} $, respectively. In Tables 5 and 7, the convergence rates $ h^2 $, $ h^{4} $, and $ h^{6} $ also agree with the theoretic analysis of Theorem 4.2.

    Table 4.  Error estimate of the mid-rectangle rule $s = -\pi/2$.
    $0$ $h^4-$extra $h^6-$extra $h^8-$extra
    $8$ 5.0106e+000
    $16$ 1.2820e+000 1.9666 3.9121e-002
    $32$ 3.2236e-001 1.9917 2.4790e-003 3.9801 3.6167e-005
    $64$ 8.0707e-002 1.9979 1.5547e-004 3.9950 5.6990e-007 5.9878 4.8600e-009
    $128$ 2.0184e-002 1.9995 9.7253e-006 3.9988 8.9188e-009 5.9977 1.4339e-011 8.4156
    $256$ 5.0464e-003 1.9999 6.0796e-007 3.9997 1.3873e-010 6.0088 -6.3238e-013 4.0491

     | Show Table
    DownLoad: CSV
    Table 5.  A posteriori estimate of the mid-rectangle rule $s = -\pi/2$.
    $0$ $h^4-$extra $h^6-$extra $h^8-$extra
    $8$
    $16$ 1.2429e+000
    $32$ 3.1988e-001 1.9581 2.4428e-003
    $64$ 8.0551e-002 1.9896 1.5490e-004 3.9791 5.6504e-007
    $128$ 2.0174e-002 1.9974 9.7164e-006 3.9948 8.9044e-009 5.9877 1.9003e-011
    $256$ 5.0458e-003 1.9993 6.0782e-007 3.9987 1.3937e-010 5.9976 5.8710e-014 8.3268

     | Show Table
    DownLoad: CSV
    Table 6.  Error estimate estimate of the mid-rectangle rule $s = 0$.
    $0$ $h^4-$extra $h^6-$extra $h^8-$extra
    $8$ -5.6517e+00
    $16$ -1.4432e+00 1.9694 -4.0361e-02
    $32$ -3.6271e-01 1.9924 -2.5567e-03 3.9806 -3.6452e-05
    $64$ -9.0799e-02 1.9981 -1.6033e-04 3.9951 -5.7436e-07 5.9879 -4.8673e-09
    $128$ -2.2707e-02 1.9995 -1.0029e-05 3.9988 -9.0075e-09 5.9947 -3.3666e-11 7.1899
    $256$ -5.6773e-03 1.9999 -6.2699e-07 3.9996 -1.6669e-10 5.7493 -2.6361e-11 -

     | Show Table
    DownLoad: CSV
    Table 7.  A posteriori estimate of the mid-rectangle rule $s_j = s+(\tau+1) h_j/2$.
    $0$ $h^4-$extra $h^6-$extra $h^8-$extra
    $8$
    $16$ -1.4028e+00
    $32$ -3.6016e-01 1.9616 -2.5203e-03
    $64$ -9.0638e-02 1.9904 -1.5976e-04 3.9796 -5.6949e-07
    $128$ -2.2697e-02 1.9976 -1.0020e-05 3.9949 -8.9738e-09 5.9878 -1.8955e-11
    $256$ -5.6766e-03 1.9994 -6.2682e-07 3.9987 -1.4033e-10 5.9990 -2.8645e-14 9.6078

     | Show Table
    DownLoad: CSV

    In Tables 8 and 10, we present the error of the middle rectangle rule as $ h $, $ h^{2} $, $ h^{3} $, and $ h^{4} $, respectively. In Tables 9 and 11, the convergence rates $ h $, $ h^{2} $, and $ h^{3} $ also agree with the theoretic analysis of Theorem 2.1.

    Table 8.  Error estimate of the mid-rectangle rule $s = -\pi/2$.
    $0$ $h^2-$extra $h^3-$extra $h^4-$extra
    $32$ 8.1836e+00
    $64$ 3.7710e+00 1.1178 -6.4164e-01
    $128$ 1.7978e+00 1.0687 -1.7536e-01 1.8714 -1.9940e-02
    $256$ 8.7613e-01 1.0370 -4.5544e-02 1.9450 -2.2699e-03 3.1350 2.5439e-04
    $512$ 4.3227e-01 1.0192 -1.1587e-02 1.9747 -2.6881e-04 3.0779 1.7053e-05 3.8989
    $1024$ 2.1467e-01 1.0098 -2.9214e-03 1.9879 -3.2661e-05 3.0410 1.0752e-06 3.9874

     | Show Table
    DownLoad: CSV
    Table 9.  A posteriori estimate of the mid-rectangle rule $s_j = s+(\tau+1) h_j/2$.
    $0$ $h^2-$extra $h^3-$extra $h^4-$extra
    $32$
    $64$ 4.4126e+00
    $128$ 1.9732e+00 1.1611 -1.5542e-01
    $256$ 9.2167e-01 1.0982 -4.3274e-02 1.8447 -2.5243e-03
    $512$ 4.4386e-01 1.0542 -1.1319e-02 1.9348 -2.8587e-04 3.1425 1.5822e-05
    $1024$ 2.1760e-01 1.0284 -2.8887e-03 1.9702 -3.3736e-05 3.0830 1.0652e-06 3.8927

     | Show Table
    DownLoad: CSV
    Table 10.  Error estimate of the mid-rectangle rule $s = -\pi$.
    $0$ $h^2-$extra $h^3-$extra $h^4-$extra
    $32$ -4.5518e+00
    $64$ -1.9715e+00 1.2071 6.0874e-01
    $128$ -9.0311e-01 1.1263 1.6531e-01 1.8807 1.7498e-02
    $256$ -4.3016e-01 1.0700 4.2802e-02 1.9494 1.9665e-03 3.1535 -2.5230e-04
    $512$ -2.0964e-01 1.0369 1.0874e-02 1.9768 2.3104e-04 3.0894 -1.6885e-05 3.9013
    $1024$ -1.0345e-01 1.0190 2.7394e-03 1.9889 2.7950e-05 3.0472 -1.0634e-06 3.9890

     | Show Table
    DownLoad: CSV
    Table 11.  A posteriori estimate of the mid-rectangle rule $s_j = s+(\tau+1) h_j/2$.
    $0$ $h^2-$extra $h^3-$extra $h^4-$extra
    $32$
    $64$ -2.5803e+00
    $128$ -1.0684e+00 1.2720 1.4781e-01
    $256$ -4.7296e-01 1.1757 4.0835e-02 1.8559 2.2188e-03
    $512$ -2.2051e-01 1.1008 1.0643e-02 1.9400 2.4793e-04 3.1618 -1.5694e-05
    $1024$ -1.0619e-01 1.0542 2.7115e-03 1.9727 2.9013e-05 3.0951 -1.0548e-06 3.8952

     | Show Table
    DownLoad: CSV

    Example 6.3. In this example, we will not only test our method on a hypersingular integral, but also solve a hypersingular integral equation $ g(s) = -2\cos(2s)- 2\sin(2s), c = -\pi, c+2\pi = \pi $, while the analysis solution is $ f(t) = \cos(2t)+ \sin(2t) $. The good numerical results indicate that our quadrature is efficient and accurate, which matches the theoretical analysis.

    In Table 12, we list the numerical results at $ s = 1.45122657606971 $. Numerical results show that the convergence rate of the hypersingular integral equation has the order $ O(h^2) $. With extrapolation methods, the order can reach $ O(h^4) $, $ O(h^6) $, and $ O(h^8) $ in Table12.

    Table 12.  Error estimate of a hypersingular integral equation.
    $h^2-$ $h^4-$extra $h^6-$extra $h^8-$extra
    $16$ 3.70E-02
    $32$ 9.13E-03 1.67E-04
    $64$ 2.27E-03 1.03E-05 -1.70E-07
    $128$ 5.68E-04 6.39E-07 -2.61E-09 4.19E-11
    $256$ 1.42E-04 3.99E-08 -4.06E-11 1.72E-13

     | Show Table
    DownLoad: CSV

    The article mainly focuses on a class of hypersingular integral equations in the boundary element method. A solution centered on the extrapolation algorithm is proposed, which is based on the asymptotic expansion of the error of the composite rectangle rule. By leveraging the asymptotic characteristics of the error of the composite rectangle rule, an extrapolation algorithm is constructed, providing a new approach to solving hypersingular integral equations. The algorithm is verified through in-depth theoretical and numerical examples. It is confirmed that the main part of the error function of the composite rectangle rule has an asymptotic expansion. Numerical experiments show that special functions significantly affect the convergence rate of the algorithm. This discovery has important guiding significance for subsequent algorithm optimization and improvement of computational efficiency. The adoption of this algorithm has two prominent advantages. First, it can obtain high-precision calculation results, meeting the precision requirements of practical applications; second, it is convenient for deriving a posteriori error estimates.

    $ \int_{a}^{b} \hskip-.22in = $ — —- —- hypersingular integrals

    $ \int_{a}^{b} \hskip-.22in - $ — —- —- Cauchy principal integral

    $ {I}_{n}(f; s) $ — —- —- quadrature for hypersingular integrals

    $ {E}_{n}(f; s) $ — —- —- error functional of the quadrature for $ {I}_{n}(f; s) $

    $ \gamma(\tau) $ — —- —- distance of a singular point to the mesh point

    $ \eta(y) $ — —- —- distance of singular point $ s $ to the boundary point

    $ \omega_i(s) $ — —- —- Cote coefficients of $ {I}_{n}(f; s) $

    $ \phi_{ik}(t) $ — —- —- special function of $ {E}_{n}(f; s) $

    $ a_{i}(\tau) $ — —- —- special function in $ {E}_{n}(f; s) $

    $ \gamma_{0n} $ — —- —- the regularization factor

    $ T(h_j) $ — —- —- extrapolation value

    $ \mathcal{H}_j(t) $ — —- —- errors of

    $ Hj(t)=f(t)fC(t)i=0k=i(1)i+1f(k+1)(s)h(i+1)!Mjik(t,s)(ki)!,  t(tj1,tj), $

    subinterval $ [t_{m}, t_{m+1}] $.

    Qian Ge performed the data analysis; Jin Li performed the formal analysis and wrote the manuscript. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

    The work of Jin Li was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2022MA003).

    The authors declare no conflicts of interest.

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