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<p>This paper presents a new numerical algorithm with extrapolation for solving the three-dimensional axisymmetric elasticity problems with Dirichlet boundary conditions. Firstly, by computing the fundamental solutions of the axisymmetric elasticity problems, the problems can be transformed to axisymmetric boundary integral equations with weakly singular kernels. Secondly, the numerical solutions are obtained by discretizing the integral equations using the trapezoidal rule and a singular integral formula, which have the accuracy of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(h^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Thirdly, the convergence of the numerical solutions is proved by estimating eigenvalues of the discrete matrix and using the collectively compact convergence theory, and the single parameter asymptotic error expansion with odd power <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O(h^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is obtained. Moreover, based on the asymptotic expansion, an extrapolation algorithm can be constructed to improve the accuracy of the numerical solutions. Finally, numerical examples support our theoretical analysis.</p>

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An efficient numerical algorithm with extrapolation for three-dimensional axisymmetric elasticity problems

  • 期刊论文

  • 发表日期：

    2025年10月9日

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  • Hu Li,
  • Jin Huang

摘要

This paper presents a new numerical algorithm with extrapolation for solving the three-dimensional axisymmetric elasticity problems with Dirichlet boundary conditions. Firstly, by computing the fundamental solutions of the axisymmetric elasticity problems, the problems can be transformed to axisymmetric boundary integral equations with weakly singular kernels. Secondly, the numerical solutions are obtained by discretizing the integral equations using the trapezoidal rule and a singular integral formula, which have the accuracy of \(O(h^3)\) O ( h 3 ) . Thirdly, the convergence of the numerical solutions is proved by estimating eigenvalues of the discrete matrix and using the collectively compact convergence theory, and the single parameter asymptotic error expansion with odd power \(O(h^3)\) O ( h 3 ) is obtained. Moreover, based on the asymptotic expansion, an extrapolation algorithm can be constructed to improve the accuracy of the numerical solutions. Finally, numerical examples support our theoretical analysis.