1.
Introduction
Equilibrium equations in elasticity are classical equations to solve plane problems. There are displacement and stress methods that are used to solve equilibrium equations. The displacement method takes displacement as an unknown quantity, as there is only the displacement component to deduce the equations and boundary conditions. For the stress method, there is only the stress component to deduce the equations and boundary conditions as the unknown quantities.
In the area of in-plane crack problems, heat transfer, nuclear reactor dynamics and so on, the impact of system memory is often dependent on the nonlinear fraction equation and nonlinear time-dependent Burgers' equations. These problems have been studied by using the Galerkin finite element method [1], localized collocation schemes [2] and singular boundary method (SBM) [3]. Lots of numerical methods such as finite element methods (FEM)[4,5,6,7], finite difference methods, spectral method [8,9] and the differential quadrature method and so on are developed to solve plane elastic problems [10,11].
In what follows, we consider the equilibrium equations
where σx,σy and τyx are the stress components.
Geometric equations:
where ϵx,ϵx and τyx are the strain components and u and v are displacement variables.
Constitutive relations of plane stress problem:
Constitutive relations of the plane strain problem:
Displacement boundary equation:
Stress boundary equations:
Combining Eqs (1.4) and (1.2), we have the stress components of displacement:
and the equilibrium equations:
and the displacement boundary equations:
Barycentric formulae have been studied by [12,13,14,15,16,17,18] to avoid the Runge phenomenon. Volterra equations (VE) and Volterra Integro-Differential equations (VIDE) [19,20,21,22,23] have been investigated through the use of linear barycentric collocation methods (LBCMs). The LBCM types include the linear barycentric Lagrange collocation method (LBLCM) and linear barycentric rational collocation method (LBRCM). By comparing the with LBLCM and LBRCM, we can get the error estimate of linear rational barycentric interpolation; then, the convergence rate of the LBRCM can be obtained. Initial value and boundary value problems[24], plane elasticity problems [25], incompressible plane problems [26] and non linear problems [27] have been the focus of barycentric interpolation and rational collocation method in recent years. In previous studies [28,29], heat conduction and telegram equations were solved by LBRCM. In other studies [30,31], biharmonic equation and fractional differential equations were solved by using the LBRCM.
In this paper, first, the polar coordinates of the equilibrium equations are obtained via the transformation of x=ρcosθ,y=ρsinθ. Second, the LBRCM for equilibrium equations is constructed and the matrix equation of the LBRCM is also presented. Third, the convergence rate of LBRCM is proved for the equilibrium equations. At last, some numerical examples are given to validate the proposed theorem.
2.
Polar coordinates of equilibrium equations
In order to get the polar coordinates of the equilibrium equations, let us take x=ρcosθ,y=ρsinθ and (ρ,θ) at some point P(ρ,θ); the displacement components are uρ and uθ, stress components are σρ, σθ and τθr and the physical components are fρ and fθ. The equilibrium equations of the polar coordinates can be represented as:
Geometric equations:
Constitutive relations of plane stress problem:
Combining Eqs (2.1)–(2.3), the displacement of the equilibrium equations for the plane stress problem is expressed as:
The displacement of th stress components is expressed as follows:
where we have used
then we have
and
with the equilibrium condition, we get ϕ(r,θ) below:
and
where ∇2ϕ=(∂2∂ρ2+1ρ∂∂ρ+1ρ2∂2∂θ2) is the Laplace operator.
3.
Collocation method for equilibrium equations
We partition the area [ρa,ρb]×[θ0,θ2π] into ρa=ρ0<ρ1<⋯<ρm=ρb,h=ρb−ρam and [θa,θb] into θ0=θ0<θ1<⋯<θn=θ2π,τ=θ2π−θ0n with [ρa,ρb]×[θ0,θ2π] and (ρi,θj),i=0,1,⋯,m;j=0,1,,⋯,n.
where ri(ρ) and rj(θ) are the barycentric rational interpolation basis functions [24] for ρ and θ, respectively, which can be given as
Ji={k∈Im:i−d1≤k≤i},Im={0,⋯,m−d1}, and
Jj={k∈In: j−d2≤k≤j},In={0,⋯,n−d2}.
Combining Eqs (3.1), (2.4) and (2.5), we get the discrete equilibrium equations, which be expressed as
and
Equations (3.4) and (3.5) can be written in matrix form
and
where ⊗ is the Kronecher product of the matrix and R(0,k)=(R(0,k)ij)m×m, R(k,0)=(R(k,0)ij)n×n,k=1,2, U=[u00,u01,⋯,u0n,u10,u11,⋯,u1n,⋯,um0,um1,⋯,umn]T, Fρ=[f00,f01,⋯,f0n,f10,f11,⋯,f1n,⋯,fm0,fm1,⋯,fmn]T, fij=ρ2if(ρi,θj) and
Taking the notations as
then we have
4.
Convergence and error analysis
Replacing the barycentric rational interpolants of the function u(ρ,θ) with rm,n(ρ,θ) in Eq (3.1), we have
where
Then the error function is defined as
Now we give the theorem as below
Theorem 1. For e(ρ,θ) defined in Eq(4.3) and u(ρ,θ)∈Cd1+2[0,ρ]×Cd2+2[0,θ], we have
{Proof}. For (ρ,θ), the function wi,j(ρ,θ) is defined as Eq (4.2); then, we get
where
see [24].
By the error formula
it follows that
By a similar method of analysis as that of Floater and Kai [15], we have
and
Combining Eqs (4.7)–(4.9) together, the proof of Theorem 1 is completed.
Corollary 1. For e(ρ,θ) defined in (4.3), we have
This corollary can be obtained similarly as Theorem 1, so it is omitted here.
Theorem 2. Let
and
where Ω=[ρa,ρb]×[θ0,θ2π] and g(ρ,θ) is consistent. Then we get
where u(\rho, \theta) \in C^{d_{1}+4}[\rho_a, \rho_b]\times C^{d_{2}+4}[\theta_{0}, \theta_{2\pi}], d_{1} \ge 2, d_{2} \ge 2.
Theorem 3. Let
and
where \Omega = [\rho_a, \rho_b] \times[\theta_{0}, \theta_{2\pi}] g(\rho, \theta) is consistent and
also, \Omega_{kl} = [\rho_k, \rho_{k+1}] \times[\theta_{l} and \theta_{l+1}], \phi(\rho, \theta) \in C^{d_{1}+6}[\rho_a, \rho_b]\times C^{d_{2}+6}[\theta_{0}, \theta_{2\pi}], d_{1} \ge 4, d_{2} \ge 4.
Proof. Let \phi(\rho, \theta) and \phi_{i, j} be the analysis solution and numerical solution of Eq (4.14) respectively:
where
for R_{1}(\rho, \theta) , we have
where
For R_{2}(\rho, \theta) , we have
and
For R_{3}(\rho, \theta) we have
and
Similarly, for R_{4}(\rho, \theta) , R_{5}(\rho, \theta) and R_{6}(\rho, \theta) , we also get
Combining Eqs (4.18) and (4.19)–(4.25), the proof of Theorem 4.3 is completed.
5.
Numerical examples
In the following part, we present some examples to illustrate our numerical scheme analysis.
We define the absolute error estimate and relative error estimate as
and
Example 1. Consider the following elastic polar curved bar bending
and
where A = \frac{P}{2N}, B = -\frac{Pa^2b^2}{2N}, D = -\frac{P}{N}(a^2+b^2), L = \frac{D\pi}{E}, N = a^2-b^2+(a^2+b^2)\ln\frac{a}{n} , K = -\frac{1}{E}\left[D(1-\mu)\ln \rho_{0}+A(1-3\mu)\rho^2_{0}+\frac{B(1+\mu)}{\rho^2_{0}} \right], \rho_{0} = \frac{a+b}{2}, a < \theta < b and 0 < \theta < \frac{\pi}{2} and the boundary conditions are given as \sigma_{\rho}|_{\rho = a} = 0, \sigma_{\rho}|_{\rho = b} = 0, \tau_{\rho \theta}|_{\rho = a} = 0, \tau_{\rho \theta}|_{\rho = b} = 0, \sigma_{ \theta} = 0, \int_{a}^{b}\tau_{\rho \theta}d\rho = P, \theta = 0, u_{\rho} = 0, u_{\theta} = 0, \theta = \pi/2 and
In Tables 1 and 2, the error estimates of displacement and stress are presented for the barycentric rational interpolation collocation methods (BRICMs) with d = 5 and barycentric Lagrange interpolation collocation methods with n = 11 and n = 19 . From the table, the displacement and stress have higher accuracy for the Lagrange interpolation collocation methods than for the BRICMs.
In Tables 3 and 4, the error estimates of \sigma_{\rho} and \sigma_{\theta} are presented for barycentric rational interpolation with d_{1} = d_{2} = 2, 3, 4, 5 for equidistant nodes.
In Tables 5 and 6, the error estimates of \sigma_{\rho} and \sigma_{\theta} are presented for barycentric rational interpolation with d_{1} = d_{2} = 2, 3, 4, 5 for quasi-equidistant nodes.
In Tables 7–11, the errors of u_{\rho} , u_{\theta} , \sigma_{\rho} , \sigma_{\theta} and \tau_{r\theta} are shown for barycentric rational interpolation with d = 2, 3, 4, 5 for equidistant nodes. The convergence rate is O(h^{d}) for u_{\rho} , u_{\theta} , \sigma_{\rho} and \sigma_{\theta} , and O(h^{d-1}) for \tau_{\rho \theta} which agrees with our theorem analysis.
Example 2. Consider the the following elastic thick circular:
with
Then we get the displacement equation as
and the boundary conditions can be given as
which means that
and the matrix equations can be given as
and
with a = 0.5 m, b = 1 m, P_{a} = 1000 Pa, P_{b} = 2000 Pa, E = 10^7 Pa, \mu = 0.3.
In Tables 12 and 13, the error estimates of the BRICM with d = 5 and Lagrange interpolation collocation methods with n = 11 and n = 19 for displacement and stress are given.
In Tables 14–16, the errors of barycentric rational interpolation, i.e., u_{\rho} , \sigma_{\rho} and \sigma_{\theta} are shown with d = 2, 3, 4, 5 for equidistant nodes and O(h^{d}) for u_{\rho} , \sigma_{\rho} and \sigma_{\theta} which agrees with our theorem analysis.
In Tables 17–19, for quasi-equidistant nodes, the errors of u_{\rho} , \sigma_{\rho} and \sigma_{\theta} are given for the barycentric rational interpolation with d_{1} = d_{2} = 2, 3, 4, 5 . The convergence rate is O(h^{2d-2}) for u_{\rho} and O(h^{d}) for \sigma_{\rho} and \sigma_{\theta} which coincides with our theorem analysis.
6.
Concluding remarks
In this paper, first, the equilibrium equations were transformed into polar coordinates; then, the LBRCM was presented to solve the equilibrium equations. Third, the matrix equations of the equilibrium equations were obtained and the convergence rate of the LBRCM was also proved. At last, some numerical examples were given to validate the proposed theorem. The plane elasticity problems under the irregular domain can also be solved by using the LBRCM, as will be discussed it in the near future.
Acknowledgments
The work of Jin Li was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2022MA003) and Natural Science Foundation of Hebei Province (Grant No. A2019209533).
Conflict of interest
The author declares that there are no conflicts of interest.