Curved Boundary Element Methods

• Curved Boundary Element Methods are numerical schemes that use curved, higher-order panels to accurately represent complex geometries in boundary integral equations.
• They leverage advanced quadrature techniques, including polar coordinate transformations and singularity subtraction, for robust evaluation of challenging integrals.
• These methods achieve superior convergence and efficiency compared to planar discretizations, reducing computational effort for high-accuracy solutions.

Curved boundary element methods (Curved BEM) constitute a class of numerical schemes for the discretization and solution of boundary integral equations where geometric panels or elements are taken to be curved—often of quadratic or higher order—in order to more accurately conform to the true geometry of a physical boundary. Such methods were originally developed to overcome the well-recognized geometric error, convergence limitations, and inefficiencies inherent in piecewise-linear (planar) panel discretizations, especially when modeling complex shapes or requiring high-order accuracy. Curved BEM techniques are notable for their ability to robustly and efficiently handle the singular, weakly singular, or near-singular integrals that arise in fundamental PDEs—including Laplace, Helmholtz, and elasticity equations—by leveraging advanced geometric mapping and quadrature strategies.

1. Geometric Representation and Element Definition

Curved boundary elements are defined by higher-order mappings—most prominently second-order (quadratic) interpolation—of reference triangles or quadrilaterals to the physical domain surface. The classic approach for a quadratic triangular element utilizes six nodes (three at the vertices and three at edge midpoints), with the geometry parameterized by quadratic shape functions:

$$\mathbf{y}(\xi,\eta) = \sum_{i=1}^{6} L_i(\xi, \eta) \, \mathbf{y}_i, \quad 0 \leq \xi \leq 1, \; 0 \leq \eta \leq 1-\xi$$

(where $L_i$ are the standard quadratic shape functions).

Each edge of the triangle is itself a quadratic curve, parameterized for instance as:

$$\mathbf{y}(\gamma) = \mathbf{y}_i J_1(\gamma) + \mathbf{y}_j J_2(\gamma) + \mathbf{y}_{i+3} J_3(\gamma)$$

with

$$J_1(\gamma) = 2\gamma^2 - 3\gamma + 1, \quad J_2(\gamma) = 2\gamma^2 - \gamma, \quad J_3(\gamma) = -4\gamma^2 + 4\gamma$$

This geometric flexibility improves domain conformity over planar panels, allowing for a more accurate representation of complex surfaces and internal or external boundaries. Such elements support more accurate interpolation of both geometry and (potentially) solution variables (Carley, 2013).

2. Quadrature and Singular Integration on Curved Elements

A central challenge in curved BEM is the accurate and robust evaluation of singular, weakly singular, and near-singular surface integrals. To address this, several advanced quadrature strategies have been developed.

Polar Coordinate Transformation:

For the Laplace equation and similar kernels, a curved triangle is first reoriented and projected so that the three vertices lie in $z = 0$; the field point is projected onto the origin. The surface integral is then rewritten in polar coordinates:

$$I = \iint f(\xi,\eta) \, J(\xi,\eta) \, d\eta d\xi = \iint f(r,\theta) \, r \, dr \, d\theta$$

Integration limits in $\theta$ are carefully constructed from the angular directions to vertices, tangents to curved edges, and intersections of rays with the (curved) triangle boundary. For each angular interval, Gaussian quadrature rules are applied in both $\theta$ and $r$, mapping each quadrature node back to reference coordinates—employing Newton's method for inversion as necessary. The weights account for both the Jacobian and polar transformation (Carley, 2013).

Singularity Subtraction and Continuation:

For higher order singular or nearly-singular integrals (as in Helmholtz or elasticity), tailored subtraction of analytically derived singular terms—using high-order Taylor expansions in the reference element—removes the leading-order singularities. This is followed by a "continuation approach" that collapses 2D singular integrals onto 1D edge integrals, which are further evaluated using transplanted Gauss quadrature with conformal mappings, providing exponential convergence even when the singularity is near an edge. This framework is effective for both weakly and strongly singular integrals (Montanelli et al., 2021, Montanelli et al., 2023).

3. Error Analysis and Convergence

Curved BEM achieves higher convergence rates compared to planar BEM, provided the solution and data are sufficiently smooth and the geometric approximation matches the order of the basis functions. Using quadratic elements and polar quadrature, the empirical error for the Laplace equation in a challenging geometry such as a "cat's eye" domain demonstrates:

• For quadratic (second-order) curved elements:

$\text{error} \sim P^{-1.6}$

where $P$ is the number of elements/panels.

• For linear panels:

$\text{error} \sim P^{-1.1}$

This demonstrates that curved elements offer a substantially better convergence rate and per-degree-of-freedom efficiency than linear ones (Carley, 2013). Further, in recent mathematical analyses, the theoretical consistency errors of curved BEM for the 3D Laplace and Helmholtz equations have been precisely characterized, leading to improved convergence rate guarantees in associated norms, and confirmed with numerical experiments up to order-four basis functions (Faria et al., 18 Jul 2025).

4. Implementation Considerations

Curved BEM requires:

• Evaluation of shape functions and their derivatives on curved elements, often requiring inversion from global coordinates (for quadrature) to reference coordinates.
• Resolution of possible "multiple entry-exit" and tangency cases in integration, particularly prominent in nonconvex or highly-curved panels.
• Tailored mesh generation capable of producing meshes where edges and faces align with geometric curvature or higher-order definitions (Carley, 2013).

In modern codes, the modular design of quadrature (polar mapping, geometric operations, and nested Gaussian quadrature) allows replacement of earlier planar panel routines with curved-element routines with minimal refactoring. For challenging cases, integrating robust root-finding (e.g., Newton's method for inversion) and analytic handling of singular behavior is crucial (Montanelli et al., 2021, Montanelli et al., 2023).

Curved boundary element routines can be integrated with direct solvers and hierarchical matrix techniques, providing scalable computations for boundary element systems with millions of unknowns (Montanelli et al., 2023).

5. Comparative Advantages and Practical Impact

The primary advantages of curved BEM over planar BEM include:

• Higher accuracy per element, especially in problems where high geometric fidelity is critical.
• Significantly improved convergence rates, reducing the degrees of freedom required to achieve a prescribed accuracy.
• Superior efficiency in representing smooth boundaries and in handling high-frequency or high-order problems.

Curved elements are particularly beneficial in applications with complex shapes and when simulating phenomena sensitive to geometric features, such as acoustic scattering, electromagnetic wave propagation, or aerodynamic analysis.

Curved BEM also supports adaptive quadrature, facilitating local mesh refinement and error control, and is compatible with a broad range of boundary PDEs—Laplace, Helmholtz, elasticity, and beyond (Carley, 2013, Faria et al., 18 Jul 2025).

6. Future Research Directions and Extensions

Promising directions for further development in curved BEM include:

• Generalizations to three-dimensional higher-order surface elements and multipatch surfaces with arbitrary topology and smoothness (Carley, 2013).
• Further theoretical analysis of convergence and consistency for singular and near-singular integration, particularly in strongly singular cases (Faria et al., 18 Jul 2025).
• Integration with adaptive mesh refinement, error estimators based on geometric and quadrature measures, and extension to time-dependent and nonlinear PDEs.
• Application to hybrid coupling schemes, such as the coupling of BEM with finite element or virtual element methods on curved domains and boundaries (Desiderio et al., 2021).

The modular design of advanced quadrature strategies—combining polar transformation, singularity subtraction, continuation, and transplanted quadrature—is expected to facilitate robust application of curved BEM in increasingly challenging scientific and industrial contexts (Carley, 2013, Montanelli et al., 2021, Montanelli et al., 2023, Faria et al., 18 Jul 2025).