Matthew J. Zahr

Nonlinear Model Reduction for CFD Problems Using Local Reduced Order Bases

A model reduction framework based on the concept of local reduced-order bases is presented. The offline phase of the method builds the local reduced-order bases using an unsupervised learning algorithm. In the online phase of the method, the choice of the local basis is based on the current state of the system. Inexpensive rank-one updates to the local bases are performed during the online phase for increased accuracy. Applications to nonlinear CFD simulations show that the method is effective in producing small and accurate reduced order models.

Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction

Projection-based model reduction techniques rely on the definition of a small dimensional subspace in which the solution is approximated. Using local subspaces reduces the dimensionality of each subspace and enables larger speedups. Transitions between local subspaces require special care and updating the reduced bases associated with each subspace increases the accuracy of the reduced-order model. In the present work, local reduced basis updates are considered in the case of hyper-reduction, for which only the components of state vectors and reduced bases defined at specific grid points are available. To enable local reduced basis updates, two comprehensive approaches are proposed. The first one is based on an offline/online decomposition. The second approach relies on an approximated metric acting only on those components where the state vector is defined. This metric is computed offline and used online to update the local bases. An analysis of the error associated with this approximated metric is then conducted and it is shown that the metric has a kernel interpretation. Finally, the application of the proposed approaches to the model reduction of two nonlinear physical systems illustrates their potential for achieving large speedups and good accuracy.

Construction of Parametrically-Robust CFD-Based Reduced-Order Models for PDE-Constrained Optimization

A method for simultaneously constructing a reduced-order model and using it as a surrogate model to solve a PDE-constrained optimization problem is introduced. A reducedorder model is built for the parameters corresponding to the initial guess of the optimization problem. Since the resulting reduced-order model can be expected to be accurate only in the vicinity of this point in the parameter space, emphasis is placed on constructing this model by searching for regions of high error. These are determined by solving a small, nonlinear program with the objective defined as a linear combination of a residual error indicator and the objective function of the original PDE-constrained optimization problem. The reduced-order model is updated with information from the high-dimensional model in the regions of large error, and the process is repeated with more emphasis placed on solving the PDE-constrained optimization problem. The iteration terminates when the optimality conditions of the surrogate PDE-constrained optimization problem are satisfied. Application to a standard, nonlinear CFD shape optimization problem shows that the proposed method effectively solves a PDE-constrained optimization problem with few full CFD simulation queries.

An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems

The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high-order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary LagrangianEulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit RungeKutta scheme is used for the temporal discretization. Quantities of interest that take the form of spacetime integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. These quantities of interest and their gradients are used in the context of gradient-based PDE-constrained optimization.The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible NavierStokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradients computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration. Globally high-order discretization of deforming domain PDE (DG-in-space, RK-in-time).Solver-consistent integration of quantities of interest ensures globally high-order.Derived corresponding fully discrete adjoint equation/method for computing gradients.Enables gradient-based optimal control, shape, time-morphed geometries of PDEsEnergetically optimal time-morphed geometry and control for foil in viscous flow.

Performance tuning of Newton-GMRES methods for discontinuous Galerkin discretizations of the Navier-Stokes equations

In this work, we investigate numerical solvers and time integrators for the system of Ordinary Differential Equations (ODEs) arising from the Discontinuous Galerkin Finite Element Method (DG-FEM) semi-discretization of the Navier-Stokes equations to explore potential speedup opportunities. DG-FEMs have many desirable properties such as stability, high-order accuracy, and ability to handle complex geometries. However, they are notorious for their high computational cost and storage. In this document, our problems are spatially discretized with DG-FEM, temporally discretized with various implicit ODE solvers, and the nonlinear systems are solved using a Newton-GMRES method [1, 2]. We study the effects of varying several parameters, including the ODE solver, predictors for Newton’s method, GMRES tolerance, and Jacobian recycling. We show that by properly choosing these parameters, speedup factors of between 5 and 14 can be achieved over nonoptimal choices. The numerical experiments are performed on two model flow problems: An Euler vortex and the viscous flow over a 2D NACA wing at a high angle of attack.

A fully discrete adjoint method for optimization of flow problems on deforming domains with time-periodicity constraints

Abstract A variety of shooting methods for computing fully discrete time-periodic solutions of partial differential equations, including Newton–Krylov and optimization-based methods, are discussed and used to determine the periodic, compressible, viscous flow around a 2D flapping airfoil. The Newton–Krylov method uses matrix-free GMRES to solve the linear systems of equations that arise in the nonlinear iterations, with matrix-vector products computed via the linearized sensitivity evolution equations. The adjoint method is used to compute gradients for the gradient-based optimization shooting methods. The Newton–Krylov method is shown to exhibit superior convergence to the optimal solution for these fluid problems, and fully leverages quality starting data. The central contribution of this work is the derivation of the adjoint equations and the corresponding adjoint method for fully discrete, time-periodically constrained partial differential equations. These adjoint equations constitute a linear, two-point boundary value problem that is provably solvable. The periodic adjoint method is used to compute gradients of quantities of interest along the manifold of time-periodic solutions of the discrete partial differential equation, which is verified against a second-order finite difference approximation. These gradients are then used in a gradient-based optimization framework to determine the energetically optimal flapping motion of a 2D airfoil in compressible, viscous flow over a single cycle, such that the time-averaged thrust is identically zero. In less than 20 optimization iterations, the flapping energy was reduced nearly an order of magnitude and the thrust constraint satisfied to 5 digits of accuracy.

High-Order, Time-Dependent Aerodynamic Optimization using a Discontinuous Galerkin Discretization of the Navier-Stokes Equations

The fully discrete adjoint method, corresponding to a globally high-order accurate discretization of the compressible Navier-Stokes equations on deforming domains, is introduced. A mapping-based Arbitrary Lagrangian-Eulerian description transforms the governing equations to a fixed reference domain. A high-order discontinuous Galerkin spatial discretization and diagonally implicit Runge-Kutta temporal discretization are employed to obtain the globally high-order discretization of the Navier-Stokes equations. Relevant quantities of interest, to be used as the objective function in aerodynamic trajectory optimization problems, are discretized in a solver-consistent manner. Gradients of these quantities of interest are computed via the adjoint method and verified against a secondorder finite difference approximation. The proposed fully discrete adjoint method is coupled with state-of-the-art, gradient-based numerical optimization software to solve aerodynamic trajectory optimization problems. The first example is an inverse design problem with a known, global optimum that the solver is able to recover in fewer than 20 iterations. In a second problem, a trajectory is determined that successfully completes a prescribed mission while harvesting energy from the flow.

An Optimization Based Discontinuous Galerkin Approach for High-Order Accurate Shock Tracking

This work presents a high-order accurate, nonlinearly stable numerical framework for solving steady conservation laws with discontinuous solution features such as shock waves. The method falls into the category of a shock tracking or r-adaptive method and is based on the observation that numerical discretizations such as finite volume or discontinuous Galerkin methods that support discontinuities along element faces can perfectly represent discontinuities and provide appropriate stabilization through approximate Riemann solvers. The difficulty lies in aligning element faces with the unknown discontinuity. The proposed method recasts a discretized conservation law as a PDE-constrained optimization problem whose solution is a (curved) mesh that tracks the discontinuity and the solution of the discrete conservation law on this mesh. The discrete state vector and nodal positions of the high-order mesh are taken as optimization variables. The objective function is a discontinuity indicator that monotonically approaches a minimum as element faces approach the shock surface in a neighborhood of radius O(h), where h is the mesh size parameter. The discretized conservation law on a parametrized domain defines the equality constraints for the optimization problem. A full space optimization solver is used to simultaneously converge the state vector and mesh to their optimal values. This ensures the solution of the discrete PDE is never required on meshes that are not aligned with discontinuities and it increases the nonlinear stability. The method is demonstrated in one and two dimensions: transonic flow through a nozzle and supersonic flow around a bluff body. In both cases, the framework tracks the discontinuity closely with curved mesh elements and provides accurate solutions on extremely coarse meshes, e.g., O(10) degrees of freedom to resolve supersonic flow at Mach 4 in two dimensions.

Energetically Optimal Flapping Flight via a Fully Discrete Adjoint Method with Explicit Treatment of Flapping Frequency

This work introduces a fully discrete, high-order numerical framework for solving PDEconstrained optimization problems using gradient-based methods in the case where one or more of the optimization parameters affects the time domain; a canonical example being optimization of the frequency of a flapping wing. In a fully discrete setting, this effective parametrization of the time domain leads to a parametrization of the time discretization, e.g., to maintain a fixed number of timesteps per period, the timestep size is parameterdependent. Gradients of quantities of interest in this work are computed using the adjoint method, which must take into account the parametric dependence of the time discretization. As this work considers energetically optimal flight, a globally high-order discretization of conservation laws on deforming domains is employed: an Arbitrary Lagrangian-Eulerian formulation maps the conservation law to a fixed reference domain and a high-order discontinuous Galerkin method and diagonally implicit Runge-Kutta method are used for the spatial and temporal discretizations. This framework is applied to study energetically optimal flapping subject to a minimum required thrust, including frequency and pitching/heaving trajectories as optimization parameters. This marks a distinct departure from other adjoint-based approaches to optimal flapping that fix the frequency.

Gradient based aerodynamic shape optimization using the FIVER embedded boundary method

A computational approach is presented for evaluating shape sensitivities using an innovative embedded boundary method for computational fluid dynamics that mitigates the well-known difficulties associated with meshing, mesh updating and/or re-meshing. It is based on the analytical derivation of relevant semi-discrete gradients of the underlying comprehensive flow solver and associated computational geometry support. The proposed approach is integrated in a gradient-based optimization procedure and exploited to perform aerodynamic shape optimization of complex rigid and aeroelastic configurations. It demonstrates the potential of embedded boundary methods for the multi-disciplinary shape optimization of complex aerodynamic systems, and highlights their practical advantages in terms of flexibility, robustness, and performance.