An Adjoint-Based Data Assimilation Algorithm using the Hybridizable Discontinuous Galerkin Framework*

CFD and experiments are often used as separate lenses to study the same flow physics. Data assimilation integrates both views by finding a solution to the modeled equations that best fits the measurements. This work presents an adjoint-based data assimilation algorithm for flows governed by the incompressible Navier-Stokes (N-S) equations, aiming to minimize discrepancies between simulations and experiments due to uncertainties in the initial and boundary conditions.

The problem is formulated as minimizing the velocity differences between numerical and experimental data, with the N-S equations as equality constraints using Lagrange multipliers. The optimization parameters are the initial condition and time-varying Dirichlet boundary conditions for velocity.

The algorithm consists of three components: First, a novel Hybridizable Discontinuous Galerkin (HDG) method solves the incompressible N-S equations with arbitrary spatial order on an unstructured mesh (Anantharamu & Mahesh, 2024, IJNMF, In Revision). Second, the adjoint of the discretized N-S system is solved using the same HDG framework to compute the gradient of the objective function. Third, a quasi-Newton method solves the minimization problem. 

For various canonical and realistic flow test cases, the algorithm successfully finds optimal initial and boundary conditions for velocity and obtains a solution to the N-S equations that best matches the discrete velocity measurements. Results will be discussed in detail.