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Shape Sensitivity Analysis in Aerodynamics Using an Isogeometric Discontinuous Galerkin Method

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A new discontinuous Galerkin (DG) method is introduced that seamlessly merges exact geometry with high-order solution accuracy. This new method is called the blended isogeometric discontinuous Galerkin (BIDG) method. The BIDG method contrasts with existing high-order accurate DG methods over curvilinear meshes (e.g. classical isoparametric DG methods) in that the underlying geometry is exactly preserved at every mesh refinement level, allowing for intricate and complicated real-world mesh design to be streamlined and automated using computer-aided design (CAD) software. The BIDG method is designed specifically for easy incorporation into existing code architecture. This paper discusses specific details of implementation using two examples: (1) the acoustic wave equations, and (2) Maxwell’s equations. Basic tests of accuracy and stability are demonstrated, including optimal convergence, and supplemental theoretical results are provided in the appendix along with links to fully operational working code examples.
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Chapter
In this chapter we apply the mathematical tools presented in Chap. 2 to analyse some of the basic properties of the time–dependent Euler equations. As seen in Chap. 1, the Euler equations result from neglecting the effects of viscosity, heat conduction and body forces on a compressible medium. Here we show that these equations are a system of hyperbolic conservations laws and study some of their mathematical properties. In particular, we study those properties that are essential for finding the solution of the Riemann problem in Chap. 4. We analyse the eigenstructure of the equations, that is, we find eigenvalues and eigenvectors; we study properties of the characteristic fields and establish basic relations across rarefactions, contacts and shock waves. It is worth remarking that the process of finding eigenvalues and eigenvectors usually involves a fair amount of algebra as well as some familiarity with basic physical quantities and their relations. For very complex systems of equations finding eigenvalues and eigenvectors may require the use of symbolic manipulators. Useful background reading for this chapter is found in Chaps. 1 and 2.
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We present an uncertainty analysis for free convection in corn syrup. One shape-and three valueparameters are considered: height of the enclosure, viscosity and conductivity of corn syrup, and the temperature applied to the heating element. A general formulation of the continuous sensitivity equation that accounts for complex parameter dependence in physical fluid properties is used. In this case we are interested in temperature dependent viscosity, conductivity, and specific heat. We use the parameter uncertainties reported in the work of Chu and Hickox. Uncertainty bars of predicted and measured velocity and temperature show significant overlap. Thus, our computational predictions are considered to be in excellent agreement with measurements. © 2001 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc.
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A Sensitivity Equation Method (SEM) for shape parameters is presented and used for fast evaluation of nearby flows via linear Taylor series in parameter space. A detailed methodology is described for the extrapolation of surface and global quantities of pratical interest for engineers (Cp, Cf, CL and CD) for parameter dependent geometries. The proposed approach is first verified on a problem with an exact solution. The method is then applied to viscous flows around a cylinder and several NACA airfoils to investigate the ability of the method to deal with non trivial geometrical changes.
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The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a hybrid scheme that captures some of the strengths of each strategy. With this compromise, discrete aerodynamic sensitivity derivatives are calculated with the efficient incremental iterative solution algorithm of the original flow code. Moreover, the principal advantage of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accuracy). The basic equations for second-order sensitivity derivatives are presented, which results in a comparison of four different methods. Each of these four schemes for second-order derivatives requires that large systems are solved first for the first-order derivatives and, in all but one method, for the first-order adjoint variables. Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems are solved, the equations and solution procedures are analogous to those for the first-order derivatives. From a practical viewpoint, implementation of the second-order methods is feasible only with software tools such as automatic differentiation, because of the extreme complexity and large number of terms. First- and second-order sensitivities are calculated accurately for two airfoil problems, including a turbulent-flow example. In each of these two sample problems, three dependent variables (coefficients of lift, drag, and pitching-moment) and six independent variables (three geometric-shape and three flow-condition design variables) are considered. Several different procedures are tested, and results are compared on the basis of accuracy, computational time, and computer memory. For first-order derivatives, the hybrid incremental iterative scheme obtained with automatic differentiation is competitive with the best hand-differentiated method. Furthermore, it is at least two to four times faster than central finite differences, without an overwhelming penalty in computer memory.
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Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.
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Over the last fifty years, the ability to carry out analysis as a precursor to decision making in engineering design has increased dramatically. In particular, the advent of modern computing systems and the development of advanced numerical methods have made computational modelling a vital tool for producing optimized designs. This text explores how computer-aided analysis has revolutionized aerospace engineering, providing a comprehensive coverage of the latest technologies underpinning advanced computational design. Worked case studies and over 500 references to the primary research literature allow the reader to gain a full understanding of the technology, giving a valuable insight into the world's most complex engineering systems. Key Features: Includes background information on the history of aerospace design and established optimization, geometrical and mathematical modelling techniques, setting recent engineering developments in a relevant context. Examines the latest methods such as evolutionary and response surface based optimization, adjoint and numerically differentiated sensitivity codes, uncertainty analysis, and concurrent systems integration schemes using grid-based computing. Methods are illustrated with real-world applications of structural statics, dynamics and fluid mechanics to satellite, aircraft and aero-engine design problems. Senior undergraduate and postgraduate engineering students taking courses in aerospace, vehicle and engine design will find this a valuable resource. It will also be useful for practising engineers and researchers working on computational approaches to design.
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In this paper we present a synthetic method to differentiate with respect to a parameter partial differential equations in divergence form with shocks. We show that the usual derivatives contain the differentiated interface conditions if interpreted by the theory of distributions. We apply the method to three problems: the Burgers equation, the shallow water equations and Euler equations for fluids. To cite this article: C. Bardos, O. Pironneau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 839–845.
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This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier–Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least-squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite-element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd.
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In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.
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A new approach for optimal shape design based on a CAD-free framework for shape and unstructured mesh deformations, automatic differentiation for the gradient computation and mesh adaption by metric control in 2D is presented. The CAD-free framework is shown to be particularly convenient for optimization when the mesh connectivities and control space size are variable during optimization. Constrained optimization for a transonic regime has been investigated in both 2D and 3D. Copyright © 1999 John Wiley & Sons, Ltd.
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This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.
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The use of gradient-based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design. A typical procedure uses a simulation scheme to evaluate the objective function (from the approximate states) and its gradient, then passes this information to an optimization algorithm. Once the simulation scheme (CFD flow solver) has been selected and used to provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients. Although this black-box approach has many advantages in shape optimization problems, one must compute mesh sensitivities in order to compute the design sensitivity. In this paper, we present an alternative approach using the PDE sensitivity equation to develop algorithms for computing gradients. This approach has the advantage that mesh sensitivities need not be computed. Moreover, when it is possible to use the CFD scheme for both the forward problem and the sensitivity equation, then there are computational advantages. An apparent disadvantage of this approach is that it does not always produce consistent derivatives. However, for a proper combination of discretization schemes, one can showasymptotic consistencyunder mesh refinement, which is often sufficient to guarantee convergence of the optimal design algorithm. In particular, we show that when asymptotically consistent schemes are combined with a trust-region optimization algorithm, the resulting optimal design method converges. We denote this approach as thesensitivity equation method.The sensitivity equation method is presented, convergence results are given, and the approach is illustrated on two optimal design problems involving shocks.
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A new mesh movement algorithm for unstructured grids is developed which is based on interpolating displacements of the boundary nodes to the whole mesh with radial basis functions (RBF’s). A small system of equations, only involving the boundary nodes, has to be solved and no grid-connectivity information is needed. The method can handle large mesh deformations caused by translations, rotations and deformations, both for 2D and 3D meshes. However, the performance depends on the used RBF. The best accuracy and robustness with the highest efficiency are obtained with a C2 continuous RBF with compact support, closely followed by the thin plate spline. The deformed meshes are suitable for flow computations as is shown by performing calculations around a NACA-0012 airfoil.
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The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h- and p-refinement schemes are presented and a new, more efficient, higher-order concept, k-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A k-refinement strategy is shown to converge toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.
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this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L
Uncertainty management in Simulation-Optimization of Complex Systems: Algorithms and Applications
  • B Iooss
  • P Lemaitre
B. Iooss and P. Lemaitre, Uncertainty management in Simulation-Optimization of Complex Systems: Algorithms and Applications, Springer, 2015, ch. A Review on Global Sensitivity Analysis Methods, pp. 101-122.
Application of a sensitivity equation method to the k − model of turbulence
  • É Turgeon
  • D Pelletier
  • J Borggaard
É. Turgeon, D. Pelletier, and J. Borggaard, Application of a sensitivity equation method to the k − model of turbulence, in 15th AIAA Computational Fluid Dynamics Conference, Anaheim, CA, Jun. 2001.