On the Application of an Output-based Adaptive, Higher-order Finite Element Method to Sonic Boom Propagation

Abstract

The reduction of sonic boom loudness to within acceptable limits is a crucial factor for the viability of supersonic aircraft. This thesis presents a computational framework for simulating sonic boom propagation using an output-based adaptive, higher-order finite element method. The research employs the Variational Multiscale with Discontinuous Subscales (VMSD) method, integrating Continuous Galerkin (CG) and Discontinuous Galerkin (DG) features, referred to as VMSD-BR2. This approach leverages static condensation to manage computational cost while utilizing DG stabilization techniques for enhanced stability and adjoint consistency. A key component of this work is the application of the dual weighted residual (DWR) method for output error estimation, which in turns drives the mesh optimization process. The method’s efficacy is validated using smooth solutions for the viscous Burgers equation and the adjoint PDE for a volume output functional. Additionally, artificial viscosity is incorporated via a shock sensor PDE approach to handle shock presence, with necessary corrections applied to the DWR error estimate. The VMSD-BR2 method is applied then to a real-world scenario solving the augmented Burgers equation, which models the propagation of sonic booms. The results include the pressure perturbation field, adapted meshes, ground-level B-SEL filtered pressure, and perceived loudness at ground, demonstrating the method’s practical application.