James C. Newman

Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables

The use of complex variables for determining sensitivity derivatives for turbulent flows is examined. Although a step size parameter is required, the numerical derivatives are not subject to subtractive cancellation errors and, therefore, exhibit true second-order accuracy as the step size is reduced. As a result, this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude. This behavior is in contrast to the use of finite differences, which suffer from inaccuracies due to subtractive cancellation errors. In addition, the complex-variable procedure is easily implemented into existing codes

Overview of Sensitivity Analysis and Shape Optimization for Complex Aerodynamic Configurations

This paper presents a brief overview of some of the more recent advances in steady aerodynamic shape-design sensitivity analysis and optimization, based on advanced computational fluid dynamics (CFD). The focus here is on those methods particularly well-suited to the study of geometrically complex configurations and their potentially complex associated flow physics. When nonlinear state equations are considered in the optimization process, difficulties are found in the application of sensitivity analysis. Some techniques for circumventing such difficulties are currently being explored and are included here. Attention is directed to methods that utilize automatic differentiation to obtain aerodynamic sensitivity derivatives for both complex configurations and complex flow physics. Various examples of shape - design sensitivity analysis for unstructured-grid CFD algorithms are demonstrated for different formulations of the sensitivity equations. Finally, the use of advanced, unstructured-grid CFDs in multidisciplinary analyses and multidisciplinary sensitivity analyses within future optimization processes is recommended and encouraged.

Three-dimensional aerodynamic shape sensitivity analysis and design optimization using the Euler equations on unstructured grids

Developed is an approach whereby aerodynamic shape sensitivity analysis and design optimization are pedormed on three-dimensional unstructured meshes. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater eficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. The nonlinear Euler equations are solved using a fullyimplicit, upwind, cell-centered, finite-volume scheme. The discrete, linearized systems which result from this scheme are solved iteratively by a preconditioned conjugate-gradient-like algorithm known as GMRES; a similar procedure is also used to solve the accompanying linear aerodynamic sensitivity equalions in incremental iterative form. As shown, this particular form of the sensitivity equations makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory eSJicient methods to construct exact matrix-vector products. Wing-planform parameterization is trccornplished via scaling and translation factors at pre-selected locations along the wing span, then linearly varying these factors between locations. Once the surface has been deformed, the unstructured grid is adapted by considering the mesh as a system of interconnected springs. Grid sensitivities are obtained by differentiating the surface parameterization and the grid adaptation algorithms with ADIFOR (which is an advanced automatic-diferentiation software tool). To evaluate this shape optimization procedure, the planform shape of an initially rectangular wing with uniform NACA-0012 cross-sections is optimized in a compressible, inviscid flow. 1. Introduct ion As recently noted by Reuther et al. [ 11 “while flow analysis has maturecl to the extent that Navier-Stokes calculations are routinely carried out over very complex configurations, direct CFD based design is only just beginning to be used in the treatment of moderately complex three-dimensional coizfgurations”. This is primarily due to the fact that to generate a single structured grid about such a configuration is difficult, if not impossible. Thus, to handle geometry of practical interest, some sort of domain decomposition scheme must be incorporated into the design code. For structured grid solvers, these techniques would include multiblocked, zonally patched, and overlapped (sometimes referred to as Chimera) grid algorithms. However, as the geometric flexibility of the method increases, so does the complexity of the underlying algorithm. Since the use of sensitivity analysis, to evaluate the needed gradients for a numerical optimizer, is still evolving, little work has been done toward extending these algorithms to include these domain decomposition methods. The research which has been accomplished has mostly concentrated on the use of niultiblocked grids. On this, Reuther et al. [ I ] have developed a multiblock-multigrid adjoint solver (“variational” or “control theory” approach [ 2 ] ) which was applied for the wing redesign of a transonic business jet. Eleshaky and Baysal [3] developed a multiblock “discrete” adjoint solver which was applied to a simple axisynlmetric nozzle near a flat plat. As for the use of the more advanced domain decomposition methods (zonal and overlapped grids), and combinations of the three various types, Taylor [4] has differentiated an advanced flow-analysis code to perform the discrete sensitivity analysis. * Graduate Research Assistant. Student Member, AIM # Associate Professor. Member, AIAA. Copyright

Aerodynamic Shape Sensitivity Analysis and Design Optimization of Complex Configurations Using Unstructured Grids

A three-dimensio nal unstructured grid approach to aerodynamic shape sensitivity analysis and design optimization has been developed and is extended to model geometrically complex configurations. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater efficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. In this work the nonlinear Euler equations are solved using an upwind, cell-centered, finite-volume scheme. The discrete, linearized systems which result from this scheme are solved iteratively by a preconditioned conjugate-gradient-like algorithm known as GMRESfor the two-dimensional geometry and a Gauss-Seidel algorithm for the three-dimensional; similar procedures are used to solve the accompanying linear aerodynamic sensitivity equations in incremental iterative form. As shown, this particular form of the sensitivity equation makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory efficient methods to construct exact Jacobian matrix-vector products. Simple parameterization techniques are utilized for demonstrative purposes. Once the surface has been deformed, the unstructured grid is adapted by considering the mesh as a system of interconnecte d springs. Grid sensitivities are obtained by differentiati ng the surface parameterization and the grid adaptation algorithms with ADIFOR (which is an advanced automatic-differentiation software tool). To demonstrate the ability of this procedure to analyze and design complex configurations of practical interest, the sensitivity analysis and shape optimization has been performed for a two-dimensiona l high-lift multielement airfoil and for a three-dimensio nal Boeing 747-200 aircraft.

High-Order Finite-Element Method and Dynamic Adaptation for Two-Dimensional Laminar and Turbulent Navier-Stokes

A dynamic adaptation algorithm has been implemented within a streamline/upwind Petrov-Galerkin (SUPG) finite-element method. The proposed adaptation is able to perform multi-level h-, p-, and hp-refinement/derefinement and can be utilized within any continuous Galerkin formulation. To consistently account for hanging nodes, constrained approximation method is utilized. To demonstrate the developed methodology, the Euler and Reynolds Average NavierStokes (RANS) equations, equipped with a modified Spalart-Allmaras (SA) turbulence model, are used. A fully implicit linearization is used to advance each iteration or time-step, for steady-state or unsteady simulations, respectively. Adjoint-based and feature-based adaptations are employed in several numerical examples to assess the capability of the current approach. These examples include the comparison of adjoint-based h-, p-, and hp-adaptation for steady inviscid flow over a four element airfoil, adjoint-based h-adaptation for steady turbulent flow over a three element airfoil, and dynamic feature-based hand p-adaptation for laminar flow over a cylinder. Results illustrate consistent accuracy improvement of the functional outputs and also capability enhancement in capturing typical viscous effects such as flow separation, vortex shedding, and turbulent flow structures.

Investigation of Unstructured Higher-Order Methods for Unsteady flow and Moving Domains

An existing Petrov-Galerkin finite-element method has been extended to include highorder temporal accuracy using several schemes. Unlike finite-volume and discontinuous Galerkin finite-element methods, the mass matrix is not constant in stabilized finite-element formulations. Therefore, within the existing literature, it is uncommon to find semi-discrete stabilized finite-element schemes with time accuracy greater than second-order. The temporal integration methods investigated are the backward differentiation formula (BDF), the modified extended backward differentiation formula (MEBDF), the two implicit advanced step-point (TIAS) method, and a stiffly accurate singly diagonally implicit RungeKutta (SDIRK) scheme. Furthermore, it is shown that these methods may be recast into a single algorithm whereby only the number of stages and coefficients need be specified. Using the method of manufactured solutions, results are shown that demonstrate that each method achieves its design order of accuracy. Additionally, the convergence of each scheme is examined for a time-dependent, moving boundary problem.

Extension of the Petrov-Galerkin Time-Domain Algorithm for Dispersive Media

The extension of an implicit, high-order, Petrov-Galerkin, time-domain, finite-element method for application to dispersive materials is derived and implemented. The resulting scheme does not require additional source terms to be added to Maxwells curl equations. While the emphasis of this research is the continued development of the Petrov-Galerkin algorithm for electromagnetic applications, the current formulation can also be used in a discontinuous-Galerkin scheme.

Stabilized Finite Elements in FUN3D

A streamlined upwind Petrov–Galerkin (SUPG)–stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of ...

Computational optimization and sensitivity analysis of fuel reformer

Abstract In this study, the catalytic partial oxidation of methane is numerically investigated using an unstructured, implicit, fully coupled finite volume approach. The nonlinear system of equations is solved by Newton’s method. The catalytic partial oxidation of methane over rhodium catalyst in a coated honeycomb reactor is studied three-dimensionally, and eight gas-phase species (CH 4 , CO 2 , H 2 O, N 2 , O 2 , CO, OH and H 2 ) are considered for the simulation. Surface chemistry is modeled by detailed reaction mechanism including 38 heterogeneous reactions with 20 surface-adsorbed species for the Rh catalyst. The numerical results are compared with experimental data and good agreement is observed. Effects of the design variables, which include the inlet velocity, methane/oxygen ratio, catalytic wall temperature, and catalyst loading on the cost functions representing methane conversion and hydrogen production, are numerically investigated. The sensitivity analysis for the reactor is performed using three different approaches: finite difference, direct differentiation and an adjoint method. Two gradient-based design optimization algorithms are utilized to improve the reactor performance.

An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

In this study, an adjoint-based hp-adaptation algorithm has been developed within a Petrov-Galerkin finite-element method. The developed mesh adaptation algorithm is able to perform non-conformal mesh adaptations. To consistently account for hanging nodes, the constrained approximation method has been utilized. Hierarchical basis functions have been employed to facilitate the implementation of the constrained approximation. The methodology has been demonstrated on numerous cases using the Euler and Reynolds Average Navier-Stokes (RANS) equations, equipped with a modified SpalartAllmaras (SA) turbulence model. Also, a PDE-based artificial viscosity has been added to the governing equations, to stabilize the solution in the vicinity of shock waves. For accurate representation of the geometric surfaces, high-order curved boundary meshes have been generated and the interior meshes have been deformed through the solution of a modified linear elasticity equation. Fully implicit linearization has been used to advance the solution toward a steady-state. Dirichlet boundary conditions have been imposed weakly and the functional outputs have been modified according to the weak boundary conditions in order to provide a smooth adjoint solution near the boundaries. To accelerate the error reduction in presence of singularity points, an enhanced hrefinement, based on solution’s smoothness, has been used. Numerical results illustrate consistent accuracy improvement of the functional outputs for both hand hpadaptation, and also capability enhancement in capturing complex viscous effects such as shock-wave/turbulent boundary layer interaction.