AIAA

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics370 L’Enfant Promenade, S.W., Washington, D.C. 20024

AIAA-2003-1119Implicit Approaches for Moving Boundariesin a 3-D Cartesian Method

Scott M. MurmanELORETMoffett Field, CA

Michael J. AftosmisNASA Ames Research CenterMoffett Field, CA

Marsha J. BergerCourant InstituteNew York, NY

41st AIAA Aerospace Sciences MeetingJanuary 6-9, 2003 / Reno, NV

AIAA-2003-1119

Implicit Approaches for Moving Boundaries

in a 3-D Cartesian Method

Scott M. Murman∗

ELORETMoffett Field, CA 94035smurman@nas.nasa.gov

Michael J. Aftosmis†

NASA Ames Research CenterMS T27B Moffett Field, CA 94035

Marsha J. Berger∗

Courant Institute251 Mercer St.

New York, NY 10012

Abstract

A method for simulating moving impermeable boundaries within a fixed Cartesian mesh is described.The scheme leverages the automated volume mesh generation process which has previously been demon-strated for static geometries. An implicit dual-time method is used for the time advance, which limitsthe number of times the geometry must be intersected with the Cartesian volume mesh over a com-plete simulation. A general motion is decomposed into a rigid-body motion of the entire computationaldomain, with a relative-body motion superimposed. The rigid-domain motion is treated using an ALEformulation, which confines the required geometry processing only to the regions of relative motionwithin the domain. A detailed space-time analysis is used to present and discuss the moving-boundaryscheme, with particular attention given to complexities arising in multiple dimensions. A hierarchy ofconservative approximations for the evolution of the moving geometry over a timestep is presented.Preliminary results are discussed in one, two and three dimensions using CFL numbers based upon themoving wall velocity of between 1 and 20.

1 Introduction

This work considers numerical simulation ofthree-dimensional flows with moving boundaries.Such problems pose a variety of challenges for nu-merical schemes, and have received a substantialamount of attention in the recent literature. Sincesuch simulations are unsteady, time-accurate solu-tion of the governing equations is required. In spe-cial cases, the body motion can be treated by auniform rigid motion of the computational domain.For the more general situation of relative-body mo-

∗Member AIAA†Senior Member AIAACopyright c©2003 by the American Institute of Aero-

nautics and Astronautics, Inc. No copyright is asserted inthe United States under Title 17, U. S. Code. The U. S.Government has a royalty-free license to exercise all rightsunder the copyright claimed herein for Governmental pur-poses. All other rights are reserved by the copyright owner.

tion, however, this simplification is unavailable andthe simulations require a mechanism for ensuringthat the mesh evolves with the moving boundaries.This involves a “remeshing” of the computationaldomain (either localized or global) at each physicaltimestep, and places a premium on both the speedand robustness of the remeshing algorithms. Thiswork presents a method which includes unsteadyflow simulation, rigid domain motion, and relativebody motion using a time-evolving Cartesian gridsystem in three dimensions.

While 3-D moving-boundary simulations havebeen performed on body-fitted structured and un-structured grid systems for some time [1–4], theliterature on Cartesian methods has been largelyrestricted to two dimensions and/or simplified con-figurations. Early approaches to the 3-D Cartesianmoving-body problem included both the Volume ofFluid (VOF) [5] and level-set approaches [6]. More

1

recently, there has been interest in the immersed-boundary [7–9] and cut-cell Cartesian approaches[10–13]. Non-body-fitted, Cartesian approacheslike these are particularly interesting since they canbe made both extremely fast and robust [14]. More-over, they are comparatively insensitive to the com-plexity of the input geometry since the surface de-scription is decoupled from the volume mesh, andcan therefore handle complex geometries with rel-ative ease.

This work builds upon the inviscid, Cartesiancut-cell solver in Ref. [15]. In this method, thecells which are cut by the boundary geometrycan be arbitrarily small, making explicit updateschemes overly restrictive for time-dependent prob-lems. Approaches to overcome this restriction usu-ally either extend the difference stencil of the spa-tial terms [16], or use a cell-merging approach [10],so that cut-cells can be advanced with the ex-plicit timestep of a full uncut cell. The coupling ofcell size and allowable timestep with explicit meth-ods implies that the boundary motion will be re-stricted by the size of the finest boundary inter-secting Cartesian cell during a single timestep.

In a Cartesian moving-boundary scheme, themost delicate operation is the re-intersection of thebody with Cartesian grid at each time step. Notonly is this the most computationally expensivepart of Cartesian mesh generation, but it requiresspecial procedures to ensure that the floating-pointintersection calculations are robust [14]. In a twodimensional mesh with O(N2) cells, the bound-ary geometry only intersects O(N) cells. In threedimensions, however, the boundary is a surfaceinstead of a line and the number of intersectioncalculations is squared with respect to the 2-Dcase. Moreover, these intersection calculations arehigher-dimensional, and therefore each is more ex-pensive and more difficult to compute robustly.Arguments for both efficiency and robustness inthree dimensions weigh heavily in favor of mov-ing the body as rarely as possible to minimizethe re-cutting of Cartesian cells. These argumentsare amplified in parallel-computing environments,where mesh modifications often imply rebalancingthe load distributed to the various processors byexchanging cells across subdomain boundaries.

Following this line of reasoning, the present workadopts a fully-implicit temporal operator to de-couple the timestep from the local mesh scale. Itleverages the same multigrid smoother used by thesteady-state solver [15] by embedding it in a dual-

time framework. The implicit approach means thatfiner mesh simulations do not automatically requiremore remeshings than coarse simulations. Simula-tions proceed at a timestep dictated by the appro-priate physics rather than stability constraints.

As noted earlier, special subclasses of arbitrarybody motion can be treated by rigid motion ofthe entire domain. The current work imple-ments a rigid domain motion within the ArbitraryLagrangian-Eulerian (ALE) formulation of Hirt etal. [17]. In this Lagrangian framework, a singletransformation matrix can be applied to all thefaces in a Cartesian mesh, and since these faces allhave the same orientations, the transformation canbe inexpensively precomputed, stored and applied.

Relative boundary motion is superimposed ontop of the rigid domain motion using an adaptivere-meshing strategy which does not rely on explicitcell merging. Since the geometry moves relativeto the volume cells, this aspect of the implementa-tion is Eulerian. A space-time analysis is used toensure conservation, and to develop a hierarchy ofapproximations to the moving boundary. The ba-sic analysis is presented in detail in 1-D, with thecomplexity which arises when extending to multi-ple dimensions also discussed. The final sectionpresents initial numerical results in one, two, andthree dimensions using the implicit, relative-motionscheme, including comparison with analytic solu-tions and experimental data.

2 Dual-time formulation

In order to leverage the infrastructure of thesteady-state flow solver outlined in Ref. [15], a dual-time formulation (cf. Refs. [18, 19]) was developedfor the time-dependent scheme,

dQdτ

+ R∗ (Q) = 0

R∗ (Q) =∂Q∂t

+ R (Q)(1)

where τ is referred to here as “pseudo-time”, and isthe iterative parameter, and t is the physical time.Q is the vector of conserved variables, and R (Q)is an appropriate numerical quadrature of the fluxdivergence, 1V

∮Sf · ndS. As dQdτ → 0 the time-

dependent formulation is recovered. The multi-gridsmoother described in [15] is used to converge theinner pseudo-time integration. An explicit, multi-stage, pseudo-time-integration scheme is utilized to

2

θ ξ φ Method Order1 0 0 Backward Euler 1

1/2 0 0 Trapezoidal 21 1/2 0 2nd-order backward 2

3/4 0 -1/4 Adam’s type 21/3 -1/2 -1/3 Lee’s type 21/2 -1/2 -1/2 Two-step trapezoidal 25/9 -1/6 -2/9 A-contractive 2

Table 1: Partial list of A-stable, two- and three-level meth-ods (Eqn. 2) from Beam and Warming [22].

converge the “inner loop” in Eqn. 1. This is similarto the scheme outlined by Jameson [20], however,the semi-implicit approach of Melson et al. [21] isused here for the physical time-derivative term.

Various time-dependent schemes can be con-structed for Eqn. 1 by appropriately discretizingthe time derivative. As noted in the Introduc-tion, it’s desirable to utilize an unconditionally-stable, implicit scheme to allow a large timestepto be chosen based upon physical considerationsrather than a potentially smaller stability-limitedtimestep. Beam and Warming [22] outline a fam-ily of consistent time-dependent schemes that uti-lize three time levels, Qn−1, Qn, Qn+1. These aregiven by

R∗ (Q) =(1 + ξ)Qn+1 − (1 + 2ξ)Qn + ξQn−1

∆t+ θR

(Qn+1

)+ (1− θ + φ)R (Qn) (2)

− φR(Qn−1

)Table 1 contains a partial list of the A-stable, threetime level methods that can be formulated usingEqn. 2.

3 ALE formulation

In many applications with moving geometry, themotion of the geometry can be decomposed intoa “uniform” rigid-body motion, with relative mo-tion confined to a subset of the domain. Examplesinclude rotating airframes with dithering canards[23], rotorcraft, or stage separation from space ve-hicles. It’s desirable to simulate such a motion us-ing a rigid-body motion of the entire computationaldomain, and treat the relative motion within themoving domain separately. This again limits theamount of computational work that is required toprocess the moving geometry.

An ALE formulation (cf. Hirt et al. [17]) was uti-lized in order to account for the rigid-body motionof the computational domain. This is accomplishedby modifying the flux through a boundary to ac-count for the motion of the boundary. For the in-viscid flux vector used here, this becomes

f · n =

ρunρunu + pnρune + pu · n

(3)where

un = (u− uΩ) · n

is the velocity relative to the moving boundary, anduΩ is the velocity of the moving domain. Hence theconvective part of the flux is modified to accountfor the motion of the boundary, compared to thetreatment for a fixed domain.

A modified form of van Leer’s flux-vector split-ting (FVS) [24] is used with the ALE formulation.This modification generalizes the scheme by usingthe Mach number relative to the moving boundarywhen determining the characteristic speeds of thesystem. Using the relative Mach number makes thescheme Galilean invariant, so that simulations witha static domain provide the same numerical resultsas those which use a moving domain when comput-ing the same physical problem. The subsonic fluxfor van Leer’s FVS can be written as a combina-tion of convective and acoustic terms (for a staticdomain) as

f̂± ·n = M±n

ρcρcuρcH

±M±n (2∓Mn) 0pn0

∓ γ

γ − 1M±n (2∓Mn)

2

00pc

(4)where Mnc = u · n and M±n = ± 14 (1±Mn)

2.The numerical flux across a boundary which sep-arates two domains (left and right) then becomesf̂(QL,QR) = f̂+(QL) + f̂−(QR).

With an ALE formulation, the Mach number rel-ative to the moving boundary is Mnc = (u− uΩ) ·n. If this relative Mach number is simply substi-tuted into van Leer’s FVS when computing on amoving domain, the energy equation will not beconsistent, in the sense of f̂(Q,Q) = f(Q). Thisis due to combining the total energy and pressurework terms into the total enthalpy H in the numer-ical energy flux in Eqn. 4. Physically, the pressure

3

work due to the domain motion (puΩ) is identicallyzero, hence this term does not appear in the energyflux in Eqn. 3.

A modified form of van Leer’s FVS was devel-oped for the ALE scheme. The mass and momen-tum flux are unmodified from the original form,only the energy flux is changed. Examining theenergy flux in Eqn. 4, it’s seen that the last termdisappears when QL = QR. Neglecting this lastterm in the energy flux, the total enthalpy is splitinto the total energy and pressure work, and theseterms are treated separately in the same manneras the momentum equations. The modified schemethen becomes

f̂± ·n = M±n

ρcρcuρce

±M±n (2∓Mn) 0pn

pu · n

(5)

where M±n is now based upon the relative Machnumber. The convective terms and acoustic termsare treated consistently in all 3 equations of thesystem. This numerical flux is similar to the WPSscheme developed by Agarwal and Halt [25]

The modified scheme for the energy flux main-tains the smoothness property of van Leer’s origi-nal FVS. It is continuously-differentiable throughMn = ±1, as can be seen in Fig. 1, where thenormalized energy flux ( fρc3 ) is plotted against theMach number. Both FVS schemes smoothly ap-proach the asymptotic limit of pure upwinding.

-1 -0.5 0 0.5 1Mach Number

-3

-2

-1

0

1

2

3

Nor

mal

ized

Ene

rgy

Flux

(f/

ρc3 )

van Leer F+

van Leer F-

Mod. van Leer F+

Mod. van Leer F-

Sum F+, F

-

Figure 1: Normalized energy flux for the FVS schemes,Eqns. 4 and 5. Both schemes are continuously-differentiablethrough Mn = ±1

The geometry of the domain is expressed inthe moving coordinate system, and must be trans-formed to the inertial system where the equationsof motion are specified. With a Cartesian scheme,the faces of each computational cell have normals

-3 -2 -1 0 1 2 3Angle of Attack (deg)

-0.5

-0.25

0

0.25

0.5

Nor

mal

For

ce C

oeff

icie

nt

Exp. (AGARD 702)Comp. - Rigid-Domain Motion

pitch u

p

pitch

down

Figure 2: Variation of normal force coefficient with an-gle of attack for oscillating NACA 0012. (M∞ = 0.755,α(t) = 0.016 + 2.51 sin (2π62.5t)). The simulation uses 100timesteps per complete cycle of the airfoil, and an isotropicmesh with a finest resolution of 0.001 chord. Experimentaldata from [26].

pointing in one of the Cartesian directions. Thesenormals all transform to the inertial system simi-larly, and are simply pre-computed and stored priorto each timestep.

The ALE formulation was utilized to simulate atransonic NACA 0012 pitching airfoil (cf. AGARDReport 702 [26]). The 2nd-order backward time-integration scheme was used to compute the un-steady motion starting from a converged steady-state solution. The simulation uses 100 timestepsper complete cycle of the airfoil, and an isotropicmesh with a finest resolution of 0.001 chord. Asthe airfoil oscillates, the shock transitions betweenthe upper and lower surfaces of the airfoil, and thefluid state is path-dependent. This hysteresis is ev-ident in the normal force history plotted in Fig. 2.After an initial transient of approximately 1/2 cy-cle, the solution is periodic. At a given angle ofattack, the normal force is multi-valued dependingupon whether the airfoil is pitching up or down.The computed variation of normal force is in goodagreement with the experimental data.

4 Relative motion

As an introduction to computing moving bound-aries with a non-body-fitted Cartesian scheme,Fig. 3 contrasts three methods of moving geome-try during a timestep: overset meshes, deformingmeshes, and the current Cartesian approach. Boththe overset and deforming mesh approaches usebody-fitted volume meshes. Overset approaches(cf. [1, 27, 28]) are relatively easy to implement

4

and efficient, as the volume mesh processing is donea priori; only (lower-dimensional) boundary inter-polations are updated at each step. Overset meth-ods cannot maintain conservation across compositegrid boundaries without complex mesh construc-tions in the overlap region. It also can be diffi-cult and labor-intensive to maintain a compatiblespatial resolution across composite grids, and toprocess the boundaries when components are closetogether. With deforming meshes (cf. [3, 29, 30]),the volume mesh deforms in response to the surfacemotion, and after large deformations a volume re-meshing and (often non-conservative) interpolationto the new mesh is performed. Deforming-mesh ap-proaches can be conservative over a timestep, mak-ing them attractive for small deformations. How-ever, for gross boundary motions associated withlarge timesteps the quality of the difference sten-cil can degrade severely due to this distortion. Inthe current Cartesian approach, the moving bound-ary “sweeps” through a fixed Eulerian mesh overa timestep. The cells within the swept region ofthe domain change volume and shape over thetimestep, and cells can appear or disappear (orboth) as well. The space-time geometry associatedwith these swept cells is complicated, however it ispossible to formulate conservative schemes as theboundary is impermeable, hence the flux throughall of the moving faces of a cell can easily be ap-proximated. Formulating a non-body-fitted Carte-sian scheme which maintains conservation for largetimesteps (large motions) of a moving boundary,while still retaining efficiency and robustness is thechallenge for the current (and future) research.

4.1 Governing Equations

The motion of a solid body through an inviscidfluid discretized by a fixed Cartesian mesh is gov-erned by the same ALE set of conservation equa-tions (a Lagrangian body moves through an Eule-rian mesh) as the rigid-domain motion of the pre-vious section,

d

dt

∫V (t)

QdV = −∮

S(t)

f · ndS S = ∂V (6)

f · n =

ρunρunu + pnρune + pu · n

un = (u−w) · n

t = n

t = n+1

Composite Grid Boundary

(a) Overset

(b) Deforming Mesh

Uncut

Swept

Cut

t = n

t = n + 1

(c) Cartesian

Figure 3: Approaches for moving boundary simulations.Deforming mesh figures courtesy T. Baker [29].

5

Here w is the velocity of the moving boundary withrespect to the Eulerian frame, and is used to dif-ferentiate from the rigid domain velocity uΩ. Forthe current discussion the rigid-body motion of thedomain will be ignored, and only the relative mo-tion will be formulated. No changes to the currentscheme are required when the rigid-domain motionis superposed.

For a cell in a Cartesian mesh swept by a movingboundary, Eqn. 6 can be simplified as the bound-aries of the cell are fixed, hence w·n = 0, except forthe motion of the solid surface through the volume,for which u · n = w · n holds (i.e. the convectiveportion of the flux is zero through an impermeablesurface). Eqn. 6 is preferred over a simplified formhowever, as it emphasizes the deforming-cell natureof the problem.

Applying Liebniz’ rule to the left side of Eqn. 6gives

d

dt

∫V (t)

QdV =∫

V (t)

∂Q∂t

dV +∮

S(t)

Qw ·ndS (7)

and the right-hand-side flux through the boundarysurface can be decomposed as

−∮

S(t)

f · ndS = −∮

S(t)

[Q (u−w) + P] · ndS

(8)

where P =

0pδpu

is the acoustic portion of theflux. Balancing terms gives∫

V (t)

∂Q∂t

dV +∮

S(t)

[Qu + P] · ndS = 0 (9)

Equation 9 can be recast into a space-time diver-gence form by applying Gauss’s thereom to the spa-tial volume integral. Defining the 4-D space-timenormal as ñ =

{t̂,n

}, where t̂ is the normal in

the time direction, n is the unit spatial normal,and Q̃ = {Q,Qu + P}, the space-time conserva-tion equation is ∮

Q̃ · ñdΩ = 0 (10)

where Ω is the boundary of the space-time volume.Expanding Q̃ for clarity gives

Q̃ =

ρρu

ρe

t̂, ρuρuu + pδ

ρeu + pu

n (11)

t = n

t = n+1

x

t

tc

W (closed boundary)

Qu + PQu + PQ

Q t̂

1w

n

Figure 4: One-dimensional space-time cell. The imperme-able moving boundary (shown in red) crosses the fixed leftface of the cell at time tc. The impermeable boundary hasa normal with elements in both space and time.

Note that the velocity of the moving solid surfaceno longer explicitly appears in Eqn. 10 as the mo-tion of the boundaries is accounted for in the direc-tion and “area” of the space-time boundary. Fig. 4shows a 1-D space-time cell volume for a Carte-sian cell as a moving-boundary crosses the left cellface. The impermeable portion of the space-timecell boundary (red boundary in Fig. 4 with slope1/w) has a normal with elements in both space andtime, and is analogous to the role of the bound-ary velocity in Eqn. 6. In essence, the mixed La-grangian/Eulerian formulation of Eqn. 6 has beenconverted to an Eulerian formulation in space-time.

Equations 6 and 10 explicitly satisfy theso-called Geometric Conservation Law (GCL)(cf. Thomas and Lombard [31]). The GCL is usu-ally presented as

d

dt

∫V (t)

dV =∮

S(t)

w · ndS (12)

which states that the change in cell volume is equiv-alent to the area swept by the moving boundary.The supplementary information that the cell mustclose,

∮ndS = 0, is also required. The space-time

analog to the GCL is∮ñdΩ = 0 (13)

which states that the space-time cell must close.Whether numerical schemes are built for Eqn. 6

or Eqn. 10 is largely a matter of convenience, asboth are mathematically equivalent. In the cur-rent work, Eqn. 6 is discretized directly, since itmaintains a similar formulation as the governingequations for both static and rigid-domain motions.This leverages the existing flow solver infrastruc-ture for multigrid, dual-time, etc., for solving the

6

relative motion equations. Equation 10 is used inthe discussion for deriving or clarifying expressions,and often provides more a more intuitive form.

4.2 One-dimensional Motion

While the form of Eqn. 6 is familiar, the space-time geometry of the deforming cells becomes quitecomplex for a boundary moving through a Carte-sian mesh. In order to focus the discussion, the ex-amples here begin with the motion in 1-D, which il-lustrates many of the same properties as the multi-dimensional equations. Points of interest or em-phasis in expanding from 1-D to multiple dimen-sions are discussed. Further, only discretizationsfor the momentum equation are presented, thesebeing representative. Example simulations will bepresented for motions in 1, 2, and 3 dimensions inSec. 5.

4.2.1 CFLw < 1

A representative space-time cell for a one-dimensional motion is shown in Fig. 5. In thisexample the CFL number based upon the wall nor-mal velocity is less than unity, so that the geome-try moves less than one uncut hexahedron∗ duringthe timestep. The velocity of the wall is not con-stant over the timestep, hence the boundary motionis non-planar in space-time. Linearizations of thisboundary motion will be discussed in Sec. 4.2.3.While this example shows small motions by way ofan introduction, and the focus of the current workis implicit schemes which allow large motions, smallmotions still occur often in practice. An exampleis pure rotation, where close to the axis of rotationthe body movement is small.

In Fig. 5a, the moving boundary remains in thesame hexahedra at tn and tn+1. If it is assumedthat the cell volume, V , is known exactly for allcells in the domain at both time levels, then thetemporal contribution to the space-time flux canbe evaluated directly. A semi-discrete momentum

∗Borrowing from 3-D, the Cartesian mesh is said to bemade up of hexahedra in 1- and 2-D to maintain a commonnomenclature.

equation for cell j can then be written as†

(ρuV )n+1j − (ρuV )nj

+∫ tn+1

tnf̃Rj dΩ−

∫ tn+1tn

f̃Lj dΩ = 0 (14)

where ∫f̃dΩ =

∫Q̃ · ñdΩ (15)

is an integral over space and time. The super-scripts R and L in Eqn. 14 refer to the right and leftfaces of the 1-D cell respectively. In the Cartesianscheme the right boundary of cell j is fixed relativeto the inertial frame, and is not cut by the geom-etry, hence it is evaluated with the same formula-tion as in a static domain. The flux through theleft boundary, however, has contributions in boththe temporal and spatial directions. The tempo-ral (convective) contribution is identically zero, asu · n = w · n for an impermeable boundary. Thisleaves only the pressure contribution to the moving-boundary flux, which in the current 1-D examplebecomes ∫ tn+1

tnf̃Lj dΩ =

∫ tn+1tn

pwdt (16)

where pw represents an approximation for the wallpressure to some order of accuracy. In other words,pressure acts only in the spatial directions, hencethe space-time area over which the surface pressureacts is the projection along the spatial axes of themoving wall area. In the vector notation of theprevious section(

Q̃ · ñ)

w= (P · n)w (17)

An example in multiple dimensions is shown inFig. 6 where a boundary moves through a group ofhexahedra in 2-D. Examining the space-time cell k,the pressure acts on the projection of the movingfront along the −x and +y axes. This is the pro-jection of the red triangular region in Fig. 6b alongthe spatial coordinates.

For cells such as sketched in Fig. 5a, where thesolid boundary does not cross a cell vertex overthe timestep, it is unnecessary to calculate anyspace-time geometry. Conservation is satisfied ifthe change in cell volume is known, and the accu-racy of the scheme for Eqn. 14 is determined by the

†Similar equations can be written for cell i behind themoving front.

7

t = n

t = n+1

i j

i j

x

t

(a) Behind(Emerging)

Ahead(dissappearing)

(b)

t = n

t = n+1

i j

i j’

x

t

k

k

tctc’

Figure 5: One-dimensional motions of an impermeableboundary for motion less than the uncut hexahedron spacing(CFLw < 1).

form of the flux quadrature through the space-timeareas.

Fig. 5b again shows a boundary motion which isless than the uncut hexahedron spacing, howeverin this example the boundary crosses a cell vertexduring the timestep. The cell labeled j disappearsahead of the moving front, while the new cell j′

emerges behind the front. Examining the situationahead of the moving front first, the cell j does notexist on the mesh at time level n + 1, hence noth-ing needs to be computed in this cell. The cell kis the first cell at time level n + 1 ahead of thefront. This cell receives a contribution from theuncut face to the right, and two contributions onthe left: a flow contribution from tn to tc, the timeof the vertex crossing, and a solid wall contribu-tion from tc to tn+1. In general, the flux througha space-time face is thus composed of a flow por-tion, Ωf , and a boundary contribution Ωw. Thesemi-discrete equations for cell k are written as

(ρuV )n+1k − (ρuV )nk +

∫ tn+1tn

f̃Rk dΩ

−∫ tc

tnf̃Lk dΩf −

∫ tn+1tc

pwdt = 0 (18)

where the simplification for the solid wall contribu-tion Ωw from tc to tn+1, Eqn. 16, has been made.The flux entering cell k from tn to tc is identical tothe flux out of cell j. The semi-discrete conserva-

t = n + 1 t = n

x

y

(a)

t = n + 1

t = n

xy t

j

k

(b)

Figure 6: 2-D cells swept by a moving boundary. The mov-ing boundary is shaded in red, and begins at time level n inhex j. The boundary both rotates and translates, moving inthe +x and −y directions over the timestep. At time leveln + 1 hex j is completely interior to the boundary, and cellk is now cut.

tion equations for cell j are

− (ρuV )nj +∫ tc

tnf̃Rj dΩf −

∫ tctn

pwdt = 0 (19)

solving for∫

f̃Rj dΩf and substituting for−

∫f̃Lk dΩf in Eqn. 18 gives

(ρuV )n+1k − (ρuV )nk − (ρuV )

nj

+∫ tn+1

tnfRdΩ−

∫ tn+1tn

pwdt = 0 (20)

as the semi-discrete equations for cell k in Fig. 5b.Comparing with Eqn. 14, the only required modifi-cation is a conservation correction from cell j. This

8

“cell-merging” correction will be discussed furtherin the next section.

The conversion of the flux in cell k from tn to tc

to a pressure contribution in Eqn. 20 is not general,and in multiple-dimensions the spatial flux for a cellahead of the front will contain both wall and flowcontributions. This is seen in Fig. 6b, where thecell j closes as the boundary moves, however cell kahead of the moving boundary has both wall andflow components of the space-time flux through its−x and +y space-time faces. In order for the flowcomponent of the flux to telescope completely to apressure component all of the spatial faces of a cellmust become closed.

Cell j′ in Fig. 5b emerges behind the body at tc′.

Again, the temporal (convective) contribution dueto the moving boundary is zero for an impermeableboundary, and the only contribution due to pres-sure is a projection of the wall along the spatialaxes. The left face of cell j′ receives a flux con-tribution beginning at tc

′so that the semi-discrete

governing equations for cell j′ are

(ρuV )n+1j′ +∫ tn+1

tc′pwdt−

∫ tn+1tc′

f̃Lj′dΩf = 0 (21)

Unlike the cell k ahead of the front, for cells behindthe moving boundary in 1-D it is necessary to de-termine the time tc

′in some manner. Various levels

of approximation for determining this vertex cross-ing time will be outlined in Sec. 4.2.3, after thecurrent discussion is extended to arbitrarily-largeboundary motions in the next section.

4.2.2 Arbitrary CFL

The implicit framework of the current methodallows large geometry motions during a timestep,such as shown in Fig. 7, again in 1-D. Hexahedrasuch as j, k, and l in Fig. 7 are referred to as “time-split” hexahedra, in a similar manner as spatially-split hexahedra (cf. Aftosmis et al. [14]). The cellsbehind the moving front are processed as describedin Eqn. 21 without modification. Examining thecells ahead of the moving front which no longer ap-pear at time level n+1, the semi-discrete equationfor cell m can be written as

(ρuV )n+1m − (ρuV )nm −

∑ahead

(ρuV )n

+∫ tn+1

tnf̃RmdΩ−

∫ tn+1tn

pwdt = 0 (22)

t = n

t = n+1

x

t

i j’ k’ l’ m

t4

t3t2

t1t0

t5

i j k l mFigure 7: One-dimensional motion of an impermeableboundary for displacement greater than the uncut hexahe-dron spacing (CFLw > 1).

The term∑

ahead (ρuV )n represents a conservation

correction for cell m which is an agglomeration ofthe conserved quantity in all the cells ahead of themoving front. In 1-D, determining this conserva-tion correction term is unambiguous. An exampleof the complexity in multi-dimensions is shown inFig. 6b, where the correction from cell j must beapportioned among its three neighbors (those shar-ing the −y face, +x face, and the diagonal neighbork). This apportionment is similar to the flux redis-tribution used by Pember et al. [32].

The “flux telescoping” used here is analogous tothe cell-merging technique used in the explicit, 2-D simulations by Bayyuk et al. [10, 11]. In cellmerging the cells surrounding the moving body atboth time levels are physically merged into a singlecell, and then integrated forward in time. After thetimestep a reconstruction is performed back to theunmerged mesh. Flux telescoping avoids the com-plex constructions required to physically merge thecells in multiple dimensions. Ahead of the front,the cell merging and flux telescoping techniquesare mathematically equivalent, however behind thefront they are not necessarily the same. For ex-ample, since the current method maintains the dis-cretization through the timestep, a reconstructionstep is not required behind the front. This recon-struction can be problematic for large boundarymotions and in multi-dimensional simulations.

4.2.3 Levels of Space-timeGeometry Approximation

As described in the previous sections, in general,some details of the space-time geometry must bedetermined in order to evaluate the spatial fluxterms. As a first step, the current article focuseson two-time-level schemes with the flow and ge-ometry states synchronous. These require only asingle computation of the cut-cell intersection per

9

t = n + 1

t = n

xy

t

k

area

Ω

w

x

=

area

Ω

f

x

=

Figure 8: Isolated view of cell k from the two-dimensionalmotion in Fig. 6.

timestep (time level n simply being saved from theprevious step). A similar approach staggers the ge-ometry and fluid states in time [8], which also onlyrequires a single geometry intersection per step. Asdescribed previously, the change in the conservedquantity can be discretized directly since the cellsvolumes are known at time levels n and n + 1 asa result of the volume mesh generation. This sec-tion describes levels of approximation in determin-ing the required space-time geometry. These ap-proximations affect the accuracy of the numericalscheme, however conservation is maintained dis-cretely regardless of how the space-time geometryis approximated.

With the Cartesian approach, determining thedetails of the moving boundary motion is not ex-plicitly required, and non-planar wall motions canbe evaluated. Fig. 8 shows an isolated view of the2-D cell k from Fig. 6. If the flow contributionsto the space-time area (Ωf ) are known, then thewall contribution can be determined by using thespace-time GCL, Eqn. 13. Since Eqn. 13 is a vectorequation it can be applied to each Cartesian direc-tion independently. Thus, enforcing closure (direc-tionally) of the space-time cell determines the wallcontribution. In this manner it is not necessaryto explicitly linearize the wall motion in order todetermine the wall flux contribution, though withmost methods an implicit linearization does occuras will be described.

The space-time GCL also simplifies evaluation ofthe pressure work due to the moving boundary in

the energy equation. The pressure work term isgiven by ∫

w

pu · ndΩw (23)

where here the integral is taken over the movingboundary within the space-time cell. If the wallpressure is approximated with a constant value pw,and the substitution u·n = w·n for an impermeableboundary is made, then Eqn. 23 becomes

pw

∫w

w · ndΩw (24)

The integral∫

ww · ndΩ is the area swept by the

moving boundary over the timestep, which is equiv-alent to the change in volume within the space-timecell (cf. Eqn. 7). Hence the pressure work term canbe evaluated using the known cell volumes at n andn + 1 as ∫

w

pu · ndΩw ≈ pw∆V (25)

Since the contribution to mass conservation due toan impermeable boundary is identically zero, thisleaves only the pressure contribution in the momen-tum equations yet to be evaluated. The next sec-tions include different levels of approximation forthis pressure flux.

Sequential-staticThe lowest-fidelity approach for the space-time

geometry is to simply ignore the time-dependentnature of the problem, and solve the govern-ing equations with a steady-state solver at eachtime level. The steady-state solver can easily beaugmented with a moving-wall boundary condi-tion. While this sequential-static approach ap-pears crude, there are many applications havingtimescales where this approach can be effective, andexamples can be found in the literature. The ad-vantage is that no specialized time-dependent ormoving-body algorithms are required in order toperform the simulations, however the applicabilityand accuracy is limited (and unknown in general).

Staircase-in-timeThe sequential-static simulation approach can be

improved by including a “history” of the fluid evo-lution, i.e. including the time derivative of the fluidstate ∂Q∂t . The simplest way to accomplish this isto use a staircase approximation to the space-timegeometry, analogous to using a staircase geometryin space. The geometry is considered fixed over a

10

timestep [tn, tn+1], for example by holding the ge-ometry in its state at tn, or tn+1/2, etc. In thismanner the cell volume is held constant, so thata history of the motion is not provided, howeverthe correct moving-wall boundary condition is stillapplied.

Linear-in-timeIn the current scheme, the intersection of the ge-

ometry with a fixed Cartesian mesh is available attime levels n and n + 1 so that a history of themotion can be provided through the change in cellvolume over a timestep. The integral of the fluxthrough a face which is cut by the geometry duringthe timestep must still be evaluated however. Asdiscussed in Sec. 4.2.1, in general the flux througha space-time face is composed of both flow and wallcontributions∫ tn+1

tnQ̃ · ñdΩ =

∫ tn+1tn

Q̃ · ñdΩf

+∫ tn+1

tnQ̃ · ñdΩw (26)

For the fixed Cartesian flow faces the space-time in-tegral simplifies as

(Q̃ · ñ

)f

= ([Qu + P] · n)f . Ingeneral the integrals on the right-side in Eqn. 26 arecomposed of multiple components. For example,the flow contribution on the −x face in Fig. 8 canbe broken into two integrals; one up to the vertexcrossing and one after. The space-time flux termsin Eqn. 26 can be evaluated with a backward-Eulerquadrature, i.e.∫ tn+1

tnQ̃ · ñdΩ ≈ Q̂n+1 · ñfΩn+1f

+ Q̂n+1w · ñn+1w Ωn+1w (27)

with Q̂ a numerical approximation for Q̃ andΩn+1 = Sn+1∆t. In this manner the state ofthe flow at time tn+1 is held over the entiretimestep [tn, tn+1]. A 2nd-order spatial approxima-tion for Q̂n+1 is used. The impermeable moving-wall boundary condition is implemented in the wallflux term,

(Q̃ · ñ

)w

= (P · n)w (Eqn. 17). Thisprovides a straightforward means to approximatethe moving geometry and provide a history of thewall motion. Example results using this backward-Euler approximation will be presented in Sec. 5.Comparisons of the sequential-static and backward-Euler approximation for prescribed and 6-DOF mo-tions are presented in [23] and [33] respectively.

An improvement can be made to the backward-Euler approximation, essentially at no cost, by im-proving the approximation to the space-time ge-ometry. This approximate geometry approach ismotivated by the observation that it isn’t neces-sary to determine the complete details of the space-time geometry in order to implement a numeri-cal scheme, it is only required to determine thespace-time area for the spatial flux terms. If thisarea can be approximated in some manner (and anappropriate quadrature determined for the flux),an improvement in accuracy is possible. This ap-proach is similar in spirit to using an agglomer-ated wall within the cut-cell geometry, rather thanseparately computing a contribution due to eachpolygonal contribution from the surface triangula-tion (cf. Aftosmis et al. [14]). A simple example isto approximate the flow portion of the space-timegeometry using Ωf ≈ 12

(Snf + S

n+1f

)∆t. The wall

portion of the space-time geometry is then deter-mined by forcing the space-time cell to close, as dis-cussed above. Combining this approximation witha trapezoidal quadrature gives∫ tn+1

tnQ̃ · ñdΩ ≈ 1

2

(Q̂n ·ñSn + Q̂n+1 ·ñSn+1

)f

∆t

+(Q̂ · ñ

)w

Ω̂w (28)

Ω̂w represents the approximation to the wall space-time area. More complex approximations for thespace-time geometry are also possible. Evaluatingthe approximate space-time geometry approach for3-D simulations is a focus of current research.

Full space-time intersectionThe highest-fidelity approach is to determine the

actual space-time geometry. In 3-D, each spatialface is a 2-D area which evolves to a 3-D volume inspace-time. If the motion of the moving boundaryis planar in space-time, or can be linearized, thenthe same techniques which are used to determinethe cut-cell geometry for spatial meshes can be usedto determine the space-time geometry of each spa-tial face. This requires six 3-D boundary/cell inter-sections to be computed for each swept Cartesiancell using methods similar to those in [14].

5 Numerical Results

The previous sections described moving-boundaries within a Cartesian scheme, along

11

with outlining implicit algorithms for solvingsuch problems. The current section presentsnumerical results in one, two, and three dimen-sions to demonstrate the implicit approach. Allresults were obtained using the backward-Eulerscheme discussed in the previous section. Beforepresenting the numerical results however, thefull three-dimensional implementation is brieflydiscussed.

Currently, a fast global re-meshing is performedat each timestep with the same volume mesh gener-ation package as used for static simulations. Workon incorporating solution and moving-geometryadaptation capability, similar to the static, steady-state method outlined in Aftosmis and Berger [34],is in progress. Note that if the motion is prescribedall the meshes can be processed a priori, and inparallel. Integrating the deforming-cell governingequations, Eqn. 6, for a representative cell j usingthe backward-Euler scheme gives

(QV )n+1j − (QV )nj −

∑ahead

(QV )n =

−∆t∑ (

f̂ · n∆S)n+1

j(29)

The summation on the right side includes thewall and flow contributions within cut cells. Thiscan be numerically integrated using the dual-timescheme outlined in Sec. 2. The terms (QV )nj and∑

ahead (QV )n become fixed source terms in the

dual-time scheme. (QV )n however is only availableon the mesh at time level n, while it is required onthe mesh at time level n + 1 in order to integrateEqn. 29. (QV )n is conservatively transferred fromthe mesh at time level n to the new mesh at n + 1external to the flow solver. The transfer of the solu-tion between two volumes meshes takes advantageof the space-filling-curve ordering of the cells in theCartesian meshes (cf. Ref. [15]). This allows thetransfer to be performed very efficiently, requiringonly two sweeps over the mesh cell list.

5.1 1-D Piston

A 1-D piston instantaneously moving at Mp =2.0 into an initially quiescent fluid is simulatedto demonstrate the conservation of the scheme.While the physical problem is one-dimensional, a3-D mesh and solver are used, with suitable bound-ary conditions to ensure no lateral flow develops.The piston is originally centered at x = 0, andhas width 4∆x, where ∆x is the uncut hexahedron

spacing. The piston moves in the +x direction, anda shock forms ahead of the piston, with an expan-sion region to the rear (cf. Liepmann and RoshkoSec. 3.2 [35]). If conservation is not maintained,a shock will not form ahead of the piston, or willhave the wrong speed, as mass, momentum, andenergy continually “leak” through the piston face.Numerical simulations moving the piston relativeto a fixed domain are compared with the exact an-alytic solution. Figure 9 plots the pressure on thecompression-side face of the piston as a function oftime for two CFL numbers based upon the pistonvelocity, CFLw =

wp∆t∆x = 1.0 and 10.0. After an

initial transient, both numerical simulations reachthe same pressure on the piston face, which is inagreement with the analytic solution. The pressureand density variation along the axis are plotted inFig. 10 after the piston has traveled 22CFLw, i.e.the same number of timesteps for each simulation.Both simulations predict the correct shock locationahead of the traveling piston, and the agreement inthe expansion region behind the shock is likewisevery good. As expected, the shock is smeared overseveral cells using the backward-Euler time integra-tion scheme and remains monotone.

0 2 4 6 8 10Time/CFL

3

6

9

12

Com

pres

sion

-Fac

e W

all P

ress

ure

p w/p

inf

AnalyticCFL

w = 1

CFLw

= 10

Figure 9: Pressure on the compression-side face of a pistonmoving at Mp = 2.0 into an initially quiescent fluid.

5.2 2-D Oscillating Airfoil

The oscillating NACA 0012 airfoil presented inSec. 3 is used to examine the behavior of therelative-motion scheme in 2-D. The experimentalcase was simulated using both the 2nd-order back-ward scheme with the moving-domain ALE scheme(cf. Sec. 3), and the 1st-order (in time) relative-motion scheme. Both schemes utilize the samespatially 2nd-order numerical flux formulation, and

12

-20 -10 0 10 20 30 40 50Axial Location

0

1

2

3

4

5

6

7

Density Pressure

∆xwall

= 22*CFL

(a) CFLw =wp∆t

∆x= 1.0

175 200 225 250 275 300 325 350Axial Location

0

1

2

3

4

5

6

7

Density Pressure

∆xwall

= 22*CFL

(b) CFLw =wp∆t

∆x= 10.0

Figure 10: Density and pressure variation along the axisfor a piston moving at Mp = 2.0 into an initially quiescentfluid. The mesh has a uniform spacing of ∆x = 0.25.

100 timesteps per cycle were used in both simula-tions. The computed normal force variations withangle of attack are shown in Fig. 11. Both simula-tions capture the hysteresis caused by the unsteadyshock formation, and are in good agreement witheach other and the experimental data. The con-vergence and stability properties of the ALE andrelative-motion formulations are similar for thisproblem. Using two coarsening levels of multigrid,each physical timestep converges roughly 2 ordersof magnitude in the L1-norm of density in 25 W-cycles using the dual-time formulation.

Snapshots of the 2-D pressure contours com-puted with the relative motion scheme are shownin Fig. 12 for the NACA 0012 oscillating airfoil.In general, the contours are smooth, with sharpdefinition of the shock location, i.e. no numericalartifacts from the relative motion scheme are visi-ble. The hysteresis is evident comparing Fig. 12aand Fig. 12c, where the airfoil passes through zeroangle of attack on the upstroke and downstroke re-spectively. The sonic “bump” which forms behind

-3 -2 -1 0 1 2 3Angle of Attack (deg)

-0.5

-0.25

0

0.25

0.5

Nor

mal

For

ce C

oeff

icie

nt

Exp. (AGARD 702)Comp. - Rigid-Domain MotionComp. - Relative Motion

pitch u

p

pitch

down

Figure 11: Variation of normal force coefficient with angleof attack for oscillating NACA 0012. Compare with pres-sure contours in Fig. 12. (M∞ = 0.755, α(t) = 0.016 +2.51 sin (2π62.5t)). The simulation uses 100 timesteps percomplete cycle of the airfoil, and an isotropic mesh with afinest resolution of 0.001 chord. Experimental data from[26].

the upstream shock as it moves from the lower toupper surface is seen in frame (b). The maximumshock strength is shown in Fig. 12d, and the lagbetween maximum pitch angle and the angle of at-tack which produces the maximum shock strengthis clear.

5.3 3-D Rolling Missile

In order to determine the conservation correctionterm

∑ahead (QV )

n in multiple dimensions, it isnecessary to simulate in some manner the physicalconvection process which is no longer discretizedwith a large timestep. Determining the conserva-tion correction in 3-D, and the manner in which todistribute it, is an area of ongoing research. The3-D simulations presented here do not include aconservation correction.

The complete three-dimensional time-dependentflow solver, using both a rigid-domain motion andlocalized relative motion, is demonstrated by sim-ulating a supersonic rolling-missile with ditheringcanards. The complete details of these numeri-cal simulations are contained in [23], and only abrief overview will be presented here. The entirecomputational domain rotates with the body rollrate using the ALE formulation, while the canardsconcurrently rotate relative to the body using therelative-motion scheme. As the relative motion isconstrained to the region near the canards, all cellsaway from this region do not require any specialtreatment.

The missile body of Fig. 13 rotates at a constant

13

(a) α = 0.08◦ ↑

(b) α = 1.47◦ ↑

(c) α = 0.02◦ ↓

(d) α = −1.84◦ ↑

Figure 12: Snapshots of pressure during oscillation ofNACA 0012 airfoil. Arrows in caption indicate whetherairfoil is pitching nose up, or nose down. Compare withnormal force variation in Fig. 11. (M∞ = 0.755, α(t) =0.016 + 2.51 sin (2π62.5t)).

Figure 13: Rotating missile with dithering canards. Theentire computational domain rotates at the body roll rateusing the ALE formulation, while the canards rotate relativeto the body using the relative-motion scheme.

prescribed rate of 8.75 Hz. As the body rolls, thetwo canards change positions synchronously to af-fect controlled movements, such as yaw or pitch.The computed force and moment variations withroll angle for one complete roll cycle are presentedin Fig. 14, along with the canard deflection an-gles. The simulations use a mesh containing 3.4Mcells, and a timestep which rolls the body 1◦ dur-ing a step (360 steps/cycle). The computed re-sults are compared to high-resolution (40M cells,10,000 steps/cycle) overset, viscous simulations ofNygaard and Meakin [36]. The current results com-pare well with the viscous results in terms of bothroll-averaged values, and the roll-dependent varia-tions. As expected, the viscous results predict aconsistent axial force increment compared with thecurrent inviscid results. The load variation withroll angle shows the “dynamic overshoot” whichoccurs when the canards instantaneously stop theirmotion. Snapshots of the velocity magnitude at 5axial cutting-planes along the body as the missilerolls are presented in Fig. 15. The canards changeposition from their maximum to minimum deflec-tion through the three snapshots. The change inshock pattern on the canards as they pitch down,and the change in sense of rotation of the canardtip vortices are both visible. The twist in the ca-nard tip vortices as the body rotates is also evident,though difficult to discern at this low spin rate.

Roll-averaged loads for the rolling missile withdithering canards were measured experimentally[37]. Numerical simulations are compared to theexperimental data in Table 2 for a body roll rateof 18Hz using the experimentally-measured canarddither schedule (cf. [23] for more details). The com-puted forces and moments are all within the 95%

14

CN CY Cl Cm Cn

Experiment 0.45 – 0.61 0.15 – 0.20 -0.036 – -0.019 -1.5 – -0.40 0.93 – 1.5Computed 0.55 0.20 -0.034 -0.48 1.46

Table 2: Roll-averaged forces and moments for experimentally-measured dither schedule (M∞ = 1.6, α = 3.0◦, φ̇ = 18Hz).Experimental range corresponds to a 95% confidence level [37].

0 90 180 270 360Roll Angle (deg)

-1

-0.5

0

0.5

1

1.5

2

For

ce C

oeff

icie

nt

Normal - 3M cell Cartesian EulerAxial - 3M cell Cartesian EulerLateral - 3M cell Cartesian EulerAxial - 40M cell viscousLateral - 40M cell viscousNormal - 40M cell viscous

0 90 180 270 360-20

-10

0

10

20

Can

ard

Pit

ch A

ngle

(de

g)Canard Dither

(a) Forces

0 90 180 270 360Roll Angle (deg)

-6

-4

-2

0

2

4

6

8

Mom

ent

Coe

ffic

ient

abo

ut I

niti

al C

.G. L

ocat

ion

Pitch - 3M cell Cartesian EulerYaw - 3M cell Cartesian EulerPitch - 40M cell viscousYaw - 40M cell viscous

0 90 180 270 360Roll Angle (deg)

-20

-10

0

10

20

Can

ard

Pit

ch A

ngle

(de

g)

Canard Dither

(b) Moments

Figure 14: Force and moment comparison for rotating mis-sile with dithering canards. The canard pitch angle is shownas a solid black line with scale at right. Viscous simula-tions by Nygaard and Meakin [36]. (M∞ = 1.6, α = 3.0◦,φ̇ = 8.75Hz).

confidence level for the experimental data.

5.4 F/A-18C Store Separation

The final simulation example presented is acoupled six-degree-of-freedom(6-DOF)/CFD tra-jectory prediction of a U.S. Navy GBU-31/JDAMseparating from an F/A-18C. The complete detailsof this simulation are contained in [33], and only asummary is presented here. Time-dependent simu-lations are coupled with a 6-DOF method to predictthe trajectory of the store after it is ejected fromthe F/A-18 wing pylon. The trajectory is com-puted through 0.45 sec., using a constant timestep

(a) φ = 45.5◦

(b) φ = 57.6◦

(c) φ = 71.8◦

Figure 15: Velocity magnitude contours for rotating missilewith dithering canards. Red corresponds to a large magni-tude, and blue low. (M∞ = 1.6, α = 3.0◦, φ̇ = 8.75Hz).

15

of 0.0075 sec. This provides a CFL number basedupon the wall motion of 10-20 (depending upon thestore’s location in the trajectory). Figure 16 showssurface pressure on the F/A-18 aircraft, along witha composite image of the JDAM separation, at anaircraft velocity of M∞ = 1.055. The store fallsdownward under the influence of gravity, and con-currently is pitching nose down and outboard. Thispitching of the store is due to the shock from theleading edge of the wing which impinges directly onthe nose of the store. The computed trajectory isin good agreement with the flight-test data, and iscommensurate with previous coupled 6-DOF/CFDsimulations for this configuration. Complete quan-titative details of the trajectory are included in [33].Figure 17 presents snapshots of the JDAM dur-ing the trajectory simulation, along with cuttingplanes through the computational volume mesh ateach step. The mesh automatically adapts to thechange in geometry, as well as coarsens after thestore has passed.

Figure 16: Surface pressure on the F/A-18C and GBU-31/JDAM. A composite of the JDAM at several positions issuperimposed. (M∞ = 1.055, α = −0.65◦, γ = 44◦).

(a) t = 0.0 sec.

(b) t = 0.1 sec.

(c) t = 0.2 sec.

(d) t = 0.3 sec.

Figure 17: Cutting planes through the volume mesh duringJDAM separation. (M∞ = 1.055, α = −0.65◦, γ = 44◦).

16

6 Summary

A method for simulating moving impermeableboundaries within a fixed Cartesian mesh hasbeen developed, and initial results are demon-strated. This scheme leverages the automated vol-ume mesh generation process which has previouslybeen demonstrated for static geometries. A majorgoal in this work is to limit the amount of geome-try processing which is required during a completesimulation, in order to obtain an efficient and ro-bust scheme. This is especially relevant for three-dimensional applications. An implicit dual-timemethod is used for the time advance, which allowsa (large) timestep to be chosen based upon physicalconsiderations and not stability restrictions. Thislimits the number of times the geometry must beintersected with the Cartesian volume mesh over acomplete simulation. A general motion is decom-posed into a rigid-body motion of the entire com-putational domain, with a relative-body motion su-perimposed. Since the rigid-domain motion can betreated using an ALE formulation, this confines thegeometry processing only to the regions of relativemotion within the domain.

The details of the relative-motion scheme arepresented from a space-time analysis. This anal-ysis covers the key ideas first in 1-D, and then dis-cusses the complexity which arises when extendingthe scheme to higher dimensions. Features of rela-tive motion which are unique to Cartesian schemesare highlighted. A hierarchy of approximations tothe boundary motion during a timestep are pre-sented, along with preliminary results in one, two,and three dimensions. As a first step, only schemeswhich evaluate the geometry at two time-levels areconsidered for the relative motion.

Acknowledgments

The authors would like to thank Dr. Tor Ny-gaard of ELORET and Dr. Robert Meakin of theU.S. Army AFDD for providing the viscous rolling-missile results for comparison. Also, Dr. TimBaker of Princeton University provided permis-sion to reuse the deforming mesh figures. MarshaBerger was supported by AFOSR grant F19620-00-0099 and DOE grants DE-FG02-00ER25053 andDE-FC02-01ER25472.

References

[1] Meakin, R.L. and Suhs, N., “Unsteady Aero-dynamic Simulation of Multiple Bodies in Rel-ative Motion,” AIAA Paper 89-1996-CP, June1989.

[2] Batina, J., “Unsteady Euler Algorithm withUnstructured Dynamic Mesh for Complex Air-craft Aeroelastic Analysis,” AIAA Paper 89-1189, June 1989.

[3] Löhner, R., “Adaptive Remeshing for Tran-sient Problems,” Computer Methods in Ap-plied Mechanics and Engineering, 75:195–214,1989.

[4] Venkatakrishnan, V. and Mavriplis, D. J.,“Computation of Unsteady Flows over Com-plex Geometries in Relative Motion,” in 1stAFOSR Conference on Dynamic Motion CFD,June 1996.

[5] Hirt, C.W. and Nichols, B.D., “Volume ofFluid (VOF) Method for the Dynamics ofFree Boundaries,” Journal of ComputationalPhysics, 39:201–225, 1981.

[6] Osher, S. and Sethian, J.A., “Fronts Propa-gating with Curvature Dependent Speed: Al-gorithms based in Hamilton-Jacobi Formula-tions,” Journal of Computational Physics, 79:12–49, 1988.

[7] Peskin, C.S., “Numerical Analysis of BloodFlow in the Heart,” Journal of ComputationalPhysics, 25:220–252, 1997.

[8] Forrer, H. and Berger, M.J., “Flow Simu-lations on Cartesian Grids Involving Com-plex Moving Geometries,” in 7th InternationalConference on Hyperbolic Problems, Zurich,Switzerland, 1998.

[9] Roma, A.M., Peskin, C.S., and Berger,M.J., “An Adaptive Version of the ImmersedBoundary Method,” Journal of ComputationalPhysics, 153:509–534, 1999.

[10] Bayyuk, S.A., Powell, K.G., and van Leer, B.,“A Simulation Technique for 2-D Unsteady In-viscid Flows around Arbitrarily Moving andDeforming Bodies of Arbitrary Geometry,”AIAA Paper 93-3391-CP, 1993.

17

[11] Bayyuk, S.A., Powell, K.G., and van Leer, B.,“Computation of Flows with Moving Bound-aries and Fluid-structure Interactions,” AIAAPaper 97-1771, 1997.

[12] Falcovitz, J., Alfandary, G., and Hanoch,G., “A Two-Dimensional Conservation LawsScheme for Compressible Flows with Mov-ing Boundaries,” Journal of ComputationalPhysics, 138:83–102, 1997.

[13] Lahur, P.R. and Nakamura, Y., “Simulationof Flow around Moving 3D Body on Unstruc-tured Cartesian Grid,” AIAA Paper 2001-2605, June 2001.

[14] Aftosmis, M.J., Berger, M.J., and Melton,J.E., “Robust and Efficient Cartesian MeshGeneration for Component-Based Geometry,”AIAA Paper 97-0196, Jan. 1997. Also AIAAJournal 36(6): 952–960,June 1998.

[15] Aftosmis, M.J., Berger, M.J., and Adomavi-cius, G., “A Parallel Multilevel Method forAdaptively Refined Cartesian Grids with Em-bedded Boundaries,” AIAA Paper 2000-0808,Jan. 2000.

[16] Berger, M.J. and LeVeque, R., “Stable Bound-ary Conditions for Cartesian Grid Calcula-tions,” in Symposium on Computational Tech-nology for Flight Vehicles, Pergamon Press,1990. Also ICASE Report 90–37.

[17] Hirt, C.W., Amsden, A.A., and Cook, J.L.,“An Arbitrary Lagrangian-Eulerian Comput-ing Method for All Flow Speeds,” Journal ofComputational Physics, 14:227–253, 1974.

[18] Merkle, C.L. and Athavale, M., “A TimeAccurate Unsteady Incompressible AlgorithmBased on Artificial Compressibility,” AIAAPaper 87-1137, June 1987.

[19] Rogers, S.E., Kwak, D., and Kiris, C., “Nu-merical Solution of the Incompressible Navier-Stokes Equations for Steady-State and Time-Dependent Problems,” AIAA Paper 89-0463,1989. Also AIAA Journal 29(4): 603–610,1991.

[20] Jameson, A., “Time Dependent CalculationsUsing Multigrid with Applications to Un-steady Flows Past Airfoils,” AIAA Paper 91-1596, June 1991.

[21] Melson, N.D., Sanetrik, M.D., and Atkins,H.L., “Time-accurate Navier-Stokes Calcula-tions with Multigrid Acceleration,” in Proceed-ings of the Sixth Copper Mountain Conferenceon Multigrid Methods, Apr. 1993.

[22] Beam, R.M. and Warming, R.F., “ImplicitNumerical Methods for the CompressibleNavier-Stokes and Euler Equations,” VKI Lec-ture Series 81-01, Mar. 1981.

[23] Murman, S.M., Aftosmis, M.J., and Berger,M.J., “Numerical Simulation of Rolling-Airframes Using a Multi-Level CartesianMethod,” AIAA Paper 2002-2798, June 2002.

[24] van Leer, B., “Flux-Vector Splitting for theEuler Equations,” in Lecture Notes in Physics,vol. 170, pp. 507–512, Springer Verlag, Berlin,1982.

[25] Agarwal, R.K. and Halt, D.W., “A Modi-fied CUSP Scheme for Wave/Particle SplitForm for Unstructured Grid Euler Flows,” inFrontiers of Computational Fluid Dynamics(Caughey, D.A. and Hafez, M., ed.), pp. 155–168, John Wiley & Sons, 1994.

[26] Landon, R.H., “Compendium of UnsteadyAerodynamic Measurements,” AGARD Re-port No. 702, Aug. 1982.

[27] Belk, D. and Maple, R., “Automated Assem-bly of Structured Grids for Moving Body Prob-lems,” AIAA Paper 95-1680-CP, June 1995.

[28] Löhner, R., Sharov, D., Luo, H., and Rama-murti, R, “Overlapping Unstructured Grids,”AIAA Paper 2001-0439, Jan. 2001.

[29] Baker, T.J. and Cavallo, P.A., “DynamicAdaptation for Deforming TetrahedralMeshes,” AIAA Paper 99-3253-CP, June1999.

[30] Lesoinne, M. and Farhat, C., “Geometric Con-servation Laws for Flow Problems with Mov-ing Boundaries and Deformable Meshes, andTheir Impact on Aeroelastic Computations,”Computer Methods in Applied Mechanics andEngineering, 134:71–90, 1996.

[31] Thomas, P.D. and Lombard, C.K., “Geomet-ric Conservation Law and its Application toFlow Computations on Moving Grids,” AIAAJournal, 17:1030, 1979.

18

[32] Pember, R.B., Bell, J.B., Colella, P., Crutch-field, W.Y., and Welcome, M.L., “An Adap-tive Cartesian Grid Method for UnsteadyCompressible Flow in Irregular Regions,”Journal of Computational Physics, 120:195–214, 1995.

[33] Murman, S.M., Aftosmis, M.J., and Berger,M.J., “Simulations of 6-DOF Motion with aCartesian Method,” AIAA Paper 2003-1246,Jan. 2003.

[34] Aftosmis, M.J. and Berger, M.J., “MultilevelError Estimation and Adaptive h-Refinementfor Cartesian Meshes with Embedded Bound-aries,” AIAA Paper 2002-0863, June 2002.

[35] Hans W. Liepmann and Anatol Roshko, Ele-ments of Gasdynamics. John Wiley & Sons,1960.

[36] Nygaard, T. and Meakin, R., “An Aerody-namic Analysis of a Spinning Missile withDithering Canards,” AIAA Paper 2002-2799-CP, June 2002.

[37] “Defensive Missile Wind Tunnel Test for theValidation and Verification of CFD Codes,”Dynetics Technical Report, Jan. 2002.

19

IntroductionDual-time formulationALE formulationRelative motionGoverning EquationsOne-dimensional MotionCFLw < 1Arbitrary CFLLevels of Space-time Geometry Approximation

Numerical Results1-D Piston2-D Oscillating Airfoil3-D Rolling MissileF/A-18C Store Separation

SummaryReferences