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

Stefan Schoder

TL;DR

The paper tackles the problem of transferring structured finite-difference flow fields to unstructured finite-element acoustic meshes by assembling the FE right-hand side $b_i = \int_{\Omega} N_i(\boldsymbol{x}) f(\boldsymbol{x})\,d\Omega$. It compares two schemes: (i) high-order Gaussian quadrature with high-order FD-to-FE interpolation using B-splines or Lagrange polynomials, and (ii) cut-cell (supermesh) integration with geometric clipping that computes intersections between meshes. The study shows that high-order quadrature converges toward machine precision as interpolation and quadrature orders increase, but interpolation errors can dominate if not carefully balanced; in contrast, the cut-cell supermesh method can converge to machine precision universally, independent of local field smoothness. The findings indicate that the supermesh approach provides a robust, energy-conserving transfer for both smooth and highly oscillatory sources, with significant accuracy and performance advantages demonstrated on aeroacoustic source terms and subsonic flow models; practical implementations are available in openCFS and pyCFS.

Abstract

This work presents two integration methods for field transfer in computational aeroacoustics and in coupled field problems, using the finite element method to solve the acoustic field. Firstly, a high-order Gaussian quadrature computes the finite element right-hand side. In contrast, the (flow) field provided by the finite difference mesh is mapped by higher-order B-Splines or a Lagrangian function. Secondly, the cut-cell or supermesh integration with geometric clipping. For each method, the accuracy, performance characteristics, and computational complexity are analyzed. As a reference, the trapezoidal integration rule was computed from the finite difference results. The high-order quadrature converges as the B-Spline interpolation order increases, and the finite difference results and mesh resolutions are consistent. The supermesh approach eliminates interpolation and approximation errors at the grid-to-mesh level and improves accuracy. This behaviour is universal for smooth or strongly oscillating field quantities, which will be shown in a comparative study between the Lighthill-like source term and the source term of the perturbed convective wave equation for subsonic flows.

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

TL;DR

The paper tackles the problem of transferring structured finite-difference flow fields to unstructured finite-element acoustic meshes by assembling the FE right-hand side . It compares two schemes: (i) high-order Gaussian quadrature with high-order FD-to-FE interpolation using B-splines or Lagrange polynomials, and (ii) cut-cell (supermesh) integration with geometric clipping that computes intersections between meshes. The study shows that high-order quadrature converges toward machine precision as interpolation and quadrature orders increase, but interpolation errors can dominate if not carefully balanced; in contrast, the cut-cell supermesh method can converge to machine precision universally, independent of local field smoothness. The findings indicate that the supermesh approach provides a robust, energy-conserving transfer for both smooth and highly oscillatory sources, with significant accuracy and performance advantages demonstrated on aeroacoustic source terms and subsonic flow models; practical implementations are available in openCFS and pyCFS.

Abstract

This work presents two integration methods for field transfer in computational aeroacoustics and in coupled field problems, using the finite element method to solve the acoustic field. Firstly, a high-order Gaussian quadrature computes the finite element right-hand side. In contrast, the (flow) field provided by the finite difference mesh is mapped by higher-order B-Splines or a Lagrangian function. Secondly, the cut-cell or supermesh integration with geometric clipping. For each method, the accuracy, performance characteristics, and computational complexity are analyzed. As a reference, the trapezoidal integration rule was computed from the finite difference results. The high-order quadrature converges as the B-Spline interpolation order increases, and the finite difference results and mesh resolutions are consistent. The supermesh approach eliminates interpolation and approximation errors at the grid-to-mesh level and improves accuracy. This behaviour is universal for smooth or strongly oscillating field quantities, which will be shown in a comparative study between the Lighthill-like source term and the source term of the perturbed convective wave equation for subsonic flows.
Paper Structure (17 sections, 25 equations, 8 figures, 1 table, 3 algorithms)

This paper contains 17 sections, 25 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. High-Order Gaussian Quadrature with Higher-Order Accurate Field Interpolation
  4. Algorithm
  5. Convergence Analysis
  6. Cut-Cell-based Supermesh Integration
  7. Domain Decomposition
  8. Integration
  9. Physical space to target element reference mapping
  10. Geometry Processing
  11. Algorithm
  12. Comparative study
  13. Conclusions
  14. Reference integration by the trapezoidal rule
  15. Target element vectorized Newton-Raphson
  16. ...and 2 more sections

Figures (8)

  • Figure 1: Interpolation error analysis as a function of finite difference grid spacing and interpolation order.
  • Figure 2: Interpolation error (left) and right-hand side assembly time of algorithm 1 (right) as a function of the Gauss quadrature order.
  • Figure 3: Interpolation error (left) and right-hand side assembly time of algorithm 1 (right) as a function of the Gauss quadrature order.
  • Figure 4: Source fields used for integration.
  • Figure 5: H-refinement analysis with CFD data for the material derivative $D\Phi_p/Dt$. Comparison between supermesh integration, cubic interpolation, and quintic interpolation.
  • ...and 3 more figures