David L. Darmofal

Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows

An anisotropic, unstructured grid adaptive method is presented for improving the accuracy of functional outputs of viscous, compressible flow simulations for general discretizations. The procedure merges output error control with Hessian-based anisotropic grid adaptation. An adjoint formulation is used to relate the estimated functional error to the local residual errors of both the primal and adjoint solutions. This relationship allows local error contributions to be used as indicators in a grid adaptive method designed to produce specially tuned grids for accurately estimating the chosen functional. Element stretching and orientation information is obtained from interpolation error estimates for linear triangular finite elements. The proposed adaptive method is implemented using a standard second-order upwind finite volume discretization, although the procedure is applicable to other types of discretizations such as the finite element method. A series of airfoil test cases, including separated, high-lift flows, are presented to demonstrate the approach; the functionals considered are the lift and drag coefficients. The proposed adaptive method is shown to be superior in terms of reliability and output accuracy relative to pure Hessian-based adaptation.

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.

Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation

Artificial viscosity can be combined with a higher-order discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a non-smooth artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a higher-order, state-based artificial viscosity with an associated governing partial differential equation (PDE). In the governing PDE, a shock indicator acts as a forcing term while grid-based diffusion is added to smooth the resulting artificial viscosity. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDE-based artificial viscosity is less susceptible to errors introduced by grid edges oblique to captured shocks and boundary layers, thereby enabling accurate heat transfer predictions.

Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations

A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton-Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.

Impact of Geometric Variability on Axial Compressor Performance

A probabilistic methodology to quantify the impact of geometric variability on compressor aerodynamic performance is presented. High-fidelity probabilistic models of geometric variability are derived using a principal-component analysis of blade surface measurements. This probabilistic blade geometry model is then combined with a compressible, viscous blade-passage analysis to estimate the impact on the passage loss and turning using a Monte Carlo simulation. Finally, a mean-line multistage compressor model, with probabilistic loss and turning models from the blade-passage analysis, is developed to quantify the impact of the blade variability on overall compressor efficiency and pressure ratio. The methodology is applied to a flank-milled integrally bladed rotor. Results demonstrate that overall compressor efficiency can be reduced by approximately 1% due to blade-passage effects arising from representative manufacturing variability.

Higher-Dimensional Integration with Gaussian Weight for Applications in Probabilistic Design

Higher-dimensional Gaussian weighted integration is of interest in probabilistic simulations. Motivated by the need for variance calculations with functions being at least quadratic, the family of degree 5 formulae is considered. Using an existing formula for the integration over the surface of an n-sphere, an efficient, new formula for Gaussian weighted integration is obtained. Several other formulae that have appeared in the numerical integration literature are also given. The number of function evaluations required by the formulae is compared to a minimal bound result. The degree 5 formulae are applied to simple test problems and the relative errors are compared.

An optimization-based framework for anisotropic simplex mesh adaptation

We present a general framework for anisotropic h-adaptation of simplex meshes. Given a discretization and any element-wise, localizable error estimate, our adaptive method iterates toward a mesh that minimizes error for a given degrees of freedom. Utilizing mesh-metric duality, we consider a continuous optimization problem of the Riemannian metric tensor field that provides an anisotropic description of element sizes. First, our method performs a series of local solves to survey the behavior of the local error function. This information is then synthesized using an affine-invariant tensor manipulation framework to reconstruct an approximate gradient of the error function with respect to the metric tensor field. Finally, we perform gradient descent in the metric space to drive the mesh toward optimality. The method is first demonstrated to produce optimal anisotropic meshes minimizing the L^2 projection error for a pair of canonical problems containing a singularity and a singular perturbation. The effectiveness of the framework is then demonstrated in the context of output-based adaptation for the advection-diffusion equation using a high-order discontinuous Galerkin discretization and the dual-weighted residual (DWR) error estimate. The method presented provides a unified framework for optimizing both the element size and anisotropy distribution using an a posteriori error estimate and enables efficient adaptation of anisotropic simplex meshes for high-order discretizations.

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

Using concept maps and concept questions to enhance conceptual understanding

Conceptual understanding is the ability to apply knowledge across a variety of instances or circumstances. Several strategies can be used to teach and assess concepts, e.g., inquiry, exposition, analogies, mnemonics, imagery, concept maps, and concept questions. This paper focuses on the last two -concept maps and concept questions. Concept maps are two-dimensional, hierarchical diagrams that show the structure of knowledge within a discipline. Concept questions are questions posed to students to encourage higher order thinking and help them understand the basic principles of a discipline. This paper describes current progress at MIT in the development and use of concept maps and concept questions in aerospace engineering.