Krzysztof J. Fidkowski

Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjoint-based techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering. The definition of the adjoint as the sensitivity of an output to residual source perturbations is used to derive both the adjoint equation, in fully discrete and variational formulations, and the adjoint-weighted residual method for error estimation. Assumptions and approximations made in the calculations are discussed. Presentation of the discrete and variational formulations enables a side-by-side comparison of recent work in output-error estimation using the finite volume method and the finite element method. Techniques for adapting meshes using output-error indicators are also reviewed. Recent adaptive results from a variety of laminar and Reynolds-averaged Navier-Stokes applications show the power of output-based adaptive methods for improving the robustness of computational fluid dynamics computations. However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics.

A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equations

This paper presents a mesh adaptation method for higher-order (p>1) discontinuous Galerkin (DG) discretizations of the two-dimensional, compressible Navier-Stokes equations. A key feature of this method is a cut-cell meshing technique, in which the triangles are not required to conform to the boundary. This approach permits anisotropic adaptation without the difficulty of constructing meshes that conform to potentially complex geometries. A quadrature technique is proposed for accurately integrating on general cut cells. In addition, an output-based error estimator and adaptive method are presented, appropriately accounting for high-order solution spaces in optimizing local mesh anisotropy. Accuracy on cut-cell meshes is demonstrated by comparing solutions to those on standard, boundary-conforming meshes. Robustness of the cut-cell and adaptation technique is successfully tested for highly anisotropic boundary-layer meshes representative of practical high Re simulations. Furthermore, adaptation results show that, for all test cases considered, p=2 and p=3 discretizations meet desired error tolerances using fewer degrees of freedom than p=1.

Non-linear model reduction for uncertainty quantification in large-scale inverse problems

SUMMARY We present a model reduction approach to the solution of large-scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non-linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non-linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient-function approximation. The resulting model reduction methodology is applied to a highly non-linear combustion problem governed by an advection‐diffusion-reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non-linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three-dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full-order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright q 2009 John Wiley & Sons, Ltd.

An Entropy Adjoint Approach to Mesh Refinement

This work presents a mesh refinement indicator based on entropy variables, with an application to the compressible Navier-Stokes equations. The entropy variables are shown to satisfy an adjoint equation, an observation that allows recent work in adjoint-based error estimation to be leveraged in constructing a relatively cheap but effective adaptive indicator. The output associated with the entropy-variable adjoint is shown to be the entropy production in the computational domain, including physical viscous dissipation when present, minus entropy transport out of the domain. Adaptation using entropy variables, which is equivalent to adapting on the integrated residual of the entropy transport equation, thus targets areas of the domain responsible for numerical, or spurious, entropy production. Adaptive results for inviscid and viscous aerodynamic examples in two and three dimensions demonstrate performance efficiency on par with output-based adaptation, as measured by errors in various engineering quantities of interest, with the comparative advantage of the proposed approach that no adjoint equations need to be solved.

DEVELOPMENT OF A HIGHER-ORDER SOLVER FOR AERODYNAMIC APPLICATIONS

We present the results from the development of a higher-order discontinuous Galerkin nite element solver using p-multigrid with line Jacobi smoothing. The line smoothing algorithm is presented for unstructured meshes, and p-multigrid is outlined for the non- linear Euler equations of gas dynamics. Analysis of 2-D advection shows the improved performance of line implicit versus point implicit relaxation. Through a mesh renemen t study, we determine that the accuracy of the discretization is the optimal O(h p+1 ) for three dieren t smooth problems. The multigrid convergence rate is found to be indepen- dent of the polynomial order but does depend weakly on the grid size. Timing studies for each problem indicate that higher order is advantageous over grid renemen t when high accuracy is required. HILE CFD has achieved signican t maturity during the past decades, computational costs are extremely large for aerodynamic simulations of aerospace vehicles. In this applied aerodynamics con- text, the discretization of the Euler and/or Navier- Stokes equations is almost exclusively performed by nite volume algorithms. The pioneering work of Jameson began this evolution to the status quo. 1{3 During the 1980s, upwinding mechanisms were in- corporated into these nite volume algorithms leading to increased robustness for applications with strong shocks, and perhaps more importantly, to better res- olution of viscous layers due to decreased numerical dissipation in these regions. 4{8 The 1990s saw major advances in the application of nite volume methods to Navier-Stokes simulations (in particular the Reynolds- Averaged Navier-Stokes equations). Signican t gains were made in the use of unstructured meshes and solu- tion techniques for viscous problems. 9{12 While these algorithmic developments have resulted in an ability to simulate aerodynamic o ws for very complex prob- lems, the time required to achieve sucien t accuracy in a reliable manner places a severe constraint on the application of CFD to aerospace design. The accuracies of many of the nite-v olume meth- ods currently used in aerodynamics are at best p = 2, i.e. the error decreases as O(h p ) where h is a measure of the grid spacing. As a practical matter, however, the accuracy of these methods on more realistic prob- lems appears to be less than this, ranging between 1 p 2. The development of a practical higher- order solution method could result in a signican t decrease in the computational time required to achieve an acceptable error level. To better demonstrate the

Anisotropic hp-Adaptation Framework for Functional Prediction

This paper presents a method for concurrent mesh and polynomial-order adaptation with the objective of direct minimization of output error using a selection process for choosing the optimal refinement option from a discrete set of choices that includes directional spatial resolution and approximation order increment. The scheme is geared toward compressible viscous aerodynamic flows, in which various solution features make certain refinement options more efficient than others. No attempt is made, however, to measure the solution anisotropy or smoothness directly or to incorporate it into the scheme. Rather, mesh anisotropy and approximation order distribution arise naturally from the optimization of a merit function that incorporates both an output sensitivity and a measure of solution cost on the new mesh. The method is applied to output-based adaptive simulations of laminar and Reynolds-averaged compressible Navier-Stokes equations on body-fitted meshes in two and three dimensions. Two-dimensional resul...

Output-based space-time mesh adaptation for the compressible Navier-Stokes equations

This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space-time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space-time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier-Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space-time meshes are shown to be relatively insensitive to the starting mesh.

Output-based mesh adaptation for high order Navier-Stokes simulations on deformable domains

We present an output-based mesh adaptation strategy for Navier-Stokes simulations on deforming domains. The equations are solved with an arbitrary Lagrangian-Eulerian (ALE) approach, using a discontinuous Galerkin finite-element discretization in both space and time. Discrete unsteady adjoint solutions, derived for both the state and the geometric conservation law, provide output error estimates and drive adaptation of the space-time mesh. Spatial adaptation consists of dynamic order increment or decrement on a fixed tessellation of the domain, while a combination of coarsening and refinement is used to provide an efficient time step distribution. Results from compressible Navier-Stokes simulations in both two and three dimensions demonstrate the accuracy and efficiency of the proposed approach. In particular, the method is shown to outperform other common adaptation strategies, which, while sometimes adequate for static problems, struggle in the presence of mesh motion. We present output-based mesh adaptation for Navier-Stokes on deforming domains.Space-time adaptation is driven by unsteady state and GCL adjoints.Significant reductions in mesh size and CPU time are obtained in 2 and 3 dimensions.If the GCL is used, a GCL adjoint is necessary for accurate error estimates.However, accurate outputs can often be obtained without the GCL.

An adaptive simplex cut-cell method for discontinuous Galerkin discretizations of the Navier-Stokes equations

A cut-cell adaptive method is presented for high-order discontinuous Galerkin discretizations in two and three dimensions. The computational mesh is constructed by cutting a curved geometry out of a simplex background mesh that does not conform to the geometry boundary. The geometry is represented with cubic splines in two dimensions and with a tesselation of quadratic patches in three dimensions. High-order integration rules are derived for the arbitrarily-shaped areas and volumes that result from the cutting. These rules take the form of quadrature-like points and weights that are calculated in a pre-processing step. Accuracy of the cut-cell method is verified in both two and three dimensions by comparison to boundary-conforming cases. The cut-cell method is also tested in the context of output-based adaptation, in which an adjoint problem is solved to estimate the error in an engineering output. Two-dimensional adaptive results for the compressible Navier-Stokes equations illustrate automated anisotropic adaptation made possible by triangular cut-cell meshing. In three dimensions, adaptive results for the compressible Euler equations using isotropic refinement demonstrate the feasibility of automated meshing with tetrahedral cut cells and a curved geometry representation. In addition, both the two and three-dimensional results indicate that, for the cases tested, p = 2 and p = 3 solution approximation achieves the user-prescribed error tolerance more efficiently compared to p = 1 and p = 0.

Output-Driven Anisotropic Mesh Adaptation for Viscous Flows using Discrete Choice Optimization

This paper presents a mesh adaptation scheme for direct minimization of output error using a selection process for choosing the optimal refinement option from a discrete set of choices. The scheme is geared for viscous aerodynamic flows, in which solution anisotropy makes certain refinement options more efficient compared to others. No attempt is made, however, to measure the solution anisotropy directly or to incorporate it into the scheme. Rather, mesh anisotropy arises naturally from the minimization of a cost function that incorporates both an output error estimate and a count of the additional degrees of freedom for each refinement option. The method is applied to output-based adaptive simulations of the laminar and Reynolds-averaged compressible Navier-Stokes equations on body-fitted meshes in two and three dimensions. Two-dimensional results for laminar flows show a factor of 2-3 reduction in the degrees of freedom on the final adapted meshes when the discrete choice optimization is used compared to pure isotropic adaptation. Preliminary results on a wing-body configuration show that these savings improve in three dimensions and for higher Reynolds-number flows.