Mesh adaptation for simulation of unsteady flow with moving immersed boundaries

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2013; 72:453477Published online 3 Dec 2012 in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.3751

Mesh adaptation for simulation of unsteady flow with movingimmersed boundaries

C. H. Zhou1,*, and J. Q. Ai21Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

2The First Aircraft Institute, Aviation Industry Corporation of China, Xian 710089, China

SUMMARY

In this work, an approach for performing mesh adaptation in the numerical simulation of two-dimensionalunsteady flow with moving immersed boundaries is presented. In each adaptation period, the mesh is refinedin the regions where the solution evolves or the moving bodies pass and is unrefined in the regions wherethe phenomena or the bodies deviate. The flow field and the fluidsolid interface are recomputed on theadapted mesh. The adaptation indicator is defined according to the magnitude of the vorticity in the flowfield. There is no lag between the adapted mesh and the computed solution, and the adaptation frequencycan be controlled to reduce the errors due to the solution transferring between the old mesh and the new one.The preservation of conservation property is mandatory in long-time scale simulations, so a P1-conservativeinterpolation is used in the solution transferring. A nonboundary-conforming method is employed to solvethe flow equations. Therefore, the moving-boundary flows can be simulated on a fixed mesh, and there is noneed to update the mesh at each time step to follow the motion or the deformation of the solid boundary.To validate the present mesh adaptation method, we have simulated several unsteady flows over a circularcylinder stationary or with forced oscillation, a single self-propelled swimming fish, and two fish swimmingin the same or different directions. Copyright 2012 John Wiley & Sons, Ltd.

Received 25 February 2012; Revised 22 August 2012; Accepted 21 October 2012

KEY WORDS: mesh adaptation; unsteady flow; immersed boundary; time-dependent problems; movingboundary; nonboundary-conforming method

1. INTRODUCTION

Despite the constant development of computers and numerical methods, the scale and complexityof flow problems encountered in science and engineering still surpass the capacity of modern com-puters. Techniques of mesh adaptation have been extensively explored over the past 20 years as anindispensable method to not only pursue an accurate solution of the governing equations but alsoreduce the cost of computation because the nodes/elements of an adapted mesh are always placedin the locale regions of interest. For time-dependent problems, mesh adaptation is more crucial asthe physical phenomena may progress arbitrarily in the whole computational domain. Without meshadaptation, a uniform fine mesh is required to preserve the accuracy of numerical solution. The useof such a mesh is computationally prohibitive, especially for three-dimensional problems.

Until now, a large number of papers have been published for the mesh adaptation in the simulationof steady flows, whereas the number of papers addressing the time-dependent problems is relativelysmall. For time-independent problems, the mesh adaptive to the steady solution is constructed viaseveral solution-adaptation iterations. For time-dependent simulations where the solution progresses

*Correspondence to: C. H. Zhou, Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China.

E-mail: [email protected]

Copyright 2012 John Wiley & Sons, Ltd.

454 C. H. ZHOU AND J. Q. AI

in time, evolving flow structures must be captured. In most of the existing methods of mesh adapta-tion for unsteady flows, the approach taken is to simply adapt the mesh per n time steps using thesolution at the first time step in the current period to construct the adaptation indicators. Therefore,the adapted mesh always lags behind the computed unsteady solution. The features of interest maymove outside the refined region, and the accuracy of solution will decrease quickly. To reduce the lagand contain the feature evolution within the resolved region, the mesh was adapted frequently [13].But, in this case, an important source of error due to solution transferring (by interpolation) from theold mesh to the current adapted mesh is introduced, and the accuracy of solution depends stronglyon the number of mesh adaptations. On the other hand, frequent adaptations may be excessive forthe flow fields that develop slowly. In the work of [1,2], two layers of cells adjacent to each markedcritical region are refined to ensure that the refined region contains the traveling feature during theshort adaptation period. In the aforementioned methods, the number of adaptations cannot be con-trolled, and the introduction of errors due to the interpolation of solution from the old mesh to thenew one cannot be prevented effectively. In [4], Cavallo et al. proposed a mesh adaptation methodfor transient flows, which is based on a new projection and error-wake concept. Mesh refinement isperformed by projecting the error estimate ahead of its current position. Mesh coarsening is carriedout in the wake region where the errors have propagated since last adaptation. A wake excess ismonitored to determine when the next adaptation is necessary. Alauzet et al. [5, 6] demonstrate anapproach in which the flow field and mesh are recomputed several times over a given period untiltheir coupling is converged. The mesh adaptation is based on a metric intersection in time proce-dure and reflects the flow field evolution during that period. The applications in [5, 6] are mainly toshock-dominated problems.

Recently, nonboundary-conforming methods are attracting more and more attention in the numer-ical simulation of moving-boundary problems, especially of complex fluidstructure interactions.Using this kind of methods, one can simulate flows over moving bodies on a fixed mesh, and thereis no need to update the mesh at each time step to follow the motion or deformation of the solidboundary. In nonboundary-conforming methods, the body surface is discretized by a Lagrangemesh, and the Eulerian mesh for solving flow field is independent of the geometry of body. Theeffect of solid boundary is reflected by the solution at the nodes in the vicinity of wall. To reflectaccurately the boundary effect, the mesh in the near-wall region should be dense enough. Cartesianmeshes are usually used in nonboundary-conforming methods. In many applications [79], a uni-form fine mesh is used in a small region containing the solid object, and the mesh is stretchedfrom the boundary of this small region to the outer boundary of the computational domain in someappropriate way. In this case, clustering of Cartesian mesh in the small region must be maintainedto the outer boundary and lead a large number of nodes. This problem can be addressed by localmesh refinement in the vicinity of wall. However, when the body is moving, the mesh within theregion where the body will pass during the whole simulated period will be refined. Even not for theentire computational domain, such a refinement increases the number of mesh nodes dramatically,especially for three-dimensional computations. Furthermore, in the complex fluidstructure inter-action problems, the path of a moving body is not known a priori, and the region where the meshrefinement is needed cannot be determined in advance of computation.

In [1013], Shu and Zhou proposed and developed a nonboundary-conforming method, calledthe local domain-free discretization (DFD) method, to solve partial differential equations. In thismethod, a partial differential equation is discretized at all mesh points inside the solution domain,but the discrete form at an interior mesh point may involve some points outside the solution domain.The functional value at an exterior dependent point is computed via some approximate form of solu-tion in the vicinity of boundary. The readers can refer to the work of [1013] for more details of thismethod and its applications.

In this work, we present an approach for performing mesh adaptation in the simulation of two-dimensional unsteady flow over moving bodies. The flow equations are solved by the local DFDmethod. In the present approach, the mesh is adjusted per specified number of time steps (oneadaptation period). In each adaptation period, the mesh is refined in the regions where the solutionevolves or the moving bodies pass and is unrefined in the regions where the phenomena or the bodiesdeviate since the last adaptation. The unsteady flow field and the dynamically evolving fluidsolid

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2013; 72:453477DOI: 10.1002/fld

MESH ADAPTATION FOR UNSTEADY FLOW WITH MOVING BOUNDARIES 455

interface are recomputed on the new adapted mesh. There is no phase shift in time between theadapted mesh and the computed solution, and the adaptation frequency can be controlled to reducethe errors due to solution interpolation between the old mesh and the new one. A P1-conservativeinterpolation is used for the solution transferring to preserve the conservation property of solution.The applications aim to incompressible viscous flows, which are dominated by vortex shedding. So,the mesh adaptation indicator is defined according to the magnitude of the vorticity in the flow field,which is precomputed over the current adaptation period.

For the remainder of this paper, the content is arranged as follows. In Section 2, the governingequations and the local DFD method are described in brief. In Section 3, the mesh adapta-tion approach for unsteady flow with moving boundaries is presented. In Section 4, we describethe P1-conservative interpolation for the solution transferring between two different meshes. InSection 5, we perform numerical experiments for several flows over a circular cylinder stationary orwith forced oscillation, a single self-propelled swimming fish, and two fish swimming in the sameor different directions. Comparisons with experimental data or the published results obtained onconventional meshes are also provided in this section. Finally, in Section 6, we summarize this workand present the conclusions.

2. GOVERNING EQUATIONS AND BASIC NUMERICAL METHODS

We will consider a homogeneous, laminar, and constant viscosity flow without body force effect.The governing equations are the time-dependent incompressible NavierStokes equations, whichcan be written in nondimensional form as follows:

@ ui

@xiD 0 (1)

Dui

DtD @p

@xiC @ij

@xj(2)

In these equations, ui denotes the Cartesian velocity component, D =Dt the material derivative, pthe pressure divided by the density, and ij the dimensionless viscous stress tensor

ij D 1Re

@uj

@xiC @ui

@xj

(3)

where Re is the Reynolds number. We are interested in solving the governing equations in a domaincontaining moving bodies.

The dynamically evolving fluidsolid interface .t/ is discretized with material points, whichlie on .t/ at all times and can be tracked with their global Lagrangian position vectors X.t/.The fluid and structure dynamics are coupled together at the interface by the following no-slipboundary conditions

ui D Vi D dXidt

at .t/ (4)

where Vi denotes the Cartesian velocity component of material points on the interface and Xi theCartesian component of the position vector X.t/.

The NavierStokes equations (1) and (2) are solved using the local DFD method. This methodhas been described and discussed in detail in [1014], so only a brief description is given herein. Inthe DFD method, a partial differential equation is discretized at all mesh points inside the solutiondomain, but the discrete form at an interior point may involve some mesh points outside the solutiondomain, which serve as the role to implement the boundary condition. The implementation of theboundary condition and the discretization of the partial differential equation are treated separately.The critical issue for successful implementation of the DFD method is how to evaluate the functional



values at the exterior dependent points, that is, to construct some approximate form of solution inthe local region. In the original DFD method [14], the approximate form of solution is pursued alongthe whole mesh line that only involves two boundary points. This way is not applicable for complexdomains. To make the method be more general, the local DFD was developed in [1013]. In the localDFD, the low-order schemes are adopted for spatial discretization and also for the approximate formof solution near the wall boundary. For an incompressible viscous flow, the functional values at theexterior dependent points are updated at each time step by proper extrapolation along the normaldirection to wall in conjunction with the no-slip boundary condition and the simplified momentumequation in the vicinity of the solid wall. The Galerkin finite-element approximation [15] is usedfor spatial discretization, and the discrete equations are integrated in time via a dual-time-steppingscheme based on artificial compressibility [16].

The DFD method belongs to the category of nonboundary-conforming methods. All the nodes ofthe Eulerian mesh are classified as interior points, exterior dependent points, and exterior indepen-dent points that are blanked out of computation. When the body is moving or deforming, only thestatus of points is changed, and the mesh can stay fixed. The body surface can be treated as sharpinterface. In the previous work [1013], the method has been validated extensively for various flowswith stationary or moving boundaries. The body motion may be prescribed or determined by thefluid force.

3. MESH ADAPTATION FOR UNSTEADY FLOW WITH MOVING BOUNDARIES

In the classical mesh adaptation algorithm for time-independent problems, the obtained mesh isadapted to the steady state of solution. For the simulation of time-dependent problems, it is ideal butnot practical to generate one adapted mesh for one instantaneous solution at each time step. If wegenerate an adapted mesh per specified number n of time steps (one adaptation period) only accord-ing to the solution at the first time step of this period, the mesh is always left behind the unsteadysolution. If the mesh is adapted frequently to diminish the lag between mesh and solution, an impor-tant source of error due to solution interpolation from the old mesh to the new one is introduced. Fora long-time simulation, the interpolation errors may accumulate, and the accuracy of solution willdecrease greatly.

We may refine the mesh in the region where the solution evolves during the whole simulatedperiod, but this will result in a mesh with a very large number of nodes/elements. In such an adjustedmesh, the refined region is also too large for the instantaneous solution at each time step.

In the present work, mesh adaptation is also performed per n time steps, but the mesh is adaptedto the phenomenon progression in each adaptation period. This is the principle of the proposedalgorithm we will discuss subsequently.

3.1. Adaptation scheme

We consider a time period with n time steps, in which the solution evolves and an adapted meshwill be generated. Starting from the initial solution at time t , we compute the instantaneous solu-tion and fluidsolid interface at each time step t D t C t , t C 2t , : : : , t C nt . Hence, thesolution behavior and the motion of solid body throughout this mesh adaptation period can bepredicted. Then, a new mesh is generated according to the adaptation indicator defined at eachelement and vertex, which takes into account the solution progression. All elements in the regionsthe phenomena pass through in this period will be refined, and some vertices in the regions thephenomena deviate from since the last adaptation will be deleted. Consequently, the new mesh isadapted to the solution progression. After having generated the adaptive mesh, the initial solutionof this period on the old mesh is transferred by interpolation to the new one, and then, the computa-tion is resumed on the new adapted mesh to obtain the numerical solution of unsteady flow and theevolving fluidsolid interface within this time period. Using the adapted mesh as the initial meshand the solution at the last time step in the current period as the initial solution, one can start thenext adaptation period.



This mesh adaptation scheme is illustrated in Figure 1.To reduce the computational cost, only 20 dual iterations are required per physical time step

in the computation to predict the solution behavior, whereas 50 dual iterations are required in therecomputation for the final solution on the adapted mesh.

In the present approach, the mesh is adaptive to the solution progression, so there is no phaseshift in time between mesh and solution and it is not necessary to adapt the mesh frequently. Alarge number n of time steps within one adaptation period can be chosen, and then, the errors due tothe solution transferring can be reduced. For a large n, the time cost of mesh adaptation decreases,but the number of nodes/elements of each adapted mesh and the time cost of flow computationsincrease as a price (because the area the phenomena are sweeping during n time steps is larger andthe refined region will be larger). Therefore, n cannot be taken too large in practical computations.Choosing the value of n is a compromise between efficiency and accuracy. In our numerical experi-ments, n is determined empirically to be 50100, which is much larger than the values .n D 5 10/taken in the methods based on frequent mesh adaptation [13].

In the transient fixed point mesh adaptation method proposed by Alauzet et al. [5], numerical solu-tion and mesh are recomputed several times within each adaptation period until the convergence ofthe solution-mesh couple is achieved. In the present work, we concentrate on the simulation of theincompressible viscous flows. Large gradients exist in the boundary layer near the body surface,which can be identified directly. Outside the boundary layer, the vorticity that distributes in a rela-tively wide region is continuous, and the gradient is not steep. The prediction of physical phenomenaprogressing in an incompressible flow is easy and may not be as stringent as in a compressible flowwhere steep gradients or genuine discontinuities (shocks) exist. Therefore, we transform the internalloop of iterations in the fixed point method [5] into the simpler predictorcorrector steps in thepresent method. Consequently, the cost of computation can be reduced. In the section of numericalexperiments, we will justify the rationality of the choice of predictorcorrector steps.

Figure 1. Mesh adaptation scheme for unsteady flow simulations. i , index for mesh adaptation period; j ,index for time step; n, number of time steps in one adaptation period; j0, initial time step for each adapta-tion period; QM , initial mesh; QS , solution on initial mesh; Q , fluidsolid interface on initial mesh; M , adaptedmesh; S , solution on adapted mesh; , fluidsolid interface on adapted mesh; E, mesh adaptation indicator.



3.2. Criteria for mesh adaptationIn this work, our applications aim to the incompressible viscous flows. The vorticity can be usedto capture physical phenomena in the flow field, such as boundary layers and vortices. For eachelement K, the magnitude of vorticity is defined as

!j .K/ DZ

K

@u1@x2

@u2

@x1

d (5)

where j is the index for time step. For the piecewise linear solution,

!j .K/ D K@u1@x2

@u2

@x1

K

(6)

where K is the area of element K.In unsteady flows, the physical phenomena develop with time. In the present mesh adaptation

approach, the adaptation indicator must take into account the solution progression. We solve thisproblem simply by defining the indicator at an element as the maximum value of all the immediatevorticity magnitudes at every time step in the adaptation period .j0 C 1/t , .j0 C n/t

Ei .K/ D maxj0C16j6j0Cn

!j .K/ (7)

where i is the index for mesh adaptation period. For each vertex P , the adaptation indicator isdefined as

Ei .P / D maxK2P

Ei .K/ (8)

where P is the subregion formed by all triangles that contain the vertex P .Two values of Emax and Emin are specified for mesh adaptation. As discussed in the next subsec-

tion, in the i th mesh adaptation period, when Ei .K/ > Emax, the element K is refined, and whenEi .P / < Emin, the vertex P is deleted. Therefore, the mesh is refined in the regions where thesolution evolves and is unrefined in the regions where the phenomena deviate.

In the nonboundary-conforming methods, the geometry of a solid body is not represented directlyby the Eulerian mesh, and the mesh in the vicinity of solid wall should be fine enough for the effectof boundary to be reflected. So, the elements intersected by the solid wall and the elements sharingthe vertices of the intersected elements are given large indicator values to ensure that the mesh innear-wall regions will be the finest. To prevent extremely large gradient of mesh density and keepthe geometric integrity of the domain as a whole, all vertices of the original mesh at the beginningof computation are also given large indicator values and will not be deleted in the mesh adaptation.

3.3. Refinement and unrefinementIn the present work, the original triangular mesh is generated by dividing the square cells of a uni-form Cartesian mesh. Therefore, the original mesh consists of equilateral right-angled triangles. Themesh adaptation is based on refinement/unrefinement techniques without node displacement to keepall the triangles similar to each other and also reduce the interpolation errors.

We begin with a brief summary of the refinement procedure, which is based on the longest edgebisection algorithm of Rivara [1719]. All current elements are placed in a heap data structureaccording to the value of their adaptation indicators. The element with the largest value is at theroot of the heap. This element is selected for refinement and is bisected along its longest edge. Theneighbor element sharing the longest edge is also bisected along its longest edge. If the result is atriangulation, the process stops. Otherwise, it is recursively applied to the longest edge neighbors ofall refined elements. An example is shown Figure 2. This process has a finite termination, typicallyin a very small number of steps. When the process stops, the newly generated elements are added totriangulation data structure. New elements inherit vorticity values from their parents, so the heap canbe updated. The refinement process continues until a new mesh with indicator values at all elementsless than Emax is created.



Figure 2. Element is refined by the longest edge bisection algorithm [17,18]. Left: original mesh; middle:the first step does not yield a compatible triangulation; right: the final compatible triangulation.

Figure 3. Vertex P is eliminated from the mesh. Left: the subregion P associated with the vertex P ;middle: the vertex and all its incident edges are removed; right: the subregion is triangulated using the

boundary vertices.

A value of hmin is specified to control the mesh density and prevent numerical instability due tothe elements with very small volume. In the refinement process, when the length of the shortest edgeof an element is smaller than hmin, the value of its indicator is set to be zero. This element will notbe bisected anymore.

By using the longest edge bisection algorithm, all the elements of the refined mesh are still theequilateral right-angled triangles.

In the case of unrefinement, the basic step consists of deleting vertices from the mesh, rather thandirectly deleting elements [17,18]. Each vertex P is associated with a subregion P . This subregionis formed by all triangles that contain the vertex P , as illustrated in Figure 3. The value of adaptationindicator at the vertex is defined as the maximum of those defined at the elements contained in P .All the vertices are also placed in a heap data structure according to the values of their indicators,with the vertex of the smallest value at the root. In the unrefinement process, the root vertex andall its incident edges are eliminated from the mesh. The subregion P associated with this vertexis then triangulated, as illustrated in Figure 3. The created elements inherit vorticity informationfrom the original elements in P , and the vorticity values are computed for the new elements. Thevertices lying on the boundary of P have their indicators updated, and then, the heap is updated.The unrefinement process continues until a new mesh with indicator values at all vertices greaterthan Emin is created.

To keep the newly generated elements always to be equilateral right-angled triangles, only whenits associated subregion P is equilateral, the vertex P is allowed to be deleted in the unrefinementprocess. So, the mesh quality will not degrade after unrefinement. As a price, a small fraction of thevertices, which are originally marked for unrefinement, will not be deleted.

In the process of refinement and unrefinement, it is not necessary to store the history, and there isno extra requirement of memory.

4. P1-CONSERVATIVE INTERPOLATION

After having generated a new adapted mesh (current mesh), the solution obtained on the old mesh(background mesh) must be recovered on the current mesh to restart the computation from the pre-vious state. This stage is critical in the mesh adaptive simulation of unsteady flows as the errors dueto solution transferring may accumulate throughout the time-dependent computations. In the spatialapproximation of our basic numerical method, the solution is given at the vertices and is continuous



piecewise linear by elements. Moreover, the preservation of conservation property is mandatory inlong-time scale simulations. Therefore, a P1-conservative interpolation is used in the present meshadaptation method to transfer solution from the background mesh to the current mesh. Followingthe idea of Alauzet and Mehrenberger [20], we develop the P1-conservative interpolation operatorin this section.

A projection operator c1 from the background mesh M back to the current mesh M is P1-conservative means that

RM back u D

RM

c1u and c1 is P1 exact. For each background triangle

Kback, we have the constant gradient ruKback and the mass mKback D jKbackju.G/ where G is thebarycenter of the triangle.

For each triangle K of the current mesh, we compute the intersection with all triangles of thebackground mesh

Kbackj

it overlaps. Each couple of triangles K and Kbackj provides a simplicial

mesh of their intersection region denoted by Tj D K\Kbackj , as shown in Figure 4. For each triangleK, the mass and the gradient can be obtained as follows:

mK DZ

K

c1u DX

j

ZTj

u andrc1uK D

Pj

RTj

ru

jKj (9)

Denote by uK the P1-conservative interpolated solution on the triangle K; the value at thebarycenter and the gradient on K are given by

uK.GK/ D mKjKj and ruK Drc1uK (10)

The value of interpolated solution at each vertex Pi of the element K can be obtained byTaylor expansion

uK.Pi / D uK.GK/ C ruK !GKPi (11)To avoid the loss of monotony, we apply the correction proposed by Alauzet and Mehrenberger

[20] to the nodal values at each element to enforce the maximum principle. Let be the set of thevertices of the elements of the background mesh that the element K of the new mesh overlaps. Foreach vertex Pi of K, the nodal values satisfy the maximum principle if

umin D minQ2 u.Q/6 uK.Pi /6 maxQ2 u .Q/ D umax (12)

First, the indices are reordered such that uK .P1/6 uK.P2/6 uK .P3/. We set

uMK .P3/ D min .uK .P3/ ,umax/ (13)

uMK .P2/ D minuK .P2/ C 1

2max .0,uK .P2/ umax/ ,umax

(14)

1

24

3

56

1

24

3

56

Figure 4. Left: intersection region (polygon 123456) of the triangle K (dash-dot lines) of the current meshwith one (dashed lines) of the triangles Kback

jof the background mesh; right: the simplicial mesh of the

intersection region.



uMK .P1/ D 3uK.GK/ uMK .P2/ uMK .P3/ (15)and then, the new corrected nodal values QuK .Pi / can be obtained as follows:

QuK.P1/ D maxuMK .P1/ ,umin

(16)

QuK.P2/ D maxuMK .P2/

1

2max

0,umin uMK .P1/

,umin

(17)

QuK.P3/ D 3uK.GK/ QuK.P1/ QuK.P2/ (18)This interpolated solution on the new mesh is conservative and piecewise linear by elements, but itis discontinuous. One more stage is required to retrieve a solution at vertices of the new mesh. Thesolution can be redistributed to each vertex P of the new mesh by averaging

Qu.P / D

PP 2Ki

jKi j QuKi .P /P

P 2KijKi j (19)

where the summations are over all triangles that contain vertex P . The final interpolated solution iscontinuous piecewise linear and conservative, and still satisfies the maximum principle.

5. NUMERICAL EXPERIMENTS

In this section, numerical experiments for unsteady incompressible viscous flows with stationaryor moving boundaries are performed to validate the present mesh adaptation method. In our work,the original mesh will not be coarsened. The original mesh must be fine enough to identify thegeometry of solid objects correctly in the nonboundary-conforming numerical methods. So, themaximal mesh size depends on the original mesh. To prevent the mesh to be too dense and obtainhigh resolution of boundary layer at the same time, the minimal size of adapted mesh hmin is setin advance. The value of hmin is taken empirically according to the Reynolds number of flow. Inthe following computations, hmin D 102 for the test cases of circular cylinder (Re 6 300) andhmin D 3 103 for the cases of swimming fish (Re 4000). As for the adaptation criteria, Emaxis related to the mesh size and the absolute value of vorticity by Equation (5). The value of Emaxand the ratio of Emax to Emin control the extent of refined region and the density of adapted mesh,and are also determined empirically. We set Emax D 2.5 103 in the computations for the circularcylinder and Emax D 5104 in the computations for the swimming fish, corresponding to differentminimal mesh sizes. In all computations, we always set Emin D Emax=10.

5.1. Unsteady flow over a stationary circular cylinderThe test case is low Reynolds number two-dimensional flow over a stationary circular cylinder inunbounded domain, which exhibits periodic vortex shedding at Reynolds numbers above 43. TheReynolds number is defined as Re D UD= where U is the free-stream velocity, D is the diameterof the cylinder, and is the kinematic viscosity of fluid. This test is one of the benchmark problemsused to validate incompressible flow solvers. A large domain size of 50D 30D has been usedto minimize the outer boundary effects, and the cylinder is placed at .10D, 0/. The initial meshhas 126 76 nodes. In computations, the Reynolds numbers are set to be Re D 100, 200, 300. Thenondimensional time-step size is taken to be 102. The mesh is adapted for every n D 50 time steps.

For Re D 100, the mesh for the adaptation period [135, 135.5] and the instantaneous vorticityfield at the end of this period are shown together in Figure 5. The total number of mesh nodes is33,043. The figure shows the formation of alternating vortices, their shedding, and the downstreamconvection. Information of vortices can be well represented as the refined triangles concentrate inthe region of vortex traveling. If the errors due to solution interpolation between the old mesh andthe new one accumulate, nonconstant and generally damped amplitude can be observed when the



4

2

0

-2

-10 -5 0 5

Y

X

Figure 5. The adapted mesh and vorticity distribution in the wake region for Re D 100.

0 20 40 60 80 100 120 140

-0.4

-0.2

0

0.2

0.4

v

t

Figure 6. Time history of the vertical velocity at a point 12D downstream of the cylinder.

0

-2-10 -8 -6

Y

X

Figure 7. The adapted mesh and vorticity distribution in the near wake region for Re D 200.

behavior of the vertical velocity component at a given point downstream of the cylinder is plotted[21]. Figure 6 shows the periodic behavior of the vertical velocity component at a point 12D down-stream of the cylinder. Note that the problem regarding damping and nonuniformity of amplitudedoes not appear, and the amplitude is more uniform than that of the result obtained by Sampaio withhigher-order interpolation [21].

For Re D 200, Figure 7 shows the adapted mesh in the near-wake region for the period [120,120.5] and the vorticity field at the end of this period. The number of mesh nodes is 39,804.



In Figure 8, the initial position (blue) and the end position (red) of vorticity evolving in this adapta-tion period are plotted. It can be seen clearly from these two figures that the mesh in the region ofvorticity convection is refined, the mesh in the region where the phenomena deviate since the lastadaptation is unrefined, and the mesh density matches the intensity of the vorticity. For Re D 200,we perform one more computation with a twice longer adaptation period n D 100. The initial posi-tion and the end position of vorticity evolving and the adapted mesh in the period [120, 121] areshown in Figure 9. The number of mesh nodes is 45,933. Obviously, the refined region is largerfor n D 100, and the number of nodes of the adapted mesh is also larger. The variations of nodenumber with time, for n D 100 and n D 50, are presented in Figure 10. The node number increaseswith the formation of alternating vortices until the first generated vortex reaches the exit of thecomputational domain.

Table I presents the results for drag and lift coefficients and Strouhal number (St D fD =U , f isthe shedding frequency). For comparison, numerical results of Choi et al. [22], Mittal et al. [23], Liuet al. [24], Marella et al. [25], and Nobari et al. [26] and experiment data of Wille (reported in [27])

Figure 8. Evolving of vorticity in one adaptation period for Re D 200, n D 50.

Figure 9. Evolving of vorticity in one adaptation period for Re D 200, n D 100.

Time Step

Nod

e Nu

mbe

r

0 2000 4000 6000 8000 10000 12000

50000

Figure 10. Time history of the node number of adapted mesh for Re D 200 (dashed line: n D 100; solidline: n D 50).



and Tritton (reported in [28]) are also provided in the table. Good agreement can be observed.With mesh adaptation, numerical dissipation can be reduced, and the loss of energy due to vorticesis less compared with the computation on a conventional mesh. Therefore, the drag predictedby the present method is slightly smaller than the reference numerical results and closer to theexperimental data.

To justify the rationality of the choice of predictorcorrector steps in the present mesh adaptationmethod, we also calculate the flow over a circular cylinder at Re D 200 via replacing the predictorcorrector steps by an internal loop of four solution-mesh iterations. The mesh is adapted for everyn D 50 time steps. The solutions (and meshes) at every internal iteration are very close to each other.The results obtained with predictorcorrector steps or internal loop of four iterations are comparedin Figure 11, and the difference between them is too small to be identified.

5.2. Flow over a circular cylinder with forced transverse oscillation in a free streamThe size of computational domain and the initial mesh are the same as those in the computation forthe stationary cylinder. The nondimensional time-step size is taken to be 5 103, and the mesh isadapted for every n D 50 time steps.

To simulate oscillatory motion of the circular cylinder, the coordinates of center of the cylinderare defined as

Xc1 D 10DXc2 D Y sin(2f t/

(20)

where Y and f are the amplitude and frequency of oscillation, respectively. Simulations areperformed at Re D 300 for fixed value of frequency F D f = fs D 0.9 (fs is the natural shed-ding frequency from the stationary cylinder) and two different values of amplitude A D Y = D D0.4, 0.65.

Table I. Drag and lift coefficients and Strouhal number for unsteady flow over a circular cylinder.

Re D 100 Re D 200 Re D 300References CD CL St CD CL St CD St

Present work 1.33 0.008 0.310 0.163 1.31 0.038 0.64 0.192 1.32 0.207Choi et al. [22] 1.34 0.011 0.315 0.163 1.36 0.048 0.64 0.191Mittal et al. [23] 1.35 1.36Liu et al. [24] 1.35 0.012 0.339 0.164 1.31 0.049 0.69 0.192Marella et al. [25] 1.37 0.198 1.28 0.22Nobari et al.[26] 1.37 0.207Wille (Experiment in [27]) 1.30Tritton (Experiment in [28]) 1.28

t

CL, C

D

20 40 60 80 100-1

0

1

2

3CL iterationsCD iterationsCL predictor-correctorCD predictor-corrector

t

CL

74.2 74.4 74.6 74.8 75 75.2

0.55

0.6

0.65

Figure 11. Time history of lift and drag coefficients obtained by predictorcorrector steps and an internalloop of iterations, respectively, for Re D 200, n D 50 (left: global view; right: partial view).



Figure 12 shows the adapted mesh in the period [62.25, 62.5] for A D 0.4. Figure 13 shows thevorticity field at the end of this period. The total number of mesh nodes is 47,075. The wake issimilar to the von Karman vortex street with alternative shedding of single vortices from both sidesof the cylinder. It can be seen again that vortices are traveling in the refined region.

For A D 0.65, the adapted mesh in the period [49.75, 50] is presented in Figure 14. The vorticityfield at the end instant of this period is presented in Figure 15. The number of mesh nodes is 51,725.As the amplitude increases, the secondary vortices are formed, and the wake pattern is changed. Apair of vortices with opposite sign travels at one side and a single one at another side of the wake.The mesh in the regions where the weaker secondary vortices are passing has also been refined asexpected. The time history of lift and drag coefficients is shown in Figure 16.

2

0

-2

-10 -5 0

Y

X

Figure 12. Adapted mesh in the wake region for Re D 300, F D 0.9, A D 0.4.

2

0Y

-2

-10 -5 0X

Figure 13. Vorticity distribution in the wake region for Re D 300, F D 0.9, A D 0.4.

2

0

-2

-2 0 2 4 6 8 10 12

Y

X

Figure 14. Close up view of adapted mesh in the wake for Re D 300, F D 0.9, A D 0.65.



5

0Y

X

-5-10 -5 0 5 10 15

Figure 15. Vorticity distribution in the wake region for Re D 300, F D 0.9, A D 0.65.

t

Cl,C

D

0 10 20 30 40 50 60

0

2

Figure 16. Time history of lift (dashed line) and drag (solid line) coefficients for Re D 300, F D 0.9,A D 0.65.

The current simulations confirm the experimental observations of Griffin and Ramberg [29] andalso match the computational results of Nobari and Naderan [26]. Williamson and Roshko namedthe former pattern as 2S and the latter as PCS [30].

5.3. Simulation of self-propelled anguilliform swimmingIn this section, we apply the proposed mesh adaptation method to the numerical simulation ofself-propelled anguilliform swimming. Anguilliform swimmers propel themselves by propagatingcurvature waves along their body [31].

The deformation of the fish body is prescribed in the local body system .X1,X2/ where the cen-ter of mass of the body remains and the total angular momentum is conserved. The position andorientation of the fish body, which are determined by the fluid force acting on the body, are definedin the global computational system .x1, x2/. The fluidstructure interactions are computed in theglobal system where the swimmer is considered as a rigid body. The translation and rotation of theself-propelled body are governed by Newtons equation of motion as follows:

md2Xc

dt2D F (21)

d

dt

Id

dt

D M (22)

where F and M are the fluid force and yaw torque, Xc is the position of the center of mass ofthe body, its global angle with respect to the initial position, m the total mass, and I the inertialmoment. The time dependency of the inertial moment is considered although it is small comparedwith the moment itself. The gravitational force is neglected.



In this work, the algorithm for loose coupling of fluidstructure interaction presented in [12] isemployed to simulate the self-propelled swimming.

5.3.1. Geometrical model and description of deformation. The fish geometry is taken from thework of Kern and Koumoutsakos [32]. The width of the body is defined as an analytical function ofthe arc length s along the midline. The analytical description of width is divided into three regions:

w.s/ D

8 T

(26)

All simulations are conducted with a constant viscosity of D 1.4 104, body length L D 1,density fluid D body D 1, and undulation frequency f D 1 (undulation period T D 1/. The resultingReynolds number is about 4000. The size of computational domain is 40L 12L, and the initialmesh has 101 31 nodes. The nondimensional time-step size is taken to be 5 104.

5.3.2. A single swimming fish. The fluid forces acting on the body are measured in nondimen-sional coefficients Cjj D Fjj =

12 NV 2jj S

and C? D F? =

12 NV 2jj S

parallel and lateral to

V||, V

t0 2 4 6 8 10 12 14

-0.2

0

0.2

0.4

0.6

0.8

Figure 17. Development of the longitudinal velocity Vjj (solid lines) and the lateral velocity V? (dashedlines) as the swimming fish accelerates from rest (red lines for n D 100, green lines for n D 200).



the mean swimming direction, where S represents the circumference of the body and NVjj theasymptotic mean forward velocity of the swimming fish. The yaw torque coefficient is defined asCM D M =

12 NV 2jj SL

.

Computations are performed for two values of the mesh adaptation period: n D 100, 200.Figure 17 shows the development of longitudinal and lateral velocities of the center of mass ofthe fish body during 14 undulation cycles. The body accelerates from rest to an asymptotic meanforward velocity. The longitudinal velocity Vjj varies slightly during a cycle, whereas the lateralvelocity V? has a larger value of amplitude. Figure 18 shows the angular position of body axis withrespect to the initial position , plotted against the time. From Figures 17 and 18, it can be seenthat the computed results for n D 100 and n D 200 are very close to each other. Therefore, in thesubsequent computations for two swimming fish, the number of time steps in one adaptation periodis set to be n D 100.

t0 2 4 6 8 10 12 14

-8

-6

-4

-2

0

2

Figure 18. Development of angular position of body axis as the swimming fish accelerates from rest(red line for n D 100, green line for n D 200).

Table II. Asymptotic mean longitudinal velocity and amplitudes of lateral velocity and coefficients oflongitudinal force, lateral force, and yaw moment for a swimming fish.

References NVjj QV? QCjj QC? QCMPresent work (n D 100) 0.556 0.0383 0.0315 0.038 0.030Present work (n D 200) 0.0558 0.0382 0.0320 0.038 0.030Zhou et al. [12] 0.550 0.039 0.030 0.040 0.031Hieber et al. [34] 0.54 0.04 0.03 0.04 0.03Kern et al. [32] 0.54 0.04 0.03 0.04 0.03NVjj: mean value of Vjj; QV?, QCjj, QC?, and QCM : amplitudes of V?, Cjj, C?, and CM .

X

Y

-6 -4 -2 0 2 4

0

2

Figure 19. Global view of vorticity field around a single swimming fish at t D 12.5.



In Table II, we compare the asymptotic mean forward velocity, the amplitude of lateral velocity,and the amplitudes of the coefficients of longitudinal force, lateral force, and yaw moment with theresults of Zhou and Shu [12], Kern and Koumoutsakos [32], and Hieber et al. [34]. All numericalresults are in good agreement. Because of the reduction of numerical dissipation, the values of theasymptotic mean forward velocity of the present computations are slightly larger than those of othercomputations on conventional meshes.

The global view of vorticity field at t D 12.5 is shown in Figure 19 (n D 100). The partialview of vorticity distribution in the region near the fish tail at t D 12.5 and the adapted mesh in the

X

Y

-2.8 -2.6 -2.4 -2.2 -2 -1.8

0.2

0.4

0.6

Figure 20. Close up view of adapted mesh and vorticity distribution at t D 12.5.

X

Y

-1.2 -1 -0.8 -0.6 -0.4 -0.20

0.2

0.4

Figure 21. Motion and deformation of fish body during the adaptation period [7.9, 8] (n D 200).

V||, V,

t0 5 10 15 20

-0.2

0

0.2

0.4

0.6

Figure 22. Development of the longitudinal velocity Vjj (red line), the lateral velocity V? (green line), andangular position of body axis (blue line) as the fish takes burst and coast swimming.



corresponding adaptation period [12.45, 12.5] are shown in Figure 20 (the node number of the globalmesh is 42,560). Relative to the mesh, the solid object is moving. The flow of fluid is only inducedby the motion and deformation of the fish body, so the convection velocity of vortex is small. InFigure 20, the triangles concentrated around the vortices have been refined obviously.

Figure 21 shows the motion and deformation of the fish body within the adaptation period [7.9, 8](n D 200). The blue line represents the position and shape of body at the start, and the red linerepresents the position and shape at the end. During one adaptation period, the mesh in the regionwhere the fish is passing has been refined. So the effect of geometry of the fish can be reflectedaccurately in the nonboundary-conforming method, and the boundary layer around the body canalso be resolved.

To demonstrate the ability of the present method for nonasymptotic problems, we simulate theburst and coast swimming of a single fish as described in [35]. From Figure 17, we can see that thefish accelerates from rest to an asymptotic mean forward velocity in about seven undulation cycles.So, the burst and coast swimming starts at t0 D 7.0 in this test. At the beginning, the fish glides as a

X

Y

-5 -4 -3 -2 -1

0

1

Figure 23. Close up view of adapted mesh and vorticity distribution at t D 20 for a burst and coastswimming fish.

X

Y

-2 -1 0 1 2

0

Figure 24. Initial position of two fish swimming in the opposite directions.

V||, V

t0 2 4 6 8 10 12 14 16

0

0.5

Red: SingleGreen: LeftwardBlue: Rightward

Figure 25. Development of the longitudinal velocity Vjj (solid lines) and the lateral velocity V? (dashedlines) of two fish swimming in the opposite directions.



t0 2 4 6 8 10 12 14 16

-10

0

Red: SingleGreen: LeftwardBlue: Rightward

Figure 26. Development of angular position of body axis of two fish swimming in the opposite directions.

0

0

1

0

0

-2 -1 0 1 2

YY

YY

X

-2 -1 0 1 2X

-2 -1 0 1 2X

21 3 4X

A

B

C

D

Figure 27. Instantaneous vorticity field of two fish swimming in the opposite directions: (A) t D 4.5,(B) t D 6.0, (C) t D 9.0, and (D) t D 11.3.



A

X

Y

-1 -0.5 0 0.5

0

0.5

B

X

Y

-1 -0.5 0 0.5 1

-0.5

0

0.5

Figure 28. Close up view of adapted mesh of two fish swimming in opposite directions: (A) t 2[4.45, 4.5]and (B) t 2[5.95, 6.0].

Y

1.5 2 2.5 3 3.5 4 4.5

0

0.5

X

Figure 29. Initial position of two fish swimming in the same directions.

rigid body, and the velocity deceases to its minimal value U1 at t1. Then, the burst period starts, andthe fish takes the propulsive swimming. The velocity increases from U1 at t1 to a given maximalvalue U2 at t2, and a new coast period starts again. We denote U1 D 1Umax and U2 D 2Umax whereUmax is the maximal velocity reached by the fish using steady periodic swimming. For burst swim-ming, the undulation amplitude of the fish grows from zero to the maximal value, in the C-shapestart form described by Equations (25) and (26). For coast swimming, the amplitude decreases tozero in the converse way.

In this simulation, we choose 1 D 0.4 and 2 D 0.8. The mesh is adapted for every n D 100time steps. Figure 22 depicts the unsteady longitudinal and lateral velocities of the center of massof the fish body and the angular position of body axis. The vorticity distribution at t D 20 and theadapted mesh in the corresponding adaptation period [19.95, 20] are shown in Figure 23. The node



number of global mesh is 39,670. It can be seen obviously that the triangles concentrated aroundthe fish body and the vortices have been refined.

5.3.3. Two fish swimming in the opposite directions. In this and the next simulations for two swim-ming fish, there exist strong vortexvortex and vortexbody interactions in the flow field. Meshadaptation plays an important role in reducing numerical dissipation and preserving the strength ofvortices, particularly in the far field region away from the solid bodies.

The departing position of the two swimming fish is shown in Figure 24. The two fish deformtheir bodies in the same way. Figure 25 shows the development of longitudinal and lateral velocitiesof the two swimming fish. The angular position of body axis with respect to the initial position,plotted against the time, is shown in Figure 26. In Figures 25 and 26, the longitudinal velocityand the orientation angle of the fish swimming leftward are plotted using their negative values.For comparison, the results of the single swimming fish are also plotted in these two figures. At thebeginning, the effect of interactions between the flows around the two fish is not obvious. The valuesof velocities and angles are close to those of the single swimming fish. After nearly one undulationcycle, the longitudinal velocities of both fish become smaller than that of the single one. As theswimming of the two fish continues, the rightward fish is close to the wake of the leftward one asshown in Figure 27A (t D 4.5), the flow field becomes more nonsymmetrical. The rightward fishnow is swimming faster than the leftward one (see Figure 25). From t 6, the rightward fish beginsto collide with the wake of the leftward one, as shown in Figure 27B (t D 6) and 27C (t D 9). Att D 11.3, the rightward fish is deviating from the wake downstream of the leftward fish, as shownin Figure 27D. The leftward fish is always swimming above the wake generated by the rightwardfish. Therefore, as shown in Figures 25 and 26, the effect of interactions is relatively weaker for theswimming of the leftward fish.

The adapted meshes for the periods [4.45, 4.5] and [5.95, 6] are shown in Figure 28A and 28B.The numbers of mesh nodes are 57,128 and 60,052, respectively. The vorticity distributions at theend instants of these two periods are already presented in Figure 27A and 27B.

5.3.4. Two fish swimming in the same directions. The departing position of the two swimming fishis shown in Figure 29. The two fish also deform their bodies in the same way as the single fish do.

Figure 30 shows the development of longitudinal and lateral velocities of the two swimmingfish. The variation of angular position of body with nondimensional time is presented in Figure 31.The results of the single swimming fish are also plotted in these two figures. The vorticity fields att D 2.5, 5, 10, and 16.5 are presented in Figure 32A to 32D. The adapted meshes for [2.45, 2.5]and [16.45, 16.5] are shown in Figure 33A and 33B. The node numbers of global mesh are 52,702and 82,061, respectively. The corresponding vorticity distributions at the end of these two adaptationperiods are already presented in Figure 32A and 32D, respectively.

t0 5 10 15

0

0.5Red: SingleGreen: LeadingBlue: Following

V||, V

Figure 30. Development of the longitudinal velocity Vjj (solid lines) and the lateral velocity V? (dashedlines) of two fish swimming in the same directions.



t0 5 10 15

-20

-15

-10

-5

0

5

10

Red: SingleGreen: LeadingBlue: Following

Figure 31. Development of angular position of body axis of two fish swimming in the same directions.

0

1 2

0

-3

-6 -5 -4

-2 -1 0

1 2

3 4

0

0

1

X

X

X

YY

YY

A

B

C

D

X

Figure 32. Instantaneous vorticity field of two fish swimming in the same directions: (A) t D 2.5,(B) t D 5.0, (C) t D 10.0, and (D) t D 16.5.



At the beginning, the interaction between two fish is weak. As the swimming continues, thelongitudinal velocity of the leading fish is still close to that of the single swimming fish, but themagnitudes of body angle both become larger and larger as shown in Figures 3031. At t 2.5,the following fish encounters the vortices generated by the leading one, and then, it becomes slowerthan the leader (see Figures 30 and 32A). The lateral velocity of follower is greater, and the fol-lower swims above the leader (see Figures 30, 32A, and 32B). As the distance between two fishincreases, their longitudinal velocities become closer, and the lateral velocity of the leader increasesand becomes greater than that of the follower (see Figure 30). The follower now begins to movedown relatively (see Figure 32C). After the follower falls into the wake of the leader, its longitudinalvelocity increases, and it becomes faster than the leader (see Figures 30 and 32D).

To have an idea of the efficiency of the proposed mesh adaptation method, we perform the sim-ulation of this problem for the time period 0, 9 without mesh adaptation. In this simulation, toobtain high resolution of flow near the body surface, the mesh within the region where the swim-ming fish will pass during 0, 9 is refined uniformly in advance. The vortex shedding is also almostin this prerefined region. So, the used mesh is partially uniform. The rectangular prerefined regionis specified according to the previous computation with mesh adaptation, but in general, it cannot bedetermined exactly a priori for fluidstructure interaction problems. We perform the simulation ontwo different uniform meshes. For the coarser one, the mesh spacing near the body surface is 0.006,two times larger than the minimal spacing of the previous adapted mesh, and the total number ofmesh nodes is 203,625. For the finer one, the mesh spacing near the body surface is 0.003, the sameas the minimal spacing of the adapted mesh, and the total number of mesh nodes is 749,274. Dur-ing the period 0, 9, the number of nodes of the previous adapted mesh increases from 40,470 to60,410. Although the governing equations are solved two times, the time cost of the computationwith mesh adaptation is only about one-third of that with the coarser uniform mesh and one-tenth ofthat with the finer uniform mesh. In Figure 34, the computed results of the horizontal force acted onthe follower are compared with that obtained with mesh adaptation. The results on the adapted meshand on the finer mesh are very close to each other and are more accurate and regular than that on thecoarser mesh because of the higher resolution in the boundary layer, the region of vortex shedding,

X

Y

1 2 3

-0.5

0

0.5

A

X

Y

-6 -5 -4

0.5

1

B

Figure 33. Close up view of adapted mesh of two fish swimming in the same directions: (A) t 2[2.45, 2.5]and (B) t 2[16.45, 16.5].



t

Cx

0 2 4 6 8-0.15

-0.1

-0.05

0

0.05

adapteduniform (finer)uniform (coarser)

t

Cx

5 5.2 5.4 5.6 5.8 6-0.15

-0.1

-0.05

0

0.05

adapteduniform (finer)uniform (coarser)

Figure 34. Time history of horizontal force coefficient of the follower calculated on the meshes with andwithout adaptation (left: global view, right: partial view).

and the implementation of boundary conditions [11]. The computation on the global uniform meshof the same spacing as the adapted mesh will be too costly. We did not perform a simulation on it.

6. SUMMARY AND CONCLUSIONS

In this work, we have developed a mesh adaptation method for unsteady flows with movingimmersed boundaries. The mesh constructed in each period is adaptive to the phenomena pro-gression and the motion or deformation of solid bodies. The final solution in the current periodis obtained by recomputation on the adapted mesh. The conservation property of solution is pre-served by using a P1-conservative interpolation to transfer solution from the old mesh to the newone. The local DFD method is employed to solve the flow equations, and the moving-boundaryflows can be simulated on a fixed mesh.

There is no lag between the adapted mesh and the computed solution, and the adaptation fre-quency can be controlled to reduce the errors due to the solution transferring. Several flows havebeen simulated to validate the present mesh adaptation method. The agreement between the com-puted results and the reference data is satisfactory. The number of mesh nodes can be reduced greatlywhen using a nonboundary-conforming method to simulate unsteady flows with moving boundaries.Difficulty associated with the numerical dissipation and efficiently capturing the local details of flowstructure in unsteady flows can be relieved.

ACKNOWLEDGEMENT

This work has been supported by National Science Foundation of China under grant 11072113.

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Mesh adaptation for simulation of unsteady flow with moving immersed boundaries