Sensitivity Analysis for the Compressible Navier-StokesEquations Using a Discontinuous Galerkin Method
Li Wang ∗ and W. Kyle Anderson†
SimCenter: National Center for Computational EngineeringUniversity of Tennessee at Chattanooga, Chattanooga, TN, 37403
This paper describes the formulation of adjoint-based sensitivity analysis and optimization techniques forhigh-order discontinuous Galerkin discretizations applied to viscous compressible flow. The flow is modeled bythe compressible Navier-Stokes equations and the discretization of the viscous flux terms is based on an explicitsymmetric interior penalty method. The discrete adjoint equation arising from the sensitivity derivative cal-culation is formulated consistently with the analysis problem, including the treatment of boundary conditions.In this regard, the influence on the sensitivity derivatives resulting from the deformation of curved boundaryelements is properly accounted for. Several numerical examples are used to examine the order of accuracy (upto p = 4) achieved by the current DG discretizations, to verify the derived adjoint sensitivity formulations, andto demonstrate the effectiveness of the discrete adjoint algorithm in steady and unsteady design optimizationfor both two- and three-dimensional viscous design problems.
I. Introduction
The use of computational flow simulations in conjunction with numerical optimization techniques has becomean indispensable tool in modern aerodynamic designs.1–5 This is not only because of the economic benefits fromeliminating massive wind tunnel testing before a final design is obtained, but also the improved accuracy and efficiencyof the optimization method in attaining a target-guided design process. While the majority of such design workhas relied on second-order finite-volume methods, the need for a a higher-order algorithm has become apparent dueto the difficulties in delivering asymptotically grid converged solutions.6,7 High-order discontinuous Galerkin (DG)methods8–11 have emerged as a competitive alternative in solving a variety of computational fluid dynamics problems.Moreover, the robustness and efficiency of the DG methods have been improved significantly in the past decade. Thismotivates the investigation of high-order DG methods in applications of aerodynamic design optimization. As anextension of previous work12 on sensitivity analysis for inviscid flows, this paper continues on the study of an adjointsensitivity algorithm for high-order DG discretizations in compressible viscous flow, focusing on both steady andunsteady design problems in two and three space dimensions.
To numerically solve viscous flow problems governed by the compressible Navier-Stokes (NS) equations, the DGdiscretization of the viscous flux terms must be carried out, together with the subsequent adjoint problem. The currentwork employs an explicit symmetric interior penalty (SIP) method described in references8,13,14 due to the fact that thescheme is capable of preserving optimal error convergence rates for the flow solution and also being dual consistent13
for the adjoint solution. For unsteady viscous flow problems, a backward difference formula (BDF)15 is used to avoidrestrictions on the selection of time-step sizes. Since the primal flow and adjoint solutions are required in a typicaloptimization iteration, the efficiency of a design process is related closely to the solution strategy of these solvers. Tomake the proposed algorithm efficient and competitive, we consider a multigrid approach,11,15,16 driven by a linearizedelement Gauss-Seidel smoother15 or a Generalized Minimal Residual (GMRES) algorithm.17
It is known that the use of high-order curved boundary elements is essential for high-order schemes to deliveran overall high-accuracy solution.18,19 Therefore, the deformation of curved boundary elements and computation ofthe resulting mesh sensitivities remain a topic of considerable importance in the representation of smoothed surfacegeometries and the accuracy of sensitivity derivative calculations. Due to the fact that the present paper focuses onviscous laminar flow with small and moderate Reynolds numbers, curvilinear elements are applied only on physicalboundaries, while straight-sided elements are used in the interior meshes. In this context, the mesh sensitivities mustaccount for the contributions from both linear and higher-order geometric mapping coefficients12 occurring in all
∗Research Assistant Professor, University of Tennessee at Chattanooga, AIAA Member, email: [email protected]†Professor, University of Tennessee at Chattanooga, AIAA Associate Fellow, email: [email protected]
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20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii
curved elements. These key ingredients are automatically included in the adjoint sensitivity formulation developed inthe present work and more attention is paid on the effects of the viscous discretization terms to the mesh sensitivities.
The remainder of the paper is structured as follows. In Section II the governing equations are introduced and thespatial discontinuous Galerkin discretizations along with an implicit time-integration scheme are formulated. SectionIII describes the mesh parameterization and the formulation of the adjoint-based sensitivity derivative calculation.Several numerical examples are presented in Section IV for demonstrating the accuracy of the current DG schemesand the performance of the adjoint techniques in steady and unsteady aerodynamic shape optimization. Finally, SectionV summarizes the conclusion and discusses the future work.
II. Governing Equations and Discretizations
A. Governing Equations
The governing equations that we consider exclusively in this work are the compressible Navier-Stokes equations thatcan be written in the following conservative form:
¶U(x, t)¶t
+Ñ · (Fc(U)− Fv(U,ÑU)) = 0 in W (1)
where W is a bounded domain. The vector of conservative flow variables U, the inviscid and viscous Cartesian fluxvectors, Fc and Fv, are defined by:
U =
r
rurvrwrE
, Fxc =
ru
ru2 + pruvruw
(rE + p)u
, Fyc =
rvruv
rv2 + prvw
(rE + p)v
, Fzc =
rw
ruwrvw
rw2 + p(rE + p)w
(2)
Fxv =
0
txxtxytxz
utxx + vtxy +wtxz +k¶T¶x
, Fyv =
0
txytyytyz
utxy + vtyy +wtyz +k¶T¶y
, Fzv =
0
txztyztzz
utxz + vtyz +wtzz +k¶T¶z
(3)
where the notations r , p, and E denote the fluid density, pressure and specific total energy per unit mass, respectively.u = (u,v,w) represents the Cartesian velocity vector. t represents the fluid viscous stress vector and is defined, for aNewtonian fluid, as,
ti j = µ(
¶ui
¶x j+
¶u j
¶xi− 2
3¶uk
¶xkdi j
)(4)
where di j is the Kronecker delta and subscripts i, j,k refer to the Cartesian coordinate components for x = (x,y,z). µrefers to the fluid dynamic viscosity and is obtained via the Sutherland’s law. The pressure p is determined by theequation of state for an ideal gas,
p = (g − 1)(
rE − 12
r(u2 + v2 +w2))
(5)
where g is defined as the ratio of specific heats, which is 1.4 for air. k and T denote the thermal conductivity andtemperature, respectively, and are related to the total energy and velocity as,
kT =µg
Pr
(E − 1
2(u2 + v2 +w2)
)(6)
where Pr is the Prandtl number and is set as 0.72.For the purpose of discretization, we rewrite the Cartesian viscous fluxes as the following equivalent form:
Fxv = G1 j
¶U¶x j
, Fyv = G2 j
¶U¶x j
, Fzv = G3 j
¶U¶x j
(7)
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where G denotes the homogeneity tensor and its components Gi j(U) are determined by G1 j = ¶Fxv/¶(¶U/¶x j), G2 j =
¶Fyv/¶(¶U/¶x j) and G3 j = ¶Fz
v/¶(¶U/¶x j) for j = 1,2 and 3 so that they are purely dependent on the conservativeflow variables.
B. Discretizations
The computational domain W is partitioned into an a tessellation of non-overlapping elements (triangular elements intwo dimensions and tetrahedral elements in three dimensions), such that W =
Sk Wk, where Wk refers to the volume
of an element k in the computational mesh. The Galerkin finite-element approximation is expanded as a series oftruncated basis functions,20 {f j, j = 1, · · · ,M}, and solution coefficients as,
Uh =M
åi=1
Uhi fi(x). (8)
The discontinuous Galerkin discretization proceeds by formulating a weak statement of the governing equations, bymultiplying Eq. (1) by a set of test functions, with the maximum polynomial order of p, and integrating within eachelement, e.g. k, as: Z
Wk
f j
[¶Uh(x, t)
¶t+Ñ · (Fc(Uh)− Fv(Uh,ÑUh))
]dWk = 0. (9)
Integrating this equation by parts and implementing the symmetric interior penalty method13,14 for the viscous fluxesyields the following weak formulation,
ZWk
f j¶Uh
¶tdWk −
ZWk
Ñf j · (Fc(Uh)− Fv(Uh,ÑhUh))dWk +Z
¶Wk
[[f j]]Hc(U+h ,U−
h ,n)dS (10)
−Z
¶Wk\¶W
{Fv(Uh,ÑhUh)} · [[f j]]dS −Z
¶Wk\¶W
{(Gi1¶f j
¶xi,Gi2
¶f j
¶xi,Gi3
¶f j
¶xi)} · [[Uh]]dS +
Z¶Wk\¶W
n{G}[[Uh]] · [[f j]]dS
−Z
¶Wk∩¶W
f+j Fb
v(Ub,ÑhU+h ) · ndS −
Z¶Wk∩¶W
(Gi1(Ub)¶f
+j
¶xi,Gi2(Ub)
¶f+j
¶xi,Gi3(Ub)
¶f+j
¶xi) · (U+
h − Ub)ndS
+Z
¶Wk∩¶W
nG(Ub)(U+h − Ub)n · f
+j ndS = 0
where the unit normal vector n is outward to the boundary. Hc(U+h ,U−
h ,n) represents an approximate Riemann convec-tive flux function, such as Lax-Friedrichs21 or HLLC22 to resolve the solution discontinuities (represented by U+
h andU−
h ) at shared elemental interfaces. The notations {}, [[ ]] and [[ ]] indicate the respective average and jump operators,defined as:
{j} =j+ +j−
2, [[j]] = j
+n+ +j−n−, [[j]] = j
+ − j− (11)
The sixth and the last integrals in Eq. (10) are referred to as penalty terms, where the penalty parameter n is explicitlyevaluated by the element geometry and the order of discretization,8 given by:
n =(p+1)(p+d)
(2d)max(
S+k
V +k
,S−
k
V −k
) (12)
where d represents the space dimensions; Vk and Sk represent the volume and surface of elements k± which sharethe interface. The boundary conditions on ¶W are imposed weakly by constructing a boundary state, denoted byUb. At solid walls, an adiabatic wall with no-slip boundary condition is imposed, which yields ÑT · n = 0 andUb = (U1,0,0,0,U5), and thus the component of the boundary viscous fluxes Fb
v associated with the energy equationvanishes. The set of discretized equations is solved in modal space and the integrals are evaluated using Gaussianquadrature rules20 which are exact for polynomial degree 2p in volume integrals and for polynomial degree 2p+1 insurface integrals.23,24
Because the set of basis functions is defined in a master element W spanning between {0 < x,h,z < 1}, a coor-dinate mapping from the reference to a physical element is required for the computation of the first-order derivatives,solution gradients and integrals appearing in Eq. (10). The reference-to-physical transformation and the correspondingJacobian Jk associated with each element k are given by:
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xk =M
åi=1
xkifi(x,h,z), Jk =
¶x¶x
¶y¶x
¶z¶x
¶x¶h
¶y¶h
¶z¶h
¶x¶z
¶y¶z
¶z¶z
. (13)
where xk represent the element-wise geometric mapping coefficients. In the simple case of straight-sided elementsthe transformation is linear thus the geometric mapping coefficients can be evaluated only by using the element vertexcoordinates because the higher-order mapping modes are zero. However, in the more complex cases of high-ordercurved elements, which are often required at physical boundaries, additional surface quadrature nodes16 must beincluded for determining the higher-order modes (p > 1) of the geometric mapping coefficients, obtained by:
xk = F−1xpk (14)
where xpk = {xck ,xqk } refers to the coordinates of physical points in the element k, consisting of the element verticesxck as well as additional surface quadrature points xqk . F denotes the projection mapping matrix which is constituted bythe basis functions evaluated at the aforementioned physical points in the master element (xpk
← xpk ). The additionalsurface points are initially created based on linear interpolation of the vertex coordinates, and then projected onto thesurface of the original geometry. After all elements with curved boundary faces have been associated with appropriatequadrature points, additional neighboring elements are also required to be considered as curvilinear elements if theycontain at least one edge on the curved surface.
Returning to Eq. (10), we rewrite it as the following ordinary differential equation (ODE) form,
MdUh
dt+R(Uh) = 0 (15)
where R represents the discretized steady-state residual (including both convective and viscous terms) and M denotesthe mass matrix. A time integration is then performed via an implicit temporal scheme, such as a second-orderbackward difference formula (BDF2), formulated as:
BDF2 : Rn+1e (Un+1
h ) =MDt
(32
Un+1h )+R(Un+1
h )− MDt
(2Unh − 1
2Un−1
h ) = 0 (16)
(17)
where Rn+1e represents the unsteady flow residual at time step n+1. The use of a higher-order implicit time-integration
scheme such as a sixth-stage, fourth-order Implicit Runge-Kutta (IRK4) scheme can be referred to our previouswork,12,15 and the references cited therein.
The high-order discontinuous Galerkin solver described in this work uses the standard MPI message-passing li-brary for inter-processor communication,25 and the mesh is partitioned based on the METIS graph partitioner26 oper-ating on the dual graph of the mesh.16 The current parallel algorithm results in exactly the same residual values andconvergence as the corresponding sequential algorithm.
III. Discrete Adjoint-Based Sensitivity Analysis
In this section, we derive a discrete adjoint-based sensitivity algorithm for a design optimization process in thecontext of high-order discontinuous Galerkin discretizations for the compressible Navier-Stokes equations.
A. Mesh Parameterization and Deformation
Assuming a computational mesh and geometry are given, and an objective functional L for obtaining a specific de-sign purpose is prescribed, a set of design variables are then identified for reshaping the surface geometry. Sincethe additional surface quadrature points are important components for representing surface shapes in high-order DGdiscretizations, the surface quadrature points must deform in a similar manner as the standard surface nodes.
In a two-dimensional design iteration, surface nodes and additional surface quadrature points are set to deformsimultaneously through the superposition of the Hicks-Henne bump function27 placed at a set of designated surfacenodes. The magnitudes of the Hicks-Henne function are determined by the values of the corresponding design vari-ables, D, and thus the displacements and new coordinates of the surface points are expressed as
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Dxsi = nsi
Nd
åm=1
Dmbi(xsi,xm) and xsnewi = xs
oldi +Dxsi (18)
Dxqi = nqi
Nd
åm=1
Dmbi(xqi,xm) and xqnewi = xq
oldi +Dxqi (19)
where Nd refers to the number of design variables; xsi and xqi denote the Cartesian coordinates of surface node i andsurface quadrature point i in the normal directions, nsi and nqi respectively; bi(xsi,xm) denotes the Hicks-Henne bumpfunction27 placed at xm for the surface node i with the x-coordinate of xsi. In a three-dimensional design problem, onthe other hand, design variables are placed at designated surface nodes as well as surface quadrature points, and thedisplacements at these surface points are purely determined by the values of the corresponding design variables. Othersurface nodes and quadrature points not associated with any design variables are kept fixed during a mesh deformation.
In response to changes of surface points, interior mesh points are deformed to prevent generation of overlappingelements. Here, we employ the linear tension spring analogy,28 in which each edge of the mesh is represented by aspring whose stiffness is related to the length of the edge. The governing equations for the mesh motion are expressedas
[K]Dx = Dxs (20)
where [K] denotes the stiffness matrix obtained from the discrete mesh motion equations. The spring tension analogyapproach requires a stencil with only the nearest neighbors, such that the [K] matrix can be represented by a blockd × d matrix and corresponds to identity diagonal blocks and zero off-diagonal blocks for boundary nodes. Here,we remark that due to the viscous problems of low Reynolds numbers considered in the current work, the meshesmainly contain regular elements and the linear tension spring approach is found to be capable of avoiding generationof negative-Jacobian elements. However, a more sophisticated mesh motion method is required for designs with highReynolds-number viscous flow.
B. Adjoint-based Sensitivity Formulation
The adjoint sensitivity derivation starts with the formulation of a forward linear problem, in which the discretizedsystem of equations is linearized and the sensitivity derivatives of an objective functional are formulated. The discreteadjoint formulation is then derived by transposing each matrix of the tangent problem and performing the operationsin reverse order. The discrete adjoint sensitivity formulation is shown in Eqs. (21)-(23), while a detailed derivation ofthis procedure is described in reference12.(
dLdD
)T
=
[(¶xs
¶D
)T
[K]−T(
¶x¶x
)T
+(
¶xq
¶D
)T (¶x¶xq
)T](
¶L¶x
)T
(21)
where the sensitivities of the objective functional with respect to the geometric mapping coefficients for the entiremesh elements are expressed as (
¶L¶x
)T
=(
¶L¶x
)T
−[
¶R¶x
]T
Uh
lllu (22)
for a steady-state flow problem, or (¶L¶x
)T
=(
¶L¶x
)T
−N
ån=1
[¶Rn
e
¶x
]T
Uh
lllnu (23)
for an unsteady flow problem. In the above equations, [¶R/¶x] and [¶Rne/¶x] denote sensitivities of the respective
steady and unsteady (at time step n) residual with respect to the modal geometric mapping coefficients, evaluatedusing appropriate computed flow states.12 Note that due to the fourth term appearing in the DG weak formulation(c.f. Eq. (10)) where the surface integral involves the solution gradients from both sides of the elements sharingthe interface, it leads to non-zero off-diagonal block components in the [¶R/¶x] matrix (similarly in [¶Rn
e/¶x]) thatmust be included to ensure accuracy of the computed adjoint sensitivity derivatives. In addition, the computationalcost for evaluating these mesh sensitivity matrices is comparable to that for evaluating the flow-Jacobian matrix. Anoticeable difference in the adjoint sensitivity formulation for steady and unsteady viscous flow problems lies in thelast terms appearing in Equations (22) and (23), where the complete mesh sensitivities in an unsteady problem require
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contributions of those arising from each time step, while the ones in a steady flow problem consist of the sensitivitiespurely from the converged flow states. lllu denotes the steady or unsteady flow-adjoint solution, computed by solvingthe following flow-adjoint equations:[
¶R¶U
]T
lllu =(
¶L¶U
)T
or[
¶Re
¶U
]T
lllu =(
¶L¶U
)T
(24)
where Re represents the full unsteady residual vector (Re = {Rne ,n = 1,2, · · · ,N}) spanning the entire time domain.
We note that the transpose of the Jacobian of the discretized flow equations is used in the definition of the flow-adjointvariables, evaluated using the computed flow states. Therefore, the flow-adjoint solution in a steady design iterationcorresponds to a single linear problem, while a series of linear problems are required in an unsteady design step toobtain the flow-adjoint solution for all discrete time step locations. More details about the solution procedure ofunsteady flow-adjoint problems can be referred to29 for a standard backward-difference temporal scheme and12 fora higher-order multistage Runge-Kutta scheme. Current work employs a linear multigrid method23 or a GMRES17
algorithm to efficiently solve the linear problems.To avoid a direct solve for the inverse of the transposed mesh stiffness matrix [K]−T , we further introduce the mesh
adjoint variables, lllx, satisfying:
[K]T lllx =¶L¶x
T
where¶L¶x
T
=¶x¶x
T¶L¶x
T
(25)
The mesh-adjoint vector is a size of d ×Nn (d and Nn refer to the space dimensions and the total number of mesh points,respectively), and it corresponds to the solution of a single linear problem and can be solved by several hundred sweepsof a Gauss-Seidel scheme due to relatively coarse meshes that are generally employed with high-order discretizations.Substituting Eq. (25) into Eq. (21) yields the final expression for the discrete adjoint sensitivity formulation, shown as(
dLdD
)T
=(
¶xs
¶D
)T
lllx +(
¶xq
¶D
)T (¶x¶xq
)T (¶L¶x
)T
(26)
Since the terms relevant to the design variables (i.e. (¶xs/¶D)T and (¶xq/¶D)T ) are evaluated at the last step in thisformulation, the evaluation of the adjoint-based sensitivity derivatives is essentially independent of the number ofdesign variables, which makes the adjoint method well-suited for cases with a large number of design variables.
The computational cost for the evaluation of adjoint-based sensitivity derivatives depends primarily on the solu-tions of the analysis and flow-adjoint problems, and the cost of the linear flow-adjoint problem is somewhat lower thanthat of the analysis problem in our previous experiences.12 Furthermore, since the mesh adjoint problem is related to amuch smaller system (containing d ×Nd degrees of freedom) than the primal or flow-adjoint system, the computationalcost is negligibly small, especially for the relatively coarse meshes generally used in high-order discretizations. Othermatrices and vectors required in Equations (21)- (26) are obtained automatically by a direct differentiation approach,and thus the contribution to the total computational cost is trivial.
C. Design Optimization Procedure
Once the objective sensitivities have been evaluated, they are used to drive a design optimization process to seek aminimum in a specified objective functional. It is well known that the overall efficiency of an optimization processrelates closely to the particular optimization algorithm. While a gradient-based steepest descent method is simple toimplement, this approach tends to deliver slow convergence when a large number of design variables are employed.The current work employs the PORT trust region optimization strategy,30 which requires the initial bounds to bespecified for the set of design variables, which are selected so that subsequent geometry changes are reasonable forthe flow solver and avoid the generation of nonsensical shapes.
For a typical design optimization cycle in the context of high-order DG discretizations, five sequential steps arerequired:
1. Solve the discretized (steady or unsteady) flow equations.
2. Solve the flow-adjoint variables followed by the mesh adjoint variables.
3. Evaluate gradients or objective functional sensitivities dL/dD denoted by Eq. (26).
4. Compute a new set of design variables, Dnew, using the PORT optimization algorithm based on the computedsensitivity derivatives.
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5. Deform the geometry and additional quadrature points for curved elements based on the new design variablesand specific mesh parameterization, and then deform the interior mesh using the mesh motion equations, fol-lowed by recomputation of geometric mapping coefficients, mesh Jacobians and surface normals.
This procedure is repeated until the objective functional is sufficiently minimized. The total computational cost for acomplete design task depends on the physical problem, the number and type of design variables and the acceptableconvergence level.
IV. Numerical Results
In this section, we first examine the order of accuracy of the DG discretizations for the compressible Navier-Stokesequations, and then we present a series of numerical experiments to demonstrate the performance of the discreteadjoint-based shape optimization strategy in two-dimensional laminar flow problems and to establish the accuracy ofsensitivity derivatives for three-dimensional viscous flows.
A. Verification of Order of Accuracy
Using the Method of Manufactured Solutions,31 we first examine the spatial error convergence of the DG discretiza-tions applied to viscous flows in both two and three dimensions. Verification of the temporal error convergence can befurther referred to reference.15
1. Viscous Flow in a Square
We specify the square domain with [0,1]2 and supplement the steady-state compressible Navier-Stokes equations (c.f.Eq. (1)) with an inhomogeneous forcing function, S, such that the analytical solution is given by
where the coefficients (r0,u0,v0,Et0) are set to be (1,0.5,0.5,3) and the parameter k which specifies the frequency ofthe velocity solution is set to be 2. The Reynolds number is set to be 1. The analytical solution for the momentumequations can thus be simply obtained by using the solution profiles of r, u and v expressed in the Eq. (27). Dirichletboundary conditions are specified with the exact solution at the square boundaries. A series of four grids, consistingof 348, 1134, 4118 and 14508 unstructured triangular elements, are used to evaluate the spatial error convergence ofthe DG schemes for polynomial orders ranging from p = 1 to p = 4.
Figures 1(a)-(b) illustrate the second coarse mesh used in this test as well as the exact solution for density andu-velocity. We can see that the frequency of the velocity solution is actually twice of that of the density solution. Fig.1(c) shows the L2(W)-norm of the solution error, computed using all field variables, as a function of grid spacing (i.e.related to the square root of the number of elements). We observe that the spatial discretization error converges at theexpected optimal rate (∼ hp+1) as the mesh is refined for each fixed order of p. In particular, the asymptotic slopes forthe p = 1, p = 2, p = 3, and p = 4 DG schemes are 1.96, 2.99, 4.02, and 5.02 respectively.
2. Viscous Flow in a Cube
Next we consider an examination of the spatial discretization error for the three-dimensional steady-state compressibleNavier-Stokes solver. To this end, we let W = [0,1]3 and perform a similar test as the previous example. By supplyingthe proper forcing function, the analytical solution to the steady-state form of Eq. (1) is given by
where the coefficients (r0,ru0,rv0,rw0,Et0) are set to be (1,0.5,0.5,0.1,3) and the parameter k is equal to 1.5. TheReynolds number is set to be 1. Dirichlet boundary conditions are specified with exact solution values applied at allouter boundaries. Figures 2(a)-(b) show the exact solution contours for density and x-momentum at certain constantx- or z-planes.
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(a) Computational mesh with 1134 triangular elements andexact density contours
(c) L2 norm of the solution error versus grid spacing
Figure 1. Accuracy examination on the DG discretizations for the two-dimensional compressible NS equations.
A sequence of grids, containing 5788, 20185 and 68323 unstructured tetrahedral elements, are used to evaluatethe spatial error convergence of the DG discretizations for orders ranging from p = 1 to p = 4. In Fig. 2(c), theL2(W)-norm of the solution error, ‖Uex − Uh‖L2 , is plotted as a function of grid spacing (i.e. cube root of the numberof elements). We again observe that the optimal error convergence rate (∼ hp+1) is achieved for all orders of DGdiscretization. Particularly, the asymptotic slopes for the p = 1, p = 2, p = 3, and p = 4 DG schemes are 2.09, 3.03,4.20, and 5.25 respectively.
B. Laminar Flow Over a Circular Cylinder
In this example, we consider a validation case describing laminar flow over a circular cylinder, computed by the presenthigh-order DG discretizations. The incoming freestream flow is uniform with a Mach number of 0.2 and a Reynoldsnumber of 40 based on the cylinder diameter.
The computational domain is subdivided into 1622 unstructured triangular elements, as displayed in Fig. 3(a)for the mesh near the cylinder. One can observe that the mesh tends to become much coarser as the distance fromthe cylinder surface increases. The cylinder geometry is represented with piecewise polynomials of an order that isconsistent with the discretization order for the solution approximation. Dirichlet boundary conditions are set at outerboundaries and an adiabatic, non-slip boundary condition is imposed at the cylinder surface. The HLLC approximateRiemann flux function22 is employed for computing the convective flux terms.
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(a) Exact density contours (b) Exact x-momentum contours
(c) L2 norm of the solution error versus grid spacing
Figure 2. Accuracy examination on the DG discretizations for the three-dimensional compressible NS equations.
The computation is performed using the BDF2 implicit temporal scheme with a fixed time-step size of Dt = 0.05 tosimulate the unsteady flow in a time-accurate manner. A p-multigrid approach15 driven by a linearized element Gauss-Seidel smoother is implemented to solve each implicit non-linear problem. The approach sufficiently converges theL2 residual to machine zero within 5 ∼ 8 p-multigrid iterations. Figures 3(b)-(c) show the Mach number contoursat t∗ = 3.7 and t∗ = 10.5 respectively, computed using a fifth-order accurate (p = 4) DG scheme (t∗ = denotes anon-dimensional time based on free-stream velocity and diameter of the cylinder). It is shown that the wake regionbehind the cylinder grows as time evolves, and moreover, the fifth-order DG scheme demonstrates excellent ability indelivering a very smooth solution in spite of the fact that relatively low mesh resolution is applied in the wake region.
Fig. 3(d) compares the evolution in time of the axial u-velocity distribution downstream of the cylinder, where thecurves correspond to the solution obtained by various orders of the DG discretizations and the experimental resultsdocumented in reference.32 By measuring the distance between the rear stagnation point (x/D = 0.5) and the zero-velocity point, one can clearly see that the length of the attached wake grows in time. Furthermore, the computed DGsolution (from p = 2 to p = 4) exhibits good agreement with the experimental results for both regions close to and faraway from the cylinder, which further validates the accuracy of the present DG-NS solver.
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X
Y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
(a) Computational mesh (b) Mach number contours (p = 4) at t∗=3.7
Figure 3. Laminar flow over a circular cylinder at M¥ = 0.2 and ReD = 40, computed by using various orders of the DG discretizations andthe BDF2 temporal scheme with a time-step size of 0.05. (a) computational mesh containing 1622 unstructured triangular elements; Machnumber contours at (b) t∗ = 3.7 and (c) t∗ = 10.5 using a fifth-order accurate (p = 4) DG scheme; (d) comparison of the u-velocity profileson the flow axis y = 0 with experimental data at different times.
C. Unsteady Shape Optimization for Matching Pressure Distributions
This example involves laminar flow over a cylinder discussed previously to demonstrate the performance of the un-steady adjoint-based shape optimization approach. The objective is to match a time-dependent pressure profile on thesurface of an initially deformed cylinder, using the pressures of a circular cylinder as a target. Therefore, the objectivefunctional to be minimized for this design purpose is defined as
L =
√√√√åNn=ns å
Nsj=1 å
Nqq=1
(pn
q, j − (pnq, j)∗
)2
N∗ · Ns · Nq(29)
where pnq, j represents the pressure value obtained from the current geometry configuration at the nth time step for
quadrature point q at the surface edge j, and (pnq, j)
∗ represents the pressure value for the target cylinder configurationat the same location and time step. ns and N denote the respective starting and final time steps in which the pressuredistribution is measured; Ns and Nq denote the number of surface points and quadrature points, respectively. In themesh deformation of this test case, surface grid points as well as additional surface quadrature points are allowed todeform only in the y-coordinate direction since the chord length of the optimized geometry is not desired to change.The design variables are set to be the magnitudes of the bump functions placed at surface grid points, spanning 98%
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Figure 4. Two-dimensional unsteady inverse design case for matching target pressure distributions. (a) computational mesh (containing4335 unstructured triangular elements) and baseline geometry; (b) comparison of sensitivity derivatives using the unsteady discrete adjointmethod and the finite-difference method for a DG p = 3 (i.e. fourth-order) discretization and the BDF2 implicit scheme.
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Figure 5. Objective functional convergence and surface shape comparison in the two-dimensional unsteady inverse design case.
of the chord locations on the cylinder upper and lower surfaces, which results in a total of 90 design variables.The baseline geometry and computational mesh, containing 4335 unstructured triangular elements, are illustrated
in Fig. 4(a). The baseline configuration is obtained by deforming the surface points of a circular cylinder, whichensures that the designed geometry shape should match the target as the objective functional is minimized. A fourth-order (i.e. p = 3) spatial discontinuous Galerkin scheme and the BDF2 implicit temporal scheme are employed for therespective spatial and temporal discretizations. A fixed time-step size of Dt = 0.05 is used and the flow is simulatedfrom t = 0 to t = 3 while the actual objective time interval is set to be [1,3]. The cylinder surfaces are treated as anon-slip adiabatic wall and are represented using curved boundary elements with the same order of polynomials asthat used in the analysis problem.
Before proceeding to a design optimization procedure, the sensitivity vector computed using the proposed discreteadjoint approach must be verified by comparing with finite-differenced results. This verification exercise is performedbased on the baseline geometry and initial computational mesh. Fig. 4(b) illustrates comparison of the sensitivityvalues computed using the discrete adjoint approach (for the DG p = 3 discretization and the BDF2 temporal scheme)and a finite-difference scheme with a small perturbation size of 10−7. It is shown that the computed adjoint sensitivitiesprovide an excellent match with the finite differenced results, and moreover, the average difference between the twoapproaches is within 0.04%, thereby verifying the linearization terms determined in Eq. (26). Fig. 5(a) depicts the
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Figure 6. Comparsion of pressure distributions on the target, baseline and optimized geometries at t = 2 and t = 3 in the two-dimensionalunsteady inverse design case.
computed objective functional against the number of optimization iterations, where roughly three orders of magnitudeof reduction in the objective functional are achieved within 40 design iterations. It is important to note that although theconvergence slows down in the later design steps, the objective should eventually reduce to zero since an exact solutionexists for the baseline geometry to transform to the target. Next, a comparison of surface shapes for the baseline, targetand final optimized geometries is shown in Fig. 5(b). It is seen that the consequent optimized geometry matches thetarget very well and there are no visible discrepancies.
Due to the fact that the goal of the present design optimization case is to match the target pressure distributions foreach time step in the objective time intervals, it is worthwhile to examine the agreement of the pressure distributionsat individual time steps to further understand the effectiveness of the proposed adjoint-based optimization algorithm.As a representative, Fig. 6 provides a comparison of pressure distributions on surfaces at two discrete times, t = 2and t = 3. The distributions for the baseline geometry are distinct from those produced by the target and optimizedgeometries at both times, where the baseline geometry clearly produces an unsymmetric pressure profile. However,the target unsteady pressure profile is captured very accurately by the final optimized geometry.
D. Steady Design Optimization for Viscous Flow over an Airfoil
The next example considers a steady design optimization test for viscous flow over a NACA0012 airfoil at M¥ = 0.5,0-degree angle of attack and a Reynolds number of 100 based on the airfoil chord length. The design purpose is toobtain a target pressure-based lift coefficient by changing the original airfoil configuration. The objective function forthis purpose is defined as
L = (CL −CL,target)2 (30)
where CL,target represents the pressure lift coefficient which is set to be 0.1. Two computational meshes, consistingof 2741 (displayed in Fig. 7(a)) and 8943 unstructured triangular elements, are used in this example. The HLLCapproximate Riemann solver22 is used to solve the convective flux terms and the airfoil surface is again assumed tobe an adiabatic and non-slip wall. The deformation of surface nodes and surface quadrature points is performed usingthe Hicks-Henne function (in the y-direction) and design variables are set to be placed within 90% of the chord lengthon the upper and lower surfaces. This results in a total of 57 and 86 design variables for the respective coarse and finemeshes.
A fifth-order accurate DG scheme (p = 4) is implemented and boundary elements on the airfoil surface are rep-resented by the same order of polynomials as the analysis. The flow problem is solved using a p-multigrid approachin which the linearized system at each p-level is solved using a GMRES algorithm17 with 50 search directions. Theflow-adjoint problem is solved based on the same p-multigrid approach while driven by a linearized element Gauss-Seidel smoother. Fig. 7(b) illustrates contours of Mach number computed for the initial geometry using the DG p = 4spatial discretization on the coarse mesh. It is seen that the boundary layer is relatively thick due to the low Reynoldsnumber and the flow is fully attached on the upper and lower surfaces of the airfoil.
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X
Y
-0.5 0 0.5 1 1.5
-1
-0.5
0
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1
(a) Computational mesh (b) Mach number solution for the NACA0012 airfoil (baseline)
Figure 7. Two-dimensional steady shape optimization for obtaining a target lift coefficient. (a) coarse computational mesh (containing 2741unstructured triangular elements) and baseline (NACA0012) geometry; (b) Mach number contours for laminar flow over the NACA0012airfoil at M¥ = 0.5 and a Reynolds number of 100 using a fifth-order accurate (p = 4) DG discretization scheme.
Locations of the Design Variables
dL
/dD
-0.5 0 0.5
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
AdjointFinite Difference
Figure 8. Comparison of sensitivity derivatives using the discrete adjoint method and the finite-difference method for the original meshand geometry in the two-dimensional steady shape optimization test.
Table 1. Comparison of sensitivity derivatives on the coarse mesh for the two-dimensional steady shape optimization case, using a fifth-order DG scheme.
A verification test is next performed to show the accuracy of the adjoint-based sensitivity derivatives. Table 1demonstrates a comparison of the computed adjoint-based sensitivity derivatives with the finite-difference gradients forthe first five representative design variables on the coarse mesh, using a fifth-order (p = 4) DG scheme. A perturbationsize of 10−7 is utilized to obtain the finite difference results. It is concluded that the present adjoint method providesvery accurate sensitivity derivatives with a relative difference below 0.005%. Fig. 8 further provides a completecomparison for all design variables, where excellent agreement is observed.
The convergence of the objective functional as well as gradient norms is illustrated in Fig. 9 for the present
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Design Iterations
(CL-C
L* )2
2 4 6 8 10 12 14 1610-18
10-16
10-14
10-12
10-10
10-8
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10-2
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(a) Convergence of the objective functional
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die
nt
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(b) Convergence of the gradient norm
Figure 9. Convergence of the objective functional and gradient norms in the two-dimensional steady shape optimization test case usingcoarse and fine meshes.
Figure 10. (a) Comparison of the baseline and optimized airfoil geometries in the two-dimensional steady shape optimization case usingcoarse and fine meshes; (b) comparison of the pressure coefficients on the baseline and optimized airfoil surfaces.
optimization problem, performed on both coarse and fine meshes using a DG p = 4 scheme. Since the original meshand baseline geometry (NACA0012 airfoil) correspond to a symmetric solution to the flow axis, the initial pressurelift coefficient is zero. Thus the initial objective functional is approximately 10−2. Within 12 and 16 design iterationson the respective coarse and fine meshes, the objective functional and gradient norms are sufficiently minimized.In particular, approximately 16 and 9 orders of magnitude in reduction are achieved in the objective and gradientsrespectively, demonstrating good performance of the present adjoint optimization algorithm.
Fig. 10(a) depicts the baseline and final optimized geometries obtained using the coarse and fine meshes. Wefirst observe that the final optimized airfoils on both meshes converge to a same geometry based on similar initialbounds prescribed in the optimizer. We also observe that a significant change in surface shapes occurs in the finaloptimized geometries, compared to the original NACA0012 airfoil. A relatively high camber is currently associatedwith the front portion of the airfoil and the camber line curves back up near the trailing edge, and moreover, theupper surface of the optimized airfoils has relatively bigger curvatures than the lower surface, which accelerates theflow on the upper surface, thereby increasing the lift. Fig. 10(b) compares the pressure coefficients distributed onthe baseline NACA0012 airfoil and final optimized airfoils. It is clearly seen that the original airfoil produces zeropressure lift since the pressure is distributed symmetrically on the lower and upper surfaces. On the other hand, thepressure distribution is substantially modified by the optimized airfoil. In regions ranging from 10% to 70% of chordlength locations, a pressure increase on the lower surface of the airfoil is clearly observed, while the upper side of the
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(a) Mach number contours for the optimized airfoil (b) Pressure contours for the optimized airfoil
Figure 11. Mach number (a) and pressure (b) contours for laminar flow over the optimized airfoil at M¥ = 0.5 and a Reynolds number of100 using the coarse mesh (N = 2741) and a fifth-order accurate (p = 4) DG scheme.
airfoil produces a significant pressure decrease, thereby resulting in an increase in lift.Mach number and pressure contours are illustrated in Fig. 11 for the final optimized airfoil, obtained by the DG
p = 4 scheme on the coarse mesh. It is shown that the flow remains well attached on the upper and lower surfaceswith only minor separation occurring at 99% of chord length locations. Fig. 11(b) depicts that low-pressure regionsare mainly concentrated on the front portion of the upper airfoil surface.
E. Verification of Sensitivity Derivatives for Three-dimensional Viscous Flow
In this section, we concentrate on examining the accuracy of discrete adjoint-based sensitivity derivatives in three-dimensional viscous flow problems. Here we consider, as a representative example, three-dimensional viscous flowsover a sphere with 0-degree angle of attack and low Reynolds numbers.
X
Y
Z
(a) Sphere surface (b) Close-up view
Figure 12. Computational mesh (containing 6608 tetrahedral elements) for the three-dimensional steady and unsteady sensitivity derivativecalculations, where the dots indicate locations of the design variables.
Fig. 12 displays the (surface) computational mesh consisting of 6608 unstructured tetrahedral elements. Thedesign variables in the following tests are designated to be placed at the surface mesh points, spanning the upper andlower surfaces within 60% of the chord length locations and 40% of spanwise locations. This results in a total of 46design variables, as indicated by the solid dots in Fig. 12(a). Deformation of the surface nodes is specified to occur innode normal directions and the magnitudes are determined by the values of the corresponding design variables.
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(a) Mach number contours around the sphere
Design Variables
dL/d
D
5 10 15 20 25 30 35 40 45
-4
-2
0
2
4
6 AdjointFinite Difference
10-4×
(b) Comparison of sensitivity derivatives
Figure 13. (a) Mach number contours for laminar flow over a sphere at M¥ = 0.25 and ReD = 50 using a third-order accurate (p = 2) DGdiscretization scheme; (b) comparison of sensitivity derivatives obtained from the discrete adjoint method and the finite-difference methodfor objective functional of pressure drag.
The triangles on sphere surface are represented by polynomials of an order consistent with the flow analysisproblem. Therefore, the locations of additional surface quadrature points must be prescribed to attain the geometricmapping coefficients for high-order curved elements. Fig. 12(b) shows a close-up view on a part of the sphere surface,which is represented by a p = 3 geometric mapping transformation. Although the surface mesh appears to be relativelycoarse, the actual geometry can continually be represented by a higher-order geometric mapping in a more accuratemanner.
1. Viscous Flows Over a Sphere at Re = 50 and M¥ = 0.25
This numerical example involves three-dimensional laminar flow over a sphere at M¥ = 0.25 and a Reynolds numberof 50 based on diameter of the sphere. The sphere surface is modeled as a nonslip adiabatic wall and the HLLC ap-proximate Riemann solver22 is used to solve the convective flux terms. A third-order (p = 2) DG spatial discretizationscheme is implemented for the sensitivity verification test.
The non-linear flow analysis problem is solved using a p-multigrid approach15 along with a GMRES method17 tosolve the linearized system at each p-level. For such a low Reynolds number, a steady-state flow solution is obtainedby a third-order DG scheme where a stationary wake appears behind the sphere. Fig. 13(a) illustrates Mach numbercontours around the sphere and we observe that the third-order DG scheme is capable of resolving a qualitativelyreasonable solution on the current mesh.
To establish the accuracy of code for computing adjoint-based sensitivity derivatives in three-dimensional viscousflow problems, we set the objective functional to be the pressure drag computed using the converged flow solution(L2(R) ∼ 10−15) although the present objective is not for a design purpose. As discussed previously, the flow-adjointproblem is solved after the analysis solution is attained, with the same solution strategy as the analysis. Then thesensitivity derivatives can be computed based on the formulation shown in Eq. (26). It is noted that the computationalcost depends primarily on the solutions of the flow analysis and flow-adjoint problems, and the contribution of themesh adjoint solution is negligibly small due to the current grid size.
Fig. 13(b) plots the sensitivity derivatives computed by the proposed discrete adjoint method for the third-orderaccurate DG scheme and a finite difference method for all design variables. A perturbation size of 10−7 for each designvariable is selected in the finite difference approach. Since the locations of design variables can be randomly distributedon the sphere surface, the shape of the curve is not as smooth as that shown in the two-dimensional problems. However,it is seen that the adjoint sensitivity derivatives are in good agreement with the finite-difference gradients and theaveraged relative difference between the two methods is about 0.03%.
2. Unsteady Flows Over a Sphere at Re = 300 and M¥ = 0.2
In the last numerical example, we attempt to establish the accuracy of the adjoint-based sensitivity calculation pre-sented in this work for three-dimensional unsteady viscous flow. To this end, we reconsider the previous numerical
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(a) t = 1 (b) t = 14
(c) t = 25 (d) t = 100
Figure 14. Entropy contours at various times for viscous flow over a sphere at M¥ = 0.2 and ReD = 300 using a fourth-order accurate DGdiscretization and the BDF2 temporal scheme with a time-step size of 0.05.
Design Variables
dL/d
D
5 10 15 20 25 30 35 40 45-5
-4
-3
-2
-1
0
1
AdjointFinite Difference
10-4×
Figure 15. Comparison of sensitivity derivatives obtained from the unsteady discrete adjoint method and the finite-difference method forobjective functional of pressure drag at the final time step (N = 10), using a fourth-order DG scheme.
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example while setting the Reynolds number to be 300 and M¥ = 0.2 and solving the flow problem in a time-accuratemanner. For a demonstration purpose, an objective functional is selected as the pressure drag at the final time step.
The sphere surface is treated as a nonslip adiabatic wall and the HLLC approximate Riemann solver22 is used tosolve the convective flux terms. The unsteady flow is simulated by a fourth-order (p = 3) spatial DG discretizationand a second-order backward difference scheme (BDF2) with a fixed time step size of Dt = 0.05. The flow analysisarising from each time step is solved by a p-multigrid method driven by a linearized element Gauss-Seidel smoother.15
Typically 5 ∼ 6 p-multigrid iterations can drive the unsteady residual (in L2 norm) to 10−14. Fig. 14 illustrates thecomputed entropy contours in the present problem for four different times (here t is a non-dimensional quantity basedon free-stream speed of sound and diameter of the sphere). As time evolves, flow separates from the surface of thesphere and symmetric recirculation zones are formed, which become unsteady at later times.
To access the accuracy of code for computing discrete adjoint sensitivity derivatives in three-dimensional unsteadyviscous flow problems, a time interval of [0,0.5] (corresponding to a total of 10 time steps) is chosen as the time-integration period. This is sufficient for the verification purpose because the proposed discrete adjoint method reliesonly on the discretization system. Table 2 provides, as an example, a comparison of the sensitivity derivatives for thelast six design variables, computed by the finite difference method and the discrete adjoint method with a fourth-orderDG discretization scheme. A perturbation size of 10−7 is chosen in the former approach. The results show very goodagreement between these methods and the relative difference is below 0.003%, thus again verifying the accuracy ofthe proposed method as well, including the treatment for high-order curvilinear boundary elements. Fig. 15 furtherprovides a complete comparison for all design variables.
Table 2. Comparison of sensitivity derivatives in the three-dimensional unsteady sensitivity derivative calculation, using a fourth-order DGscheme.
V. Conclusions
A discrete adjoint approach for high-order discontinuous Galerkin discretizations is developed in the present workand aerodynamic design optimization problems are investigated for steady and unsteady viscous flows in both twoand three space dimensions. The evaluation of sensitivity derivatives for meshes involving curved boundary elementsrequires accounting for the mesh sensitivities arising from both mesh points and additional surface quadrature points.Moreover, the formulation of the discrete adjoint system must be consistent with the analysis problem since the formeris based on linearization and a transpose operation to the forward linear problem. A similar deformation strategyis implemented for the additional surface quadratures as well as for standard surface grid points to ensure a smoothand accurate representation of the new surface geometry, and the current approach has shown success for designoptimization in low Reynolds number viscous flow. Designed order of accuracy is achieved by the DG discretizations(up to p = 4) for the two- and three-dimensional compressible NS equations and the present work also shows capabilityof the current DG-NS solver in delivering smooth and accurate viscous flow solution.
Since the current paper focuses on flow problems with low and moderate Reynolds numbers, high-order curvedelements have been applied only on physical boundaries. In order to capture flow features in the boundary layer forhigher Reynolds-number flow, highly stretched elements may be required, possibly along with curved interior elementsin the boundary layer. Further work will incorporate these effects and implement a more sophisticated deformationmethod to handle design optimization problems in high Reynolds number viscous flow.
VI. Acknowledgments
The work was supported by the Tennessee Higher Education Commission (THEC) Center of Excellence in Ap-plied Computational Science and Engineering (CEACSE). The support is greatly appreciated. The first author wouldalso like to thank Mr. Nicholas Burgess and Dr. Cristian Nastase from the department of Mechanical Engineering,
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University of Wyoming, for valuable discussions on DG methods.
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