Inlet Shape Optimization Based on POD Model

Reduction of the Euler Equations

Jennifer Goss∗ and Kamesh Subbarao†

The University of Texas at Arlington, Arlington, TX 76019-0018, USA

In this paper we derive and implement a reduced order model of the Euler equationsfor inlet shape optimization. The basis functions in the reduced order model are computedusing the method of snapshots form of Proper Orthogonal Decomposition. The correspond-ing weights are then derived using an optimization in the design parameter space to matchthe reduced order model to the original snapshots. A continuous map of these weights interms of the design variables is obtained via a response surface approximation . This map isthen utilized in another optimization framework to determine the optimal inlet-shape. Totest the methodology a reduced order model is generated for the quasi-1-D duct problem.It is tested for efficiency and accuracy by performing an inverse optimization to match thepressure along the inlet to a desired pressure profile. The same approach is then used togenerate a reduced order model for the 2-D inlet. In this case, we optimize the inlet shapeto minimize the mass weighted pressure loss.

I. Introduction

Model reduction techniques have been a large area of study for many years. The first presentations of thePrincipal Orthogonal Decomposition (POD) method for model reduction were independent works by Loeve(1945) and Karhunen (1946). after which the method was known as the Karhunen-Loeve (K-L) expansion.The method has been called many names over the years including Principal Component Analysis (PCA),Principal Factor Analysis (PFA), and Hotalling transformation. It wasn’t until 1967 when Lumley firstnamed the method as “Proper Orthogonal Decomposition” in his studies with turbulent structures. Theearly uses of the POD method involved many studies of coherent structures in turbulent flows which alsoled Sirovich1 to introduce the method of snapshots in 1987. Today POD model reduction has been used in awide range of scientific fields including; image processing, signal analysis, data compression, oceanography,processes identification, and control in chemical engineering. More details on the history and a thoroughdevelopment of the POD method can be found in Ref. 2.

Within the field of optimal control and optimization reduced order modeling has been studied to greatextent. Early work by Ito and Ravindran3 investigated the use of various reduced basis methods (RBM).With these methods the basis functions are generated from the problem being solved through either a La-grange or Taylor method or a hybrid of both known as the Hermite method. The RBM’s often require manybasis functions which typically contain redundant information and there is no systematic way to reduce thenumber of basis. Later Ravindran4 goes on to compare the RBM with the POD method. The POD methodprovides a means of sorting the basis functions which allows the dominant modes to be clearly identified andextracted to obtain a reduced-reduced basis model. Other methods of determining dominant modes are alsoavailable including a centroid Voronoi tessellation (CVT) method presented by Burkardt, Du, Gunzburgerand Lee.5 The CVT method naturally introduces the concept of clustering in the construction of the basisfunctions and is very useful in a variety of applications including; optimal representation, quantization, celldivision, optimal distribution of sensors and actuators, grid generation, etc. See Ref. 5 for more details on

∗AIAA student member, Graduate Research Assistant, Department of Mechanical & Aerospace Engineering, 500 W. FirstSt., Box 19023, 211 Woolf Hall, jennifer.goss@uta.edu

†AIAA life member, Assistant Professor, Department of Mechanical & Aerospace Engineering, 500 W. First St., Box 19018,211 Woolf Hall, subbarao@uta.edu

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Copyright © 2008 by Subbarao, Goss. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

the CVT method.

Once the basis functions are generated there are variety of methods available to project the governingequations onto the reduced order space. The most common method is the Galerkin approach of which thereare several minor variations including the discontinuous Galerkin approach. In Ref. 6 Iollo studies the stabil-ity and accuracy of the Galerkin projection methods for several 1-D problems. The generally accepted ruleis that any POD-Galerkin scheme based on an underlying stable finite difference or finite volume scheme isalso stable. This is not necessarily the case. It is found that Galerkin POD schemes for Euler equationsneed additional stabilization from that provided by a straight forward discretization of the projection ofthe equations in finite dimensional space. This result is corroborated by Lucia and Beran7 who comparedGalerkin projection with a direct projection based on a Taylor series expansion. This work found the directprojection to require less stabilizing influence. In both cases as with other projection methods in generalthey are very difficult to apply to nonlinear systems such as the Euler and Navier Stokes equations, and theresults are not very robust. LeGresley and Alonso8,9 present a finite volume projection that is solved usinga nonlinear least squares methodology. This method has similarities to the method presented in this paperin that we also use a sub-optimization routine to determine the approximate flow solution given the set ofbasis modes.

Optimization and optimal control problems have been addressed using POD model reduction on variousflow systems such as; the heat equation,10–12 Burgers equation,10,13 Euler equations8,9 and the Navier-Stokesequations.4 The heat equation studies have involved optimal control problems with LQR controllers11 andadaptive critic neural networks12 in the solution of the control input. The Burgers equation studies haveincluded suboptimal feedback control implementation13 and studies into design-then-reduce Vs. reduce-then-design perspectives. Optimization studies have mainly concerned airfoil shape optimization governedby the Euler equations.8,9

This paper investigates the use of a POD reduced order model for shape optimization of the 2-D inlet. Toour knowledge this is the first instance of the use of POD based optimization for this specific problem. Themethod of developing the POD reduced order model and using it in an optimization problem is first testedon the quasi 1-D duct. As the methodology is explained later, the basis functions for the POD model arederived from snapshots of the converged flow for various values of the control variable. Weights correspond-ing to each basis function are determined in a simple and unique way by minimizing the variation betweenthe individual snapshot solutions and the POD model output. Finally, a response surface approximationis obtained for the weights as a function of the design variables. This smooth approximation enables theoverall optimization process that is verified in simulation studies.

Problem Statement: Succinctly stated, the problem we seek to solve is to derive a reduced order modelthat sufficiently captures the dynamics of the desired Euler system of equations, then use that model in anoptimization framework for applications pertaining to shape optimization of inlets.

The paper is laid out as follows; The POD methodology for Euler equations is explained, followed bythe details of the optimization sub-problem to find the weights of the modal expansion. Finally, the shapeoptimization problems are set up. We conclude by verifying the results obtained from the POD basedoptimization with the finite difference based optimization.

I.A. Notation

We follow a similar notation as in Ref. 4, given below.L2(Ω) denotes the collection of all square-integrable functions defined on a flow region Ω ⊂ R2 and theassociated norm is denoted by ‖ · ‖0; also,

H1(Ω) = v ∈ L2(Ω) :∂v

∂xi∈ L2(Ω) for i = 1, 2

and the norm on it be ‖ · ‖1. In addition, L2(0, T ;H1) denotes the space of all measurable functionsf : (0, T ) → H1 such that,

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‖f‖L2(0,T ;H1) =

T∫0

‖f‖21dt

1/2

< ∞

Bold-face symbols denote the vector valued counterparts of the above-mentioned spaces. The innerproducts are denoted by < ·, · >Γ where, Γ denotes the boundary of Ω.

II. The Proper Orthogonal Decomposition of Euler Equations

There are several methods available to generate reduced order models of infinite dimensional distributedparameter systems (DPS) and all of them can essentially be categorized as reduced basis methods. Finitedifference and finite element solutions are also considered reduced order models in that they approximate asystem governed by partial differential equations. Whilst finite difference methods use grid functions, thefinite element methods use piecewise polynomials as basis functions.3

In this paper we are interested in a reduced order model which reduces the computational burden and as suchwe consider a set of basis functions Φi, (i = 1, . . . , N) where N is much less than the number of grid pointsused in an equivalent finite difference approximation or the number of functions used in a finite elementmethod. The approximation to the state variables U ∈ V , spanΦ1, . . . ,ΦN can then be written as alinear combination U =

∑Ni=1 aiΦi where ai are weighting coefficients corresponding to the basis functions

Φi. Selection methods for the weights will be discussed later in this section.

Numerous methods are available for selection of the basis functions mentioned above, such as Taylor, La-grange, Hermite, and snapshot set based approaches. All of these methods use data from the high fidelityapproximation (simulations) to generate a set of basis functions and have become known as reduced-basismethods. This paper will use the snapshot set method with POD for selection of the basis functions. Themain advantage of the POD method over the others is that it reduces the number of redundant basis func-tions and retains those that contain all the essential information. This results in a somewhat optimal reducedbasis model.

II.A. The proper orthogonal decomposition method

The essential approach to POD following the method of snapshots described here is a summary of the workby Ravindran.4 We reproduce the basic ideas and the formulation for the sake of completion. The problemthat the POD method seeks to solve is to identify a structure in a random vector field. The objective isto seek a function Φ that has a structure typical of the members of an ensemble of random vector fields,U(i). To resolve this problem, one would project the field ensemble on Φ and find that Φ that maximizes< U(i),Φ > while ensuring that the amplitude effects are removed through normalization (Φ is being madeparallel to the ensemble). The solution Φ is sought from the space of functions for which the inner-productexists i.e. Φ must be L2(Ω). To include the statistics, we maximize

< U(i),Φ >√< Φ,Φ >

in some average sense. Also, since the maximization only needs to consider the magnitude and not the sign,we would consider the mean of the square of the above.Now consider a set of N snapshots, that form the ensemble set.

S = U(i) : 1 ≤ i ≤ N

In case of control of fluid flows or any DPS, these snapshots are the solutions at N different time steps, ti.The objective in that case is to seek a function Φ ∈ L2(Ω) that maximizes

1N

N∑i=1

| < U(i),Φ > |2

< Φ,Φ >(1)

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It has been shown in Ref. 1 that when the number of degrees of freedom required to describe U(i) is largerthan the number of snapshots N, it is effecient to express the basis functions as a linear combination of thesnapshots. Thus, it is proposed that Φ has a special form in terms of the original data as

Φ =N∑

i=1

wiU(i) (2)

where, wi need to be determined such that Φ maximizes Eq. 1. This maximization problem can be castin an equivalent eigenvalue problem. To this effect, let’s define,

KΦ =1N

N∑i=1

∫Ω

U(i)(x)U(i)(x′)Φ(x′)dx′ (3)

then

< KΦ,Φ > =1N

N∑i=1

∫Ω

∫Ω

U(i)(x)Φ(x)dxU(i)(x′)Φ(x′)dxdx′

=1N

N∑i=1

| < U(i),Φ > |2

Moreover,< KΦ,Φ >

< Φ,Φ >=

1N

∑Ni=1 | < U(i),Φ > |2

< Φ,Φ >= λ

The maximization of the above can be performed using straightforward calculus of variations, i.e. assumingΦ∗ as the function that maximizes λ and using small perturbations to expand, Φ = Φ∗ + εΦ′, one can findthe necessary conditions and show that

< KΦ∗,Φ′ > = λ < Φ∗,Φ′ >

It is hence clear that the maximization of Eq. 1 is equivalent to finding the eigenvalues of the eigenvalueproblem,

KΦ∗ = λΦ∗ (4)

Therefore if Eq. 2 and 3 are used in Eq. 4, we have

CW = λW (5)

where

Cij =1N

∫Ω

U(i)(x)U(j)(x)dx, and W =

w1

w2

...wN

C is a spatial correlation matrix which is nonnegative and Hermitian so it can be decomposed into a completeset orthogonal eigenvectors

W1 =

w1

1

w12...

w1N

, W2 =

w2

1

w22...

w2N

, . . . . . . , WN =

wN

1

wN2...

wNN

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along with a set of eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λN ≥ 0. We can then obtain the solutions to Eq. 1 as,

Φ1 =N∑

i=1

w1i U

(i), Φ2 =N∑

i=1

w2i U

(i), . . . , ΦN =N∑

i=1

wNi U(i), (6)

We then normalize these by requiring,

< Wj ,Wj > =1

Nλj

This now provides us a set of orthonormal basis functions Φ1, Φ2, . . . ,ΦN i.e.,

< Φl,Φm > =

1 l = m

0 l 6= m

Also, the POD subspace is then essentially defined as, VPOD = spanΦ1, Φ2, . . . ,ΦN. The energy of agiven data set associated with the corresponding mode Φi can be quantified based on Eq. 3.

λi =1N

N∑j=1

< Φi,U(j) >2

Ravindran4 then also shows that the POD subspace calculated above is optimal in the sense that theapproximation of the snapshots,

U(l) =Nk∑i=1

aliΦi, al

i =< Φi,U(l) >

maximizes the captured energy

E =1N

N∑i=1

< U(i),U(i) >

=Nk∑i=1

λi for all Nk < N

To accurately capture the underlying dynamics of the system, N has to be large. In such a case, using aGalerkin procedure, one can obtain a high fidelity model for large N . In many cases the majority of theenergy capture is contained in the first few modes. If this is the case then the number of functions requiredto accurately describe the system may be significantly less than N . The set can be truncated to Nk to obtainan optimal set.

II.B. Application of POD to Euler equations

The governing equations are the 2-D Euler equations are summarized below. (see Eq. 7)

∂U∂t

+∂F

∂x+

∂G

∂y= 0 (7)

where the state variables U and the flux vectors F and G are given as

U =

ρ

ρu

ρv

ρE

, F =

ρu

ρu2 + p

ρuv

u(ρE + p)

, G =

ρv

ρuv

ρv2 + p

v(ρE + p)

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where ρ, p, u and v are the density, pressure and x and y velocity components respectively. To close thesystem we assuming the perfect gas relationship holds

E =p

ρ(γ − 1)+

12(u2 + v2)

The corresponding 1-D equations for a duct like configuration can be obtained from the above. The flowequations in this case are written in conservative form as follows

∂U∂t

+∂F

∂x= 0 (8)

where the state variables, flux and pressure terms are given as follows

U =

ρ

u

p

, F =

ρu

ρu2 + p

ρu(E + pρ )

, P =

0p

0

,

E =p

ρ(γ − 1)+

12(u2)

For the case of the Euler equations (2-D as well as 1-D), we apply the procedure discussed in theprevious sections for each of the primitive variables, (ρ, u, v, p) that would in turn be used to compute theconservative state vector (ρ, ρu, ρv, ρE) if need be. Having obtained the basis functions (modes) withinacceptable accuracy, we expand the flow solution about an arbitrary geometry. To illustrate, the flow fieldsfor a double ramp 2-D inlet case would be expressed as

ρ(x, y, α1, α2) =Nk∑i=1

aρi(α1, α2)Φ(ρ)i

u(x, y, α1, α2) =Nk∑i=1

aui(α1, α2)Φ(u)i

v(x, y, α1, α2) =Nk∑i=1

avi(α1, α2)Φ(v)i

p(x, y, α1, α2) =Nk∑i=1

api(α1, α2)Φ(p)i

(9)

where α1, α2 are the two angles that characterize the double ramp inlet shown in Fig. 7

To generate the basis functions for these cases, we first convert the flow solutions (snapshot solutions) for ajudiciously chosen grid of design parameters into groups of vectors. If the number of nodes on the computa-tional grid are [IMAX, JMAX], then U(i) is a [IMAX ∗ JMAX, 1, 4] matrix. The last index refers to numberof primitive variables i.e., sizeof (ρ, u, v, p). Thus, for Np design parameter combinations, the ensemble ofsnapshots will be a [IMAX ∗ JMAX, Np, 4] matrix.

The snapshots are then obtained following the approach outlined in the previous section. Since the com-putation of the basis functions is done separately for each of the primitive variables, all calculations shownbelow are repeated for each of the primitive variables. To obtain the spatial correlation matrix,

Cij =1N

∫Ω

U(i)(x)U(j)(x)dx, and W =

w1

w2

...wN

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we approximate the above for the discrete computational domain as follows,12

Cij =1N

IMAX∗JMAX∑k=1

U(i)(k)U(j)(k)∆Ak

where, ∆Ak is corresponding cell area. This evaluation poses a problem with situations wherein unstructuredmeshes would be utilized and when the domain of the ith and jth snapshots are different. We use a commondomain for all snapshots thereby avoiding this problem. This requires us to apply changes to the boundaryconditions to account for the change in the boundary.14

II.C. Determination of the Weights for the Reduced Order Model (ROM)

Having obtained the basis functions as outlined earlier, we can now seek to obtain the weights aρi, etc (seeEq. 9). The most common methods used to calculate the weights are projection based methods, Galerkinprojection being the most popular among them. In these cases the governing partial differential equations(PDE) are recast into a system of ordinary differential equations (ODE). While the Galerkin projection is oneof the most common methods, it is generally limited to incompressible flows due to its sensitivity to errorsin boundary conditions and the need for simple, smooth geometries.7 The Galerkin method can be appliedto the compressible case if these issues are addressed carefully and the projection is of minimal order (smallnumber of basis functions). Ref. 7 compares the Galerkin projection method to their own direct projectionmethod for computational savings in the compressible Euler equations.

If the system is well known and well behaved then projection methods can work well although they do getcomputationally expensive for more complex systems. However, for purely data driven systems and reducedorder modeling of such, the projection methods would no longer be possible as the governing PDEs areunavailable. This necessitates that the weights be determined in a different manner. We consider it as asystem identification problem and approach it as such. Since the actual snapshots (solutions) are availableat discrete design values, we propose to utilize the following approach to generating the weights. The weightsare derived by minimizing the following objective function for each snapshot condition

JPOD =∑Nk

‖U(i) − U(i)‖2 (10)

where the approximation U(i) to the desired data U(i) can be written in terms of the orthonormal basisΦi(x), i = 1, 2, . . . , Nk as

U(i)(x, α1, . . .) =Nk∑i=1

ai(α1, . . .)Φi(x) (11)

where, again the ai’s are the weighting coefficients of the ith mode in the function expansion. This defines thesolution for the given set of basis most closely resembling the snapshot data. Given a set of basis functionsand the generated snapshot data, the cost of the reduced order model approximation is simply the cost ofthe optimization of weighting coefficients as given in Eq. 10.

III. POD Based Shape Optimization

While most POD approaches have been used for flow control problems8,11,13 and in other applicationssuch as prediction of flutter,14 control of DPS12 in general, POD based optimization studies are limited.Ref. 8 studied the airfoil optimization problem and showed the tremendous reduction in the degrees offreedom for the 2-D problem. In this paper, we study the shape optimization of two geometries, the quasi1-Dimensional duct flow and the double ramp 2-D inlet.

III.A. POD for the quasi 1-D duct shape optimization

The “quasi 1-D” term is used because we assume that any y-components of the flow are negligible (see Fig. 1).This is a valid assumption if the cross-section of the duct is smooth and does not change dramatically. Formost duct flow problems the quasi 1-D duct is a reasonable approximation of the full 3-D flow in a duct.

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In what follows, we will first summarize the governing equations for the quasi 1-D duct, followed by theboundary conditions’ specification, the reduced order model derivation, the formal optimization procedureand the results.

h = h(x)IN

h(in)

ρ(in)

u(in)OUT

h(out)

ρ(out)

u(out)

------

------

-X

Figure 1. 1-D duct schematic

III.A.1. Governing equations for the quasi 1-D duct flow

The governing equations for the above system is derived from the full 1-D system of equations beginningwith the continuity equation. Consider a duct such as the one shown in Fig. 1. The flow variables are thevelocity, u, density, ρ and pressure, p and the duct shape is defined by h = h(x). Given a small controlvolume of area ν and with a boundary S, the quasi 1-D continuity equation is derived as follows

∂

∂t

∫ν

ρdν +∮S

ρu · dS = 0

⇒ ∂

∂t

∫ν

ρdν +∮

h(in)

ρu · dS +∮

h(out)

ρu · dS +∮

h(x)

ρu · dS = 0

Restricting the solution to steady state and imposing the wall condition (u · dS = 0), the following termsare eliminated. ∮

h(x)

ρu · dS = 0,∂

∂t

∫ν

ρdν = 0

With dS defined as the outward normal being positive the equation can be written as

⇒ −h(−1)ρ(−1)u(−1) + h(1)ρ(1)u(1) = 0⇒ h(x)ρ(x)u(x) = constant

⇒ d

dx(ρuh) = 0 (12)

Considering the same control volume the quasi 1-D momentum equation is derived as follows

∂

∂t

∫ν

ρudν +∮S

(ρu · dS)u = −∮S

pdS

Again restricting the solution to steady state and imposing the wall condition (u · dS = 0), the equationsimplifies to

⇒∮

h(in)

(ρu · dS)u +∮

h(out)

(ρu · dS)u = −∮

h(in)

pdS−∮

h(out)

pdS +∮

h(x)

pdS

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⇒ −h(−1)ρ(−1)u2(−1) + h(1)ρ(1)u2(1) = h(−1)p(−1)− h(1)p(1) +∮

h(x)

pdS

⇒ d

dx(h(x)ρ(x)u2(x)) +

d

dx(h(x)p(x)) = p

d

dx(h(x))

⇒ d

dx(hρu2) +

d

dx(hp)− p

d

dx(h) = 0 (13)

For notational convenience, from here on we will denote h(x) by h. The energy equation is similarly derivedas follows

∂

∂t

∫ν

ρEdν +∮S

ρEu · dS = −∮S

pu · dS

which in steady state

⇒∮S

ρEu · dS = −∮S

pu · dS

∴d

dx(hρEu) +

d

dx(hpu) = 0 (14)

Assembling the conservation equation for mass Eq. 12, momentum Eq. 13, and energy Eq. 14 together givesthe final system of equations for the duct

d

dx(hρu) = 0

d

dx(hρu2 + hp)− p

d

dx(h) = 0

d

dx(hρEu + hpu) = 0

where E is the total energy and is defined by the equation of state for an ideal gas.

E =p

ρ(γ − 1)+

12u2

The flow equations are written in conservative form as follows

R(U, h) =d

dx(hF )− P

dh

dx= 0 (15)

where h(x) represent the height of the duct and the state variables, flux and pressure terms are given asfollows

U =

ρ

u

p

, F =

ρu

ρu2 + p

ρu(E + pρ )

, P =

0p

0

,

III.A.2. Boundary conditions for the quasi 1-D duct problem

The boundary conditions of any flow problem governed by the Euler equations are set by the characteris-tics of the flow. The characteristics are essentially the eigenvalues of the Jacobian matrix dF

dU , that can beanalytically computed to be u, u − a, and u + a where a is the speed of sound. The number of necessaryconditions to be specified at a boundary is set by the number of characteristics that point into the system

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from that boundary.

For instance, in subsonic flow the IN-flow boundary (see Fig. 1) would have two characteristics pointing in (uand u+a) and one pointing out (u−a). In this paper, we study the supersonic case with no shocks present.The resulting boundary conditions require that three IN-flow conditions need to be specified and the outletconditions are all left unspecified. The inlet conditions are fixed at a given enthalpy of H = E + p/ρ = 4,total pressure Po = 2, and Mach number M = 3.0.

III.A.3. POD based reduced order model for the quasi 1-D duct flow

An inverse pressure matching optimization problem is solved using a desired pressure profile that correspondsto some optimal shape of the duct. The shape of the duct is parameterized through a parameter α andgoverned by the following equation.

h(x, α) = α x2 − (√

0.8α) x + 1, 0 < α ≤ 1 (16)

Without loss of generality the optimal pressure profile is chosen corresponding to α = 0.8

The reminder of this section describes the algorithm for applying the POD based model reduction to thegoverning equations. Let U(i)(x, α) be a steady state solution of the flow for a given α defining the ductshape from Eq. 16. The following algorithm describes the process of determining the complete POD reducedorder model.

1. Generate a set of N snapshots U(i) from steady state flow solutions of Eq. 15 for a random set of α’s.The steady state duct flow is computed for 12 discrete values of the control variable α ranging between0.55 and 0.95

2. Compute the correlation matrix C from Eq. 5

3. Solve the eigenvalue problem CW = λW to get a complete set of eigenvectors wi. The eigenvalues forthe correlation matrix are shown in Fig. 2. The amount of information contained in each mode dropsoff quickly and only the first 2 basis functions are required to capture 99.99% of the energy.

0 2 4 6 8 10 1210

−20

10−15

10−10

10−5

100

105

Eigenvalues of the Correlation Matricies

Logs

cale

of t

he E

igen

valu

es

density ρvelocity, upressure, p

Figure 2. Eigenvalues for correlation matricies for each of the primitive variables

4. Obtain the basis functions Φi using Eq. 6. Fig. reffig:basisFunctions shows the first two basis functions.The first mode has the same shape as the true response from the system and contains the majority ofthe information of the system.

5. Determine the corresponding weights in Eq. 11 by minimizing the cost function given in Eq. 10

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

0.9

1

1.1

1.2

1.3Basis Functions for Reduced Order Model

Φ1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4

Φ2

x

densityvelocitypressure

Figure 3. First two basis function generated from the eigenvalues of the correlation matrix

6. Figure 4 shows the relationship between the weights and the design parameter α. We notice a nicecorrelation between the control variable and calculated weights. The weights bear a simple linearrelationship with the design variable α.

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950

0.5

1

1.5

2

2.5

w1

Linear Curve Fit of the Basis Function Weights

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95−0.04

−0.02

0

0.02

0.04

α

w2

wρ

wu

wp

Figure 4. Weights corresponding to the basis functions as a function of the design parameter α

With the basis functions and weights computed the reduced order model is complete. To illustrate how wellthe reduced order model captures the original snapshots, we show a comparison of the POD model withthe CFD results for some specific control (design) values. Fig. 5 shows this comparison for the two extremevalues of α that were used to generate the snapshots, i.e. α = 0.55 and α = 0.95.

III.A.4. Optimization of the quasi 1-D Duct shape

As mentioned earlier, the duct shape optimization problem is posed as an inverse optimization for pressurematching along the duct. The cost function is chosen such that the pressure, p along the duct is to match somedesired pressure distribution p∗ subject to the flow equations given in Eq. 15 and corresponding boundaryconditions.

J =12

1∫−1

‖p− p∗‖2dx

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.7

0.8

CFD and ROM solutions for α = 0.55

Den

sity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.15

2.2

2.25

2.3

Vel

ocity

CFDROM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.25

0.3

0.35

0.4

0.45

Pre

ssur

e

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.7

0.8

CFD and ROM solutions for α = 0.95

Den

sity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.15

2.2

2.25

2.3

Vel

ocity

CFDROM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.25

0.3

0.35

0.4

0.45

Pre

ssur

e

x

Figure 5. Comparing the results of the CFD solution with the final POD reduced order model for the extremum valuesof the control variable.

With only one control variable the optimization routine used here is very straight forward. We use aLevenberg-Marquardt based approach (that combines the quasi-Newton and steepest descent) to optimizethe pressure matching scenario with a target pressure profile. The target profile is generated using thesame CFD solver as that used to generate the snapshot data. As mentioned before, we chose a target profilecorresponding to α = 0.8 and then began the optimization based on the POD model with an initial conditionof α0 = 0.2. This initial condition is a good test of the POD model because is is well outside the range ofvalues used to generate the snapshots from which the reduced order model was obtained. As shown in Fig. 6the optimization was successful and was completed in only 2 iterations.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.2

0.4

0.6

0.8

1

IterationCon

trol

Var

iabl

e V

alue Final Value: 0.79953

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5x 10

−3

Iteration

Fun

ctio

n V

alue

Final Function Value: 5.25e−008

Figure 6. Results of the POD model optimization

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III.B. Application of the POD Reduced Model Method to the 2-D Inlet Flow

The POD model reduction method is now applied to the 2-D double ramp inlet problem illustrated in Fig. 7.The shape of the ramp is typically designed for a small operating range. Changing the shape of the inletwould allow it to perform optimally over a broader range of flight conditions.

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Flow

α1

α2

X (m)

Y (

m)

2−D Double Ramp Inlet

Figure 7. 2-D inlet schematic

III.B.1. Euler flow solver for the 2-D inlet

The Euler equations given in Eq. 7 are solved using the advection upstream splitting method (AUSM)scheme. The method was originally proposed by Liou and Steffen15 for typical compressible aerodynamicflows. We adopt this scheme because it’s a very efficient method for accurate shock capture. The equationsare solved on the grid given in Fig. 8 which is 240× 100 cells.

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

X (m)

Y (

m)

2−D Inlet Grid

Figure 8. Grid for the 2-D double ramp inlet.

III.B.2. Boundary conditions for the 2-D inlet

As explained in the 1-D case the boundary conditions are determined by the characteristics of the flow, u,u − a, and u + a where a is the speed of sound. The number of necessary conditions to be specified at aboundary is set by the number of characteristics that point into the system from that boundary.

In this case the IN-flow boundary is supersonic and has all characteristics pointing in and none pointing out.This requires that the entire IN-flow conditions be specified and the OUT-flow conditions are all left unspeci-fied. The inlet conditions are fixed at p = 1, ρ = 1, and Mach number M = 14.0 such that u = M a and v = 0.

All wall boundaries are treated as reflection surface for the wall normal component of the velocity vector.

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III.B.3. POD based reduced order model for the 2-D inlet

We now explore the use of a POD reduced order model in the optimization of the shape parameters. Thecontrol variables, or shape parameters, in this case are the angles of the two ramp portions of the inlet (α1, α2).

The 2-D double ramp inlet is approached in the same manner as the quasi 1-D duct. The governingequations are the 2-D Euler equations given in Eq. 7. The POD reduced order model is derived using thesame algorithm as the previous case. Let U(i)(x, y, α1, α2) be a steady state solution of the flow equationsfor a set of design variables (α1, α2).

1. Generate a set of N snapshots U(i) from steady state flow solutions of Eq. 7 for a random set of αi’s.The steady state inlet flow is computed for 88 permutations of the control variables with α1 rangingbetween 4.1 and 5.15 and α2 ranging between 5.2 and 6.7.

2. Compute the correlation matrix C from Eq. 5

3. Solve the eigenvalue problem CW = λW to get a complete set of eigenvectors wi. The eigenvalues forthe correlation matrix are shown in Fig. 9. The amount of information contained in each mode dropsoff quickly initially then becomes very gradual. The first 3 modes contain 99.0% of the energy in thesystem, where the first 10 modes contain 99.98%. To gain one more 0.01% to reach 99.99% we need tojump from 10 to 53 modes. Thus to maintain accuracy yet keep the computation time low, we choseto keep 10 modes.

0 10 20 30 40 50 60 70 80 9010

−25

10−20

10−15

10−10

10−5

100

105

1010

Eigenvalues of the Correlation Matricies

Logs

cale

of t

he E

igen

valu

es

density ρvelocity, upressure, p

Figure 9. Eigenvalues for correlation matrices for each of the primitive variables

4. Obtain the basis functions Φi using Eq. 6.

5. Determine the corresponding weights in Eq. 11 by minimizing the cost function given in Eq. 10

6. Obtain a response surface to determine a relationship for the weights as a function of (α1, α2).

pth order response surface The weights are computed as response surfaces to the Np combinationof the design parameters. A 3rd and 5th order surface is chosen.

For the approach mentioned above, we suppose that the desired ROM will best approximate the snap-shot solutions.We obtain the functional representation for the weights using a least squares polynomialapproximation for any specified order.

For the sake of illustration, we will outline the procedure to obtain the least squares based response surfacefor the vector weight function, aρ (see Eq. 9); for each of the Np combinations of the parameters, we firstassume that the function can be represented by the following,

aρi,j = (cρ,0 + cρ,1α1,j + cρ,2α21,j + · · ·+ cρ,kαk

1,j)⊗ (bρ,0 + bρ,1α2,j + bρ,2α22,j + · · ·+ bρ,kαk

2,j)= HjX

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where, j = 1 . . . Np and X = [cρ,0, cρ,1, · · · , cρ,k, bρ,0, bρ,1, · · · , bρ,k]T . The matrix H consists of theproducts of αl

1,jαm2,j with l = 1 . . . k and m = 1 . . . k. To compute the parameters cρ,i and bρ,i. The

parameters are then obtained as a weighted least squares solution,

X =[HT ΛH

]−1HT Λaρi

III.B.4. Optimization of the 2-D inlet using the POD model

The optimization problem for this flow problem is different from that of the quasi 1-D duct problem. Weare no longer looking at an inverse optimization problem, instead we seek to minimize the mass weightedpressure loss at the exit plane of the double ramp 2-D inlet. The minimum pressure loss represents theoptimal shape configuration with minimal energy losses due to the shocks cancelling on the shoulder of theinlet. The cost function is summarized as follows

FPressureLoss =∑

ρi,jui,j∆yi,j (Pi,j − P∞)mP∞

(17)

where, Pi,j is the static pressure at node (i, j) and P∞ is the pressure in the far-field. Figures 10 and 11 showthe POD model optimization results for 3rd and 5th order response surfaces respectively. The top figure ineach shows the evolution of the control variable for each iteration of the optimization and the lower figuregives the evolution of the cost function. The optimization is completed in only 6 iterations. The resultsof the POD model optimization are summarized in Table 1 and are compared with the results of a finitedifference optimization. Note, that the results due to the POD based optimization matches very closely withthe finite difference optimization run on the high fidelity model.

Case Angles (deg)α1 α2

Finite difference optimization 4.8446 5.6458POD optimization (3rd order response surface) 4.7551 5.6407POD optimization (5rd order response surface) 4.73 5.619

Table 1. Final values of the control variables for double ramp inlet optimization using the proper orthogonal decom-position reduced order model.

0 1 2 3 4 5 64.5

5

5.5

6

Iteration

Con

trol

Var

iabl

e V

alue

Final Values: 4.7551 5.6407

0 1 2 3 4 5 62.3

2.35

2.4

2.45

2.5

2.55

Iteration

Fun

ctio

n V

alue

Final Function Value: 2.3041

Figure 10. Results of the POD model optimization with 3rd order response surface

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0 1 2 3 4 5 64.5

5

5.5

6

Iteration

Con

trol

Var

iabl

e V

alue Final Values: 4.73 5.619

0 1 2 3 4 5 62.2

2.3

2.4

2.5

2.6

Iteration

Fun

ctio

n V

alue

Final Function Value: 2.2619

Figure 11. Results of the POD model optimization with 5th order response surface

IV. Conclusions and Future Work

In this paper, we developed and implemented a proper orthogonal decomposition method to obtainreduced order models of 1-D as well as 2-D Euler equations. The reduced order models were utilized tooptimize inlet geometries for a double ramp supersonic inlet as well as the shape of a quasi 1-D duct. Toobtain the weights for the POD based models, we developed an optimization based scheme that is lessexpensive computationally as opposed to the more traditional projection based schemes.5 In all the casesthe optimization process for the inverse pressure matching of the quasi 1-D duct as well as the double rampinlet took only a few iterations. The procedure illustrates a lot of promise for optimization of DPS in general.

References

1Sirovich, L., “Turbulence and the dynamics of coherent structures: Part I-III,” Quarterly of Applied Mathematics,Vol. 45(3), 1987, pp. 561–590.

2Berkooz, G., Holmes, P., and Lumley, J. L., “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,”Annual Review of Fluid Mechanics, Vol. 25, 1993, pp. 539–575.

3Ito, K. and Ravindran, S. S., “A Reduced Basis Method for Control Problems Goverened by PDEs,” Control andEstimation of Distributed Parameter Systems, edited by K. K. Wolfgang Desch, F. Kappel, Vol. 126, International Series ofNumerical Mathematics, Birkhause Verlag, Basel, 1998, pp. 173–191, ISBN 3764358351.

4Ravindran, S. S., “Proper Orthogonal Decomposition in Optimal Control of Fluids,” Tech. rep., NASA, Langley ResearchCenter, Hampton, Virginia, March 1999, TM-1999-209113.

5Burkardt, J., Du, Q., Gunzburger, M., and Lee, H.-C., “Reduced Order Modeling of Complex Systems,” 20th BiennialConference on Numerical Analysis, University of Dundee, Scotland, UK , June 24-27 2003, pp. 29–38.

6Iollo, A., “Remarks on the Approximation of the Euler Equations by a Low Order Model,” Research Report RR-3329,Inria, December 1997.

7Lucia, D. J. and Beran, P. S., “Projection Methods for Reduced Order Models of Compressible Flows,” Journal ofComputational Physics, Vol. 188, No. 1, June 2003, pp. 252–280.

8LeGresley, P. A. and Alonso, J. J., “Airfoil Design Optimization Using Reduced Order Models Based on Proper OrthogonalDecomposition,” AIAA Fluids 2000 Conference and Exhibit , June 12-22 2000, AIAA Paper 2000-2545.

9LeGresley, P. A. and Alonso, J. J., “Investigation of Non-Linear Projection for POD Based Reduced Order Models forAerodynamics,” 39th AIAA Aerospace Sciences Meeting and Exhibit , January 8-11 2001, AIAA Paper 2001-0926.

10Atwell, J. A., Proper Orthogonal Decomposition for Reduced Order Control of Partial Differential Equations, Ph.D.thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, April 10 2000.

11Camphouse, R. C., “Boundary Feedback Control Using Proper Orthogonal Decomposition Models,” Journal of Guidance,Control, and Dynamics, Vol. 28, No. 5, September-October 2005, pp. 931–938.

12Padhi, R. and Balakrishnan, S. N., “Proper Orthogonal Decomposition Based Feedback Optimal Control Sysnthesis ofDistributed Parameter Systems Using Neural Networks,” American Control Conference, May 8-10 2002.

13Kunisch, K. and Volkwein, S., “Control of the Burgers Equation by a Reduced-Order Approach Using Proper OrthogonalDecomposition,” Journal of Optimization Theory and Applications, Vol. 102, No. 2, August 1999, pp. 345–371.

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14Pettit, C. and Beran, P., “Reduced-order modeling for flutter prediction,” AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics and Materials Conference, Atlanta, Georgia, USA.

15Liou, M. and Steffen, C., “A new flux splitting scheme,” Journal of Computational Physics, Vol. 107, No. 1, July 1993.

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