STABILIZED FINITE ELEMENT METHOD WITH AN ALESTRATEGY TO SOLVE MOVING BOUNDARIES PROBLEMS

M. Viale and N. NigroUniversidad Nacional de RosarioPelegrini 250, (2000) Rosario, Argentina

nnigro@intec.unl.edu.ar

Abstract. Fluid flow problems with moving boundaries are one of the challenge problems to be solved in thisdecade. Our target is oriented to develop a finite element code to simulate the fluid flow inside the cylinder of aninternal combustion engine in order to understand how this flow field is related with the combustion, the heat releaseand the intake and exhaust charging process. In this paper we only present some preliminary results using the ALEtechnique coupled with a stabilized finite element method applied to compressible fluid flow problems with movingboundaries.

Keywords. Fluid mechanics Computational fluid dynamics, moving boundaries, in-cylinder flows

1. Introduction

Fluid flow problems with moving boundaries are one of the challenge problems to be solved in this decade.They are inherently unsteady problems where the domain is changing according to a user-defined law commonlyassociated with a prefixed boundary motion or in a coupled way with the flow itself. The former case is simplerthan the later, commonly named fluid-structure interaction, because of many reasons. One of them is associatedwith the grid generation. While in the former the mesh may be produced in advance with no enough care aboutthe evolution of the flow field in the later the boundaries move in an unpredictable way and the grid generationshould be solved at flow field computation time. Other important subject is related to the different time scalesthat may characterize the behavior of fluid and solid materials. Flow with moving boundaries can be encounteredin many practical situations. For example flow motion around a ship with a changing free surface has attracteda great deal of research interest in recent years. In this case the free surface is an unknown moving boundaryand the ship may be treated as a rigid body. The landing of an aircraft with the air flowing around the movingcomposed flaps is another example of great interest. Other field with a rapid development is that associatedwith the simulation of the complex haemodynamics in order to solve the blood flow in moving arteries.

Recently, with the enormous development of computer capabilities, the simulation of in-cylinder flow insidean internal combustion engine is feasible with the target in lower emissions and higher performances. In thiscase the combustion chamber is a variable volume vessel with moving boundaries represented by the piston andthe intake and exhaust valves. This project is oriented to simulate the in-cylinder flow in order to understandhow this flow field is related with the combustion, the heat release and the intake and exhaust charging process.

The numerical solution of this problem was originally performed via a finite volume method using an ALE(Arbitrary Lagrangian Eulerian) strategy. The list of references is huge and only some of the most cited ref-erences are included here (I. Demirdzic (1990),J. Trepanier and Camarero (1991),J. Trepanier and Camarero(1993),C. Farhat and Stern (1997)). One of the main difficulties of the finite volume formulation is the satis-faction of the geometric conservation law (GCL). Several different proposals were published in order to bypassthis difficulty. On the other hand the finite element formulation has the ability of being completely consistentwith the ALE strategy satisfying the GCL with no special care (I. Lomtev (1999)). Therefore a compressibleNavier-Stokes code based on finite elements with an ALE strategy is adopted (J. Donea (1982),T. Nomura(1992),T. Sheu (1999),I. Lomtev (1999),W. Tworzydlo (1992)). An SUPG formulation for the spatial dis-cretization (Brooks and Hughes (1982); Hughes and Tezduyar (1984)) and a finite difference scheme for thetime evolution is chosen. This work plan is divided in three steps, the first considers the development of anscalar transport element with an ALE strategy, the second adds the inviscid flow equations in a moving domainformulation and the third step considers the original target, the compressible Navier-Stokes solved via a finiteelement and ALE code.

In the first stage the model originally presented by Kershaw and coworkers (Kershaw and Milovich (1998))based on the usage of a transformation to a reference grid selected as the initial mesh was used. Even thoughthis strategy is valid it presents several disadvantages specially when vector fields having tensor diffusion termsare involved. In the scalar case in an isotropic medium this inconvenience is not relevant because the diffusioncoefficient is transformed in a diffusion tensor. In Euler equations this problem is not present because thereis no diffusion term. However in the Navier-Stokes equations the stress tensor playing the role of diffusionintroduces Christoffel symbols when the transformation to the reference grid is performed. This fact increasesthe computational cost in a drastic way. To avoid this situation it is possible to use a standard ALE strategy

where the solution is computed over a moving grid in a lagrangian way but with an arbitrary velocity. In generalthis velocity is different from the fluid one in order to keep the mesh in a good shape. For the solution of ascalar field a model presented by Sheu et.al. (T. Sheu (1999)) was adopted and numerical results over a testcase is presented. This development was extended to the inviscid flow model (Euler equations) and some basictests were solved to demonstrate mainly the geometric conservation law. In future works it is planned to applythis development to in-cylinder flow in an internal combustion engine .

2. The mathematical model

Our target is the simulation of compressible viscous flow inside the cylinder of an internal combustion engine.Then, the mathematical model is defined by compressible Navier-Stokes equations in a variable domain. In thefollowing sections we present first the mathematical formulation in a fixed domain and then the ALE strategyto consider the boundary movement.

2.1. Compressible viscous flow in a fixed domain

Compressible Navier-Stokes equations are one of the most used models to predict the behaviour of aninternal gas flow. This set of partial differential equations may be seen as an advective-diffusive system ofequations like

U

t+F aixi

=F dixi

+ S (1)

where the first term is the rate of change of the state variable U expressed in conservative form followed bythe divergence of the advective and diffusive fluxes and a source term. The conservation variables are definedas:

U = [, u1, u2, u3, e] (2)

where is the density, e is the total energy composed by the internal energy i plus the kinetic energy,

e = i+12

i

u2i , i = CvT

and [u1, u2, u3] represents the linear momentum vector. The advective fluxes and the diffusive fluxes aredefined as:

F ai =

ui

u1ui + pi1u2ui + pi2u3ui + pi3

hui

, F di =

0i1i2i3

qi + ikuk

(4)being h = e+ p/ the enthalpy, q = T the heat flux according to the Fourier law and ij an element

of the deviatoric stress tensor. The source term depends in particular on the applications. In general masssource term is not included, the momentum sources are external forces applied to the system and finally, energysources are the work of these external forces plus some body heat addition or release between the system andthe environment.

S = [0; b1; b2; b3; b u +Q] (5)2.2. ALE strategy

In this section we only present a brief summary of the main expressions about the original work of ArbitraryLagrangian Eulerian (ALE) in a finite element context for the inviscid case written by Donea and coworkers. Formore details about this formulation we refer to the original work (J. Donea (1982)). It is well known that thereare two viewpoints mostly used in the description of the flow motion equations, one is called the Lagrangianapproach where the mesh moves with the fluid and the other is the Eulerian approach in which the mesh isfixed and the fluid moves around it. The arbitrary Lagrangian-Eulerian description is a generalization of thesetwo approaches where the computational grid moves with an arbitrary velocity w in the laboratory system. Ifw = 0 we recover the Eulerian approach and if w = u we recover the Lagrangian one considering u as the fluidflow velocity. A very simple physical scenario is given by figure 1.

Figure 1: Different approaches for the description of the flow motion

The boat number 1 is attached to the ground by an anchor and therefore its velocity is null and the observerinside this boat has an Eulerian description of the flow. The boat number 2 is free to follow the river streamand its observer has a Lagrangian description of the flow motion. Finally there is a third boat which is poweredby an engine with a velocity w non aligned with the river stream, therefore its observer has an ALE descriptionof the flow.

!! "

$#%&

')(+* ,-(./* 0 (.

13242

,50.(67

8

9;:=

x

?A@B

Figure 2: Definition of different reference frames

In order to gain some insight about this general description figure 2 shows the three reference frames andtheir corresponding coordinates. A particular point P of a portion of fluid with a position a relative to thematerial domain is plot for both the initial time t = 0 and for some time after. By definition these materialcoordinates do not change in time. It is also possible to follow the motion of the same point P in the spatialdomain by its position x and by an arbitrary motion of the reference domain where its position may be definedby .

Initially the material domain (Lagrangian coordinates), the spatial domain (Eulerian coordinates) and thereference domain (mixed coordinates) coincide among them. So, x = a and the position of any point in thereference domain relative to the original configuration is expressed by = 0.

Some time after x 6= a and 6= 0. In the ALE description a particle is identified by its material coordinatesa in the initial configuration but this process of identification is indirect and takes place through the mixedposition vector . There is a link between the material coordinates and the mixed coordinates in the form:

i = fi(ai, t) (6)

ALE description is similar to a mapping between the initial configuration of the continuum in the currentconfiguration of the arbitrary reference frame, whose jacobian determinant is computed by

J = | iaj| (7)

Other two important properties of this jacobian are expressed by:

dV = J(a, t)dV0 (8)

J

t= J 5 w (9)

2.2.1. Kinematics in the ALE description

Following J. Donea (1982) we describe a physical property associated with the flow of a continuum mediaby g(i, t). By (6) we have

g(i, t) = g[fi(ai, t)] = g(ai, t) (10)

The time derivative of (10) keeping the material coordinates constant is written as:

g(a, t)t

a

=g(, t)t

+g(, t)i

it

(11)

Noting that wi = it is the grid velocity and using the following identity

(gw) = g w + w g (12)and (9) we arrive to

J (gw) = Jg w + Jw g= g Jt + Jw g

(13)

Therefore (11) may be rewritten as:

Lagrangian g(a, t)t

a

=

Eulerian g(, t)t

+g(, t)i

it

Jg(a, t)t

= J [g(, t)t

+g(, t)i

it

]

Jg(a, t)t

+ g(a, t)J

t= g(a, t)

J

t+ J [

g(, t)t

+g(, t)i

it

]

(Jg)t

= J [1JgJ

t+g

t+ w g]

(Jg)t

= J [g w + gt

+ w g](14)

(Jg)t

= J [g

t+ (gw)] (15)

Expression (15) appears to be a fundamental relationship which enables us to transform any law expressedin spatial (Eulerian) variables into an equivalent law expressed in mixed variables. The gt in (15) is a spatialderivative, i.e., taken with the point fixed in position.

2.2.2. Conservation laws in the ALE description

We rewrite our original set of equations (1) in expanded form as:

t+

xj(uj) = 0

uit

+

xj(uiuj) = bi p

xje

t+

xj(euj) = ujbj

xj(puj)

(16)

and using (15) they may be transformed to the following set of equations:

t(J) = J xj ((wj uj))

t(uiJ) = J xj (ui(wj uj)) +J(bi

p

xj)

t(eJ) = J xj (e(wj uj)) +J(ujbj

xj(puj))

(17)

The above set of equations (17) may be rewritten again in a compact form as:

U

t+F aixi

w 5U = Fdi

xi+ S (18)

similar to (1) but now for the ALE description. In the algebraic manipulations we have dropped the viscousfluxes because they do not change in the ALE description relative to those in fixed grids.

2.2.3. Weak variational form of the ALE conservation equations

We follow the paper of J. Donea (1982) but instead of assuming constant density we work with compressibleflow formulation.

Mass equationAfter some algebraic manipulations using (9) and (15) and the first equation in (17) we arrive to:

t= (w u) u= (u) + w

(19)

Even though both expressions are equivalent and useful we have chosen the last one. Multiplying thisequation by an arbitrary weighting function and integrating over a control volume V (t) we obtain thefollowing weak form of the mass conservation equation:

V (t)

tdV =

V (t) (u)dV +

V (t)w dV

(20)

Energy equationIn the same way for the energy conservation equation we have its weak variational form written as:

V (t)

t(e)dV +

V (t)

((e+ p)u)dV V (t)

w (e)dV = V (t)

b u

(21)

Momentum equationsFinally the weak variational formulation of the momentum conservation equation is written as:

V (t)

t(u)dV +

V (t)

(u u + pI)dV V (t)

w (u)dV = V (t)

b

(22)

where means for tensor product and I represent an identity second order tensor.

2.2.4. Some remarks

We have to remark that this formulation satisfies naturally the geometric conservation law (GCL) as itwas demonstrated by I. Lomtev (1999).

Another remark is due to the spatial stabilization technique. Equation (18) restricted to inviscid flowsmay be written in a quasi-linear form as:

U

t+F aixi

w 5U = 0U

t+ [Ai wiI]5 U = 0

(23)

Therefore, the spatial stabilization (upwind) scheme to be used may be computed from the modifiedadvective jacobian instead of the original ones. In the next section we present some details about thenumerical method employed with special emphasis on the spatial discretization scheme.

3. The numerical method

In order to get a numerical solution of the continuum problem presented in the above section we have todiscretize the problem using a particular numerical method. In this work we have used a SUPG technique thatis very popular in the context of finite element method and is one of the most referenced in the CFD area. Thereare a lot of references concerning with this technique that may be included here but for simplicity reasons weforward the readers towards two of the most cited references (Brooks and Hughes (1982),Hughes and Tezduyar(1984)).

This technique is based on a Petrov-Galerkin weighted residual method which allows to use test functionsthat can be different from the interpolation ones and not necessarily continuous. This method introduces thenumerical dissipation needed to stabilize the system in advection-dominated problems, keeping the consistencywith the continuum problem. For each node a there is an interpolation function Na (hat type in 1D, bilinearin 2D, and multilinear in general) and a test function Wa = Na + Pa, where Pa is called the perturbationfunction. The standard Galerkin method is recovered when we impose Pa 0. The Pa (and, of course, Wa)are not necessarily continuous through the inter-element boundaries.

3.1. Perturbation function

Following the procedure employed in the fixed grid Euler equations and adopting for the ALE descriptionthat the advective jacobians are transformed according to (23) we define the SUPG perturbation function as:

Pa = [AwI] Na =

h

2||AwI||1

(24)

In this work we have used an extra way for computing the intrinsic time scale matrix that is less expensivethan (24) and works reasonably good. This procedure is due to G. Le Beau and Tezduyar (1993) and may bewritten as:

= I

=h

2|max|1

max = |uw|+ c(25)

Rewriting the weak variational formulation presented above (20),(21) and (22) and after using the compactnotation with the SUPG method just described we arrive to:

V (t)

WaU

tdV +

V (t)

Wa F adV V (t)

Waw UdV =V (t)

WaF ddV +V (t)

WaSdV (26)

4. Validation tests

4.1. Test 1: Shock-reflection problem

This two dimensiona, inviscid, steady problem involves three flow regions separated by an oblique shockand its reflection from a wall. It is a standard benchmark problem (G. Le Beau and Tezduyar (1993), Shakib(1988)). The importance of such a test is to check the ability of our code to resolve flows involving shocks. Thecomputational domain is a rectangular region of dimensions 4.1 in the x direction and 1.0 in the y direction.The mesh consists of 60 20 rectangular elements. The boundary conditions are the following:

{[, u1, u2,M ] = [1, 1, 0, 2.9] left,[, u1, u2,M ] = [1.7, 0.9033,0.1746, 2.3781] top.

(27)

While figure (3) shows at left the isobars and at right the density at y = 0 25 where it may be viewed thethree different regions above mentioned.

Figure 3: Shock reflection problem. Results

4.2. Test 2: ALE for scalar advection-diffusion problems.

This problem proposed by T. Sheu (1999) consists of solving the standard scalar advection-diffusion equationwritten here as:

t + 2yx + xy y2(xx + yy) = f(x, y, t)f(x, y, t) = ety(2x3 2y3 3x2y + 4xy2 2x2 2y2 + 3xy) (28)

whose analytic solution is

(x, y, t) = etx(1 x)y2 (29)The domain is defined by x [0, 1] y [0.5, 1.5] and the bottom side is changing in time. To compute this

variation the grid velocity is specified by:

w = [0, pi/2 sin(2pi) cos(pit)( 1)] (30)

The following table shows the evolution of the error norm with time.

Time L2 norm Time L2 norm0.1 1.7094104 0.6 4.3216104

0.2 2.6002104 0.7 3.4826104

0.3 3.7011104 0.8 2.4646104

0.4 4.5049104 0.9 1.5422104

0.5 4.7182104 1.0 9.6967104

The results in the above table and in the figure (4) show a very good agreement with those reported byT. Sheu (1999).

Figure 4: Scalar advection-diffusion with ALE. Test

4.3. Test 3: Random grid

This test proposed by J. Trepanier and Camarero (1991). The test consists in verifying the geometricconservation law (GCL) and for that we move the mesh keeping the boundaries fixed. The initial solution isspatially uniform and should remain steadily frozen. This was verified by the simulation and here we only showsome meshes used in the computation, see figure 5. The solution is constant in time and uniform in space atmachine tolerance.

5. Conclusions and future trends

These are only some preliminary results about solving CFD problems involving variable domain. Wehave chosen ALE strategy instead of other methodologies like space-time finite element method because ofcomputational cost reasons. We have found our results very promising, specially in the geometric conservationlaw point of view. We plan to continue our work solving more challenge problems involving inviscid or viscousflow with our final target of simulating the heat and fluid flow inside a cylinder of an internal combustion engine.

6. References

Brooks, A. and Hughes, T., 1982. Streamline upwind petrov-galerkin formulation for convection dominatedflows with particular emphasis on the incompressible navier-stokes equations. Comp. Meth. Applied Mech.and Engineering, volume 32, pp. 199259.

0 2 4 6 8 101

0

1

2

3

4

5

6

0 2 4 6 8 101

0

1

2

3

4

5

6

0 2 4 6 8 101

0

1

2

3

4

5

6

0 2 4 6 8 101

0

1

2

3

4

5

6

Figure 5: Random grid. Meshes

C. Farhat, M. L. and Stern, P., 1997. High performance solution of three-dimensional nonlinear aeroelasticproblems via parallel partitioned algorithms: methodology and preliminary results. Advances in Engin-nering Software, volume 28, pp. 4361.

G. Le Beau, S. Ray, S. A. and Tezduyar, T., 1993. Supg finite element computation of compressible flowswith the entropy and conservation variables formulations. Comp. Meth. Applied Mech. and Engineering,volume 104, pp. 2742.

Hughes, T. and Tezduyar, T., 1984. Finite element methods for first order hyperbolic systems with particularemphasis on the compressible euler equations. Comp. Meth. Applied Mech. and Engineering, volume 45,pp. 217284.

I. Demirdzic, M. P., 1990. Finite volulme method for prediction of fluid flow in arbitrarily shaped domainswith moving boundaries. Int. Journal for Num. Meth. in Fluids, volume 10, pp. 771790.

I. Lomtev, R. Kirby, G. K., 1999. A discontinuous galerkin ale method for compressible viscous flows in movingboundaries. Journal of Comp. Physics, volume 155, pp. 128159.

J. Donea, S. Giuliani, J. H., 1982. An arbitrary lagrangian-eulerian finite element method for transient dynamicfluid-structure interactions. Comp. Meth. Applied Mech. and Engineering, volume 33, pp. 689723.

J. Trepanier, M. Reggio, H. Z. and Camarero, R., 1991. A finite volume method for solving the euler equationson arbitrary lagrangian-eulerian grids. Computer and Fluids, volume 20(4), pp. 399409.

J. Trepanier, M. Reggio, M. P. and Camarero, R., 1993. Unsteady euler solutions for arbitrarily moving bodiesand boundaries. AIAA Journal, volume 31(10), pp. 18691874.

Kershaw, Prasad, S. and Milovich, 1998. 3d unstructured mesh ale (arbitrary lagrangian-eulerian) hidrody-namics with the upwind discontinuos finite element method. Comp. Meth. Applied Mech. and Engineering,volume 158, pp. 81116.

Shakib, F., 1988. Finite element analysis of the compressible Euler and Navier-Stokes equations. Ph.D. thesis,Department of Mechanical Engineering, Stanford University.

T. Nomura, T. H., 1992. An arbitrary lagrangian-eulerian finite element method for interaction of fluid and arigid body. Comp. Meth. Applied Mech. and Engineering, volume 95, pp. 115138.

T. Sheu, H. C., 1999. A transient analysis of incompressible fluid flow in vessels with moving boundaries. Int.J. of Num. Meth. for Heat and Fluid Flow, volume 9(8), pp. 833846.

W. Tworzydlo, C. Huang, J. O., 1992. Adaptive implicit/explicit finite element methods. Comp. Meth.Applied Mech. and Engineering, volume 97, pp. 245288.