High order methods have the potential of delivering higher accuracy with less CPU time than lower order

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1 21st AIAA Computational Fluid Dynamics Conference June 24-27, 2013, San Diego, CA AIAA Adjoint Based Anisotropic Mesh Adaptation for the CPR Method Lei Shi and Z.J. Wang Department of Aerospace Engineering, University of Kansas, Lawrence, KS, Adjoint-based adaptive methods have the capability of dynamically distributing computing power to areas which are important for predicting an engineering output such as lift or drag. In this paper, we apply an anisotropic h-adaptation method for simplex meshes using the correction procedure via reconstruction(cpr) method with the target of minimizing the output error. An adjoint-based error estimation with a local refinement sampling process is utilized to drive the anisotropic mesh refinement without making any assumption about the solution features. The accuracy and efficiency of the isotropic and anisotropic adaptation 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 order methods. However, for a compact scheme, the number of degrees of freedom (DOF) rises rapidly with the 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 desired area to achieve required accuracy with minimal costs.[1 3] For this reason, adaptive high-order methods have received considerable attention in the high-order CFD community. [4 9]. The truncation error of a spatial discretization is determined by the mesh size and the order of the polynomial approximation. P-adaptations can be applied to smooth regions while h-adaptations are preferred for shock waves or singularities[10, 11]. The discretization of a compact scheme only depends on a small stencil of 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 discontinuous Galerkin(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 reconstruction stencil, the above methods employ local DOFs to support high-order piecewise solution polynomials in each element, and the interaction between the local cell and its neighbors is provided by the common flux at the boundary. Recently, the flux reconstruction or the correction procedure via reconstruction (CPR) formulation was developed in 1D [17], and further extended in Ref. [18 23]. It is a nodal differential formulation which can unite several well known high-order methods such as DG, SV and SD. The CPR formulation combines the compactness and high accuracy with the simplicity and efficiency of the finite difference method, and can be easily implemented for mixed unstructured meshes. The effectiveness of adaptation methods highly depends on the accuracy of error estimations. There are at least three major types of adaptation criteria: gradient or feature based [24 27], residual-based [28 33], and adjoint-based [4 8, 34 42]. Heuristic feature-based criterion such as large gradient cannot provide an universal and robust error estimation [5, 43]. The residual-based error indicator which is defined locally on each element has had some successes; however, it can lead to false refinements in convection-dominated flow. Adjoint-based error estimations relates a specific functional output directly to the local residual by solving an additional adjoint problem, and is gaining a lot of research attention, which relates a specific functional output directly to the local residual by solving an additional adjoint problem. It can capture the 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. Copyright 2013 by the, Inc. All rights reserved. 1 of 18

2 hp-adaptation procedure to obtain a very accurate prediction of the functional output. Recently Fidkowski and P.L Roe developed a new error indicator based on the entropy variables to drive an hp-adaptation for inviscid and viscous flow. Entropy variables can be interpreted as the dual solution for the output of entropy balance in the whole domain. It can be obtained directly from the state variables without solving extra adjoint equations.[44, 45] Compressible viscous flow may produce strong directional phenomena, such as boundary layers, shear layers and shock waves. For isotropic adaptation, each cell is subdivided into four elements in 2D and eight elements in 3D, which is very costly to resolve these behaviors. In contrast, stretched elements with high aspect ratio are preferred for optimal resolution of anisotropic features. Considerable work has been devoted to the adjoint based anisotropic adaptation. A common and simple approach to incorporate the directional information for adaptation is to use the Hessian-based metric field of a solution variable, which represents the interpolation error[2, 46]. However, it does not provide any information for the functional error. Venditti and Darmofal [5] have extended the Hessian-based metric of the Mach number to the dual weighted metrics with size information. While similar techniques have been applied in the Ref. [3, 7, 47 49], their anisotropy decision requires a priori knowledge of the solution. Furthermore, the directional information does not directly relate to the functional error. Recently, a popular approach to drive anisotropic adaptation is to perform a output error sampling procedure from a discrete set of refinement choices. The idea of guiding anisotropy adaptation for the engineering output by solving local problems has been previously proposed in the Ref. [39, 41, 42, 50]. During the trial refinements process, the elemental functional error is directly estimated 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 continuous adjoint equation which is a partial differential equation using any numerical method or directly solve the discrete adjoint equation derived from the discretized primal equation. It has been shown that the discrete adjoint solution leads to a more accurate error estimation for the fine grid functional, while continuous adjoints 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 and adjoint approximations. There are several possible sources of dual inconsistency that can be introduced into a high-order discretization. A dual consistent discretization with semilinear forms such as the finite element and 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, which is 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 are described 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 was originally developed by Huynh in Ref. [17, 54] under the name of flux reconstruction, and extended to simplex and hybrid elements by Wang & Gao in Ref. [18] under lifting collocation penalty. In Ref. [55], CPR was further extended to 3D hybrid meshes. The method is also described in two book chapters [56]. CPR can be derived from a weighted residual method by transforming the integral formulation into a differential one. 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 Ω is discretized into N non-overlapping triangular elements { } N i=1 Let W be an arbitrary weighting function or test function. The weighted residual formulation of Eq. (1) on element can be expressed as ( Q t + F(Q))W dv = 0. (2) Copyright 2013 by the, Inc. All rights reserved. 2 of 18

3 After applying integration by parts to the flux divergence, we can get Q t W dv + W F(Q) n ds W F(Q) dv = 0. (3) Let Q i be an approximate solution to the analytical solution Q on. On each element, the solution belongs to the space of polynomials of degree k or less, i.e., Q i P k ( ) (or P k if there is no confusion) with no continuity requirement across element interfaces. Let the dimension of P k be K = (k + 1)(k + 2)/2. In addition, the numerical solution Q i, for the moment, is required to satisfy Eq. (3) as Q i t W dv + W F(Q i ) n ds W F(Q i ) dv = 0. (4) Obviously the surface integral is not properly defined because the numerical solution is discontinuous across element interfaces. Following the idea used in the Godunov method [57, 58], the normal flux term in Eq. (4) is replaced with a common Riemann flux, e.g., in Ref. [59 61] F n (Q i ) F(Q i ) n F n com(q i, Q i+, n), (5) where Q i+ denotes the solution outside the current element. Instead of Eq. (4), the approximate solution is required to satisfy Q i t W dv + W Fcom n ds W F(Q i ) dv = 0. (6) Applying integration by parts again to the last term of the above LHS, we obtain Q i t W dv + W F(Q i ) dv + W [Fcom n F n (Q i )] ds = 0. (7) Here, the test space has the same dimension as the solution space, and is chosen in a manner to guarantee the existence and uniqueness of the numerical solution. Note that the quantity F(Q i ) involves no influence from the data in the neighboring cells. The interaction between the current cell and its neighbors is represented by the above boundary integral, which is also called a penalty term, penalizing the normal flux differences. The next step is critical in the elimination of the test function. The boundary integral above is cast as a volume integral via the introduction of a correction field on, δ i P k ( ), W δ i dv = W [F n ] ds, (8) where [F n ] = Fcom n F n (Q i ) is the normal flux difference. The above equation is sometimes referred to as the lifting operator, which has the normal flux differences on the boundary as input and a member of P k ( ) as output. Substituting Eq. (8) into Eq. (7), we obtain ( Q i + t F(Q i ) + δ i )W dv = 0. (9) If the flux vector is a linear function of the state variable, then F(Q i ) P k. In this case, the terms inside the square bracket are all elements of P k. Because the test space is selected to ensure a unique solution, Eq. (9) is equivalent to Q i + t F(Q i ) + δ i = 0. (10) For nonlinear conservation laws, F(Q i ) is usually not an element of P k. As a result, Eq. (9) cannot be reduced to Eq. (10). In this case, the most obviously choice is to project F(Q i ) into P k. Denote Π( F(Q i )) as a projection of F(Q i ) to P k. One choice is Π( F(Q i ))W dv = F(Q i )W dv. (11) Copyright 2013 by the, Inc. All rights reserved. 3 of 18

4 Then Eq. (9) reduces to Q i t + Π( F(Q i )) + δ i = 0. (12) With the introduction of the correction field δ i, and a projection of F(Q i ) for nonlinear conservation laws, we have reduced the weighted residual formulation to a differential formulation, which involves no explicit integrals. Note that for δ i defined by Eq. (8), if W P k, Eq. (12) is equivalent to the DG formulation, at least for linear conservation laws; if W belongs to another space, the resulting δ i is different. We obtain a formulation corresponding to a different method such as the SV method. Next, let the DOFs be the solutions at a set of solution points (SPs) { r i,j } (j varies from 1 to K), as shown in Figure 1. Then Eq. (12) holds true at the SPs, i.e., Q i,j t + Π j ( F(Q i )) + δ i,j = 0, (13) where Π j ( F(Q i )) denotes the values of Π( F(Q i )) at SP j. The efficiency of the CPR approach hinges on how the correction field δ i and the projection Π( F(Q i )) are computed. Two approaches can be used to compute this divergence as detailed in Ref. [18]. To compute δ i, we define k+1 points named flux points Figure 1. Gaussian solution points(sp) and flux points(fp) for k=2 ( -SP, -FP) (FPs) along each interface, where the normal flux differences are computed, as shown in Figure 1. We approximate (for nonlinear conservation laws) the normal flux difference [F n ] with a degree k interpolation polynomial along each interface [F n ] f I k [F n ] f l [F n ] f,l L F P l, (14) where f is a face (or edge in 2D) index, and l is the FP index, and L F l P is the Lagrange interpolation polynomial based on the FPs in a local interface coordinate. For linear triangles with straight edges, once the solution points and flux points are chosen, the correction at the SPs can be written as δ i,j = 1 f α j,f,l [F n ] f,l S f, (15) l where α j,f,l are lifting constants independent of the solution variables, S f is the face area, is the volume of. Note that the correction for each solution point, namely δ i,j, is a linear combination of all the normal flux differences on all the faces of the cell. Conversely, a normal flux difference at a flux point on a face, say (f,l) results in a correction at all solution points j of an amount α j,f,l [F n ] f,l S f /. III. Adjoint-Based Error Estimation III.A. The Continuous Adjoint Equation Adjoint-based error estimation can directly relate the local residual error from the primal equation to the engineering output. The accuracy of adjoint solution is particularly important for accurate error estimation. There are two approaches to obtain the adjoint solution. We can solve the continuous adjoint equation which Copyright 2013 by the, Inc. All rights reserved. 4 of 18

5 is a partial differential equation using any numerical method or directly solve the discrete adjoint equation derived from the discretized primal equation. As for the primal problem, a numerical scheme is defined as a consistent method if its discrete operator converges to the continuous operator, or the exact solution could satisfy the discrete numerical formulation as the mesh size approach to zero. Similarly, dual-consistency is defined as the exact adjoint solution from the continuous adjoint equation should satisfy the discrete adjoint equation. In order to analyze the dual consistency of the CPR method, we need to derive the continuous adjoint equation and its boundary conditions first. Consider a primal differential equation as a conservation law N (Q) = F(Q) = 0. (16) Given a scalar output J (Q) of interest, which may consist of surface ( Ω) and volume (Ω) integration in general as J (Q) = J Ω (Q) dω + J τ (Q) ds. (17) Ω Ω We can define a Lagrangian of the output with the constraint of the solution Q satisfies the primal equation N (Q) = 0, which leads to L = J (Q) + ψ T N (Q) dω. (18) Ω Here ψ is the adjoint solution, furthermore, it also serves as a Lagrangian multiplier [45, 62]. Let the Frechet linearization with respect to an argument in the square bracket defined as J [Q](δQ) = J (Q + δq) J (Q) = δj = J δq. (19) Q After enforcing stationary of L to a permissible variation δq, which is belong to the space of permissible state variations δq V perm, Eq. (18) yields the linearized Lagrangian or the adjoint equation L [Q](δQ) = J [Q](δQ) + ψ T N [Q](δQ) dω = 0 δq V perm. (20) Ω Plug the definition of Frechet linearization into the above equation, the adjoint Eq. (20) can be expressed as ( J Q + T N (Q) ψ dω) δq = 0. (21) Q Ω After substituting the definition of the output J and the primal differential equation N (Q) in the adjoint Eq. (21) J Ω ( Ω Q dω + J τ Ω Q ds + ψ T F i dω) δq = 0, (22) Ω x i Q then performing integration by parts leads to [ ( J Ω Ω Q ψt F i x i Q ) dω] δq + [ ( J τ Ω Q + F ψt i Q n i) ds] δq = 0. (23) Most of time, the output J only consists of boundary integrations, which means J Ω = 0, so the above equation yields the governing equation for the continuous adjoint F T i ψ = 0, (24) Q x i and the corresponding boundary conditions defined as [ ( J τ Q ds + F ψt i Q n i) ds] δq = 0 δq V perm. (25) Ω The continuous adjoint equation is a linear partial differential equation which can be solved using any numerical method. An adjoint solution can be used to perform error estimation for an engineering output. First the output error is defined as the difference between the functional evaluated with the analytical solution Copyright 2013 by the, Inc. All rights reserved. 5 of 18

6 Q and the discrete numerical solution Q h, which can be further approximated using a linear analysis after dropping the high order terms as δj = J (Q h ) J (Q) J [Q](δQ). (26) With Eq. (20), the output error can also be expressed as δj J [Q](δQ) = ψ T N [Q](δQ) dω. (27) Here, N [Q](δQ) is the residual error for the primal problem induced by the primal discretization δq = Q Q h Ω δn = N (Q h ) N (Q) = N (Q h ) N [Q](δQ). (28) So we can express the output error in the form of the adjoint solution weighted with the primal residual δj = ψ T N (Q h ) dω. (29) III.B. The Discrete Adjoint Equation Ω As shown in the previous session, the continuous adjoint equation is a partial differential equation, which is derived directly from the linearized primal equation and the linearized functional outputs. Obviously, the numerical scheme for the continuous adjoint equation and the primal governing equation can be different. For the discrete adjoint approach, the discrete adjoint equation is directly result from linearizing the discretized primal equation. In this session, discrete adjoint formulations for schemes with a variational form and differential-type schemes are presented and their difference are compared. For the fully discrete formulation, we need to consider two approximation levels. Here, H stands for a coarse level and h denotes as a fine level. In practise, the coarse and fine levels of the approximation can be achieved using h-refinement or p-enrichment. The output error between those two different approximations can be denoted as δj h = J h (Q H ) J h (Q h ). (30) Our target is to estimate the output J h (Q h ) at fine level without solving the primal solution on the fine space. With the Taylor expansion, the output J h (Q h ) and the residual R h (Q h ) at the fine space h can be expanded to a prolongated fine solution Q H h from a coarse solution Q H J h (Q h ) = J h (Q H h ) + J h Q h Q H h (Q h Q H h ) +... (31) R h (Q h ) = R h (Q H h ) + R h Q Q H h (Q h Q H h ) +... h with the assumption of R h (Q h ) = 0. The prolongated solution Q H h = IH h Q H can be obtained using the injection operator Ih H. After dropping high order terms and canceling the solution difference (Q h Q H h ) between the truth solution at the fine level and the prolongated solution, the above equation can be written as J h (Q h ) J h (Q H h ) J h Q Q H h ( R h h Q Q H h ) 1 R h (Q H h ) (32) }{{ h } J h (Q H h ) + ( ˆψ h Q H h ) T R h (Q H h ) from where we can define the equation for the discrete adjoint solution ˆψ h as ( ˆψ h ) T R h Q h = J h Q h. After transposing both sides of the above equation, we can express the fully discrete adjoint equation in the following form R T h ˆψh = J T h. (33) Q h Q h Copyright 2013 by the, Inc. All rights reserved. 6 of 18

7 For numerical methods with a weak form such as FEM methods or DG methods, after choosing a proper basis, Eq. (33) is equivalent to its variational formulation. Detailed derivations can be found in Ref. [9, 52]. The fully discrete adjoint solution for numerical schemes in semilinear form is consistent with the continuous adjoint equation. However, this is not true for numerical schemes in differential form such as the CPR method, which does not born with a variational form. Let r(q) i,j denotes a pointwise residual of a differential scheme defined at each solution point j of cell i r(q) i,j = ( F(Q i )) j. (34) If the CPR method is used for discretizing the primal equation, the pointwise residual r i,j can be expressed as r(q) i,j = Π( F(Q i )) j + 1 α j,f,l [F n ] f,l S f. (35) Substitute the pointwise residual r i,j arising from a differential scheme, the fully discrete adjoint Eq. (33) can be written as i j f l r i,j ψi,j = J, (36) Q k Q k where k is index of total DOFs in the whole domain. However, this approach is not consistent with continuous adjoint equation. If we assume the adjoint solution belongs to the same space of the primal solution and approximate the adjoint variable ψ of the cell i using the Lagrange basis L j ψ i = j L j ψˆ i,j. (37) With the above equation, directly discretizing the continuous adjoint Eq. (21) leads to i j Ω T N (Q) ψ dω = J T Q Q r i,j ω j J i,j Q ˆψ i,j = J, (38) k Q k where ω j and J i,j are the quadrature weight and the element Jacobian at the solution point j of cell i. Compared with Eq. (36), the following relation can be derived between the discrete adjoints ψ i,j and the continuous adjoint ˆψ i,j ψi,j = ω j J i,j ˆψ i,j. (39) So the fully discrete adjoint formula for a numerical scheme in differential forms is not consistent with the continuous adjoint equation. The only difference between them are the quadrature weights ω and cell Jacobian J at each solution point. The discrete adjoint formula for differential schemes should be derived in integral form. Alternatively, an explicit weak formulation should be defined for numerical schemes in differential forms for the purpose of obtaining the discrete adjoint solution only. In the next session, a discrete adjoint equation in integral form for the CPR method, which is dual consistent with the continuous adjoint, is presented and verified using several numerical tests. III.C. Numerical Verifications of the Dual Consistent CPR Method Since there is no variational or weak form of the CPR method, its discrete adjoint equation can be directly derived from the continuous adjoint equation i j r i,j ω j J i,j Q ˆψ i,j = J, (40) k Q k which results an integral equation. This integral equation can be interpreted as an explicitly defined variational form for the CPR method, whose purpose is only to find the dual consistent discrete adjoint solution. Copyright 2013 by the, Inc. All rights reserved. 7 of 18

8 First, inviscid flow over a NACA 0012 airfoil is utilized to demonstrate the smoothness of the discrete adjoint solution from the dual consistent CPR method. This test case is used in Ref. [9]. The inflow condition is set to be M = 0.4 with an angle of attack of 5. The 3 rd order CPR formulation using Gaussian quadrature points as solution points and flux points is used to ensure the integration accuracy for solving the discrete adjoint equation in integral forms. The dual solution from the fully discrete adjoint formulation of the CPR method is shown in Figure 2(a), which is not dual consistent and has an irregular mosaic like distributions in every cell. Figure 2(b) shows the adjoint solution from the discrete adjoint equation in integral forms. The adjoint solution looks much more smooth than the fully discrete adjoint. Furthermore the square root singularity of adjoint solution with respect to distance from the stagnation streamline introduced in Ref. 34 are observed near the leading edge. (a) The fully discrete adjoint Figure 2. (b) The discrete adjoint in the integral form The x-momentum component of the lift adjoint for a NACA 0012 airfoil at M = 0.4, α = 5 The problem C1.3(a) of the 1 st international workshop on high-order methods is used to further assess the accuracy of the adjoint-based error estimation with the CPR method. This test case involves subsonic flow over a NACA 0012 airfoil with a free-stream Mach number of M = 0.5 and the angle of attack, α = 2. The output of interest is chosen as the lift of the airfoil. The error in the functional J H (Q H ) J h (Q h ) is computed using p-enrichment from p = 1 to p = 2 and the effectivity of the error estimation is defined as η e H = (ψ h) T R h (Q H h ) J H (Q H ) J h (Q h ) Table 1 shows the results with 4 levels of uniformly refined meshes from the high-order workshop. Note that the error of the initial lift estimation on the very coarse meshes are kind of large; however, the effectivity index ηh e approaches unity as the mesh refined. Table 1. Adjoint-based Error Estimation for the lift of a Subsonic NACA 0012 Airfoil at M = 0.5, α = 2 Cells J H (Q H ) J h (Q h ) (ψ h ) T R h (Q H h ) ηe H e e e e e e e e (41) In this test case, subsonic inviscid flow over a Gaussian-shaped bump is used to demonstrate the implication of the boundary flux for the discrete adjoint solutions near the walls. Again, 3 rd order CPR formulation using Gaussian quadrature points as the solution points and the flux points is employed. This test case get rid of the influence from geometry singularities and stagnation points, which is first used in Ref. 62. The channel has an height of 0.8 unit and a length of 3 unit. The bump geometry is defined as y = e 25x2 (42) Copyright 2013 by the, Inc. All rights reserved. 8 of 18

9 and the output is defined as a weighted lift on bump surface J = p(x, y)n y e 50x2 ds. (43) bump The characteristic boundary conditions are used at both the inlet and outlet. The inflow Mach number is set to be M = 0.5. To impose the boundary conditions on the no-slip walls in a dual-consistent manner, the common flux defined in Ref. 34,52,62 is used. As shown in Figure 3, the resulting discrete adjoint using the inconsistent boundary conditions have significant irregularity near the wall boundary, whereas the adjoints using the dual-consistent common flux is quite smooth in the whole domain. IV.A. Figure 3. (a) The result using the dual inconsistent numerical flux at wall boundaries (b) The result using the dual consistent numerical flux at wall boundaries The x-momentum component of the weighted lift adjoint for a Gaussian-shpaed bump IV. Adjoint-based Error Indicators Adjoint-based Anisotropic h-adaptations Adjoint-based error estimation relates a specific functional output directly to the local residuals by the adjoint solution, which can capture the propagation effects inherent in the hyperbolic equations. Therefore, the adjoint-based error estimates can form an effective error indicator to drive an adaptive refinement toward any engineering output. From the Eq. (30) and Eq. (44), we can estimate the output error as δj h (Q h ) = J h (Q H ) J h (Q h ) (44) ( ˆψ h ) T R h (Q H h ), where the fine solution Q h is approximated by prolongating from the low order to the high order discretization through Q H h = I H h Q H. (45) Then the adjoint-based local error indicator can be defined as η = ( ˆψ h ) T R h (Q H h ). (46) A so-called multi-p residual-based local error indicator can be obtained by evaluating the discretization residuals on the prolongated solution Q H h only: η = Rh(Q H h ). (47) Essentially the multi-p residual-based error estimator is a unweighted version of the adjoint-based error indicator with uniform adjoint ψ = 1 everywhere. Copyright 2013 by the, Inc. All rights reserved. 9 of 18

10 IV.B. Anisotropic h-adaptations The error indicators defined above are used to drive a fixed-fraction anisotropic h-adaptation. In this approach, a certain fraction f of the current elements with the largest local error indicators η are marked for h-refinements. Then the anisotropic adaptation decision is driven by an error sampling procedure for choosing the optimal refinement from a discrete set of adaptation choices. The idea of guiding anisotropy adaptation for the engineering output by solving local problems has been previously proposed in the Ref. [39, 41, 42, 50]. The elemental functional error is directly estimated and monitored during the sampling process. For a simplex element, we consider 4 local refinement options by splitting the edges, as shown in Figure 4. (a) The original coarse mesh (b) Isotropic refinement (c) Edge 1 Split (d) Edge 2 Split (e) Edge 3 Split Figure 4. The simplex refinement options for probing the local functional error behavior Mesh refinement is performed in the original element s polynomial space using the reference coordinates. So the refined elements inherit the same geometry approximation order. However, for elements on the geometry boundaries, the newly generated vertex on the boundary edge may not be exactly on the real geometry. An extra remapping process is employed to snap the boundary points to the truth geometry during each adaptation level. As shown in Figure 5, non-conforming interfaces between cells with different h levels are created during the adaptation process. In order to maintain the smoothness of the solution, at most one level of difference is allowed for h-refinement. Special treatment is required when computing the common numerical flux on those non-conforming interfaces. Basically, a L 2 projection approach is used to preserve conservation and maintain accuracy. Detailed procedures can be found in Ref. [32]. For the simplex anisotropic adaptations, as shown in Figure 6, the hanging nodes can be completely removed by refining its face neighbors. This is in contrast to the quadlateral meshes which always generates the hanging nodes after refinements. For each refinement option denoted as κ j of the candidate element i marked by the local error indicators η, an element-wise local problem is created and solved dynamically at each adaptation step. As shown in Figure 4, all of elements in the stencil of the candidate element i are created and the current primal solution Q H and adjoint solution ˆψ h are injected by the prolongating operator Iκ H j and Q H κ j = I H κ j Q H (48) ˆψ h κ j = I H κ j ˆψh. (49) Copyright 2013 by the, Inc. All rights reserved. 10 of 18

11 Figure 5. Hanging nodes with one level restrictions (a) Remove hanging nodes, 1 edge Figure 6. Remove hanging nodes for refinements (b) Remove hanging nodes, 2 edges The residual or perturbation created by the refinement option κ j is evaluated. Then the locale functional error indicator can be obtained using Finally, a simple merit indicator m κj η κj = ( ˆψ h κ j ) T R κj (Q H κ j ). (50) defined as m κj = benefit cost are used to pick up a particular refinement option in this paper. V.A. V. Numerical Results Subsonic Flow over a NACA 0012 Airfoil = η κ j DOF 2. (51) The first test case involves subsonic flow over a NACA 0012 airfoil with a free-stream Mach number of M = 0.5 and the angle of attack, α = 2. The truth lift coefficient of and drag coefficient of 2.3e 6 are chosen from the finest level of the h-adaptation result. In order to reduce the influence from the far field, the outer boundary is located 2000 chords away. The initial mesh is shown in Figure 7. All of the simulations undergo 12 levels of isotropic and anisotropic h-adaptations. For all of cases, the solution polynomial is p = 2 and refinement fraction f=0.1 is used. The drag coefficient and lift coefficient are considered as the output of interest. The final h-adapted meshes of each strategy are shown in Figure 10. For both of the anisotropic and isotropic adaptations, the regions near the trailing edge and the leading edge are adapted consistently. Figure 8 shows the convergence of the base and corrected lift and drag coefficients for all of the adaptation strategies. When corrected by the output-based error estimates, the outputs converge much faster than the base outputs. Figure 9 compares the lift coefficient error and drag coefficient error. It is clearly to see that, for test cases with the lift and drag adjoint as the output, the anisotropic h-adaptive methods could produce more efficient error reductions in term of the DOFs. Compare with the result of the isotropic adaptation with Copyright 2013 by the, Inc. All rights reserved. 11 of 18

12 C l Figure Figure 7. The initial mesh for a NACA 0012 airfoil at M 0 = 0.5, α = 2 Uniform refinement Lift adj (iso, hanging) Lift adj( iso, no hanging) Lift adj(aniso, no hanging) Lift adj (iso, hanging) corr Lift adj(aniso, no hanging) corr DOFs (a) C L E 03 C d 1.0E E E E E E+00 Uniform refinement Drag adj(iso, hanging) Drag adj(iso, no hanging) Drag adj (aniso, no hanging) Drag adj(iso, hanging) corr Drag adj (aniso, no hanging) corr DOFs (b) C D The corrected outputs of the h-adaptations for a NACA 0012 airfoil at M 0 = 0.5, α = 2 Uniform refinement Lift adj (iso, hanging) Lift adj( iso, no hanging) Lift adj(aniso, no hanging) 10 1 Uniform refinement 10 2 Drag adj(iso, hanging) Drag adj(iso, no hanging Drag adj (aniso, no hang 10 3 C l 10 4 C d Figure sqrt_dof sqrt_dof (a) C L error (b) C D error The output error of the h-adaptations for a NACA 0012 airfoil at M 0 = 0.5, α = Copyright 2013 by the, Inc. All rights reserved. 12 of 18

13 (a) Anisotropic lift adjoint (b) Anisotropic drag adjoint (c) Isotropic lift adjoint with hanging nodes (d) Isotropic drag adjoint with hanging nodes (e) Isotropic lift adjoint without hanging nodes (f) Isotropic drag adjoint without hanging nodes Figure level adaptations for a NACA 0012 airfoil with M = 0.5, α = 2 13 of 18 Copyright 2013 by the, Inc. All rights reserved.

14 hanging nodes, the adaptation without hanging nodes waste some DOFs to refine their neighbors, which is not necessarily required in term of minimizing functional error. Since there is a geometry singularity point at the trailing edge of the airfoil and the initial mesh has relatively coarse elements near the trailing edge, the uniform mesh refinements can not achieve their expected order of accuracy. The current results show that the h-adaptations successively refine the mesh around the trailing edge; therefore it could reduce the effect of this geometry singularity and reveal the potential accuracy from the high order CPR method. V.B. (a) Mach number contours with the lift adjoint Figure 11. (b) Mesh with the lift adjoint Anisotropic h-adaptation for a NACA 0012 airfoil at M 0 = 0.8, α = 1.25 Transonic Flow over a NACA 0012 Airfoil The last test case is a NACA 0012 airfoil in inviscid transonic flow with an inflow Mach number M = 0.8 and the angle of attack, α = Again, this test is the problem C1.3(b) of the 1 st international workshop on high-order methods. The structure of the solution to this transonic problem includes shockwaves on the upper and lower surfaces of the airfoil. All of the calculations start with an uniform solution order of p=2 and consistently refined using previously described adaptation procedure. The jump indicator φ k = 1 Ω k Ω k q n ds (52) {q} introduced in Ref. [63] is used to examine the smoothness of the primal solution and determine which cell should be limited. Here, the pressure is used as the jump indicator variable q throughout this paper. The surface average operator { } and the surface jump operator are defined as {q} = 1 2 (q+ + q ) (53) q = n(q + q ), where ( ) + and ( ) notations refer to the elements on each side of the edge. The criterion of the smoothness { φ k > 1 K, h refinement φ k < 1 K, p enrichment, (54) where K = 25 suggested in Ref. [64] and Ref. [65] is used. Once a cell with large jumps is marked, the order of its solution polynomial is reduced to p=0. This approach can be treated as a spacial limiter; therefore there is no need to use artificial viscosity to stabilize the solution in the presence of shocks. So we don t need to worry about any pollution to the adjoint solution by the artificial viscosities. However, it do cause some convergence problems for the primal solver. The final mesh after 5 adaptation iterations is shown in Figure 11. The result shows that the shockwaves near the upper and lower surfaces are correctly identified using the current adjoint-based error estimation Copyright 2013 by the, Inc. All rights reserved. 14 of 18

15 without any feature based smoothness indicators. Mach number with the lift output are plotted in Figure 11(b). VI. Conclusions In this paper, we apply an anisotropic h-adaptation method on simplex meshes to minimize the functional error. An adjoint-based error estimation with a local refinement sampling process is utilized to drive the anisotropic mesh adaptation without making any assumption about the solution features. Several well-known two-dimensional inviscid flow cases are utilized to compare the effectiveness of anisotropic and isotropic adjoint-based h-adaptations. The refinement driven by the adjoint-based error indicator can directly target the error source to the engineering output. Results show some savings of degrees of freedom for the anisotropic adaptations when compared with the isotropic refinements. For the simplex meshes, to fully obtain the advantage of the DOF reductions inherited in the anisotropic meshes, grid re-generations may be required. We will further test the performance of the anisotropic h-adaptation of simplex meshes for viscous flow. Acknowledgments The authors gratefully acknowledge support by NASA under grant NNX12AK04A and AFOSR under grant FA References 1 Lhner, R., Morgan, K., Peraire, J., and Vahdati, M., Finite element flux-corrected transport (FEMFCT) for the euler and NavierStokes equations, International Journal for Numerical Methods in Fluids, Vol. 7, No. 10, 1987, pp Castro-Daz, M. J., Hecht, F., Mohammadi, B., and Pironneau, O., Anisotropic unstructured mesh adaption for flow simulations, International Journal for Numerical Methods in Fluids, Vol. 25, No. 4, 1997, pp Dompierre, J., Vallet, M., Bourgault, Y., Fortin, M., and Habashi, W., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part III. Unstructured meshes, International journal for numerical methods in fluids, Vol. 39, No. 8, 2002, pp Hartmann, R. and Houston, P., Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations, Journal of Computational Physics, Vol. 183, No. 2, 2002, pp Venditti, D. and Darmofal, D., Anisotropic grid adaptation for functional outputs: application to twodimensional viscous flows, Journal of Computational Physics, Vol. 187, No. 1, 2003, pp Venditti, D. A. and Darmofal, D. L., Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, J. Comput. Phys., Vol. 187, No. 1, May 2003, pp Fidkowski, K. and Darmofal, D., A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equations, Journal of Computational Physics, Vol. 225, No. 2, 2007, pp Wang, L. and Mavriplis, D., Adjoint-based hp adaptive discontinuous Galerkin methods for the 2D compressible Euler equations, Journal of Computational Physics, Vol. 228, No. 20, 2009, pp Fidkowski, K., Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics, AIAA Journal, Vol. 49, No. 4, 2011, pp Baumann, C. and Oden, J., A discontinuous hp finite element method for the Euler and Navier Stokes equations, International Journal for Numerical Methods in Fluids, Vol. 31, No. 1, 1999, pp Devloo, P., Tinsley Oden, J., and Pattani, P., An hp adaptive finite element method for the numerical simulation of compressible flow, Computer methods in applied mechanics and engineering, Vol. 70, No. 2, 1988, pp Copyright 2013 by the, Inc. All rights reserved. 15 of 18

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