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Part 21

Mesh Adaptation

Adjoint-Based Methodology for AnisotropicGrid Adaptation

N. K. Yamaleev, B. Diskin, and K. Pathak

Abstract A new adjoint-based grid adaptation methodology for solving steady andunsteady problems is presented. In contrast to conventional grid adaptation tech-niques utilizing the error equidistribution principle, the new approach directly solvesan error minimization problem for which grid node coordinates are used as controlvariables. A minimum of the error functional is found using a gradient method basedon the adjoint formulation. The mesh sensitivity derivatives required for solving theerror minimization problem are computed using the solution of the correspondingadjoint equations. The key advantage of this formulation is that the adjoint equa-tions are solved only once at each grid adaptation iteration regardless of the numberof grid nodes, which makes this approach well suited for mesh adaptation. The newadjoint-based grid adaptation strategy is tested on several benchmark problems gov-erned by the Euler and Poisson equations.

1 Error Minimization Problem

The proposed anisotropic grid adaptation methodology is based on minimization ofthe error in a functional output (e.g. lift, drag, torque, thrust, etc.) over each timeinterval. The problem is to find a grid at each time step, which minimizes the errorin the integral output of interest, while maintaining the same grid connectivity atall time levels. This time-dependent discrete error minimization problem can beformulated as follows:

Nail K. YamaleevNorth Carolina A&T State University, Greensboro, NC, USA, e-mail: [email protected]

Boris DiskinNational Institute of Aerospace, Hampton, VA, USA e-mail: [email protected]

Kedar PathakNorth Carolina A&T State University, Greensboro, NC, USA, e-mail: [email protected]

1

2 N. K. Yamaleev, B. Diskin, and K. Pathak

minXn

en(Qn,Xn), (1)

where en is the error in an integral output evaluated at time level n, Xn is an adap-tive mesh, and Qn is the corresponding flow solution computed on this mesh. Inthe present formulation, the mesh coordinates are control variables, and grid nodesare moved to minimize the error in the integral output, while preserving the initialgrid topology. The optimal control problem (1) is subject to the discretized flowequations. Along with the flow equations, a grid quality constraint is imposed toguarantee that no mesh degeneration occurs during the adaptation, which impliesthat there are no negative control volumes in the computational domain.

To close the error minimization problem, the error functional in Eq. (1) has tobe defined. Construction of reliable and accurate error estimates is a challengingproblem, which is beyond the scope of the present paper. In the present analysis, theerror functional is defined using the true error in an integral output

en = ( fex − f )2 , (2)

where fex and f are the exact and computed integrals of a quantity of interest (e.g.pressure). The integral output is tolerant to non-smoothness of the flow variablesand can be used for flows with strong discontinuities.

2 Adjoint-based Anisotropic Grid Adaptation Method

Finding a global extremum of the error minimization problem (1) is a difficult task.On one hand, the global minimum of this PDE-constrained minimization problemmay not be feasible. On the other hand, for large-scale aerodynamic problems, thenumber of grid points and consequently the dimensionality of the design space maybe of the order of O(107), thus making the computational cost associated with so-lution of this global minimization problem comparable or even higher than thatrequired for achieving the same error level by uniform grid refinement. The aboveconsiderations suggest that instead of seeking the global minimum, it would be morepractical to reduce the error by finding a local minimum of the error functional. Alocal extremum of the error minimization problem (1) can be efficiently found bygradient methods based on the adjoint formulation. Along with the efficiency, an-other key advantage of the adjoint-based method is that its computational cost isindependent of the number of control variables (the grid node coordinates), whichmakes this approach particularly attractive for mesh adaptation.

The discrete PDE-constrained error minimization problem (1) is solved by themethod of Lagrange multipliers which is used to enforce the governing equationsand the corresponding boundary conditions as constraints. The discrete Lagrangianfunctional is defined as follows:

L(Q,X,Λ) = en +[Λ n]T(

32V

n Qn−Qn−1∆ t +

12V

n−2 Qn−2−Qn−1∆ t +R

n +RnGCLQn−1

)(3)

Adjoint-Based Methodology for Anisotropic Grid Adaptation 3

where Λ is a vector of Lagrange multipliers, en is an error functional, Rn and RnGCLare the flow and geometric conservation law residuals, respectively.

To compute the mesh sensitivity derivatives, the Lagrangian is differentiated withrespect to Xn. This leads to the following adjoint equations for determining theLagrange multipliers (see [2] for details):

32∆ t V

nΛ n +[

∂Rn∂Qn

]TΛ n =−

[∂en∂Qn

]T(4)

The main advantage of the adjoint formulation is that at each optimization iteration,the adjoint equations (4) should be solved once regardless of the number of gridpoints. The above adjoint equation involves only the adjoint variable at the currenttime level, thus making it similar to that of the steady state adjoint formulation inthe sense that no integration of the adjoint equations backward in time is required.

With the Lagrange multiplier Λ n found from Eq. (4), the sensitivity derivative isevaluated as follows:

dLdXn

=∂en

∂Xn+[Λ n]T

(32

∂Vn

∂XnQn −Qn−1

∆ t+

∂Rn

∂Xn+

∂RnGCL∂Xn

Qn−1)

(5)

A local minimum of the error functional at each time step is found by the steepestdescent method in which each step of the optimization cycle is taken in the negativegradient direction

Xnk+1 = Xnk −δk

[dL

dXnk

]T, (6)

where δk is an optimization step size which is chosen adaptively [4], k is the op-timization iteration number. The iteration process is repeated until either the errorin the integral output becomes smaller than a user-specified error tolerance or thedifference between the corresponding grid node coordinates on the previous andupdated meshes becomes less than a given threshold value.

To avoid overlapping of neighboring control volumes during adaptive r-refinementgiven by Eq. (6), a trust region approach is used. At each optimization iteration,a trust region for the local steepest descent search algorithm is defined such thateach node remains within its control volume. Since overlapping of grid cells is notallowed, the number of grid points and grid topology remain unchanged, as fol-lows from Eq. (6). This adaptive r-refinement procedure redistributes grid nodesin anisotropic fashion to minimize the error estimate. In contrast to heuristic ap-proaches relying on the Hessian of a single flow variable to determine stretchingand orientation of mesh cells during grid adaptation [3], the proposed adjoint-basedr-refinement technique (4-6) provides a rigorous relation between the sensitivity ofthe integral output error and the size, shape, and orientation of mesh elements.


Fig. 1 Time histories of the pressure integral error (left) and the L∞ norm of the mesh sensitivityderivatives (right) obtained for the nozzle flow problem.

3 Numerical Results

The first test case is presented to evaluate the performance of the proposed grid adap-tation strategy for the 2-D Euler equations describing the steady shocked flow in adiverging nozzle. The nozzle is symmetric about the x-axis, and its cross-sectionalarea is given by the following equation: A(x) = 1.398+ 0.347tanh(0.8x− 4), 0 ≤x ≤ 10. The inflow Mach number is set equal to 1.2, and the outflow pressure hasbeen chosen to be 1.073847, so that the flow is transonic and the shock wave is lo-cated near x = 3. The functional considered for this test case is the pressure integralcomputed along the nozzle walls. The error in the functional output is estimatedusing a numerical solution obtained with the 4th-order finite difference method on afine 961×97 mesh. The initial mesh is constructed from a smooth structured quadri-lateral 81× 9 mesh by dividing each quadrilateral element in two triangles in sucha way that the mesh is symmetric with respect to the x-axis.

To assess the performance of the dynamic variant of the new adjoint-based gridadaptation procedure for unsteady flows, the steady transonic nozzle flow problem is

0 2 4 6 8 10-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 2 Adaptive grid obtained at 16th time level for the nozzle flow problem.

Adjoint-Based Methodology for Anisotropic Grid Adaptation 5

solved in a time-dependent fashion. The 2-D Euler equations are integrated in timewith a constant nondimensional time step that is set equal to 20. Sixteen time stepsare required to reduce the residual below 10−12. At each time step, the grid adap-tation iterations are repeated until the objective functional becomes smaller than auser-defined tolerance which is set equal to 10−12. Only 3-6 iterations are neededto reach convergence at each time level. Time histories of the error in the pressureintegral and the mesh sensitivities are presented in Fig. 1. The objective functionalconverges to zero, as the mesh is adapted. The optimal mesh obtained at the finaltime level is depicted in Fig. 2. Note that this test problem can be solved using

Fig. 3 Convergence of the integral error (left) and the L∞ norm of the mesh sensitivity derivatives(right) for the Poisson problem.

the static variant of the adjoint-based grid adaptation algorithm which results in adifferent optimal mesh that also nullifies the objective functional. This observationsuggests that there exist multiple solutions to the error minimization problem (1-2),and the adjoint-based steepest descent method finds an optimal mesh that is close tothe initial mesh.

The second test problem is a 2-D scalar Poisson equation with a manufacturedsolution resembling the boundary-layer profile. The equation is discretized with asecond-order node-centered finite volume scheme [1]. The right hand side and theDirichlet boundary conditions for the Poisson equation are chosen so that the prob-lem has the following exact solution:

q(x,y) = (x+1) tanh(25y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

with a thin boundary layer along the lower boundary y = 0. The integral of the nor-

mal derivative of q along the boundary y = 0, f (q) =1∫0

∂q∂y dx, is used as the quantity

of interest. The static formulation of the adjoint-based grid adaptation algorithm is


x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

,x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 4 Initial (left) and adaptive grids for the Poisson problem.

used. At each optimization iteration, the residuals of the discretized Poisson equa-tion and its adjoint are driven to the machine zero.

Convergence histories of the error functional and the maximum norm of the meshsensitivity derivatives are presented in Fig. 3. The error in the target integral and themesh sensitivities drop by more than three orders of magnitude every 10 optimiza-tion iterations and become less than 10−9 after 30 iterations, thus indicating that theoptimizer converges to a global minimum of the error minimization problem. Theinitial mesh and the final adaptive mesh obtained at the 33rd optimization iterationare shown in Fig. 4. Despite that the initial mesh is clustered near the lower bound-ary, the functional error on this mesh is of the order of one. The comparison of theinitial and adaptive grids shows that small changes in the grid node coordinates re-sult in the drastic error reduction in the integral of the normal derivative. Similarto the previous test case, the error minimization problem subjected to the Poissonequation has multiple solutions that can readily be obtained by starting from dif-ferent initial meshes. It again indicates that a solution of the error minimizationproblem is not unique.

References

1. Anderson, W. K., and Bonhaus D. L., “An implicit upwind algorithm for computing turbulentflows on unstructured grids,” Computers and Fluids, Vol. 23 pp. 1-21, 1994.

2. Nielsen, E. J., Diskin B., and Yamaleev, N. K. “Discrete adjoint-based design optimization ofunsteady turbulent flows on dynamic unstructured grids,” AIAA Journal, Vol. 48, No. 6, pp.1195-1206, 2010.

3. Perarie, J., Vahdati, M., Morgan, K., and Zenkiewicz, O. C., “Adaptive remeshing for com-pressible flow computations,” J. of Comput. Phys., Vol. 72, pp. 449-466, 1987.

4. Yamaleev, N. K., Diskin B., and Nielsen, E. J. “Adjoint-based methodology for time-dependent optimization,” AIAA Paper 2008-5857, 2008.

Transient Adaptive Algorithm based onResidual Error Estimation

Ganesh N, N Balakrishnan

Abstract In this paper, we propose a novel algorithm for transient mesh adaptationbased on residual error estimation. The error estimator, known as the ℜ–parameteris a reliable measure of the local truncation error and is used to derive local lengthscales that guide refinement and derefinement. In order to efficiently handle movingfronts, a Refinement Level Projection (RLP) algorithm is developed, which guaran-tees that the flow features remain within refined zones at all times. Dynamic adapta-tion of inviscid transonic flow past a pitching airfoil illustrates the efficiency of theproposed algorithm.

1 Introduction

Several engineering problems encountered in practice involve unsteady flow phe-nomena such as periodic vortex shedding and shock propagation among others. Ob-taining high resolution solutions for such time–dependent problems would requirea uniformly fine mesh in the entire domain if adaptive algorithms are not employed.Such an approach is obviously not cost–effective and becomes very expensive forpractical 3–D applications. This motivates the development of an efficient dynamicadaptive algorithm for accurate simulations of transient phenomena.

Ganesh NSt. Anthony Falls Laboratory, University of Minnesota, Minneapolis, USA, e-mail:[email protected]

N BalakrishnanDepartment of Aerospace Engineering, Indian Institute of Science, Bangalore, India, e-mail:[email protected]

1

2 Ganesh N, N Balakrishnan

2 Transient adaptive algorithm

The ideal transient adaptation algorithm consists of two steps: (1) Obtaining the“best” mesh for a given time level and (2) Predicting the evolution of flow phenom-ena and refining regions through which the phenomena progresses. The first stepresults in the best spatial distribution of volumes that resolves the flow features ata given time level. As the time progresses, flow phenomena on the “best” mesh getconvected out from refined zones into unrefined regions resulting in a diffusive so-lution and loss in accuracy. Ideally, one would adapt the mesh at every time level,but such a procedure is computationally expensive and can introduce interpolationerrors during solution transfer between meshes. It is therefore desirable to adapt themesh with a finite frequency (once in a few timesteps), to evolve a computationallyefficient transient adaptive algorithm with lesser interpolation errors. However, suchan approach must also guarantee uniform accuracy of the solution by ensuring thatthe flow features remain in adapted regions between successive adaptations. Thisis achieved in the second step, thanks to a simple strategy known as “RefinementLevel Projection” (RLP). In this strategy, the refinement levels of cells on the “bestmesh” are projected over a period of time which corresponds to the time intervalbetween successive adaptations. The projection of refinement levels involves thedefinition of a “projection radius” which depends on some estimate of the velocityof the flow features, the physical time step and the time interval between successiveadaptations. Effectively, the RLP strategy constructs a “buffer” of cells that guaran-tee that the flow features (which are also the error sources) remain within adaptedzones throughout and keeps the spatial error levels within acceptable bounds.

The two–step transient adaptive algorithm can be summarised below. Let G andS represent the grid and the solution respectively. Let i denote the iteration count inobtaining the “best” mesh and superscripts n, n−1 and n−2 denote the current timelevel and the two earlier time levels respectively. We shall refer to S̃ as solutions ob-tained on a given grid by mapping from a previous level mesh. The physical timestep is denoted as ∆ t. The velocity of the flow feature is denoted as Vf eature whileRmax denotes the maximum number of refinement levels. The number of iterationsbetween two successive adaptations is defined as the adaptation frequency and isdenoted as N f .

STEP 1: Determine the “best” mesh at tn given the mesh–solution pair (Gn0, Sn0),

Rmax and ∆ t.

1. Based on the solution Sni on the mesh Gni , flag the cells based on a suitable adap-

tation strategy and adapt the mesh. Refinement of the mesh is based on errorindicators and the residual error estimator, while derefinement is effected purelyusing the error estimator. The details of residual estimation for unsteady flows isdiscussed in Section 3. This gives the adapted mesh Gni+1.

Transient Adaptive Algorithm based on Residual Error Estimation 3

2. Transfer (or Map) the solutions Sn−1i and Sn−2i available on the grid G

ni to the

adapted grid Gni+1 using a constant polynomial interpolation strategy. Let themapped solutions on the grid Gni+1 be represented as S̃

n−1i+1 and S̃

n−2i+1 .

3. Solve the transient flow problem on mesh Gni+1 for a single physical time stepcorresponding to the time interval [tn-∆ t,tn]. This gives the solution Sni+1 on theadapted mesh at time level tn.

4. Set i = i + 1 and repeat 1–3 until termination, the termination criterion beingi > Rmax.

This give the “best” mesh at time level tn, denoted as Ĝn and the corresponding so-lution Ŝn.

STEP 2: Refinement Level Projection (RLP), given the mesh–solution pair (Ĝn,Ŝn) and N f

1. Define the set of cells on Ĝn whose refinement levels are to be projected.2. For every cell in this set, define a radius of projection, ∆rpro j = η ×Vf eature×

N f ×∆ t, where η is a constant to account for the uncertainty in the estimate offeature velocity.

3. For every cell in the set and whose level is l, refine all cells which have a levelless than l and that fall within its projection radius, until the cell receives the levell.

4. Transfer (or Map) the solution Ŝn from Ĝn onto the new mesh obtained afterprojection.

This give the final adapted mesh after projection, Gn and the corresponding solutionSn. The transient flow problem is now solved on the grid Gn over the time interval[tn, tn +N f ∆ t], after which the transient adaptation is again enforced. For more de-tails, refer [4].

3 Error estimation for unsteady flows

In this section, we present the theory of residual error estimation for unsteady flows.Consider the conservation law given by I[U ]=0. Splitting the exact spatio–temporaloperator I into its constituent spatial and temporal operators, represented by Ih andIt respectively, we have,

It [U ]+ Ih [U ] = 0 (1)

A numerical solution u to this conservation equation demands a suitable discretisa-tion and the discrete approximation to Eq.(1) reads,

δ 1[u] = δ 1,t [u]+δ 1,h [u] = 0 (2)


where δ 1,t and δ 1,h are the temporal and spatial operators constituting δ 1. The resid-ual estimator is the imbalance arising from the use of the exact operator with thenumerical solution. This immediately leads to,

It [u]+ Ih [u] = R1,t [u]+R1,h[u] = R1[u] (3)

Our interest lies in estimating only the spatial component of the truncation error andtherefore employing the same discrete temporal operator δ 1,t and a different spatialoperator δ 2,h as discrete approximations to It and Ih in Eq.(3), leads to,

δ 1,t [u]+δ 2,h [u] = R1,h[u]−R2,h[u] (4)

where R2,h[u] is the error due to the discretisation of the exact spatial operator Ih.Using Eq.(2) in Eq.(4), yields the final expression for the residual error estimator as,

δ 2,h [u]−δ 1,h [u] = R1,h[u]−R2,h[u] = ℜ[u]. (5)

It is evident that if R1,h[u] ∼ O(hm) and R2,h[u] ∼ O(hn) and m < n, then the ℜ–parameter will be an estimate of the local truncation error associated with the spatialdiscretisation. The definition of ℜ–parameter given by Eq.(5) can be effectively em-ployed even in a steady state problem, to account for the non–convergence of thenumerical problem to steady state. In that sense, error estimation for unsteady flowsis a generalisation of the error estimation procedure for steady flows proposed bythe authors in [1, 2, 3]. To compute the ℜ–parameter, δ 1,hu is calculated by inte-grating the fluxes on each interface with a single–point Gaussian quadrature, withthe states obtained using a linear reconstruction while δ 2,hu is calculated in a simi-lar fashion but with three–point quadrature and quadratic reconstruction. For moredetails, please refer [4].

4 Results

In order to employ the residual error estimator in the transient adaptive algorithm, itis necessary to establish the estimator to consistently estimate the truncation error.We consider, for this purpose the isentropic vortex convection in an inviscid flow[4].The initial condition is given by the exact solution at time t=0 and extrapolationboundary conditions are employed. The computational domain is [-50,50]×[-5,5]and the vortex is initially centered at the origin. The unsteady HIFUN–2D solver[4]is used for all computations, which are performed upto t=0.01 using a physical timestep of 0.001. Experiments on moving structured and unstructured meshes show thatthe ℜ–parameter indeed decreases with grid refinement, with a rate close to 2 and 1respectively.

Transient Adaptive Algorithm based on Residual Error Estimation 5

Subsequently, we attempt to solve the inviscid transonic flow past a pitchingNACA0012 airfoil using the transient adaptive algorithm based on the residual esti-mator. The sinusoidal pitch motion is defined by, α(t) = 0.016 + 2.51 sin(0.1628t)and the mach number is 0.755. A circular computational domain is chosen with afarfield of 10 chords and the entire grid is rigidly moved. Empirical studies suggesta feature velocity of 0.05 for this case. The physical time step of 0.01 and adaptationis performed once in every 100 time steps for 40 adaptation cycles. Projection usesan η value of 1.3 and maximum three levels of refinement is considered. As thegrid is moved rigidly and the slipstream moves passively with the grid, there is norefinement level projection required for the cells which are detected by the curl ofvelocity. The adapted mesh and mach contours at two different time instants (Figs.2 and 3) as well as the pressure distribution over the airfoil (Figs. 4 and 5) showexcellent resolution of the shocks and establishes the efficacy of the proposed algo-rithm.

Fig. 1 Error fall of ℜ–parameter with grid refinement for moving grid case

(a) Adapted Mesh (b) Mach contours

Fig. 2 Mesh and mach contours at α = 1.63◦(↓)


(a) Adapted Mesh (b) Mach contours

Fig. 3 Mesh and mach contours at α = 1.67◦(↑)

Fig. 4 Cp distribution for α = 2.01◦(↓) and α =−0.54◦(↑).

5 Conclusions

A new residual error estimator for unsteady flows and an associated novel transientadaptive algorithm are proposed. Dynamic adaptation of the flow past a pitchingairfoil demonstrates the efficacy of the proposed algorithm.

References

1. Ganesh, N., Nikhil, V. Shende and N.Balakrishnan,“ A residual estimator based adaptationstrategy for compressible flows”, CFD2006, Springer.

2. Ganesh, N., Nikhil, V., Shende and N.,Balakrishnan, “ℜ–parameter: A local truncation errorbased adaptive framework for finite volume compressible flow solvers”, Computers&Fluids,Vol.38, pp. 1799–1822, 2009.

3. Ganesh, N., N. Balakrishnan,“An adaptive strategy based on local truncation error”, Proc. ofEight Asian Computational Fluid Dynamics Conference (ACFD8), HongKong, 2010.

4. Ganesh, N.,“Residual error estimation and adaptive algorithms for compressible flows”, Ph.D.Thesis, Dept. of Aerospace Engg., Bangalore, India, 2010.

1

Adaptive and Consistent Properties Reconstruction for

Complex Fluids Computation

Guoping Xia1, Chenzhou Lian

1 and Charles L. Merkle

1

Abstract. An efficient reconstruction procedure on adaptive Cartesian mesh for

evaluating the constitutive properties of a complex fluid from general or

specialized thermodynamic databases is presented. Reconstruction is

accomplished on a triangular subdivision of the 2D Cartesian mesh covering

thermodynamic plane of interest that ensures function continuity across cell

boundaries to C0, C1 or C2 levels. The C0 and C1 reconstructions fit the

equation of state and enthalpy relations separately, while the C2 reconstruction

fits the Helmholtz or Gibbs function enabling EOS/enthalpy consistency also.

All three reconstruction levels appear effective for CFD. The time required for

evaluations is approximately two orders of magnitude faster with the

reconstruction procedure than with the complete thermodynamic equations.

Storage requirements are modest for today‟s computers, with the C1 method

requiring slightly less storage than those for the C0 and C2 reconstructions when

the same accuracy is specified. Sample fluid dynamic calculations based upon

the procedure show that the C1 and C2 methods are approximately a factor of

two slower than the C0 method but that the reconstruction procedure enables

arbitrary fluid CFD calculations that are as efficient as those for a perfect gas or

an incompressible fluid for all three accuracy levels.

Keywords: real fluid, property reconstruction, adaptive Cartesian method

1 Introduction

Computational fluid dynamics requires the coupled solution of the partial differential

equations that comprise the basic conservation laws (mass, momentum and energy) in

combination with an auxiliary set of constitutive relations in order to close the system.

Simple algebraic relations are enough for perfect gases and incompressible fluids.

There are, however, many engineering applications in which gases or vapors do not

follow the perfect gas laws and liquids can not be treated as incompressible. For such

applications, it is necessary to turn to more general equations of state.

One approach is to use the family of classical semi-theoretical expressions that

include the van der Waals, Peng-Robinson and Redlich-Kwong-Soave (RKS)

equations of state in which the perfect gas relation is modified by adding additional

terms and constants to improve accuracy in regions where real gas effects become

important. Although these methods provide pressure-temperature-density relations

that are accurate over a broader range, they generally do not provide an analogous

1 Purdue University, West Lafayette, Indiana, USA

Emails: [email protected]; [email protected]; [email protected]

mailto:[email protected]:[email protected]:[email protected]

2

expression for the internal energy and relations analogous to those for perfect gases

are often used for the specific heats. A second alternative is to use more complete and

accurate thermodynamic databases such as REFPROP[1], which provide highly

accurate properties across the liquid-vapor-supercritical regimes for a large variety of

fluids including full pressure-density-temperature-internal energy relations. In using

any of these equations of state, a variety of difficulties can be encountered that

increases the cost and complexity of a CFD solution. In the present paper we develop

a generalized properties evaluation procedure based upon an adaptive table look-up

method that provides flexibility, efficiency and accuracy for nearly any conceivable

property formulation. The flexibility comes because the equation of state is inverted

by separate software outside the CFD code thereby enabling the independent variables

pair to be chosen by the user while simultaneously enabling the entire variety of

equations of state to be incorporated in a CFD code in a common fashion. The

efficiency comes from using a tree-based data structure that provides fast table look-

up over large or small thermodynamic domains. The accuracy comes from an

adaptive Cartesian mesh that adjusts the table resolution to user specified input

criteria and evaluates both the thermodynamic functions and their derivatives.

Although the procedure is applicable to any properties formulation, we base our

examples on information obtained from REFPROP.

2 The Conservation Equations and Fluid Properties

The conservation form of the mass, momentum and energy equations for an arbitrary,

Newtonian fluid are[3]:

Hx

V

x

E

t

QQ

Q

Q

i

i

i

ip

p

(1)

For a flow with multiple species, the vectors, Q and E, are given by

k

j

Y

ph

uQ

0

ki

i

ji

i

i

Yu

hu

uu

u

E

0

(2)

In Eq. 1, a pseudo-time term is added to enable time-marching, where

represents the pseudo-time. Here we work with the primitive variables,

Tkip YTupQ ,,, , in which the pressure and temperature appear as the thermodynamic variables pair along with the velocity components and the mass

fractions. The Jacobian matrix that multiplies the pseudo-time term in Eq. 1 and the

product matrix from which the eigenvalues of the system are:

T

o

pp

o

p

Tp

Tp

phhuhh

uuQ

Q

1

0 and

uh

u

hu

Q

E

Q

QA

p

T

p

p

p

10

01

0 (3)

3

where /1 pTTp hh . Note that both matrices contain the density plus four thermodynamic derivatives: p , T , ph and Th . These derivatives directly

affect the fluids solution and must be specified from the fluids database. Note that

these four thermodynamic derivatives, p , T , ph and Th , can be evaluated in terms

of the first and second partial derivatives of the Gibbs function:

2

p

pp

pg

g ,

2

p

pT

Tg

g , pTpp Tggh , and TTT Tgh (4)

For compactness of notation, we have given these in terms of the one-dimensional

equations with only a single species. In mixture computing, the above

thermodynamic property of the mixture can be evaluated from the Amagat‟s law.

3 Adaptive Cartesian Grid Methods and Consistent Reconstruction

In the present paper we choose a tree-based, adaptive Cartesian grid method that

enables both fast search and local refinement for property storage. In any p-T plane

(or other thermodynamic plane[2]), we begin by defining a rectangular region that

includes the physical domain of interest. The requisite properties are then evaluated

at each of the four corners of the square and an appropriate reconstruction is used to

evaluate these properties at pre-selected points in the four quadrants of the square.

The reconstructed properties are then compared with their „exact‟ values obtained

from the complete property equations and used to assess the reconstruction error. If

the error in any exceeds a user-specified threshold, that square is subdivided into four

smaller squares.

Three different levels of reconstruction are considered, involving C0, C

1, and C

2

continuity across cell boundaries whose results range from partially to fully

consistent. In the C0 method, values for each of the six thermodynamic fluid

properties are stored at the vertices of each square in the mapped plane. The squares

are then subdivided into triangles and the values of each function at the three vertices

are used to construct independent bi-linear reconstruction functions for each property.

Subdividing the squares ensures continuity of the function across the interface of

squares of different sizes. The C1 method treats the density and enthalpy as unrelated

functions that are each handled in analogous fashion. Using the density as an

example, the goal of C1 reconstruction is to reconstruct the density function in such a

manner that the values of , p and T are consistent within any given cell and that

all three quantities are continuous across cell boundaries. This level of consistency

and continuity can be achieved by using a bi-quintic (fifth-order) polynomial. The C2

method interpolates all properties and their derivatives by reconstructing the Gibbs

function as a ninth-order polynomial.

4

Nu

m.o

fT

ab

leP

oin

ts

100

101

102

103

Perfect Gas

REFPROP

C0

C1

C2

1290

23

75

664

14

42

Fig 2. Storage Comparison of different

reconstruction methods.

4 Results

Figure 1 shows the grid structure produced for CO2. It comprises the temperature and

pressure ranges, KTK 1600500 , MPapMPa 100001.0 and lies entirely

within the vapor region. The grid of this zone is for a specified accuracy of 1% with

C0 continuity. The grid color is keyed to the magnitude of the density. From the plot

in Fig. 1 more refinement around the critical point(top left corner) is clearly observed.

In the lower pressure portion, CO2 behaves as a perfect gas and thus less grid points

are used. The 1% accuracy case requires a total of ten levels of refinement and results

in 22,000 reconstruction points. A further refinement to 0.1% accuracy(not shown

here) requires 11 refinement levels and 225,000 points. The comparison of the table

size and timing are summarized in Table I. Evaluation timing is based on the time

consumption of performing 100,000 property calculations along the two diagonal

dashed lines in Fig. 1. The evaluation times in column 4 of table I are only slightly

longer than that for a table of uniform grid size, which is considered advantageous on

searching. However, the adaptive table uses only a fraction of the storage of the

uniform table in order to achieve the same accuracy. Taking the 1% accuracy table as

an example, a uniform table refined to the same accuracy as the adaptive one will

have 1,048,576 cells, 50 times larger than that in the adaptive table. The table look-up

and evaluation time is essentially independent of the table size. Direct calculation

from the property database, REFPROP, is more than two orders slower than the

adaptive table look-up procedure, as shown in Table I.

The table sizes required for the C0, C

1 and C

2 reconstruction procedures is

compared in Fig. 2. from both a real fluid property from REFPROP and perfect gas

property. In general C1 and C

2 methods need fewer grid points due to the increased

accuracy of the interpolating functions, but they require the storage of more

values(coefficients in the bi-variate polynomials) per grid point. A significant

reduction in database size can be achieved when moving from C0 reconstruction to its

C1 counterpart. Going from C

1 reconstruction to C

2 reconstruction will result in a

modest increase of the database size.

A one-dimensional example of inviscid subsonic flow through a convergent-

divergent nozzle is presented to demonstrate the method. The nozzle is symmetric

Fig 1. Adaptive reconstruction maps for

CO2. 1% accuracy is specified, 22,000 points.

5

Fig. 4 CPU time comparison for

different interpolation methods.

about the throat with the inlet and exit areas 1.25 times the throat area. The working

fluid is chosen as H2O with an inlet total pressure of 60 MPa and stagnation

temperature of 750 K. The back pressure at the outlet is 50 MPa. A total of 200 cells

are used in the calculation. The convergence rates for this case using the C0, C

1 and

C2 methods are shown in Fig. 3. The convergence with all three of these

reconstruction methods is identical to the convergence obtained by coupling the

REFPROP routines directly into the CFD code. The convergence with discontinuous

interpolation, however, stalls.

Table I. Storage and Timing Comparisons for Properties Evaluation

#: Equivalent Equal-sized Mesh (A uniform mesh refined to the deepest level)

Although the number of iterations is essentially identical for the different property

evaluation methods, their CPU costs are significantly different. Figure 4 shows the

cumulative CPU time required for the computation as a function of iteration number

for 1000 iterations. The C0 reconstruction method results in the fastest execution.

Comparison of the results for the inconsistent and the consistent interpolation methods

indicates that the consistent interpolation method increases the CPU time by about

5%. The costs of the C1 and C

2 methods are nearly equal and are approximately twice

that of the C0 method. The calculation based on the exact property evaluation from

REFPROP is approximately 160 times slower than the C0 method and 80 times slower

than the C1 and C

2 methods.

The CPU time for a CFD calculation can be broken into two parts: the property

evaluation time and the equation solution time. The relative advantages of faster

property evaluation times clearly will decrease as the complexity of the equation

Fluid Max

Err

No.

Levels

No.

Points

Time(s)

Cartes

Time(s)

E.E.M#

Time(s)

REFPROP

Time

Ratio

CO2 1% 10 22,246 0.438 0.375 77.686 177.4

0.1% 11 225,121 0.438 0.375 77.297 176.5

H2O 1% 9 871 0.406 0.375 186.107 458.4

0.1% 11 8049 0.422 0.375 186.796 442.6

Fig. 3 Exemplary plot showing effect of

triangulation on convergence of a specific

CFD calculation.

6

solution increases, and in particular as we move from one to three dimensions. The

nominal cost of the property evaluation by REFPROP in this region of the H2O map is

about 450 times that of the C0 integration method (see Table I). For this one-

dimensional case, the CFD calculation time ratio is 160, suggesting that the ratio of

equation solution time to property evaluation time in a one-dimensional calculation is

approximately 1.8. Numerical computations show that the one-dimensional solver

requires about 61.25 iterationcells . Corresponding costs for two- and three-

dimensional solutions are 100.2 and 200.9 s cell iteration respectively.

Assuming the property evaluation time does not change when going from one to three

dimensions, this suggests that a two-dimensional calculation with C0 interpolation

method will be 115 times faster than the direct REFPROP evaluation calculation and

the three-dimensional calculation will be 66 times faster. Solutions based upon C1

and C2 reconstruction will be about half this amount.

5 Summary and Conclusions

An adaptive reconstruction method for fluid dynamics computations of complex

fluids with general equations of state is presented. The technique is based upon a

Cartesian-grid approach that uses a binary tree data structure to store property

information in an unequally spaced table whose resolution is automatically chosen to

provide user-specified accuracy. The efficiency and accuracy of the method are

assessed by comparing with the thermodynamics properties obtained from the

REFPROP database. The resulting property reconstructions are nominally two orders

of magnitude faster than property evaluations from the original database. This

properties evaluation advantage translates into solution time enhancements of

approximately one to two orders of magnitude for 3-D CFD computations.

Discontinuity in the reconstruction is removed to improve the convergence.

Reconstruction of the interpolating functions at C0, C

1 and C

2 continuity is assessed

and compared. The convergence rate in example calculations is identical for the three

reconstruction methods and for direct property evaluations from REFPROP, but the

CPU costs are approximately two orders of magnitude smaller. The CPU cost of the

C0 reconstruction method is approximately half that of the C

1 method, while the C

1

and C2 methods are about the same because the C

1 method constructs two fifth-order

polynomials as opposed to one ninth-order polynomial in the C2 method.

References

1. “NIST Standard Reference Database 23, NIST Thermodynamic properties of refrigerant mixtures database (REFPROP),” Version 4.0, Gaithersburg, MD, 1993.

2. Swesty, F.D., “Thermodynamically Consistent Interpolation for Equation of State Tables,” Journal of Computational Physics, Vol. 127, pp.118-127, 1996.

3. Merkle, C.L., Sullivan, J.Y., Buelow, P.E.O. and Venkateswaran, S., “Computational of Flows with Arbitrary Equation of State,” AIAA J., Vol. 36, No.4, pp.515-521, 1998.

Space-Filling Curve Techniques for Parallel,Multiscale-Based Grid Adaptation: Conceptsand Applications

Sorana Melian, Kolja Brix, Siegfried Müller and Gero Schieffer

1 Introduction

The numerical simulation of (compressible) fluid flow requires highly efficient nu-merical algorithms which allow for high resolution of all physical waves occurringin the flow field and their dynamical behavior. In order to use the computationalresources (CPU time and memory) in an efficient way, adaptive schemes are well-suited. By these schemes, the discretization is locally adapted to the variation of theflow field. The crucial point is the design of a criterion that allows to decide whetherto locally refine or coarsen the grid. Here we use multiscale techniques that aim atdata compression, see [5] for an overview.

Although this multiscale-based grid adaptation concept leads to a significant re-duction of the computational complexity in comparison to computations on uniformmeshes, this is not sufficient to efficiently perform 3D computations for complex ge-ometries. In addition, we need parallelization techniques in order to further reducethe computational time. For this purpose, a parallelized version of the multiscaletransformation library, developed in [4], has been embedded into the finite volumesolver Quadflow, see [2]. Here we will briefly summarize the basic concepts andpresent an application to the simulation of a Lamb-Oseen vortex. More details canbe found in [3].

Sorana Melian, Kolja Brix, Siegfried MüllerInstitut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55,52056 Aachen, Germany, e-mail: {melian,brix,mueller}@igpm.rwth-aachen.de

Gero SchiefferLehrstuhl für Computergestützte Analyse Technischer Systeme, RWTH Aachen University,Schinkelstr. 2, 52052, Aachen, Germany, e-mail: [email protected]

1

2 Sorana Melian, Kolja Brix, Siegfried Müller and Gero Schieffer

2 Multiscale-Based Grid Adaptation

Finite volume schemes have been frequently applied to the discretization of bal-ance equations arising for example in continuum mechanics. In the past decade,a new adaptive concept for finite volume schemes has been developed. This isbased on multiscale techniques, where by means of a multiscale analysis, associ-ated with a hierarchy of nested grids Gl , l = 0, . . . ,L, with increasing resolution, i.e.,Gl := {Vλ}λ∈Il with

∪λ∈Il Vλ = Ω ⊂ R

d and Vλ =∪

µ∈M0λ⊂Il+1Vµ ,λ ∈ Il , a locally

refined grid can be constructed. For this purpose, we first perform a multiscale de-composition of the data on the finest resolution level L. These data are determined bycell averages ûλ := 1|Vλ |

∫Vλ

udV,λ ∈ Il of an integrable function u. They are decom-posed into coarse scale information ûλ ,λ ∈ Il , and details dλ ,λ ∈ Jl , encoding thedifference between two refinement levels. This is realized by successively applyingthe two-scale relations

ûλ = ∑µ∈M0λ⊂Il+1

ml,0µ,λ ûµ , ml,0µ,λ :=

|Vµ ||Vλ |

, dλ = ∑µ∈M1λ⊂Il+1

ml,1µ,λ ûµ . (1)

Note that #Il+1 = #Il +#Jl , i.e. for each cell Vλ there are #M0λ −1 details.Since the details become small, if the underlying function u is locally smooth, the

basic idea is to perform data compression on the vector of details using hard thresh-olding, i.e., we discard all detail coefficients dλ ,λ ∈ Jl , whose absolute values fallbelow a level-dependent threshold value εl = 2(l−L)dε and keep only the significantdetails corresponding to the index set

DL,ε := {λ : |dλ |> εl , λ ∈ Jl , l ∈ {0, . . . ,L−1}} . (2)

In order to account for the dynamics of a flow field due to the time evolution andto appropriately resolve all physical effects on the new time level, this set is to beinflated such that the prediction set D̃L,ε ⊃ DL,ε contains all significant details of theold and the new time level. In a last step, we construct the locally refined grid, seeFigure 1 (right), and the corresponding cell averages. For this purpose, we proceedlevelwise from coarse to fine, see Figure 1 (left), and check for all cells of a levelwhether there exists a significant detail. If one is found, then we refine the respectivecell, i.e., we replace the average of this cell by the averages of its children by locallyapplying the inverse two-scale transformation

ûµ = ∑λ∈G0µ⊂Il

gl,0µ,λ ûλ + ∑λ∈G1µ⊂Jl

gl,1µ,λ dλ . (3)

Note that the mask coefficients ml,0µ,λ ,ml,1µ,λ and g

l,0µ,λ ,g

l,1µ,λ in (1) and (3), cor-

responding to the index sets M0λ , M1λ , and G

0λ , G

1λ , respectively, do not depend

on the data but on geometric information only. For specific examples we refer to[5]. The final grid is then characterized by the index set G̃L,ε ⊂

∪Ll=0 Il such that

Parallel, Multiscale-Based Grid Adaptation 3∪λ∈G̃L,ε Vλ = Ω . In order to proceed levelwise, we have to inflate the prediction set

such that it corresponds to a graded tree. This also guarantees that there is at mostone hanging node per cell edge, see [4].

Fig. 1 Grid adaptation: refinement tree (left) and corresponding adaptive grid (right).

3 Parallelization using Space-Filling Curves

When it comes to parallelization, a data parallel approach is usually appropriate. Ona distributed memory architecture, the performance of a parallelized code cruciallydepends on the load-balancing and the overhead induced by the interprocessor com-munication. As the starting point is a hierarchy of nested grids, we do not need apartition of a single uniform refined mesh, but a partition of a locally refined grid,where not all cells on all levels of refinement are active. A natural representationof a multilevel partition of a mesh is a global enumeration of the active cells. Weare aiming at the design of a method to realize this at runtime, as the adaptive meshis also created at runtime using the multiscale representation techniques. Such anenumeration is provided by a Space-Filling Curve (SFC), where the basic idea is tomap level-dependent multiindices identifying the cells in a dyadic grid hierarchy toa one-dimensional curve, i.e. the unit square (2D) or the unit cube (3D) is mapped tothe unit interval. For our type of discretization, Hilbert SFCs are well suited, helpingto reduce the overall computational time by achieving very good load-balance and inthe meanwhile saving computational resources (time and memory). Figure 2 showsthe first iterates of a 2D Hilbert SFC. For a detailed discussion on the constructionon the Hilbert space-filling curve we refer the reader to [6, 8].

Thus, each of the cells of the adaptive grid has a corresponding unique number onthe curve, due to the global enumeration. The unit interval is then split into differentparts containing approximately the same number of entries. Each of these parts ismapped to a different processor, so that we obtain a well-balanced partition, butwe also have to pay the cost of interprocessor communication, as neighbors fromthe geometrical domain may belong to different processors. For a more specificexample, Figure 4 shows a locally refined grid with three levels of refinement whereeach cell is mapped to a position on the Hilbert SFC and then a partition to threeprocessors can be determined only by ordering the numbers along the curve and thendistributing it to the desired number of processors. A well-balanced distribution of


Fig. 2 First 4 iterates of 2D Hilbert space-filling curve.

Fig. 3 Well-balanced distribution of a locallyrefined grid to 5 processors.

a locally refined grid to five processors can be seen in Figure 3. For further details,we refer to [3].

Fig. 4 Encoding of the Hilbert order for a three-level adaptive grid with the corresponding split ofthe unit interval to three processors.

4 The solver Quadflow

The above multiscale-based grid adaptation concept has been integrated into thesolver Quadflow [2]. This solver has been developed for more than one decadewithin the collaborative research center SFB 401 Modulation of Flow and Fluid-Structure Interaction at Airplane Wings, cf. [1, 7]. In order to exploit synergy ef-fects, it has been designed as an integrated tool where each of the core ingredients,namely, (i) the flow solver concept based on a finite volume discretization, (ii) thegrid adaptation concept based on wavelet techniques, and (iii) the grid generatorbased on B-spline mappings, is adapted to the needs of the others. In particular, thethree tools are not just treated as independent black boxes communicating via inter-

Parallel, Multiscale-Based Grid Adaptation 5

faces. Instead, they are highly intertwined on a conceptual level mainly linking (i)the multiresolution-based grid adaption that reliably detects and resolves all physi-cal relevant effects, and (ii) the B-spline grid generator which reduces grid changesto just moving few control points whose number is, in particular, independent ofany local grid refinement. The mathematical concepts have been complemented re-cently by parallelization techniques that are indispensable for further reducing thecomputational time to an affordable order of magnitude when dealing with realistic3D computations for complex geometries, cf. [3].

Note that the flow solver and the multiscale-based grid adaptation have totallydifferent algorithmic requirements: on one hand, there is a finite volume schemeworking on arbitrary, unstructured discretizations; on the other hand, there is themultiscale algorithm assuming the existence of hierarchies of structured meshes.The flow solver module is face-centered, since the central item is the computationof the fluxes at the cell faces, while the adaptation module is cell-centered, analyzingand manipulating cell averages. Moreover, the data structures used in the two partsare also different: while for the adaptation part hash maps are used, for the flowsolver part more simple data structures (i.e. arrays) are applied. The link betweenthese two modules is done through a data conversion algorithm, which organizes allthe data communication – the transfer of the conservative variables, volumes, cellcenters, the registration of the nodes, the construction of the faces and determinationof their neighboring cells and nodes – between the two modules is a connectivity list(see [2]). The parallelization of such an algorithm is not trivial. Due to the adaptivityof the mesh, the connectivity list might need to be reconstructed at any time in thecomputation, when also rebalancing and repartitioning is required. Thus the cellslocated at the partition’s boundary are at any time subject to special care and tointerprocessor communication in order to properly compute the flux at these faces.

5 Application

The system of vortices in the wake of airplanes continues to exist for a long pe-riod of time and this strongly influences the takeoff and landing frequency on anairport. It is possible to detect wake vortices as far as 100 wing spans behind theairplane, which is a hazard to following airplanes. In the framework of the collabo-rative research center SFB 401, one goal was to induce instabilities into the systemof vortices to accelerate their collapse. The effects of different measures taken inorder to destabilize the vortices have been examined in a water tunnel. A model of awing was mounted in a water tunnel and the velocity components in the area behindthe wing were measured using particle image velocimetry. It was possible to con-duct measurements over a length of 4 wing spans. The experimental analysis of asystem of vortices far behind the wing poses great difficulties due to the size of themeasuring system. Numerical simulations are not subject to such severe constraintsand therefore Quadflow is used to examine the behavior of vortices far behind thewing. To minimize the computational effort, the grid adaptation adjusts the refine-


ment of the grid with the goal to resolve all important flow phenomena, while usingas few cells as possible. Figure 5 shows the simulation of a Lamb-Oseen vortex.More details on the simulation can be found in [3].

Fig. 5 Parallel simulation of the Lamb-Oseen vortex on 16 processors. Slicesof the computational grid after 5466 time steps (corresponding to a computedreal time of 0.27s).

References

1. Ballmann, J.: Flow Modulation and Fluid-Structure-Interaction at Airplane Wings. Volume84 of Numerical Fluid Mechanics and Multidisciplinary Design, Springer, Berlin (2003)

2. Bramkamp, F., Lamby, Ph., Müller, S.: An adaptive multiscale finite volume solver for un-steady and steady state flow computations. J. Comp. Phys. 197(2), 460–490 (2004)

3. Brix, K., Melian, S., Müller, S., Schieffer, G.: Parallelisation of multiscale-based grid adap-tation using space-filling curves. ESAIM Proc. 29, 108–129 (2009)

4. Müller, S.: Adaptive Multiscale Schemes for Conservation Laws. Volume 27 of Lecture Noteson Computational Science and Engineering. Springer, Berlin (2003)

5. Müller, S.: Multiresolution schemes for conservation laws. In: DeVore, R., Kunoth, A. (eds.)Multiscale, Nonlinear and Adaptive Approximation. Dedicated to Wolfgang Dahmen on theOccasion of his 60th Birthday, pp. 379-408. Springer, Berlin (2009)

6. Sagan, H.: Space-Filling Curves. Springer, Berlin (1994)7. Schröder, W.: Summary of Flow Modulation and Fluid-Structure Interaction Findings-Results

of the Collaborative Research Center SFB 401 at the RWTH Aachen University, Aachen, Ger-many, 1997–2008. Volume 109 of Numerical Fluid Mechanics and Multidisciplinary Design,Springer, Berlin (2010)

8. Zumbusch, G.: Parallel Multilevel Methods. Adaptive Mesh Refinement and Loadbalancing.Advances in Numerical Mathematics, Teubner, Wiesbaden (2003)

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