Fuel Tank Geometry Optimisation For Mechanical

Damping

J. Hall ∗ , T.C.S. Rendall† and C.B. Allen‡

University of Bristol, Department of Aerospace Engineering,

Queen’s Building, University Walk, Bristol, BS8 1TR, UK

A feasible sequential quadratic optimisation is applied to the problem of minimisingsloshing motion in a two-dimensional pivoting square tank. Results are presented for casesboth with and without a constraint on the initial potential energy of the system, and thisconstraint is seen to make a significant difference to the optimised geometry. The finaldesign reduces the total variation of the angular position by 16.4% while increasing thetank volume by 3.9% and reducing the initial potential energy by only 2.6%.

I. Background and Introduction

Fuel slosh has been shown to be important in the dynamics of a number of vehicle types; spacecraft,1,42,43

tanker lorries40 and tanker ships.26 For instance in flight SA-1, the first flight of the Saturn family of launchvehicles, a high degree of coupling was observed between fuel sloshing and vehicle roll velocity.19 Anexperimental and analytical study on the effect of fuel on flutter was carried out by Sewall41 and he notesthat the sequence of emptying of internal fuel tanks is important for a highest flutter onset speeds. Kim etal.26 show that the response amplitude operator of a ship depends on sea wave slope; this result is not seenin a linear analysis implying that the non-linear effects of sloshing are important in vehicle dynamics.

Tuned liquid dampers are tanks, partially full of liquid, whose purpose is to increase the damping ofa structure by using the free surface motion of the fluid to counter any excitation. These devices havefound application in reducing the vibration in tall towers such as the Hobart Tower in Tasmania and theGold Tower in Japan.25 A related class of devices are the anti-roll tanks that are sometimes employed onships to reduce roll;9,17 these devices are able to damp a range of frequencies and require no power andlittle maintenance. The effect of fuel slosh on aerodynamic performance has been studied previously usingcomputational techniques by a number of researchers16,45 who use simplified models of the fuel, such as amass spring damper system or a potential flow based hydroelastic model, rather than solve the Euler orNavier-Stokes equations. Banim et al.6 used smoothed particle hydrodynamics (SPH) to model wing fueltank sloshing but they used a forcing function on the wing based on gust loads rather than coupling thesolution to a flow solver. They solved for tank wall pressures and noted that the calculated values for fulltanks show good agreement with analytical results. This implies that the SPH scheme produces accurateforces on containing structures. The authors have performed previous work20 using SPH to model fuel sloshwhen coupled to external aerodynamics and structural dynamics. The effect of free fuel motion was shown tobe an increase in the flutter boundary though no baffles are considered.20,45 This suggests that the internalfuel in aeroplanes and other vehicles can be used to provide damping to oscillations in much the same wayas the anti-roll tanks on ships or tuned liquid dampers in buildings, however as weight is critical the fuelmust be used rather than carry extra fluid. The objective of this paper is to optimise tank geometry suchthat the interaction between tank and fluid produces maximum mechanical damping.

∗Ph.D Student, University of Bristol, Department of Aerospace Engineering, Queen’s Building, University Walk, Bristol,BS8 1TR, UK. Tel: +44 117 331 5634†Lecturer in Aerodynamics, University of Bristol, Department of Aerospace Engineering, Queen’s Building, University Walk,

Bristol, BS8 1TR, UK. Tel: +44 117 331 5639‡Professor of Computational Aerodynamics, University of Bristol, Department of Aerospace Engineering, Queen’s Building,

University Walk, Bristol, BS8 1TR, UK. Tel: +44 117 331 5539

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10th AIAA Multidisciplinary Design Optimization Conference

13-17 January 2014, National Harbor, Maryland

AIAA 2014-1339

AIAA SciTech

There has been a small amount of previous work in this area. Eswaran et al.14 used a volume of fluidscheme to numerically study the effect of baffles on the wall pressure and free surface displacement in a 3Dcubic tank subjected to sinusoidal accelerations while Hyun-Soo and Young-Shin23 used a genetic algorithmto optimise baffle configuration. The objective function to be maximised was a sloshing reduction coefficientwhich represents decrease in sloshing amplitude from the no baffle case. Craig and Kingsley12 optimisedbaffle configuration to minimise free surface motion, baffle mass and a case where both objectives have equalweighting. To the author’s knowledge the use of optimisation to maximise mechanical damping is unique tothis work. SPH has previously been used to model problems where fluid structure interactions are importantfor stability of the overall system, such as tuned liquid dampers and anti-roll tanks; Bulian et al.9 used SPHto simulate a tank with a single rotational degree of freedom and compare their numerical predations toexperimental data. Agreement between experiment and the SPH simulation is good. Iglesias et al.24 usedSPH to model ship anti-roll tanks.

The sloshing-structural coupled part of the computational model developed by the authors20 is used tooptimise tank geometry design for maximum damping as a prelude to baffle optimisation. This model usesa 2D structural model and SPH to simulate liquid flow. The coupled nature of the model allows all relevantforces to be simulated dynamically. The use of SPH in modelling the movement of fuel in the tank is ideal fortank optimisation as it is extremely flexible when dealing with complex geometries due to being a meshlessmethod. A SPH method can also capture non-linear effects such as wave breaking and tank roof impactsthat potential flow based methods45 cannot.

In section III the formulation of the numerical techniques that have been used to produce the model willbe discussed. The derivation of the SPH method will be outlined and the specific formulation used in thiswork will be presented.

II. Coupling Options

For time accurate fluid structure simulation it is essential to solve and synchronise the fluid and structuralequations every timestep, here this is performed by iteration using an implicit scheme. It is possible to solveboth the fluid flow and the structural problem together in one time step, with a monolithic scheme.8 Howeverthis is not practical for most problems, lacks flexibility and ignores the wealth of knowledge and experiencethat exists in the computational fluid dynamics (CFD) and computational structural dynamics (CSD) fields.It is more usual to solve each of the fluid and structural domains separately using well established techniquesfor the field in question and then couple the solutions by the exchange of forces and displacements, thisbeing commonly known as the partitioned method. In this work the partitioned scheme is used for the abovereasons and also the difficulty of envisaging a monolithic scheme where part of the solution is calculatedusing meshless methods and part using mesh based methods. The solutions can be coupled either stronglyor weakly, with strong coupling sub-iteration is commonly used to achieve convergence over a time step usingimplicit integrators.5 Weak coupling alternates between structural and fluid calculations without regard forsynchronisation. A partitioned approach requires the exchange of information between solvers and this isan important and well studied problem with a number of methods for exchanging information across thedifferent mesh geometries in each domain,13,15,39 these methods include nearest neighbour interpolation,radial basis functions and Gauss interpolation. This information exchange uses a principle of virtual workin order to conserve energy. Bendiksen7 adopts a different approach and uses a formulation that allows asingle mesh for both solid and fluid. Good results are achieved with this scheme and the energy conservationacross the fluid solid interface is particularly strong. However the method will lead to an overly detailedmesh near the solid boundary in order to couple to the fluid mesh. In this work the structural model is suchthat simple integration of forces around the boundary is sufficient for the transfer of information and thusthe most important consideration is at what point in time information is exchanged.

III. Formulation Of Fluid Model

Smoothed particle hydrodynamics (SPH) is a meshless, Lagrangian particle based scheme invented inde-pendently by Gingold and Monaghan18 and Lucy.30 The technique was originally intended to study problemsin astrophysics but has been used for fluids due to the simplicity of using meshless methods to model freesurface flows which would otherwise require complex techniques such as the volume of fluid (VOF) method21

or complex schemes to track the free surface and deform the mesh. There are a number of reasons to prefer

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SPH to VOF; the first is simplicity, a basic SPH code can be written with comparative ease and no specialtechniques are required to track free surfaces. The meshless nature of the method also removes the needto create a mesh which can prove challenging in complex geometries. As the SPH method is Lagrangian itavoids false diffusion errors that can occur in Eulerian methods such as VOF. VOF also requires the solutionof a differential equation that describes the evolution of the volume fraction in the mesh volumes which isunnecessary in SPH. However, SPH is not free of problems; the solution of truly incompressible flows was notinitially examined though is becoming more common. The procedure is more complex as free surfaces mustbe identified but it can give noise free pressure fields.28 Instead the fluid is usually assumed to be weaklycompressible. Also, depending on the boundary treatment it is possible that there will be a non physical freespace between the boundary and the fluid particles. In SPH it is possible for fluid particles to interpenetrateeach other though this is not usually a problem at low Reynolds numbers.

There are two approaches for producing SPH formulations of the governing equations. The first, moregenerally used, views SPH as a method of representing any given function by summation over a set of particles.Liu and Liu29 give details of the mathematics, but the process is to take a function that approximates theDirac delta function and convolve it with the function to be represented as in equation 1. The angle bracketsdenote the integral representation.

< f(x) >=

∫Ω

f(x′)W (x− x′, h)dx′ (1)

Here f(x) is a function to be represented by integration over a volume Ω, W (x − x′, h) is a smoothing

function that depends on the difference between the location of interest and the points over which theintegral is evaluated. h is a characteristic length of W and is known as the smoothing length. This integralrepresentation is then approximated by summation over N nearby particles, shown in equation 2. Thissummation is possible because W is usually chosen to be a compact function which is non-zero only for asmall fraction of the total domain. The radius over which W is non-zero is known as the support radius,thus N is the number of particles within the support radius.

< f(x) >≈N∑j=1

f(xj)W (x− xj , h)∆Vj (2)

If each particle is assigned a mass and a density then equation 2 can be written as;

< f(x) >≈N∑j=1

mj

ρjf(xj)W (x− xj , h) (3)

Derivatives can be approximated in the same way as any other function.

< ∇ · f(x) >=

∫Ω

∇ · f(x′)W (x− x′, h)dx′ (4)

However we may not know the derivative of f but we will know the derivative of the smoothing function sowe can make use of the following identity.

∇ · [f(x′)W (x− x′, h)] = ∇ · f(x′)W (x− x′, h) + f(x′) · ∇W (x− x′, h) (5)

Hence;

< ∇ · f(x) >=

∫Ω

∇ · [f(x′)W (x− x′, h)]− f(x′) · ∇W (x− x′, h)dx′ (6)

And by applying the divergence theorem the first volume integral becomes a surface integral over the problemdomain S.

< ∇ · f(x) >=

∫S

f(x′)W (x− x′, h) · dS−∫

Ω

f(x′) · ∇W (x− x′, h)dx′ (7)

Normally, though not always, the support domain is entirely within the problem domain such that the surfaceintegral is zero and the representation can be expressed entirely in terms of the volume integral. This is not

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true in the case where the support domain overlaps the boundary, either a free surface or a fixed boundary.The particle approximation can then be performed as with any other function and is shown in equation 8.

< ∇ · f(xi) >≈ −N∑j=1

mj

ρjf(xj) · ∇W (xi − xj , h) (8)

Note that in equation 8 the gradient is evaluated at the jth particle, as most smoothing functions aresymmetric equation 8 can equivalently be written as equation 9, where the subscript i denotes evaluating thegradient at the ith particle. The sign changes as for a symmetric kernel the gradient will be antisymmetric.The method for deriving the approximation to a vector gradient is very similar. With this information it ispossible to construct the governing equations of fluid flow in the SPH formulation as all derivatives can besimply expressed in SPH form.

< ∇ · f(xi) >≈N∑j=1

mj

ρjf(xj) · ∇iW (xi − xj , h) (9)

However equation 5 can be used in conjunction with other identities to produce alternative expressions forderivatives; two commonly used identities are given in equation 10 and 11.

∇ · f(x) =1

ρ[∇ · (ρf(x))− f(x) · ∇ρ] (10)

∇ · f(x) = ρ

[∇ ·(f(x)

ρ

)+f(x)

ρ2· ∇ρ

](11)

Evaluating the derivatives on the right hand side separately gives the corresponding SPH equations.

∇ · f(xi) =1

ρi

N∑j=1

mj [f(xj)− f(xi)] · ∇iW (xi − xj , h)

(12)

∇ · f(xi) = ρi

N∑j=1

mj

[f(xj)

ρ2j

+f(xi)

ρ2i

]· ∇iW (xi − xj , h)

(13)

This technique for deriving an SPH formulation is unsatisfactory. There are multiple identities that couldbe used so this derivation leads to multiple formulations for the same derivative with no clear method forevaluating which is the most suitable. This dilemma can be overcome by a derivation that proceeds usingthe Lagrangian.38

At present the SPH formulation used is:

Wij =7

4πh2

(1− |xi − xj |

2h

)4(2|xi − xj |

h+ 1

),|xi − xj |

h≤ 2 (14)

DuiDt

= −N∑j=1

mj

[Pjρ2j

+Piρii

]∇iWij +

N∑j=1

mjνρi + ρjρiρj

xij · ∇iWij

|xij |2 + 0.001h2uij + g (15)

DρiDt

=

N∑j=1

mj(ui − uj) · ∇iWij (16)

P = B

((ρ

ρ0

)γ− 1

)(17)

Equation 14 is the quintic kernel suggested by Wendland.44 The equation of state, equation 17, is commonlyused in SPH to remove the need to solve a pressure Poisson equation. An equation of this form was firstsuggested by Cole11 for water, Monaghan32 then suggested setting B such that the speed of sound in the fluidis an order of magnitude greater than the fastest particle velocity. γ is set to 7 following Cole’s11 work. This

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is known as the weakly compressible method, which is well able to predict free surface motion27,32 thoughsolutions can have a noisy pressure field. The second term in equation 15 is a viscosity term proposed byMorris et al.36 which helps reduce the frequency with which particles non-physically penetrate each other.The density of a particle is evolved in time rather than calculated explicitly by summation to remove theproblem of the density reducing near free surfaces as a consequence of the lower number of near particles.

The question of how to deal with boundary conditions in SPH is complex and a number of schemes exist.Monaghan31 suggests three ways of enforcing a rigid boundary:

1. Forces of length scale equal to the smoothing length.

2. Reflecting particles that cross the boundary without energy loss.

3. Fixing particles on the boundary to provide a repulsive force.

In this paper the third method has been used with the modification that the solid boundary particles arefixed only relative to the others in that rigid body. The force and moment on a rigid body is thereforecalculated by summing the contribution from each of the solid boundary particles.

IV. Numerical Integration

The Newmark-Beta method22 below with γ = 1/2, β = 1/4 is used to time integrate the SPH code byapplying it to equations 15 and 16. This selection of constants gives the constant average acceleration inte-gration and is unconditionally stable. The Newmark-Beta method is implicit and synchronisation betweenthe fluid and structure is achieved by iterating until a convergence criteria has been met.

ut+∆t = ut + (1− γ)∆tut + γ∆tut+∆t (18)

ut+∆t = ut + ∆tut +1− 2β

2∆t2ut + ∆t2βut+∆t (19)

The particle acceleration and density rate of change from equations 15 and 16 are substituted intoequations 18 and 19 to perform the time integration. Fixed point iteration is carried out to converge to asolution.

V. Shape Optimization

The Group at Bristol have experience of optimization development and application, having developed amodular, generic optimization framework, that is flow-solver and mesh type independent, and applicable toany aerodynamic problem. This approach has been applied to aerodynamic optimization of two-dimensionalaerofoil problems,33 three-dimensional wing problems,34,35 and rotor blade problems.2–4

V.A. Optimization Scheme

The coupled fluid-structure system is linked here to a non-linear Feasible Sequential Quadratic Programming(FSQP)37,46,47 optimisation algorithm, and this choice is important, since it allows strict enforcement offluid and geometric constraints. Unconstrained optimizations can incorporate constraints by using a penaltyfunction for design parameters that take the design near or beyond the constraint boundary, but thesemethods can be inefficient and the method here solves the Kuhn-Tucker equations. The method requires thegradients with respect to each parameter, and has been made independent from the flow-solver by obtainingsensitivity information by finite-differences using a symmetric central-difference stencil, i.e. one positive andone negative perturbation are considered for each design variable. The code is not linked to a flow-solver,but has been written to control its own calls to the solver used, and reads the output back to compute therequired gradients. A sensitivity analysis has been performed to select perturbation size.

Since each gradient is computed by separate calls to an external flow-solver, the algorithm has beenparallelized such that one parameter (or a group of parameters) is assigned to each available CPU, allow-ing parallel evaluation of the required sensitivities. Function evaluations and optimiser updates occur onthe master process, and each CPU controls the geometry perturbation corresponding to a different designvariable, and calls the flow solver. Flow-solver results are then returned to the master for optimiser updates.

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VI. Optimisation Problem

A model problem with a small number of design variables is chosen in order to demonstrate the feasibilityof coupling an SPH solver to a non-linear FSQP optimisation scheme in order to increase liquid damping.The geometry chosen is shown in figure 1. This is a square tank with total area of 0.25m2 pivoted abouta point 0.115m from the center of the top of the tank in the direction perpendicular to the tank top. Theinitial angle of the tank to the vertical is 20 and the tank is a quarter full, figure 2 shows the equilibriumfluid configuration. This initial condition is selected to give representative non-linear flow features such asbreaking waves. No damping is used at the pivot so the forces on the tank are entirely due to gravity andthe internal fluid forces. The tank has the following inertial properties:

M 3.343 kg

Iθ 0.529 kgm2

rcg 0.288 mWhere rcg is is distance of the centre of mass from the pivot along a line that passes through the pivot

and the tank centre. The Newmark-Beta scheme given in section IV is used to integrate the governingstructural equations with fluid and structural solver coupling performed through iteration with each solverusing the implicit Newmark-Beta method. An iterative method like this yields strong coupling for increasedaccuracy. The SPH solver is advanced forward to the next time step where the fluid forces on the structureare calculated, these forces are then used to integrate the structure forward to the next time step. The newboundary position and velocity is communicated to the SPH solver which generates an improved solutionfor the boundary. This process is iterated on until the structural position has met a convergence criteria.

rcg = 0.288m

0.115m

Figure 1: Problem geometry

The objective function to be minimised by the optimiser is the total variation of angular position givenby equation 20. Here t2 is five seconds after the tank is released and ω is the angular velocity of the tank.Three control nodes are located on the bottom of the tank, one at each corner and one in the centre asshown in figure 3. Each of these control nodes can be perturbed in both the x and y directions and themotion of the control nodes is translated into tank surface geometry motion by a radial basis function basedinterpolation scheme.

F =

∫ t2

t1

|ω|dt (20)

The optimisation is constrained to not allow the tank area to change more than 5% from the initial value

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Figure 2: Tank initial geometry and position

Figure 3: Tank control point locations and degrees of freedom

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with an optional constraint on the gravitational potential energy of the fluid that it not reduce by more that10% of the initial value.

VII. Initial Conditions

The SPH solver has many advantages but it is difficult to set up a steady state hydrostatic fluid field.A typical strategy when a hydrostatic starting condition is desired is to let the solver evolve one from anon-equilibrium starting point. However this is an expensive process as the settling times can be longerthan the time taken to solve for the flow in the period of interest. Some authors have suggested a process todistribute the SPH particles in an initially hydrostatic state10 but this is both complex and still relativelycomputationally intensive.

Here an alternative strategy has been adopted which takes inspiration from a method used to deformfinite volume meshes. For a finite volume mesh the same interpolation which deforms the surface meshbased on the motion of control points can be extended to also move the finite volume mesh nodes.39 Oncea baseline hydrostatic solution has been found for the undeformed tank the SPH particles are treated asif they are finite volume nodes and moved initially using the RBF interpolation. Only the position of theparticles is enforced by the interpolation, the fluid properties are retained from the baseline solution. Thisprocess is computationally very efficient and as the movement of the surface is small also offers sufficientaccuracy. A key advantage is that the SPH particles are automatically kept within the tank.

VIII. Results

Two optimisation cases were run with the same objective function and tank initial conditions. In thefirst case the only constraint was on the tank volume and the RBF interpolation used a support radius equalto six times the tank width. This results in global geometry changes of the tank. The second optimisationwas run with the additional potential energy constraint and a support radius equal to half the tank width.This allows much more local control of the tank bottom while also fixing the top of the tank in place.

VIII.A. Tank Geometry Changes

Figure 4 shows the evolution of the tank boundary as the optimiser runs for the first case while figure 5shows the same information for the second case. For case 1 the general trend is to lengthen the left sideof the tank while lowering the right corner. Both of these changes have the effect of lowering the initialpotential energy of the system implying the optimiser could be proceeding simply to reduce potential energy.However, they also change the times at which the waves impact with the walls and the wave speeds. Withcase 2 the behaviour is more complex; both corners of the tank are moved down and out though the leftcorner is moved more, this is shown in figure 5. The middle of the tank moved slightly upward and to theleft, eventually creating a small step.

VIII.B. Comparison of Coupled Behaviour

Figure 6 shows the behaviour of the fluid in the tank at the first time where there is noticeable divergencebetween the time histories for evolution one and evolution eleven in case one. The maximum amplitude oftank motion, which occurs shortly after this point, in evolution eleven is considerably less than in evolutionone. Examining the two plots it can be seen that the dip in the fluid towards the left side of the tank islower in evolution eleven than one. This is likely due to the slight shift of the left side wall to the left. Thislower fluid level will cause surface waves to travel more slowly over this region of fluid meaning that theaccumulated mass of fluid on the right side of the tank will be larger for the evolution eleven tank. This willdirectly result in greater fluid forces on the tank as the tank must accelerate a greater amount of fluid.

In case two the behaviour is different as the time histories diverge between evolution one and elevenalmost from the start of the motion. Here one suggestion for the difference might be that the pointed regionsin the corners, and in particular the bottom right corner, act to trap fluid meaning inertial effects reducethe acceleration of the tank.

The time history for different evolution numbers is shown for case one in figure 7 and case two in figure 8.For the eleventh evolution a comparison of time histories for the cases is shown in figure 9. In case one

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Evolution 1Evolution 1Evolution 4

Evolution 1Evolution 7

Evolution 1Evolution 11

Figure 4: Case 1: Tank geometry changes with evolution number

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Evolution 1Evolution 1Evolution 4

Evolution 1Evolution 7

Evolution 1Evolution 11

Figure 5: Case 2: Tank geometry changes with evolution number

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(a) Evolution one

(b) Case 1: evolution eleven (c) Case 2: evolution eleven

Figure 6: Tank and fluid behavior at T = 0.48s

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the effect of optimisation is to reduce the size of the peak for negative angles while leaving the positivepeaks largely untouched by comparison. The fact that the positive peaks are hardly modified is due to theinteraction frequency of wave oscillation and tank motion, which results in the fluid surface being largelylevel at times just before positive peaks; this fluid state leaves few options to the optimiser. Case two evolvesin a different way. Here the peaks are not reduced as much and the behaviour is more symmetric betweenpositive and negative angles. The sharp corners in the time history are also removed by the evolutions,suggesting that the wave impacts are reduced as fluid is ’trapped’ in the corners.

-20

-15

-10

-5

0

5

10

15

20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Posi

tion

(deg

rees

)

Time (s)

Evolution 1Evolution 4Evolution 7

Evolution 11

Figure 7: Time history for case one evolutions

VIII.C. Objective Function Changes

Changes in objective function are initially small, as seen in figure 10, but become larger with more evolutions.This is due to the optimiser being conservative with geometry changes for the first few evolutions, a factwhich can also be seen in figure 4. This is a result of there being more information to populate the Hessianmatrix. Although the optimiser is not as successful in minimising the objective function in case two thegeneral gradient of reduction appears to be the same in both cases.

Figure 11 shows that in both cases the volume increases but the optimisation constraint is not encounteredso far in either case. It is interesting to see that although the geometry of the tank is quite different in eachcase the change in volume with evolution number is very similar for both cases. The volume changes arelinked to the energy changes shown in figure 12 as the increase in volume comes in part from moving thelower surface of the tank downwards. The potential energy changes, like the volume changes, are yet toencounter the optimisation constraint.

IX. Conclusions

A method has been demonstrated for optimising tank geometry to achieve maximum damping using astructure coupled time-marching SPH technique. In order to provide an initial SPH particle configurationfor each deformed surface geometry, a novel technique based on radial basis functions is demonstrated thatefficiently deforms the particle field, rather than letting the particles settle to hydrostatic equilibrium afterevery design evolution. This is especially useful for limiting the expense of the gradient computations.

The optimiser is successful in reducing the total variation of angular position while retaining the tankvolume and initial potential energy, with the final result showing physical trends consistent with greater

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Posi

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Evolution 1Evolution 4Evolution 7

Evolution 11

Figure 8: Time history for case two evolutions

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Pos

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Case 1Case 2

Figure 9: Comparison of time history for case one and two using the 11thevolution

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Ob

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Case 1Case 2

Figure 10: Objective function against evolution number for case one and two

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16 18

Vol

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nta

ge

chan

ge

Evolution number

Case 1Case 2

Figure 11: Volume against evolution number for case one and two

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-3

-2.5

-2

-1.5

-1

-0.5

0

0 2 4 6 8 10 12

Flu

idp

ote

nti

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ergy

per

centa

gech

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Evolution number

Figure 12: Fluid potential energy against evolution number for case two

dissipation of system energy through surface waves. The influence of the constraint on initial potentialenergy is significant; instead of shifting the centre of mass of the fluid, the method chooses to expand thetwo lower corners of the tank to alter the free-surface behaviour.

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