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  • Adjoint Based Anisotropic Mesh Adaptation for the

    CPR Method

    Lei Shi and Z.J. Wang

    Department of Aerospace Engineering, University of Kansas, Lawrence, KS, 66045

    Adjoint-based adaptive methods have the capability of dynamically distributing comput-ing power to areas which are important for predicting an engineering output such as lift ordrag. In this paper, we apply an anisotropic h-adaptation method for simplex meshes usingthe correction procedure via reconstruction(CPR) method with the target of minimizingthe output error. An adjoint-based error estimation with a local refinement sampling pro-cess is utilized to drive the anisotropic mesh refinement without making any assumptionabout the solution features. The accuracy and efficiency of the isotropic and anisotropicadaptation strategies are compared on several 2D inviscid flow problems.

    I. Introduction

    High order methods have the potential of delivering higher accuracy with less CPU time than lower ordermethods. However, for a compact scheme, the number of degrees of freedom (DOF) rises rapidly withthe increased order of accuracy, which affects the prevalence of high-order methods in industry applications.Solution based adaptive methods has the capability of dynamically distributing computing power to a desiredarea to achieve required accuracy with minimal costs.[13] For this reason, adaptive high-order methods havereceived considerable attention in the high-order CFD community. [49].

    The truncation error of a spatial discretization is determined by the mesh size and the order of thepolynomial approximation. P-adaptations can be applied to smooth regions while h-adaptations are preferredfor shock waves or singularities[10, 11]. The discretization of a compact scheme only depends on a small stencilof local DOFs. This compact support simplifies the task of hp-adaptations involving complex geometries.Several compact high-order methods for unstructured meshes have been developed, such as the discontinuousGalerkin(DG) method [10, 12, 13], the spectral volume (SV) method [14, 15] and the spectral difference (SD)method [16]. Unlike the finite volume method that achieves high-order by expanding their reconstructionstencil, the above methods employ local DOFs to support high-order piecewise solution polynomials in eachelement, and the interaction between the local cell and its neighbors is provided by the common flux at theboundary. Recently, the flux reconstruction or the correction procedure via reconstruction (CPR) formulationwas developed in 1D [17], and further extended in Ref. [1823]. It is a nodal differential formulation whichcan unite several well known high-order methods such as DG, SV and SD. The CPR formulation combinesthe compactness and high accuracy with the simplicity and efficiency of the finite difference method, andcan be easily implemented for mixed unstructured meshes.

    The effectiveness of adaptation methods highly depends on the accuracy of error estimations. There areat least three major types of adaptation criteria: gradient or feature based [2427], residual-based [2833],and adjoint-based [48, 3442]. Heuristic feature-based criterion such as large gradient cannot provide anuniversal and robust error estimation [5, 43]. The residual-based error indicator which is defined locallyon each element has had some successes; however, it can lead to false refinements in convection-dominatedflow. Adjoint-based error estimations relates a specific functional output directly to the local residual bysolving an additional adjoint problem, and is gaining a lot of research attention, which relates a specificfunctional output directly to the local residual by solving an additional adjoint problem. It can capturethe propagation effects inherent in hyperbolic equations and has been shown very effective in driving an

    PhD Student, Department of Aerospace Engineering, 2120 Learned Hall, Lawrence, KS 66045, AIAA Member.Spahr Professor and Chair, Department of Aerospace Engineering, 2120 Learned Hall, Lawrence, KS 66045, Associate

    Fellow of AIAA.

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    21st AIAA Computational Fluid Dynamics Conference

    June 24-27, 2013, San Diego, CA

    AIAA 2013-2869

    Copyright 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

  • hp-adaptation procedure to obtain a very accurate prediction of the functional output. Recently Fidkowskiand P.L Roe developed a new error indicator based on the entropy variables to drive an hp-adaptation forinviscid and viscous flow. Entropy variables can be interpreted as the dual solution for the output of entropybalance in the whole domain. It can be obtained directly from the state variables without solving extraadjoint equations.[44, 45]

    Compressible viscous flow may produce strong directional phenomena, such as boundary layers, shearlayers and shock waves. For isotropic adaptation, each cell is subdivided into four elements in 2D and eightelements in 3D, which is very costly to resolve these behaviors. In contrast, stretched elements with highaspect ratio are preferred for optimal resolution of anisotropic features. Considerable work has been devotedto the adjoint based anisotropic adaptation. A common and simple approach to incorporate the directionalinformation for adaptation is to use the Hessian-based metric field of a solution variable, which representsthe interpolation error[2, 46]. However, it does not provide any information for the functional error. Vendittiand Darmofal [5] have extended the Hessian-based metric of the Mach number to the dual weighted metricswith size information. While similar techniques have been applied in the Ref. [3, 7, 4749], their anisotropydecision requires a priori knowledge of the solution. Furthermore, the directional information does notdirectly relate to the functional error. Recently, a popular approach to drive anisotropic adaptation is toperform a output error sampling procedure from a discrete set of refinement choices. The idea of guidinganisotropy adaptation for the engineering output by solving local problems has been previously proposedin the Ref. [39, 41, 42, 50]. During the trial refinements process, the elemental functional error is directlyestimated and monitored. In this paper, we use this sampling procedure to drive the anisotropic adaptation.

    The adjoint solution is particularly important for the error estimation and output-based adaptation.There are two approaches to obtain the adjoint solution for primal problems. We can solve the continuousadjoint equation which is a partial differential equation using any numerical method or directly solve thediscrete adjoint equation derived from the discretized primal equation. It has been shown that the discreteadjoint solution leads to a more accurate error estimation for the fine grid functional, while continuousadjoints gives better output estimation when the primal and adjoint solutions are well resolved [51]. However,the discrete adjoint solution should be consistent with the exact adjoint from the continuous adjoint equation.It is well known that the dual consistency can significantly impact the convergence of both the primal andadjoint approximations. There are several possible sources of dual inconsistency that can be introduced intoa high-order discretization. A dual consistent discretization with semilinear forms such as the finite elementand DG methods have been well examined for the Euler and Navier-Stokes equations [34, 37, 52, 53].However, the analysis of dual consistency for differential-type methods has not been well investigated, whichis one of the focuses of the present paper.

    The rest of the paper is organized as follows: In section 2 we briefly review the high-order CPR method.The continuous and discrete adjoint equations and the dual consistent discretization of the CPR method aredescribed in Section 3. Then section 4 presents adaptation strategies and procedures for hp-adaptations.Finally, several numerical test cases are presented in Section 5, and conclusions are given in section 6.

    II. Review of the CPR Method

    For the sake of completeness, the CPR formulation is briefly reviewed. The CPR formulation wasoriginally developed by Huynh in Ref. [17, 54] under the name of flux reconstruction, and extended tosimplex and hybrid elements by Wang & Gao in Ref. [18] under lifting collocation penalty. In Ref. [55], CPRwas further extended to 3D hybrid meshes. The method is also described in two book chapters [56]. CPRcan be derived from a weighted residual method by transforming the integral formulation into a differentialone. First, a hyperbolic conservation law can be written as

    Q

    t+ ~F(Q) = 0 (1)

    with proper initial and boundary conditions, where Q is the state vector, and ~F (F,G) is the flux vector.Assume that the computational domain is discretized into N non-overlapping triangular elements {Vi}Ni=1Let W be an arbitrary weighting function or test function. The weighted residual formulation of Eq. (1) onelement Vi can be expressed as

    Vi

    (Q

    t+ ~F(Q))W dV = 0. (2)

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    Copyright 2013 by the American Institute of Aer

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