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The paper presents the formulation of a parametric integral equation system (PIES) for 2D boundary value problems with singular solutions. The proposed approach combines B-spline basis functions with the PIES formalism to approximate the results in challenging problems such as notched plates or concentrated loads. The B-spline function with a specific degree and knots is used in approximating series instead of previously applied polynomials (such as Lagrange or Chebyshev). Increasing knot multiplicity to reduce continuity at some locations or increasing the number of knots in singular regions allows for more accurate results. The proposed approach is validated through selected problems. The results confirm that the B-spline-enhanced PIES approach can effectively solve singular problems with improved accuracy.

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Using B-Spline Function Properties in the PIES Method to Handle Singularities in Boundary Value Problems

  • 会议论文

  • 发表日期：

    2025年

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  • Agnieszka Bołtuć,
  • Eugeniusz Zieniuk

摘要

The paper presents the formulation of a parametric integral equation system (PIES) for 2D boundary value problems with singular solutions. The proposed approach combines B-spline basis functions with the PIES formalism to approximate the results in challenging problems such as notched plates or concentrated loads. The B-spline function with a specific degree and knots is used in approximating series instead of previously applied polynomials (such as Lagrange or Chebyshev). Increasing knot multiplicity to reduce continuity at some locations or increasing the number of knots in singular regions allows for more accurate results. The proposed approach is validated through selected problems. The results confirm that the B-spline-enhanced PIES approach can effectively solve singular problems with improved accuracy.