Convergence of finite element right-hand-side computation from finite difference data

Technical Summary: Convergence of Finite Element Right-Hand-Side Computation from Finite Difference Data

Problem Statement
In hybrid computational aeroacoustics (CAA) and coupled field problems, accurate field transfer is required between structured finite difference (FD) grids (often used for flow simulation) and unstructured finite element method (FEM) meshes (used for acoustic propagation). The core challenge lies in assembling the FEM right-hand side (RHS) vector, defined as $b_i = \int_{\Omega} N_i(x)f(x) d\Omega$, where $f(x)$ is a source field defined on the FD grid.

Standard approaches often rely on simple interpolation, which can introduce significant errors, particularly when source terms exhibit sharp transitions or strong spatial oscillations (e.g., Lighthill-like source terms). The paper addresses the need for integration methods that ensure accuracy, conservation of energy (globally and locally), and convergence as mesh resolution increases, without being limited by the smoothness assumptions of the source field.

Methodology
The paper proposes and analyzes two distinct integration strategies for transferring FD data to the FEM RHS:

1. High-Order Gaussian Quadrature with Field Interpolation:

   • Mechanism: This method evaluates the integral using Gaussian quadrature within each FEM element. The FD field values are first interpolated to the Gauss quadrature points using either high-order B-Splines (orders $p \in \{1, \dots, 5\}$) or Lagrangian polynomials (orders $p \in \{1, 3\}$).
   • Error Analysis: The total error is decomposed into interpolation error ($\varepsilon_{interp}$), quadrature error ($\varepsilon_{quad}$), and FEM discretization error ($\varepsilon_{fem}$). The study demonstrates that for the method to converge, the interpolation order and the quadrature order must be increased consistently. If the quadrature order is increased without a corresponding increase in interpolation order, the interpolation error becomes the limiting factor, preventing convergence to machine precision.

2. Cut-Cell (Supermesh) Integration with Geometric Clipping:

   • Mechanism: This approach avoids interpolation errors at the grid-to-mesh level by treating the problem as a geometric intersection of two non-conforming meshes. The FD cells and FEM elements are overlaid to create "cut polygons" (supermesh).
   • Implementation: The domain is decomposed into intersection polygons, which are then tessellated into triangles. Integration is performed via Gaussian quadrature on these sub-triangles in physical space.
   • Complexity: A significant computational challenge is the inverse mapping required to evaluate FEM shape functions at physical Gauss points located within the cut regions. For non-linear geometry mappings, this requires solving a non-linear system (e.g., via Newton-Raphson iteration) for every quadrature point.
   • Optimization: The algorithm utilizes Axis-Aligned Bounding Box (AABB) indexing and binary search to efficiently identify intersecting FD cells, followed by polygon clipping (e.g., Sutherland-Hodgman) to define the integration domains.

Key Results

• Convergence Behavior:
  • The interpolation-based method converges as the B-Spline order and quadrature order increase, provided the FD mesh resolution is consistent. However, it exhibits a systematic error floor determined by the reconstruction order of the interpolation. It struggles to reach machine precision for highly oscillating source terms unless extremely high orders are used.
  • The supermesh method eliminates interpolation and approximation errors at the grid-to-mesh interface. It converges to machine precision across all mesh resolutions, regardless of whether the source field is smooth or strongly oscillating.
• Source Term Sensitivity: Comparative studies using Lighthill-like source terms (spatially varying, sharp transitions) and perturbed convective wave equation source terms (smoother) show that the supermesh approach is universally robust. Interpolation-based methods show higher errors for the more complex Lighthill-like terms.
• Performance: While the supermesh method involves higher geometric complexity and the cost of inverse mapping, it is highly optimized. For problems involving many elements, it can outperform unoptimized interpolation methods. The weak scaling of the integration is linear for both methods.

Significance and Claims
The paper claims that the supermesh integration approach provides a superior alternative for hybrid CAA simulations where conservation of the source term energy is critical. By eliminating the interpolation step between the FD grid and FEM Gauss points, the method ensures that the complete energy from the flow field is transferred to the acoustic mesh without reduction or artificial smoothing.

The authors emphasize that while high-order interpolation can reduce errors, it cannot guarantee sufficient convergence in practice if the field quantities are not "sufficiently smooth." Therefore, the supermesh approach is presented as a robust solution that is "universal" for both smooth and strongly oscillating field quantities. The work contributes a rigorous comparative analysis of these integration schemes, highlighting that the choice of method depends heavily on the nature of the source term and the requirement for strict conservation.

Conclusion
The study concludes that for accurate field transfer in hybrid aeroacoustics, the supermesh integration method is the preferred approach when conservation and high accuracy are paramount, as it removes the systematic errors inherent in interpolation-based schemes. The interpolation-based method remains viable for smoother source terms but requires careful balancing of interpolation and quadrature orders to minimize error.