GEOMETRIC CONSERVATION LAWS FOR AEROELASTIC COMPUTATIONS USING UNSTRUCTURED DYNAMIC

MESHES

Michel Lesoinne and Charbel Farhat*

Department of Aerospace Engineering and Sciences

and Center for Aerospace Structures - University of Colorado at Boulder

Boulder, CO 80309-0429, U.S. A.

Abs t r ac t

Numerical simulations of flow problems with mov- ing boundaries commonly require the solution of the fluid equations on unstructured and deformable dy- namic meshes. In this paper, we present a uni- fied theory for deriving Geometric Conservation La.ws (GCLs) for such problems. We consider several pop- ular discretization methods for the spatial approxi- mation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite ele- ment formulations. We show that, except for the case of the space-time discretization method, the GCLs impose import ant constraints on the algorithms em- ployed for time-integrating the semi-discrete equa- tions governing the fluid and dynamic mesh motions. We address the impa.ct of theses constraints on the so- lution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroe- lastic response of a flat panel in transonic flow.

1. In t roduct ion

In order to compute the aeroelastic response of

'Associate Professor, Senior Member AIAA Copyright 0 1 9 9 5 by Charbel Farhat Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

8 h

structures in fluid flows, the equations of motion of the structure and the fluid must be solved si- mult aneously. Because t he position of t he st ruc- ture determines at least partially the fluid domain boundaries, it becomes necessary to perform the in- tegration of the fluid equations on a moving mesh. Several methods have been proposed for this pur- pose, among which we note dynamic meshes1, the

2 Arbitrary Lagrangian Eulerian (ALE) formulation , the co-rotational approach3. 4 7 5, and the space-time

6 formulation . Although the coupled problem is usually viewed

as a two-field problem, the moving mesh can be viewed as a pseudo-structural system with its own

1 dynamics , and therefore, the coupled aeroelastic system can be formulated as a three- rather than two-field problem: the fluid, the structure, and the dynamic mesh7. The semi-discrete equations govern- ing this three-way coupled problem can be written as follows:

where t designates time, x is the position of a mov-

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ing fluid grid point, w is the fluid state vector, A results form the finite element/volume discretization of the fluid equations, FC is the vector of convective ALE fluxes, R is the vector of diffusives fluxes, q is the structural displacement vector, ent denotes the vector of internal forces in the structure, Fxt the vec- tor of external forces, M is the finite element mass - - matrix of the structure, M , D and K are fictitious mass, damping and stiffness matrices associated with the moving fluid grid and Kc is a transfer matrix that describes the action of the motion of the structural side of the fluid/structure interface on the fluid dy- - - - - R namic mesh. For example, M = D = 0, and K = K

- R where K is a rotation matrix corresponds to a rigid mesh motion of the fluid grid around an oscillating - - airfoil, and M = D = 0 includes the spring-based mesh motion scheme introduced by Batina et a1.l and the continuum based updating strategy described by Tezduyar et as pa~ticular cases.

In general, KC and K are designed to enforce con- tinuity between the grid motion and the structural displacement and/or velocity at the fluid/structure boundary

The first of Eqs. (1) involves both the position and velocity of the underlying fluid dynamic mesh. These entities are usually obtained from the solution of the second and third of Eqs. ( I ) , and optionally from the use of a predictor. When selecting a method for in- tegrating the fluid equations, it is desirable to choose one that preserves the trivial solution of a uniform flow field *. In this paper, we show that this prop- erty is verified only when the numerical scheme cho- sen for solving the fluid equations and the algorithm constructed for updating the mesh position and ve- locity satisfy a certain condition. We refer to this condition as the Geometric Conservation Law (GCL) because: (a) it can be identified as integrating exactly the area or volume swept by the boundary of a cell in a finite volume formulation, and (b) its principle is similar to the GCL condition that was first pointed out by Thomas and ~ombardlO for structured grids and finite difference schemes. In the present work, we derive the conditions imposed by the GCL in

'In the absence of other boundary conditions, a uniform flow field is a solution of the Navier-Stokes equations

terms of an appropriate choice of integration points in time, and a consistent scheme for updating the grid point velocities. This is in contrast with previous works l4, l3 where the GCL was addressed in terms of averaged normal or velocity coefficients for moving finite volume cells. The approach exposed herein for deriving and satisfying a GCL is deemed more gen- eral than those previously discussed in the literature. For example, it recovers the results of the normal av- eraging algorithm recently proposed by NKonga and ~ u i l l a r d l ~ for finite volume discretizations, and ap- plies as well to finite element methods that are not covered by this normal averaging procedure.

Throughout this paper, we focus on flow computa- tions using unstructured moving meshes. We derive several GCL conditions for these problems, and dis- cuss their various algorithmic implications. In Sec- tion 2, we consider the case where the finite volume method is chosen for the spatial approximation of the flow equations, and the ALE formulation is used for handling dynamic meshes. In Sections 3 and 4, we analyze the cases where the finite element method is employed for spatial discretization, and the moving mesh is treated with either a space-time or an ALE formulation, respectively. In particular, we show that space-time finite element methods always satisfy the fundamental geometric conservation law. In Section 5, we consider coupled fluid/structure problems and investigate the consequences of the GCL condition on the temporal integration of the structural equations of motion. In particular, we address the problem of satisfying both displacement and velocity continuity equations between the structure and fluid mesh at the fluid/structure interface, without violating the GCL. In Section 6, we highlight the importance of the GCL with an illustration of its effect on the computation of the transient aeroelastic response of a flat panel in transonic flow. Finally, we conclude this paper in Section 7.

2. The finite volume method with an ALE formulation

2.1. Semi-discretization

Let R(t) c Rn (n = 2, 3) be the flow domain of interest and r ( t ) be its moving and deforming bound- ary. We introduce a mapping function between Q(t) where time is denoted by t and the grid point coordi- nates by x, and a reference configuration O(0) where time is denoted by T and the grid point coordinates

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In the above equation, the partial time derivative is evaluated at constant (; hence, it can be moved out- side of the integral sign to obtain

Switching back to the time varying cells, Eq. (7) above can be rewritten as

Finally, integrating by parts the last term yields the governing integral equation

Figure 1: Control volume in an unstructured mesh Wdl2 ,+/ a c t ( t ) 3c(W,X) .Gdo=0 (9)

by J as follows:

In a finite volume method, the flux through the cell boundary aC;(t) is usually evaluated via $flux split- ting approximation as follows:

x = x ( J , ~ ) ; t = T (3) F;(w,x ,x) =

The conservative form of the equations describing Euler flows can be written in arbitrary Lagrangian- (3:(Wi, i) + Fz (Wj, X))C do

(10)

Eulerian (ALE) form as

where aCi,j is the intersection between the bound- a(Jw) k + JVx.Fc(W, i) = 0 (4) aries of cells C; and Cj , W; denotes the average value

at of W over the cell Ci, w is the vector formed by Fc(W,X) = F ( W ) - XW (5) the collection of Wi , and x is the vector of the time

where J = det(dx,dJ) is the jacobian of the frame dependent grid point positions. The numerical flux transformation 6 -, $, w is the fluid state vector, functions 3; and 3: are designed to make the re-

FC denotes the convective ALE fluxes, and = elc sulting system stable. An example of such functions is the ALE grid velocity that may be different from can be found in Anderson et al.'. For consistency,

the fluid velocity and from zero. these numerical fluxes must verify

The finite volume method for unstructured meshes relies on the discretization of the computational do- main into control volumes or cells Ci with bound- aries denoted by aCi (Fig. (1). Eq. (4) can then be integrated over the control cells. In an ALE formulation, these cells move and de- form in time. First, integration is performed over a reference cell in the J space as follows:

3Z(W, X ) + 3: (W, I ) = FC(W, X) (11)

Thus, the resulting discrete equation is

where

~i = La(t) d ~ x (13)

is the area/volume of cell Ci. Collecting all Eqs. (12) into a single system yields

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where A is the diagonal matrix of the cell areas, w is the vector containing all state variables Wi, and F is the collection of the fluxes Fi .

2.2. The geometr ic conservation law

Let At and t" = nAt denote respectively the cho- sen time-step and the n-th time-station. Integrating Eq. (12) between tn and tn+' leads to

The most important issue in the solution of the first of Eqs. (1) via an ALE method is the proper evalua-

- y+' tion of f tn Fi(w, x, x) in Eq. (15). In particular, it is crucial to establish where the fluxes must be inte- grated: on the mesh configuration at t = tn (xn) , on that at t = tn+' (xn+') , or in between these two con- figuration. The same questions arise as to the choice of the mesh velocity vector x .

Let W* denote a given uniform state of the flow. Clearly, a proposed solution method cannot be ac- ceptable unless it conserves the state of a uniform flow. Substituting W r = w:+' = W* in Eq. (15) gives

(A:+' - AT) W* + Fi (w* , x, x) dt = 0 (16)

where w* is the vector of the state variables when Wk = W* for all k. From Eq. (lo), it follows that

Given that the integral on a closed boundary of the flux of a constant function is identically zero

it follows that

Hence, substituting Eq. (19) into Eq. (16) yields

which can be rewritten as

= F; (W* , x)

Eq. (21) above defines a geometric conservation law(GCL) that must be verified by any proposed ALE mesh updating scheme. This law states that the change in area (volume) of each control volume be- tween t" and t"+' must be equal to the area (volume) swept by the cell boundary during At = tnf l - tn . Therefore, the updating of x and x cannot be based on mesh distorsion issues alone when using ALE so-

5 lut ion schemes. . The assumption that the numerical method per-

forms exactly the integration of Eq. (18) is referred to by Zhang et al.14 as the Surface Conservation Law (SCL). Satisfying of this condition is necessary for flow computations on static meshes and is not sye- cific to dynamic ones. Therefore, we do not discuss this condition in this paper any further and refer the reader to ref. 14 for additional details.

2.3. Implications of t h e GCL

From the analysis presented in the previous sec- tion, it follows that an appropriate scheme for eval-

n+l

uating S,, Fi(w*, x , x)dt in Eq. (12) is a scheme that respects the GCL (21). Note that once a mesh updating scheme is given, the left hand side of Eq. (21) is always exactly computed. Hence, a

n+1

proper method for evaluating St , Fi (w* , x, x) dt is a method that obeys the GCL and therefore com- putes exactly the right hand side of Eq. (21)- that

n+l

is, Jtn 2. d r dt-

2.3.1. T h e two-dimensional case Given that in two dimensions dCi is the union of segments, it suffices to consider the integration of x.n' along a seg- ment [ab]

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Figure 2: Parametrization of an edge in dimensional space

a two-

Let x , and xb denote the instantaneous positions of

Substituting Eq. (23-24) into Eq. (22) yields

tn+' 1

1l.q = + 1 ( + ( 1 - a)ib) .B 1 dadt

rtn+' ,

(a ( t )x&+l + ( 1 - 6 ( t ) ) q a ) dt

(26) where xb, = x , - xb, 1 is the length of edge [ab], and

H = ( y ilj. The mesh velocities i, and ib can

be obtained from the differentiation of Eq. (24)

two connected vertices a and b (Fig. 2). The position of any point on the edge [ab] during the time interval and I[ab] can be finally written as [tn, tn+'] can be parametrized as follows:

~ ( t ) = a x a ( t ) + ( 1 - a ) x b ( t )

~ ( t ) = a t a ( t ) + ( 1 - & ) i b ( t )

a E [0, 11 t E [in, tn+']

where

x a ( t ) = 6(t)x:+' + ( 1 - b(t) )x:

x b ( t ) = 6(t)x;+' + ( 1 - 6(t))xF

and 6 ( t ) is a real function that satisfies

a( tn ) = 0; 6(tn+') = 1

Clearly, the integrand of ILab1 is linear in 6. Therefore, I[,bl can be exactly computed using the midpoint rule, provided that Eq. (27) hold - that is

(25)

which in view of Eq. (25) can also be written as

In summary, the GCL derived herein shows that for two-dimensional problems, the integrand of

n+l Fi(w,x,k) dt in Eq. (12) must be evaluated

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(see Fig. 3): x r l

Figure 3: Parametrization of a facet in a three- x b ( t ) = 6(t)x:+' + (1 - 6 ( t ) ) x ;

dimensional space and 6 ( t ) is given in (25). Substituting the above parametrization in (32) we obtain

at the midpoint configuration, and that this integral must be computed as follows: I[a be]

I k = n for explicit schemes I

k = n + f for implicit schemes (31)

w"+* - - wn+wn+1 2

2.3.2. The three-dimensional case In a three- with

dimensional space, the boundary of each cell is polyg- xac = xu - xc; xbc = xb - x, onal and can be decomposed into a set of non overlaping triangular facets. Similarly to the two- A x , = x:+' - 2:; A x b = x:+' - x r ; (36) dimensional case, let I[abcl denote the flux crossing the facet [abc] AX, = - x:A

tn+l Noting that

b c = 1% Lbc1 1.Z d d t (32 ) xac A Xbc =

Let x,, xb and x, denote the instantaneous positions of three connected vertices a, b and c. The position of (Jx:Z1 + ( 1 - 6 ) ~ : ~ ) A (62:;' + ( 1 - J)x1t,)

any point on the facet can be parametrized as follows (37)

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is a quadratic function of 6, the integrand of I[abc] is clearly quadratic in 6 and therefore can be exactly computed using a 2-point integration rule, provided that Eq. (30) is used to compute x.

Hence, the proper met hod n+l

for evaluating St, Fi(w, x , x) dt that respects the GCL (21) in the three-dimensional case is

1 1 m l = n + l - 3

1 1 m2 = n + ~ + z

k l = n for explicit schemes

k2 = n for explicit schemes

k l = m l for implicit schemes

k2 = m2 for implicit schemes

Expanding 'I* , we obtain

'I*

which shows that the proposed GCL (20) recovers the same results as the averaged-normals method proposed by NKonga and ~ u i l l a r d ' ~ for the finite volume discretization of flow equations with moving meshes. i

L

3. The stabilized finite element method with a space-time formulation

3.1. Semi-discretization

Time-integration in space-time finite element methods is derived in a different manner than what has been presented so far. Space-time finite element methods contain the time-integration formula in the chosen shape functions. These methods are basically weighted residual formulations that perform an in- tegration in space and time of the product of the

2*4* the Euler equations and an appropriate weighting func-

tion. Stabilization is usually required for the spa- tial approximation1'. In this paper, we focus on the

In ref. 13, the convected flux accross the facet I[abc] stabilized Least-SquareIGalerkin method and time- is computed using discontinuous shape functions.

Let 0 = to < t1 < ... < tN = T be a partition of I[.bc] = $(Axa + AX, + AX,) .q the time interval I = 10, T[, and In be the subinterval

(39) Itn, tn+' [. A space-time slab is defined in In x R ~ ,

7 = $(x:~ A x:~ + +:fl A xn+l bc where d designates the spatial dimension, as follows:

while the evaluation of Eq. (35) using the two-point with boundary rule gives

Pn = {(t, r ( t ) ) I t E I n )

I[abc] = $(AX. + AX^ + AxC).'I* (40) For each space-time slab, the spatial domain is sub-

q* = $(xzl A xg l + x z 2 A x z 2 ) divided into n,l elements (t), e = 1, . . . , n,l (see Fig. 4). The following notational convention is

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it follows that W = W* is always a solution of Eq. (42). Hence, a space-time stabilized finite element method always satisfies the GCL.

4. The stabilized finite element method with an ALE formulation

4.1. Semi-discretization

The stabilized finite element method with an ALE formulation can be derived by multiplying Eq. (4) by a weighting function vh (0, integrating over R(O), and adding a consistent stabilization term s ( v h , W) to obtain

Figure 4: Space-time slabs +s(vh, W) = 0

For example, S ( v h , W ) can be selected as *

adopted W(t;)= lirn W ( t n + r ) S(Vh, W) =

c-+of

Given some finite element spaces S," and v:, the c l e i t , ( -&G)f l i , 83;. d W + vx+(w))dQx (45)

space-time (discontinuous) Least-Square/Galerkin e m

method for solving the Euler flow equations goes as follows: Consistency requires that S vanishes when W is solu-

Find wh E ~ , h such that for all vh E V: tion of the Euler equations. Integrating by parts Eq. (44) and exploiting = 0 leads to

k=3 where L = -& + C %&-, and r is a stabilization

k = l parameter .

3.2. The geometric conservation law

Provided that the spatial integration scheme can compute exactly the following quantities

Integrating the above equation between tn and tn+' yields

4.2. The geometric conservation law

Substituting a constant field W = W* in Eq. (47)

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leads to

At this point, it is essential to assume that the con- sistency of S is preserved in its discrete counterpart (at least for a uniform field), and therefore the last term in the above equation is identically zero. From Eq. (45), it can be observed that the least-square term identifies pointwise with zero, and hence the as- sumption is satisfied independently of the integration rule. One can also reasonably assume that the first and second terms of the above equation can be com- puted exactly, and that the evaluation of any term of the form

L V f h di2, = 0 (49)

where pi are constants, can also be carried out ex- actly. Indeed, the latter condition is desirable not only for ALE computations, but also for flow compu- tat ions using fixed meshes. Violating this condition will introduce artificial fluxes throughout the mesh. Therefore, this condition is the finite element form of the Surface Conservation Law introduced by Zhang et al.14 For example, if the weighting functions v h are linear polynomials over each element, V? is constant over each element and a single point integration rule will yield an exact integration formula, provided that the area/volume of the element is computed exactly.

Consequently, provided the SCL is satisfied, and for weighting functions that are zero on the boundary, it follows that

P

Hence, Eq. (48) can be rewritten as

and can be simplified to

Eq. (52) establishes the geometric conservation law for the stabilized finite element method with an ALE formulation.

4.3. Implications of the GCL

In order to find the appropriate formula for inte- grating exactly the last term of the above GCL, we proceed as follows. First, we introduce the function

and note that this function can also be written as

From the differentiation of Eqs. (53,59 it follows that

Hence, the appropriate formula for integrating ex- actly the last term in Eq. (52) and satisfy- ing the GCL is the one which computes exactly

4.3.1. The two-dimensional case Let Nk be some arbitrary mapping functions between the cur- rent and reference configurations. We have

where the subscripts 1 and 2 designate the two dif- ferent coordinates, and the subscripts k refer to the nodal vertices of the element. The jacobian J of the above transformation is given by

ac, (52)

aNk J = ldet (2 ) 1 = I d - x ) (51)

at2 at2 a<i

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where summation is assumed over repeated indices, Following the same reasoning as in the two- -

and x k , are given by dimensional case, the following conclusions can be made:

x k , ( T ) = ( b ( ~ ) + ; , + ~ + (1 - J ( T ) ) x ; , ) G ( T ) is cubic in 6 ( T ) , - $ G ( ~ ( T ) ) is quadratic

for k = l..k,,,; i=1,2 ( 5 8 ) in 6 , and therefore the GCL condition will be

satisfied if the two-point rule is used for the in-

Here, 6 ( t ) satisfies the conditions given in Eq. ( 25 ) . tegration of the last term in Ey. ( 5 2 ) .

This form shows that the matrix involved in the com- putation of J is a linear function of 6 , and therefore 4.3.3. rnterrration formulae J is a quadratic function of 6 that can be written as As discussed above, the integrand of the last term of

the geometric conservation law ( 52 ) can be linear or J(T, = Jo (t) + 6 J 1 (1) + 6 2 J ~ ( E ) ( 5 9 ) quadratic. For a linear integrand, the midpoint rule

The function G can now be rewritten as will perform an exact integration. For a quadratic integrand, the two-point rule must be employed. In all cases, Eq. (55) holds only if x is computed in a manner that is compatible with the deformation of

(Jo(E) + a J i ( E ) + a2 J ~ ( E ) ) V ~ ( E ) dE R(6) - that is, if it is obtained by derivation of Eq. e=l ~ ( 0 ) ( 58 ) . Recalling that we are interested in computing

(60) \ I

Therefore, the following conclusions can be made:

G(T) is quadratic in S ( T ) , and since

and making the change of variable depicted in Eq. ( 58 ) and that implies ii = 6(x:+' - X I ) , we obtain

1' ' & G ( ~ ( T ) ) is linear in 6 and hence, the GCL con- dition will be satisfied if the midpoint rule is used for the integration of the last term in Eq. ( 52 ) .

This in turn implies that the mesh velocity x must be computed as follows:

4.3.2. The three-dimensional case Similarly to the two-dimensional case, the mapping between p + l , Xn x = a reference and current element configuration can be At ( 6 6 ) written as

k=l kmaz

and its jacobian J is given by

In summary, the following formulae apply: two-dimensional flow problems:

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k = n for explicit schemes

k = n + ? for implicit schemes

(67 0 Three-dimensional flow problems:

H(W, X, x) = l(x) v j q ( w y w ) , q x ) ) nn,

1 1 ml = n + ~ - - 2 f i

1 m2 = n + i + - 2 f i

kl = n for explicit schemes

k2 = n for explicit. schemes

I k1 = ml for implicit schemes

k2 = m2 for implicit schemes

dition is the constraint it imposes on the mesh ve- locity computation, independently of the integration formula for the flow equations

This formula is intuitive and has been "naturally" used by several investigators independently from any geometric conservation law (see, for example, 1). However, when sophisticated time integrators are used for the structure and/or the mesh equations, - neither the computed mesh velocities i n + $ nor the computed structural velocites on the fluid/structure interface are guaranteed to obey kn+$ = - at . In that case, satisfying the GCL requires -

using the mesh velocity in++ computed by the time integrator, only for evaluating xn+' .

,, in the using the mesh velocity kn+$ = Xn+l-Xn

evaluation of the fluid fluxes. I

This means that it is not always possible to respect the continuity of both the displacement and velocity fields on the fluid structure boundary as prescribed by Eqs. (2) without violating the GCL. For example, if the displacement continuity condition x(t) = q(t) is enforced at the fluid/structure interface TF/S, - and that is usually the case - respecting the GCL implies computing a mesh velocity field on rFIs that is equal to

(70) In that case, satisfying also the velocity continuity condition x(t) = q(t) on rFIS requires that

which is not enforced by all structural time- integrators. Therefore, it is not always possible to satisfy the continuity between both the displacement and the velocity of the structure, and those of the fluid mesh at the fluid/structure interface, without violating the GCL.

Unfortunately, a discontinuity between the veloc- ity of the structure and that of the fluid mesh at

5. Impact of the GCL on the temporal solution the fluid/structure interface can perturb the energy of coupled fluid Istruct ure problems exchange between the fluid and the structure. Luck-

ily, when the implicit midpoint rule is used for ad- vancing the structure, and the displacement condi-

The most remarquable implication of the GCL con- tion r ( t ) = q(t) is enforced on rF/S using a staggered

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algorithmla, both continuity equations (2) can be en- forced without violating the GCL. The proof goes as follows.

Given some initial conditions q0 and qO, suppose that the mesh motion is initialized such that the fol- lowing holds on the fluid/structure interface

3) using the pressure computed from w n + i , corn- pute qn+' and G"+' using the midpoint rule

Defining xn as

Also suppose that a t each time-station t", the continuity of the velocity field is enforced on the fluidlstructure boundary

If the midpoint rule is used for time-integrating the structural equations of motion, and the dynamic fluid mesh is updated consistently with the GCL as in Eq. (69), it can be proved by induction that

and substituting Eq. (78) into Eq. (79) leads to

which in view of Eqs. (74, 73) yields

xn = qn ~ F I S

- (81)

and demonstrates that the continuity of the displace- ment field is also enforced on rF/S without violating the GCL.

Indeed, the above relation holds at n = 0 (see Eq. (72)). Assuming it holds at n , it follows that

Since the midpoint rule algorithm applied to the structural equations implies

it follows that

Eqs.

1)

2)

which completes the proof by induction of Eq. (74). Now, a staggered algorithm for solving the coupled

(1) can be described as follows:

6. Application to an aeroelastic computation I *

In order to highlight the impact of the GCL on cou- pled aeroelastic computations, we consider the sim- ulation of the two-dimensional transient aeroelastic response of a flexible panel in a transonic regime.

For a two-dimensional simulation, the panel is rep- resented by its cross section that is assumed to have a unit length L = 1, a uniform thickness h = x L, and to be clamped at both ends. This rectangu- lar cross section is discretized into 300 x 3 plane strain 4-node elements with perfect aspect ratio to avoid mesh locking. The two-dimensional flow do- main around the panel is discretized into 2880 trian- gles and 1504 vertices. The free stream Mach number is set to M, = 0.8, and a slip condition is imposed at the fluidlstructure boundary. Because the fluid and structural meshes are not compatible at their in- terfaces (Figure 5) the "Matcher" software15 is used to transfer the pressure load to the structure, and to transmit the structural' deformations at the surfaces of the panel to the fluid.

Initially, a steady-state flow is comput,ed around the panel at M, = 0.8. Next, this flow is per-

using the mesh position xn-f , and the mesh ve- turbed via an initial displacement of the panel that locity in that matches the structural velocity qn is proportional to its second fundamental mode, and on r F / ~ , update the mesh coordinates as follows: the subsequent panel motion and flow evolution are

computed using one of the staggered explicit/implicit xn++ = ,p-f + at;" (78) fluidlstructure finite volume/finite element solution

procedures described in ref. 12. using xn-f , xn+f and in, update the fluid st,ate TWO computed histories of the lift using the same

vector *n+$ in a manner that satisfies the GCL time-step are reported in Fig. for the case where

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the GCL is violated by updating the mesh velocity field at the fluid/structure interface via a higher-order scheme than that given in Eq. (30), and in Fig. for the case where the GCL is respected. For this exam- ple, violating the GCL is clearly shown to introduce undesirable spurious oscillations in the lift prediction.

Figure 5: Structure and fluid discretizations

Figure 6: Flat panel in a transonic regime: violating the GCL

Figure 7: Flat panel in a transonic regime: satisfying the GCL

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7. Closure

In this paper, we have considered the solution of flow problems with moving boundaries using de- formable dynamic meshes. We have presented a uni- fied theory for deriving Geometric Conservation Laws (GCLs) for these problems when either an arbitrary Lagrangian-Eulerian finite volume or stabilized fi- nite element method, or a space-time stabilized fi- nite element formulation is employed for spatial dis- cretization. We have shown that the space-time dis- cretization method always leads to numerical solution schemes that inherently satisfy the GCL. We have also discussed the implications of the GCL on the so- lution of coupled fluid/structure problems. Most im- portantly, we have addressed the issue of continuity between the structural and fluid mesh motions at the fluid/structure interface, and shown how to design a solution method that preserves both displacement and velocity continuity without violating the GCL at this coupling interface. We have illustrated these concepts and highlighted their importance with the simulation of the aeroelastic response of a flat panel in a transonic regime, where we have shown that spu- rious and potentially unstable oscillations can occur when the GCL is violated.

Acknowledgments

The authors acknowledge the support by the Na- tional Science Foundation under Grant ASC-9217394.

References

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