Closed-form perturbative integration of two-dimensional elasticity BEM kernels over curved quadratic boundary elements

The repeated numerical integration of fundamental kernels represents a significant computational cost in the boundary element method (BEM), especially when displacements and stresses are evaluated at many internal points. This work presents a closed-form perturbative procedure for integrating the two-dimensional elastostatic kernels over curved quadratic boundary elements. A straight quadratic element is adopted as the reference configuration, while the element curvature is introduced through a nondimensional perturbation parameter measuring the transverse deviation of the midside node from the straight chord. The geometric quantities entering the displacement, traction, displacement-gradient, and stress kernels are expanded analytically to first order with respect to this parameter. The kernels are first decomposed into geometry-dependent contributions and constitutive coefficients, allowing the resulting expressions to be derived independently of the elastic material parameters. Because the Jacobian of the canonical element has a unit zeroth-order value and a vanishing first-order derivative, the perturbative correction depends only on the variation of the kernel terms. All resulting functions are integrated analytically against the quadratic shape functions, leading to explicit element matrices that depend only on the local coordinates of the source point. Special attention is devoted to the branch-consistent numerical implementation of the inverse trigonometric and logarithmic terms appearing in the closed-form expressions. Numerical comparisons with high-order Gaussian and near-singular quadrature demonstrate the accuracy of the proposed first-order approximation and its potential to substantially reduce the cost of BEM post-processing without increasing the number of boundary degrees of freedom.