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Plenary Lectures - Abstracts


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Residual-driven Variational MultiscaleTurbulence Modeling for Large Eddy

Simulation of Incompressible Flow

Y. Bazilevs, V.M. Calo, J.A. Cottrell, T.J.R. HughesInstitute for Computational Engineering and Sciences, The University of Texas at Austin, USA

e-mail: {bazily,victor,jac3,hughes }@ices.utexas.edu A. Reali

Structural Mechanics Department, University of Pavia, Italy e-mail: [email protected]

G. Scovazzi1431 Computational Shock and Multi-physics Department, Sandia National Laboratories, USA

e-mail: [email protected]

Abstract

The objectives of recent variational multiscale work in turbulence have been to captureall scales consistently and to avoid use of eddy viscosities altogether. This holds thepromise of more accurate and efficient LES procedures. In this work, we describe anew variational multiscale formulation, which makes considerable progress toward thesegoals [1].

Keywords: variational multiscale methods, large eddy simulation, turbulence modeling,Isogeometric Analysis, NURBS, incompressible ows, homogeneous isotropic turbulence, tur-bulent channel ows

1 Summary

We begin by taking the view that the decomposition into coarse and ne scales is exact. Forexample, in the spectral case, the coarse-scale space consists of all Fourier modes beneath somecut-off wave number and the ne-scale space consists of all remaining Fourier modes. Con-sequently, the coarse-scale space has nite dimension whereas the ne-scale space is innitedimensional. The derivation of the coarse- and ne-scale equations proceeds, rst, by substi-tuting the split of the exact solution into coarse and ne scales into the Navier-Stokes equations,


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then, second, by projecting this equation into the coarse- and ne-scale subspaces. The projec-tion into coarse scales results in a nite dimensional system for the coarse-scale component of the solution, which depends parametrically on the ne-scale component. In the spectral case,in addition to the usual terms involving the coarse-scale component, only the cross-stress andReynolds-stress terms involve the ne-scale component. In the case of non-orthogonal bases,even the linear terms give rise to coupling between coarse and ne scales. The coarse-scalecomponent plays an analogous role to the ltered eld in the classical approach, but has theadvantage of avoiding all problems associated with homogeneity, commutativity, walls, com-pressibility, etc. The projection into ne scales results in an innite-dimensional system for the

ne-scale component of the solution, which depends parametrically on the coarse-scale com-ponent. We also assume the cut-off wave number is sufficiently large that the philosophy of LES is appropriate. For example, if there is a well-dened inertial sub-range, then we assumethe cut-off wave number resides somewhere within it. This assumption enables us to furtherassume that the energy content in the ne scales is small compared with the coarse scales. Thisturns out to be important in our efforts to analytically represent the solution of the ne-scaleequations. The strategy is to obtain approximate analytical expressions for the ne scales thensubstitute them into the coarse-scale equations which are, in turn, solved numerically. If thescale decomposition is performed in space and time, the only approximation in the procedure isthe representation of the ne-scale solution. To provide a framework for the ne-scale approx-imation, we assume an innite perturbation series expansion to treat the ne-scale nonlinearterm in the ne-scale equation. By virtue of the smallness of the ne scales, this expansionis expected to converge rapidly under the circumstances described in many cases of practicalinterest. The remaining part of the ne-scale Navier-Stokes system is the linearized operatorwhich is formally inverted through the use of a matrix Greens function. The combination of a perturbation series and Greens function provides an exact formal solution of the ne-scaleNavier-Stokes equations. The driving force in these equations is the Navier-Stokes systemresidual computed from the coarse scales. This expresses the intuitively obvious fact that if thecoarse scales constitute a good approximation to the solution of the problem, the coarse-scaleresidual will be small and the resulting ne-scale solution will be small as well. This is the casewe have in mind and it provides a rational basis for assuming the perturbation series converges

rapidly. Note that one cannot use such an argument on the original problem because in thiscase the perturbation series would almost denitely fail to converge. (If we could have used thisargument, we would have solved the Navier-Stokes equations analytically! Unfortunately, thisis not the case.) The formal solution of the ne-scale equations suggests various approximationsmay be employed in practical problem solving. We are tempted to use the word modelingbecause approximate analytical representations of the ne scales constitute the only approxima-tion and hence may be thought of as the modeling component of the present approach, but wewant to emphasize that this is very different from classical modeling ideas which are dominatedby the addition of ad hoc eddy viscosities. We will present numerical results that demonstrate


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that eddy-viscosity terms are unnecessary in the present circumstances. There are two aspectsto the approximation of the ne scales: 1) Approximation of the matrix Greens function forthe linearized Navier-Stokes system; and 2) approximation of the nonlinearities representedby the perturbation series. The rst and obvious thought for the latter aspect, nonlinearity,is to simply truncate the perturbation series. This idea is pursued in conjunction with somesimple approximations of the Greens function. It turns out there is considerable experience inlocal scaling approximations of the Greens function based on the theory of stabilized methods;Hughes [4], Hughes et al. [5], Hughes and Sangalli [6], Hughes, Scovazzi and Franca [7]. TheGreens function is typically approximated by locally dened algebraic operators (i.e., the s

of stabilized methods) multiplied by local values of the coarse-scale residual.An outline of the presentation is summarized as follows: we begin by presenting the math-ematical details of the variational multiscale theory. This represents our general approach toLES-style turbulence modeling and is independent of the specics of the discrete spaces utilizedto represent the coarse scales. The relationship between this version of the variational multi-scale method and classical stabilized methods is delineated. It is noted that that the variationalmultiscale method includes additional terms. Both conceptually and from the point of view of actual implementation, stabilized methods may be viewed as historical stepping stones leadingto the more coherent variational multiscale formulation. We then present our numerical studiesof forced isotropic turbulence at Re = 165 and Re = . (Re is the Taylor microscaleReynolds number.) We begin with a description of the approximation spaces consisting of NURBS elements (non-uniform rational B-splines, see, e.g., Rogers [14], Piegl and Tiller [13],Farin [3], and Cohen, Riesenfeld and Elber [2]). In the case of the rectilinear geometry con-sidered, NURBS reduce to B-splines, which have been advocated for turbulence calculationspreviously (see Kravchenko, Moin and Moser [8], Shariff and Moser [15], Kravchenko, Moinand Shariff [9], and Kwok, Moser and Jimenez [10]). We employ trivariate linear, quadratic,and cubic NURBS with periodic boundary conditions. Linear trivariate NURBS turn out tobe identical to trilinear hexahedral nite elements, but the higher-order NURBS are differentthan classical higher-order nite elements. We perform a dispersion error analysis for NURBSversus classical nite elements on simple, linear, one-dimensional advective and diffusive modelproblems, and conclude that NURBS have better approximation properties than classical nite

elements. We employ meshes of 323

, 643

, 1283

, and 2563

to explore convergence with meshrenement ( h-convergence). We also examine the behavior of increasing order from linear tocubic on xed meshes (k-convergence). In the case of Re = 165, we compare with the DNSspectral results of Langford and Moser [11]. Energy spectra and third-order structure functionsare presented. Sample energy spectra results are presented in Figure 1. In the case of Re = we also clearly see the development of an inertial subrange. We present results for turbulentchannel ows at Re = 395. (Re is the wall-friction Reynolds number.) We employ meshes of 323 and 643. This time the mesh is graded in the wall-normal direction to better capture theboundary layer. Again, we consider convergence from the h - and k-renement perspectives. A


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striking result is how much better quadratic elements are than linear elements. For a mesh of 643, the quadratic and cubic results are essentially identical to the DNS results of Moser, Kimand Mansour [12] for rst- and second-order statistics (see Figure 2), and for a mesh of 32 3 theyare in close agreement. We close with conclusions and suggested future directions for research.

References

[1] Y. Bazilevs, V.M. Calo, J.A. Cottrell, T.J.R. Hughes, A. Reali, and G. Scovazzi. Varia-tional multiscale residual-based turbulence modeling for large eddy simulation of incom-pressible ows. Computer Methods in Applied Mechanics and Engineering , 197:173201,2007.

[2] E. Cohen, R. Riesenfeld, and G. Elber. Geometric Modeling with Splines. An Introduction .A. K. Peters Ltd., Wellesley, Massachusetts, 2001.

[3] G.E. Farin. NURBS Curves and Surfaces: From Projective Geometry to Practical Use . A.K. Peters, Ltd., Natick, MA, 1995.

[4] T. J. R. Hughes. Multiscale phenomena: Greens functions, the Dirichlet-to-Neumannformulation, subgrid scale models, bubbles and the origins of stabilized methods. Computer

Methods in Applied Mechanics and Engineering , 127:387401, 1995.[5] T. J. R. Hughes, G. Feijoo., L. Mazzei, and J. B. Quincy. The variational multiscale

methodA paradigm for computational mechanics. Computer Methods in Applied Me-chanics and Engineering , 166:324, 1998.

[6] T. J. R. Hughes and G. Sangalli. Variational multiscale analysis: the ne-scale Greensfunction, projection, optimization, localization, and stabilized methods. SIAM Journal of Numerical Analysis , 45:539557, 2007.

[7] T. J. R. Hughes, G. Scovazzi, and L. P. Franca. Multiscale and stabilized methods. InE. Stein, R. De Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Me-chanics, Vol. 3, Computational Fluid Dynamics , chapter 2. Wiley, 2004.

[8] A. G. Kravchenko, P. Moin, and R. Moser. Zonal embedded grids for numerical simulationof wall-bounded turbulent ows. Journal of Computational Physics , 127:412423, 1996.

[9] A. G. Kravchenko, P. Moin, and K. Shariff. B-spline method and zonal grids for simulationof complex turbulent ows. Journal of Computational Physics , 151:757789, 1999.


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[10] W. Y. Kwok, R. D. Moser, and J. Jimenez. A critical evaluation of the resolution propertiesof B-spline and compact nite difference methods. Journal of Computational Physics ,174:510551, 2001.

[11] J. A. Langford and R. D. Moser. Optimal LES formulations for isotropic turbulence.Journal of Fluid Mechanics , 398:321346, 1999.

[12] R. Moser, J. Kim, and R. Mansour. DNS of turbulent channel ow up to Re=590. Physicsof Fluids, 11:943945, 1999.

[13] L. Piegl and W. Tiller. The NURBS Book (Monographs in Visual Communication), 2nd ed. Springer-Verlag, New York, 1997.

[14] D. F. Rogers. An Introduction to NURBS With Historical Perspective . Academic Press,San Diego, CA, 2001.

[15] K. Shariff and R. D. Moser. Two-dimensional mesh embedding for B-spline methods.Journal of Computational Physics , 145:471488, 1998.


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0.01

0.1

1

10

1 10

64

32

128

256

DNS

E

k

k 5/ 3

(a) C 0-continuous linear NURBS

0.01

0.1

1

10

1 10

64

32

128

DNS

E

k

k 5/ 3

(b) C 1-continuous quadratic NURBS

64

0.01

0.1

1

10

1 10

32

DNS

E

k

k 5/ 3

(c) C 2-continuous cubic NURBS

Figure 1: Energy spectra for h renement. Re = 165.


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643

DNS

0

5

10

15

20

25

0.1 1 10 1000

5

10

15

20

25

0.1 1 10 1000

5

10

15

20

25

0.1 1 10 100

P=3

P=2

P=1 U + U + U +

y+y+y+

(a) Mean stream-wise velocity

364

0.5

1

1.5

0

0.5

1

1.5

0.5

1

1.5

2

2.5

3

3.5

4

0 100 200 300 400

0.5

1

1.5

0

0.5

1

1.5

0.5

1

1.5

2

2.5

3

3.5

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0 100 200 300 400

0.5

1

1.5

0

0.5

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1.5

0.5

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1.5

2

2.5

3

3.5

4

0 100 200 300 400

P=1

DNS

P=3

P=2

DNS P=1

P=3

P=2

u + u

+ u

+

v + v

+ v

+

w + w

+ w

+

y+y+y+

(b) Velocity uctuations

Figure 2: Turbulent channel ow at Re = 395 computed on a mesh of 643 elements: k-renement interpretation of results.


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Geometric Flow for Quality Surface/Volumetric Modeling(Extended Abstract)

Chandrajit L. Bajaj

Department of Computer Science,

Center for Computational Visualization,Institute for Computational Engineering and SciencesUniversity of Texas, Austin, TX 78712

Email: [email protected]

Guoliang Xu, Qin ZhangLSEC, Institute of Computational Mathematics,Academy of Mathematics and System Sciences,Chinese Academy of Sciences, Beijing, 100080

Email: {xuguo, zqyork }@lsec.cc.ac.cn

Abstract

We present a general variational framework for a higher-order spline level-set (HLS)method and apply this to smooth surface constructions. Starting from a rst order energyfunctional, we derive the general level set formulation, and provide an efficient solution of a second order , time-dependent, geometric partial differential equation (termed geometricow), using a C 2 B-spline basis. We also present a fast cubic C 2 B-spline interpolationalgorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. We providetwo demonstrative smooth surface construction examples of our HLS method. The rst isthe construction of a smooth surface model (an implicit solvation interface) of bio-moleculesin solvent, given their individual atomic coordinates and solvated radii. The second is thesmooth surface reconstruction from a cloud of points generated from a 3D surface scanner.

1 Introduction

Level set methods are increasingly being used in the solution of time-dependent (evolutionary)partial differential equations [6]. Here we consider its application to smooth surface construction

Supported in part by NSF grant CNS-0540033 and NIH contracts P20-RR020647, R01-GM074258, R01-GM07308,R01-EB004873.

Supported in part by NSFC grant 10371130 and National Key Basic Research Project of China(2004CB318000).

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(a) (b) (c)

Fig 1.1: Variational C 2

cubic B-spline Molecular Surface Reconstruction[2, 10, 4]: (a) showsthe van der Waals surface of a molecule. (b) shows the corresponding solvent excluded molecularsurface constructed using our C 2 tri-cubic spline level-set method. (c) illustrates that the smoothsolvent excluded surface constructed tightly encloses the van der Waals surface (a).

by presenting a higher order spline level set generalization and solution of an appropriate geo-metric evolutionary partial differential equation. The level set geometric formulation is derivedby minimizing an energy functional dened with respect to the surface and its rst derivatives.

Given a non-negative function g(x ) over a domain R 3. Find a surface in , such thatthe energy functional

E () = g(x )dx + h(x , n )dx (1.1)is minimal, where x and n is a surface point and its surface normal, respectively. Further, h(x , n )is another given non-negative function dened over R 3 R 3 which is used for regularizing theconstructed smooth surface. Finally, 0 is a constant. Many solid and physical modelingproblems, such as surface (solid boundary) reconstruction, and physically based simulation of deformable interfaces could be formulated as minimizing an energy in the form of (1.1). Byminimizing the energy functional (1.1), a partial differential equation (PDE) in the level-setformulation can be generated. The PDE is solved using the higher-order spline level-set (HLS)method that we present here. Fig. 1.1 and 1.2 show two examples of smooth surface construction,an interface (surface) that separates a molecules atoms from the solvent atoms (typically water),

and a smooth surface reconstruction t to a point set generated by a 3D surface scanner.Why Use a Level-set Method? In shape deformation simulations, topology changes mayoccur. This topology change makes parametric form surface tracking difficult. However, implicitform surface deformation could overcome this difficulty. Implicit surface splines, such as tetra-hedral A-patches and prism A-patches, have been successfully used in computer graphics andsurface modeling in the past decades (see [1, 5, 12] for references), however mostly used in staticsurface modeling. The level-set method described here allows one to dynamically deform andtrack an implicit surface spline using a governing PDE, which describes various laws of motiondepending on geometry, external forces, or a desired energy minimization (see [6, 7, 9, 11]). Fur-

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(a) (b) (c)

Fig 1.2: Variational C 2 cubic B-spline Point - Sampled Surface Reconstruction[]: (a) shows thepoint set data produced by a 3D surface scanner. (b) shows the the reconstruction result usingour C 2 tri-cubic spline level-set method. (c) shows the reconstruction result for a C 0 tri-linearlevel-set method (also generated from our implementation).

thermore, the underlining data structure is simple and topological changes are handled easily,with the computation being restricted to a thin shell (traditionally called a narrow band forevolution) surrounding the level-set.

Why Use a Higher-order Spline Method? The level-set surfaces obtained from classicallevel-set methods are generally bumpy due to the use of piecewise tri-linear interpolation from

the discrete function data computed on a rectilinear grid. To produce a better quality surface,a denser grid needs to be used. However, the increased grid resolution substantially increasesthe computation costs. Another drawback of using discrete data over grids is the non-trivialrequirement of estimating derivatives for smooth interpolation. In many surface constructionproblems, such as the construction of molecular surface, the underlining surface is at least C 1

smooth. Therefore, a smooth level-set function is highly desirable. In this paper, we are solvingsecond-order geometric partial differential equations. In the solutions of these PDEs we utilizeaccurate estimates of mean surface curvature Therefore, we use C 2 tri-cubic spline as the level-setfunction basis. Note that tri-cubic is the lowest order B-spline that could achieve C 2 continuityin 3D. The advantages using C 2 spline function bases include:

1. Since the level-set function is C 2 smooth, the level-set surface is G2 smooth. There do exista nite number of critical level-set values where the level-set may have a nite numberof isolated singular points (i.e the gradient of the level-set function vanishes). Howeverworking in a nite precision numerical domain one automatically avoids these nite set of critical level-set values.

2. Derivatives up to the second order and curvatures, which appear in the governing geomet-ric partial differential equations, are easily and exactly computed from the C 2 level-setfunction.

3. Using smooth level-set functions implies that larger and higher-order spline iso-surface

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patches could be directly generated.

2 Mathematical Notation and Denitions

We rst introduce some useful notation and denitions for geometric quantities on level-sets Min terms of the corresponding level-set function . Let : R be some smooth function ona domain R 3. Suppose M c := {x : (x ) = c} is a level-set of for the level value c.For the sake of simplicity,we simply write M = M c and assume that = 0 on M . Henceby the implicit function theorem, M c is a smooth surface and the normal

n =

on the tangent space T x M is dened for every x on M . Using a co-area formula, the energyfunctional can be dened as

E[] := R

e[M c]dc =

h(x, n)d x.

Next we compute

E [], =dd

E[ + ] =0 (2.1)

= dd ( + ) h x, ( + )( + ) dx =0

=

dd

( ( + ) ) =0 h(x, r)d x

+

hxdxd

+ hnd (+ )

(+ )

d =0

dx (2.2)

The remaining task is to calculate

dd ( ( + ) ) =0 =

, (2.3)

d (+ ) (+ )

d=0

= ()/

2

= 1P, (2.4)

where operator P = I

is the projection onto the tangent space and I indicates the

identity mapping.

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Next we compute x (0) = dx( )d | =0 . Since

(x( )) + (x( )) = c,

Taking differential with respect to , and then taking to be zero, we have

()T x (0) + (x ) = 0 . (2.5)

Let

x (0) = (x )

+ (x )T (x ), (2.6)

whereT (x ) is a tangent vector at x. substituting (2.6) into (2.5), we obtain

(x ) = (x )

.

Since the motion in the tangential direction does not alter the shape of the surface, we ignorethe tangential movement and take

x (0) = (x ) 2

(2.7)

Substituting (2.3), (2.4) and (2.7) into (2.2), we obtain

(2.2) =

h(x, n)

+ (h)T

2+

( n h)T P)

dx (2.8)

Equation (2.8) is the weak formulation of Euler-Lagrange equation. If we utilize the formula of integrate by parts taking into account C0 (), Euler-Lagrange equation

div h(x, n )

+ P n h ( h)T

= 0 (2.9)

is deduced. Obviously, this equation is a second-order nonlinear partial differential equation.If we write the left hand side of (2.9) as an operator acted on function , we can constructgeometric (gradient) ow as follows:

t = L().

For further details, please see [3].

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3 Algorithm Outline

From minimizing the energy (1.1), we obtain the following evolution equation (see [3] for details).

t

= ( g + h)div

+ div(P n h)

+ 2[ (g + h)]T = L() + H (), (3.1)

where

L() = ( g + h)div

,

H () = div(P n h) + 2[ (g + h)]T ,

P = I

is a projection operator onto the tangent space and I indicates the identity

mapping. and n denote the usual gradient operator with respect to x and n . Note thatL() is a parabolic term and H () is a hyperbolic term. Hence, in solving equation (3.1) inthe following, the rst order term H () is computed using an upwind scheme (see [8] for thereason of using an upwind scheme) over a ner grid, the higher order term L() is computedusing a spline presentation dened on a coarser grid.

Consider the solution of equation (3.1) over the domain = [ a, b] [c, d] [e, f ]R 3. For

simplicity, we assume b a = d c = f e > 0. We suppose that the domain is uniformlypartitioned with vertices G0 = {x ijk }nijk =0 := {x i}ni=0 { y j }n j =0 { zk}nk=0 , where

xi = a + i x, y j = c + j y, zk = e + k z,

and x = y = z = ( b a)/n . Let Gl be the set of vertices of the grid which is generated bybinary subdividing G0 uniformly l times. Let be a piecewise tri-linear level-set function overthe grid Gl , be a tri-cubic spline approximation of over the grid G0 . In general, l is chosenas 0 or 1 or 2. In our implementation, we take l = 1. If l = 0, and are dened on the samegrid G0. This is the simplest case. The aim of the following algorithm is to compute the splinelevel set function .

Algorithm Steps .

1. Initialization . Given a initial , construct a piecewise tri-linear level-set function overthe grid Gl . If necessary, apply a re-initialization step to set to be a signed distancefunction to (see [3] for details). Convert to (see [3]).

2. Evolution . Resampling to obtain a new over the grid Gl . Compute L() and H ()in the thin shell N = {(x i , y j , zk ) Gl : |(x i , y j , zk )| < H} . Update in N for one timestep to get by an ODE time stepping method (see [3] for details).

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3. Re-initialization . Applying re-initialization step to in the shell

N = {(x i , y j , zk ) Gl : min 1 ,, 1

|i+ ,j + ,k + | H } .

to get a new (see [3] ). Convert to (see [3] for details). Go back to step 2 if thetermination condition is not satised.

4. Iso-contouring . Extract 3-sided or 4-sided iso-surface patches (vertices with normals) of = c, where c is a given iso-value.

Remark 2.1 . For the problem of molecular surface construction, the grid size G0 should beless than the radii of atoms so that atoms are distinguishable from the level set surface. In ourimplementation, the grid size is chosen to be one-half of the minimal value of the atom radii.

Remark 2.2 . The aim of using l > 0 is to make a more accurate approximation of thesigned distance function. The larger the value of l we use, the better approximation of thesigned distance function we have. Since the scanned data to be approximated in general suffersfrom noise, we use the approximation over a coarse grid G0 for denoising. Furthermore, forgenerating larger level-set surface patches, again a coarse grid needs to be used.

Acknowledgment . A substantial part of this work in this paper was done when Guoliang Xuwas visiting Chandrajit Bajaj at UT-CVC. His visit was also supported in part by the J. T.Oden ICES visitor fellowship.

References

[1] C. Bajaj, J. Chen, and G. Xu. Modeling with cubic A-patches. ACM Transactions on Graphics , 14(2):103133, 1995.

[2] C. Bajaj, V. Pascucci, A. Shamir, R. Holt, and A. Netravali. Dynamic maintenance andvisualization of molecular surfaces. Discrete Applied Mathematics , 127:2351, 2003.

[3] C. Bajaj, G. Xu, and Q. Zhang. A Higher Order Level Set Method with Applications toSmooth Surface Constructions. ICES Report 06-18, Institute for Computational Engineer-ing and Sciences, The University of Texas at Austin, 2006.

[4] C. Bajaj, G. Xu, and Q. Zhang. A Fast Variational Method for Construction of SmoothMolecular Surfaces. ICES Report 08-19, Institute for Computational Engineering and Sci-ences, The University of Texas at Austin, 2008.

[5] C. Bajaj, G. L. Xu, S. Evans R. J. Holt, and A. N. Netravali. NURBS Approximation of A-Splines and A-Patches. Intl. J. on Computational Geometry and Applications , 2001.

[6] S. Osher and R. Fedkiw. Level Set Method and Dynamic Implicit Surfaces . Springer, NewYork, 2003.

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[7] S. Osher and J. Sethian. Fronts propagating with curvature-dependent speed: Algorithmsbased on Hamilton-Jacobi formulations. Journal of Computational Physics , 79:1249, 1988.

[8] S. Osher and C.W. Shu. High-order essentially nonoscillatory schemes for hamilton-jacobiequations. SIAM Journal of Numerical Analysis , 28(4):907922, 1991.

[9] D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang. A PDE-based fast local level setmethod. Journal of Computational Physics , 155:410438, 1999.

[10] Y. Zhang, G. Xu, and C. Bajaj. Quality meshing of implicit solvation models of biomolecularstructures. Computer Aided Geometric Design , 23:510530, 2006.

[11] H. Zhao, S. Osher, B. Merriman, and M. Kang. Implicit nonparametric shape reconstructionfrom unorganized points using a variational level set method. Computer Vision and ImageUnderstanding , 80(3):295319, 2000.

[12] W. Zhao, G. Xu, and C. Bajaj. An Algebraic Spline Model for Molecular Surfaces. InProceedings of the 2007 ACM symposium on Solid and physical modeling , pages 297302,Beijing, China, 2007. IEEE pub.

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Advances in LES-based Turbulence Modeling

Venkat RamanDept. of Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin, USA.

[email protected]

Robert D. MoserDept. of Mechanical Engineering, The University of Texas at Austin, USA.

[email protected]

Abstract

Recent research progress in turbulence modeling are discussed with focus on two top-ics: optimal LES and optimal theory based performance estimation. Optimal LES is anapproach in which the subgrid model is formulated as minimum mean square error es-timate. It has the advantage of being perfectly general, but requires information aboutthe statistics of the small-scale turbulence. New advances in representing the multi-pointcorrelations in both isotropic and wall-bounded ows are discussed, as is the performanceof LES simulations based on these models. In the second part, some recent progressin analyzing the performance of models used to described turbulent combustion are dis-cussed. Based on the concept of optimal error estimation, conventional models for thesub-lter variance of mixture-fraction are analyzed. A new dynamic procedure that pro-vides improved performance is also discussed. Finally, the interaction of numerical errorswith sub-lter models is studied in an effort to identify the more suitable formulations forLES-based combustion simulations.

Keywords: large eddy simulation, optimal estimation, dynamic modeling procedure.

Large eddy simulation (LES) is now considered an attractive tool for studying turbulentows. While many applications of LES have shown very good prediction of the ow eld, manylingering questions regarding sub-lter modeling and the interaction of numerical and modelingerrors still remain. One approach to describing these errors in the optimal LES procedure.

Optimal LES is is based on the observation that the large scale elds being simulated donot provide sufficient information to reconstruct the small, scales, or even the evolution of thelarge scales [1]. The unknown small scales and therefore the LES evolution thus need to betreated statistically. Optimal LES models are formulated by postulating a model dependencyand then minimizing the mean square error in representing the exact model term. Such models


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Figure 1: Three-dimensional energy spectra from nite-volume optimal LES of innite Reynoldsnumber isotropic turbulence using a range of resolutions from 16 3 to 1283 . Also shown is thek 5 / 3 slope and the result of ltering a k 5 / 3 spectrum.

can be formulated in terms of small-separation multi-point velocity correlations. The problemof LES modeling is thus explicitly reduced to the problem of modeling these correlations.Given this information, optimal models can be constructed that account for the errors of thenumerical scheme used to solve the equations[2], and that are valid even in the presence of strong anisotropy and inhomogeneity[3]. Optimal LES is thus one approach which can addressthe shortcomings of current LES models.

The required multi-point correlations include the 2-point second order, 3-point third orderand 4-point fourth order correlations. For high Reynolds number isotropic turbulence, wherea Kolmogorov inertial range exists, models for these correlations are available or have recentlybeen developed[4]. They have been used to construct optimal LES models, which yield re-markably good results. For example, an isotropic LES based on a nite-volume discretization(lter) and the correlation models produces spectra that are consistent with the nite volumeltering of a Kolmogorov k 5 / 3 spectrum (gure 1). In a wall bounded ow, however, modelingthe correlations is more difficult. A new formulation for the anisotropy and inhomogeneity of the two-point second correlation based on the structure tensors of Kassinos et al [5] has beendeveloped. A comparison of thse model correlations with those determined from a DNS isshown in gure 2.

Details of the optimal LES and multi-point correlation modeling approaches will be dis-cussed, as will their appliaction in LES simulation.


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fig1

r 2

0.05 0 0.050.05

0

0.05

fig5

r 2

r10.05 0 0.05

0.05

0

0.05

fig2

0.05 0 0.050.05

0

0.05

fig6

r10.05 0 0.05

0.05

0

0.05

fig3

0.05 0 0.050.05

0

0.05

fig7

r10.05 0 0.05

0.05

0

0.05

fig4

0.05 0 0.050.05

0

0.05

fig8

r10.05 0 0.05

0.05

0

0.05

Figure 2: Comparison of the two-point correlation tensor components from the model andDNS in the x -y plane with no separation in z . The correlations are centered at y+ = 100 in aturbulent channel ow, with Re = 940

In the second part of the lecture, we discuss the performance of models used to describeturbulent combustion. Most combustion models use a passive scalar, termed mixture fraction,to describe the thermochemical state of the gas-phase. In LES, the ltered gas-phase propertiescan be obtained if the sub-lter variance of mixture fraction is known. This measure of sub-lterscalar energy has to be modeled and several models are available in literature. Recently, theoptimal error estimation procedure was used to evaluate sub-lter models [6, 7]. It was foundthat the dynamic models, not surprisingly, provided the least error for a range of lter-widths.However, simple Taylors series based analysis of the dynamic procedure found that certain keyterms are being neglected in the model formulation [7]. When included, the new procedure wasfound to provide lower errors compared to the conventional procedure.

To understand the impact of numerics on model performance, apriori tests were conductedusing different discretization schemes. It was found that the numerical error is of the same orderas modeling error. Further, numerical errors have a benign effect on certain models leadingto reduced overall error. These interesting ndings also indicate that mathematical structureof the model is very important for reducing the inaccuracies due to numerical discretization[8, 9].


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References

[1] J. Langford and R. Moser, Optimal LES formulations for isotropic turbulence, Journal of Fluid Mechanics 398 , 321 (1999).

[2] P. S. Zandonade, J. A. Langford, and R. D. Moser, Finite volume optimal large-eddysimulation of isotropic turbulence, Physics of Fluids 16 , 2255 (2004).

[3] S. Volker, P. Venugopal, and R. D. Moser, Optimal large eddy simulation of turbulentchannel ow based on direct numerical simulation statistical data, Physics of Fluids 14 ,3675 (2002).

[4] H. Chang and R. D. Moser, An intertial range model for the three-point third-ordervelocity correlation, Physics of Fluids 19 , 105111 (2007).

[5] S. Kassinos, W. Reynolds, and M. Rogers, One-point turbulence structure tensors, Jour-nal of Fluid Mechanics 428 , 213 (2001).

[6] A. Moreau, O. Teytaud, and J. P. Bertoglio. Optimal estimation for large-eddy simulationof turbulence and application to the analysis of subgrid models. Phys. Fluids , 18 , 1-10,2006

[7] G. Balarac, H. Pitsch and V. Raman, Modeling of sub-lter scalar variance using theconcept of optimal estimator, Physics of Fluids , 20 035114, 2008

[8] G. Balarac, H. Pitsch and V. Raman, Modeling of the sub-lter scalar dissipation rateusing the concept of optimal estimators, Submitted to Physics of Fluids , 2008

[9] V. Raman, G. Balarac, and H. Pitsch, Minimizing numerical errors in the computationalsub-lter scalar variance, To be Submitted to Physics of Fluids , 2008


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Modeling Two Phase Flow Dynamics for Deformable Interfaces(Extended Abstract)

Chandrajit Bajaj, Albert Chen, Richard Hankins, Bong-Soo SohnDepartment of Computer Science,

Center for Computational Visualization,Institute for Computational Engineering and Sciences

University of Texas, Austin, TX 78712Email: [email protected]

Abstract

We describe a Stokesian (slow viscous) ow model for producing interacting deformable sur-faces. This captures several phenomena in their nativity, such as cell membranes interacting, or softdocking of exible molecules, etc. Starting from any initial conguration of closed, compact inter-faces, the interfaces are continually evolved, as well as deformed based on the relative viscositiesand the interfacial tension. The velocity computation and the interfacial dynamics are achieved via

a Boundary Element formulation of the governing Stokesian ow equation, while the interface evo-lution and topology maintenance utilizes level set representation and underlying function updates.Effects such as coalescence, break-up, and additional near interface interactions, can also be accu-rately captured. These last three effects in particular require adaptive renement of meshed geometryand controlled coupling of the numerical errors in computation to yield topologicallyrealistic looking phenomological modeling.

1 Introduction

We present a technique to model the interaction of deformable surfaces using two-phase Stokesian (slowviscous) ows. These interfaces can represent air bubbles in a viscous liquid, oil droplets in a suspension,or cellular membranes subjected to hydrodynamics forces.

Two-phase uid simulations have been a topic of interest in computer graphics. In [5] Foster andMetaxas describe a technique for simulating free-surface water ows by solving the full Navier-Stokesequations in three dimensions via a nite difference scheme and representing the free surface with mass-less marker particles. In [4] Foster and Fedkiw implement a solution to the full Navier-Stokes equationsusing a modied version of the semi-Lagrangian technique introduced by Stam in [10] to capture thecomplex behavior of free water surfaces. To represent the air-water interface, they used a level set methodas introduced in [9]. In [3] a particle level set method was introduced for improved interface capturing.Massless marker particles are advected with the level set data and used to repair the level set in regionsof degradation due to the use of a coarse grid for animation. The combination of a semi-Langrangian

1


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Navier-Stokes solver coupled to a particle based level set method allows for complex modeling of thedynamics at the air-water interface. In all of the above treatments, the density of the air is assumed to be zero. This is a reasonable approximation as the density of air is 1000 times less than that of water.A constant pressure boundary condition is applied everywhere at the interface and the Navier-Stokesequations are solved only within the liquid region. Using these techniques, while visually realistic sim-ulations can be achieved which capture the convective (and turbulent) interaction of gases and liquids,many interesting features of air-water interfacial dynamics arising from non-convective, non-turbulent,and slow viscous ows cannot be observed. These interfacial deformations and dynamics arise frominterfacial tension and the curvature of the interface. In order to capture such interfacial deformationsin a simulation, for even visual realism, the Navier-Stokes equations must be accurately solved in both

uid regions, and pressure boundary conditions must be applied at the two phase uid interface, whilecarefully orchestrating the accuracy of interfacial geometry and the interfacial velocities. Such an accu-rate two phase uid simulation to produce realistic visual animations of deforming interfaces, is the maincontribution of this paper.

In [7], Hong and Kim with perhaps similar goals to ours, modied the semi-Lagrangian scheme [10]coupled with the volume-of-uid method (VOF) introduced in [6] to simulate bubbles in liquids. The au-thors calculate the interfacial tension, and thereby are able to capture certain interfacial deformations, theVOF method has accuracy limitations for effects such as bubble attening, coalescence, bubble necking,and break-up, and additional subtle near bubble interaction. We achieve this accuracy through our pre-cise representation of the interface geometry, coupled to a topology tracking technique and a stable andaccurate boundary element uid solver, allowing us to observe these phenomena at signicantly higher

resolution. Our solution can be broken down into three main sub-areas: accurate boundary element meshrepresentation of the interface, careful error bounded calculation of physical quantities (velocities, sur-face tension) on the interface by regularization and adaptive mesh renement, and topological trackingfor precise near-bubble interactions.

For the calculation of physical quantities we use the boundary element method (BEM) on adaptivegeometries. Given an initial conguration of bubbles, our adaptive BEM solver, estimates the veloci-ties on the interfacial boundary with greater accuracy, due to several advantages it has over competingmethods. It reduces the dimensionality of the problem by one, and focuses computational effort on the boundary which, for two phase simulations, is the region of interest thereby yielding superior accuracyof the BEM over both nite element methods (FEM) and nite difference methods (FDM), where theinterface is represented indirectly using a discretized volume domain.

We also present a dynamic remeshing algorithm for smooth evolving interfaces of an objects such as bubbles. The interface is discretized to triangular or quadrilateral mesh where the velocity of each vertexis computed by the BEM. Since an interface changes its geometric shape including surface area andcurvature distribution, meshes with xed vertex count and connectivity cannot represent the evolvinginterfaces accurately. Therefore, the number of vertices and vertex connectivity need to be adjustedaccording to given geometric properties of an interface for each timestep. The quality of triangular or quadrilateral elements, often measured with element shape, also needs to be good enough for accurateBEM calculation. The main difculty occurs when the topology of interfaces change. For instance,two bubbles may coalesce or a single bubble may break up into two bubbles. We utilize up-sampling /down-sampling methods and maintain a dynamic octree for tracking and controlling the mesh topology.

2


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F j( x0) =2

1 + (u j ( x0)

18 s f ( x)ni( x)G i j ( x, x0)d ( x)) (2.2)

2.3 Numerical Technique

In the above relations we assume N bubbles with varying viscosities immersed in a Stokes uid. Theviscosity ratio of the bubble to the uid it is embedded in is, = s , and u

j is the asymptotic Stokes

velocity that the bubbles are immersed in. The function f is the boundary condition specifying the pressure difference at the interface and is given by, f = 2 + ( fluid bubble ) z . In the above, isthe surface tension of the drop, n is the normal pointing into the suspending uid and = 12 n is theextrinsic mean curvature of the boundary. For our calculations we use the free space Greens functionsfor Stokes ow given by,

G i j( x x0) = i jr

+ xi x jr 3

, x = x x0

and

T i jk ( x x0) = 6 xi x j xk

r 5

We generate a quadrilateral mesh from our B-spline level set approximation[11], to represent theinterface separating the two uids and discretize the integral equation. We break up the integrals over each surface into a sum of integrals over each quadrilateral face of the mesh. The integration over most

faces may be done using standard quadrature techniques. However, the Greens functions appearing inthese expressions show divergent behavior as the evaluation point approaches the surface over which weare integrating. These singular and near-singular integrals must be handled carefully in order to obtainaccurate results from a BEM when two droplets are close. Details are given in [1].

2.4 Error Analysis

Proper error analysis is important for our algorithm. We use this information to determine the validityof the boundary element calculation and how to improve it by surface mesh renement. We describetwo different error tools that we use for feedback during the simulation. The rst one is a global error measure and derives from the incompressibility condition of the governing equations we are using tomodel the uids. This condition in integral form becomes,

u nd = 0 (2.3)We calculate this value and use it to determine which octree level to mesh the geometry for each timestep.The other method we implement is a local measure of the error. We use bilinear interpolation whencalculating values on the surface from the values at the vertices which were obtained through solving the boundary element system. We implement a technique to estimate the error at each face of the mesh andadd more geometry if the error is outside acceptable tolerance. The error analysis implemented is that

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of [8]. The technique is chosen based on computational speed and ease of implementation. The error onthe m-th face is estimated to be

E u(m)2 = m | u0 u |2 d m (2.4)where u0 is the predicted exact solution which we approximate by higher order interpolation and u is thenumerical solution for the velocity. Finally we measure the relative error by calculating,

E rel ,u(m)2 =E u(m)2

| u0 |2 d

(2.5)

While rigorous mathematical bounds do not exist for local errors in collocation methods this techniqueserves its purpose in quantifying the error regions of sparse geometry where the calculation could beimproved and have been implemented by authors in boundary element methods as well as nite elementmethods [12] [13]. This error analysis is done at each time step following the boundary element calcu-lation of the interfacial velocities. Given a user dened tolerance 1 > 0 we check that the velocity error computed over each face of the mesh satisfy E rel ,u < 1 . If this condition is satised then we proceedwith the interface evolution. If it is not satised then we rene the faces of the mesh where the toleranceis exceeded and recalculate the interfacial velocities using this rened mesh.

2.5 Topology Control

Assume f t is a function at time t where its level set represents deformable interfaces. f t is a piecewisetrilinear function that approximates f t . M t is a mesh that approximates the level set. The level settopology dened in f t is preserved during mesh extraction process.

Boundary Element Method (BEM) is applied to computing velocities which can be used for updatingthe function f t at time t to evolve the interface in viscous ows. This generates the function f t + 1 at timet , where level set mesh M t + 1 can be extracted.

Topology changes of bubbles may occur under various conditions such as minimum distance or con-tact surface area between two bubbles. The moment of topology changes can be also chosen manually.We introduce an oracle that is an independent procedure to decide whether the topology change occursor not based on the user-specied conditions. We also need a remeshing procedure to actually changethe topology of meshes for the bubbles if the oracle decides that. Our algorithms for the oracle and

remeshing support coalescence and breakup of bubble interfaces. Details are given in [1].

2.6 Interface Update

The interface is updated using our higher order level set method [2]. The evolution of the level set isgoverned by the level set equation,

i t

+ v i = 0

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(a) t = 1 (b) t = 40 (c) t = 51

(d) t = 66 (e) t = 100 (f) t = 200

Fig 3.1: Bubble coalescence. Several timesteps are shown here depicting bubble coalescence. Bubblesdont tend to coalesce easily in pure Stokes ow. For the simulation a term was added to the uid solver to simulate the intermolecular forces at work that cause coalescence. We see that the bubbles tend toatten out as they approach each other. A sudden joining occurs after they have been in close proximityfor enough time. The joined bubble returns to a spherical shape due to surface tension.

3 Results

3.1 Implementation

We have an implementation of the above technique that runs on a Linux platform. The implementationconsists of two main libraries: a meshing library, and a boundary element calculation library. Thesetwo sets of code are packaged in a graphical user interface where initial data can be input and physical parameters are dened. The package is also capable of outputting mesh data for analysis and rendering.

Figure 3.1 shows results of a simulation of two bubbles coalescing. The input data are two uniformspheres separated by a small distance. The viscosity ratio is = 0.5 and a small gravity eld is enabledso that the bubbles rise slightly due to buoyant forces. There is no asymptotic ow in the suspendinguid. The simulation shows that bubbles tend to exist in a attened state before intermolecular forcestake over and the bubbles ultimately join.

Figure 3.2 shows two droplets deforming as they pass each other in a shearing ow given by u =( z , 0, 0) where is a user dened parameter that controls the strength of the ow that can be used to

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(a) t = 1 (b) t = 20 (c) t = 30

Fig 3.2: Bubbles deforming in shear ow. This simulation shows bubbles deforming as they move pasteach other in a shearing ow. The viscosity ratio for the simulation is 0.5 and a small gravity eld isapplied to make the bubbles rise slightly. The shear ow is keyframed to slowly dissipate. We observethe shearing and deformations as the bubbles interact and then return to their spherical shapes as thespeed of the ow decreases.

control the strength of the eld. For this animation we set = 1 and add a small gravity eld. We observethe bubbles slowly rising and moving past each other. As the lower bubble rises it begins to move to theright as it moves to a height where the asymptotic ow is positive. The ow is keyframed to graduallydissipate over the coarse of the simulation and we see the bubbles slow down and return to their spherical

shapes.Within our simulations there are several parameters which may be adjusted to create a desired sim-

ulation based animation. The user can specify an initial conguration of droplets and a Stokes owto embed the droplets in. This ow has parameters which control the speed and variation in the pres-sure gradient along the different spatial axes. In addition, the strength of the gravity eld or other longrange body forces may also be specied by the user and short range forces may be added to simulateintermolecular interactions. Adjustment of these parameters leads to very different results.

The viscosity ratio parameter is particularly important. As a special case we set = 1 in the boundaryintegral equation eliminating the integral over the double layer potential. The result is that there is nolinear system to solve so interfacial velocities are computed simply by evaluating,

uk ( x

0) =

2

1 + (u

j( x

0)

1

8 s

f ( x)ni( x)G

i j( x, x

0)d ( x))

It should be noted that all phenomena of bubble interactions mentioned here can be observed in thisspecial case, but the computational demands are much less than the = 1 solution.

Acknowledgment . This research was supported in part by NSF grants IIS-0325550, CNS-0540033 and NIH contracts P20-RR020647, R01-EB00487, R01-GM074258, R01-GM07308

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References

[1] C. Bajaj, A. Chen, R. Hankins, and B. Sohn. A C 2 cubic B-Spline Level Set Approach to Simulat-ing Deformable Interfaces. TICAM Report 08-xx, Texas Institute for Computational and AppliedMathematics, The University of Texas at Austin, 2008.

[2] C. Bajaj, G. Xu, and Q. Zhang. A higher order level set method with applications to smooth surfaceconstructions. TICAM Report 06-18, Texas Institute for Computational and Applied Mathematics,The University of Texas at Austin, 2006.

[3] J. Ferziger D. Enright, R. Fedkiw and I. Mitchell. A hybrid particle level set method for improved

interface capturing. J. Comp. Phys. , 183:83116, 2002.

[4] N. Foster and R. Fedkiw. Practical animation of liquids. In ACM Press/ACM SIGGRAPH, E. Fiume, Ed., Computer Graphics Proceedings, Annual Conference Series , pages 2330, 2001.

[5] N. Foster and D. Metaxas. Realistic animation of liquids. Graphical Models and Image Processing ,58(5):471483, 1996.

[6] C.W. Hirt and B.D. Nichols. Volume of uid (vof) method for the dynamics of free boundaries. J.Comp. Phys. , 39:201255, 1981.

[7] J.-M. Hong and Kim C.-H. Animation of bubbles in liquid. In In Proceedings of Eurographics2003 , pages 253262, 2003.

[8] E. Kita and N. Kamiya. A new adaptive boundary element renement based on simple algorithm.Mech Res Commun. , 18(4):17786, 1991.

[9] S. Osher and J.A. Sethian. Fronts propagating with curvature dependent speed: algorithms basedon hamilton-jacobi formulations. J. Comp. Phys , 79:1249, 1988.

[10] J. Stam. Stable uids. In In Proceedings of SIGGRAPH 99, ACM SIGGRAPH/Addison Wes-ley Longman, Computer Graphics Proceedings, Annual Conference Series, ACM , pages 121128,1999.

[11] Y. Zhang and C. Bajaj. Adaptive and quality quadrilateral/hexahedral meshing from volumetricdata. Computer Methods in Applied Mechanics and Engineering (CMAME) , 195:942960, 2006.

[12] O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practicalengineering analysis. Int. J. Numer. Methods. Engng. , 24:33757, 1987.

[13] O.C. Zienkiewicz, J.Z. Zhu, and N.G. Gong. Effective and practical h-p-version adaptive analysis procedures for the nite element method. Int. J. Numer. Methods. Engng. , 28:87991, 1989.

[14] Z.A. Zinchenko, M.A. Rother, and R.H. Davis. A novel boundary integral algorithm for viscousinteraction of deformable drops. Phys. Fluids , 9:14931511, 1996.

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Isogeo m etric Analysis of Fluid-StructureInteraction

Y. Bazilevs, V.M. Calo, T.J.R. HughesInstitute for Computational Engineering and Sciences, The University of Texas at Austin, USA

e-mail: {bazily,victor,hughes }@ices.utexas.edu

Y. ZhangDepartment of Mechanical Engineering, Carnegie Mellon University, USA [email protected]

Abstract

Isogeometric analysis is a recently developed methodology based on technologies thatwere originated in the eld of computational geometry and widely used in design, graphicsand animation. It includes standard nite element analysis as a special case, but offers

other possibilities that are unique and powerful. It allows more precise and efficientgeometric modeling, it simplies mesh renement, and it possesses superior approximationproperties. Isogeometric analysis has been applied to numerous problems in solid anduid mechanics. This paper describes an isogeometric formulation for uid-structureinteraction of incompressible uid ow and nonlinear solids. The uid discretizationderives from a residual-based variational multiscale formulation, applicable to laminar andturbulent phenomena. Both uid and solid domains may undergo large motions, and thegeometry and kinematics are fully compatible across uid-structure interfaces. A stronglycoupled solution algorithm is adopted to preclude instabilities that often afflict weakly-coupled procedures. Applications to the human cardiovascular system are emphasized.

Keywords: Isogeometric Analysis, NURBS, Fluid-Structure Interaction, Vascular Model-ing, Navier-Stokes Equations, Elastic Arterial Wall, Mesh Movement, Blood Flow

1 Introduction

Isogeometric Analysis based on NURBS (non-uniform rational B-splines) was rst introducedin [1] as an attempt to improve on and generalize the standard nite element method. Fur-ther study of isogeometric analysis showed that results superior to standard nite elements areobtained in the context of structural vibrations [2]. Mathematical analysis of the isogeometric


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approach was performed in [3]. Optimal approximation estimates in p, the polynomial orderused to dene NURBS functions, were obtained for h -rened meshes. Stability and optimal con-vergence was proved mathematically and veried numerically for problems of compressible andincompressible elasticity, Stokes ow, and scalar advection-diffusion. In this paper, NURBS-based isogeometric analysis is applied to uid-structure interaction (FSI) problems with partic-ular emphasis on arterial modeling and blood ow (see [4] for a more detailed exposition). It isbelieved that the ability of NURBS to accurately represent smooth exact geometries, that arenatural for arterial systems, but unattainable in the faceted nite-element representation, andthe high order of approximation of NURBS, should render uid and structural computations

more physiologically realistic.This work adopts the arbitrary Lagrangian-Eulerian (ALE) framework. The arterial wallis treated as a nonlinear elastic solid in the Lagrangian description governed by the equationsof elastodynamics. Blood is assumed to be a Newtonian viscous uid governed by the incom-pressible Navier-Stokes equations written in the ALE form. The uid velocity is set equal tothe velocity of the solid at the uid-solid interface. The coupled FSI problem is written ina variational form such that the stress compatibility condition at the uid-solid interface isenforced weakly. The ALE equations require the specication of the uid region motion. Thismotion is found by solving an auxiliary static linear-elastic boundary-value problem for whichthe uid-solid boundary displacement acts as a Dirichlet boundary condition.

Galerkins method is employed for the structural and the uid subdomain motion parts of the formulation, while a residual-based multiscale method for the uid equations. The resultantsemi-discrete equations are advanced in time using the generalized- algorithm. The kinematicconstraint is enforced strongly by requiring basis functions to be C 0 -continuous across theuid-solid interface. The coupled nonlinear system resulting from the NURBS discretizationof the FSI equations is solved monolithically, that is, the uid, the structural, and the meshsolution increments in the Newton iteration are obtained simultaneously. The effect of thestructural and the mesh motion on the uid equations is included in the left-hand-side matrixfor robustness. The coupled system is solved iteratively by the GMRES procedure with simplediagonal scaling.

2 Nu m erical exa m ples

2.1 Blood ow in an idealized aneurys m

In this test case, taken from [5], we examine pulsatile ow in an idealized aneurysm. Theproblem setup is shown in Figure 1. A time-periodic velocity waveform, specied at the inowplane, is parabolically distributed over the circular surface. The domains proximal and distalto the aneurysm region are assumed to have rigid walls, while the aneurysm wall is elastic. Aresistance boundary condition is applied at the outow.


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Figure 1: Idealized aneurysm problem setup.

Figure 2 shows the inow and outow waveforms. Note the outow lags the inow dueto the distensibility of the aneurysm wall. This well-known phenomenon was also observed inpractice as well as computations of other researchers. Figure 2 also shows excellent agreementwith reference results of [5].

2.2 Blood ow in a patient-specic abdo m inal aorta

This computational example makes use of patient-specic geometry. Data for this model wasobtained from 64-slice CT angiography of a healthy male over 55 years of age. Some prepro-cessing, including contrast enhancement, denoising, and segmentation, was necessary in orderto nd the lumenal surface of the blood vessel. Figures 3(a) - 3(c) show the geometrical model,the control mesh, and the NURBS mesh of the abdominal aorta. As in the previous example,we specify a time-periodic velocity waveform at the inow, while all outows are assigned a re-sistance boundary condition. Figure 3(d) shows contours of the arterial wall velocity magnitudeplotted on a deformed conguration during early systole.

References

[1] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, nite elements,NURBS, exact geometry and mesh renement. Computer Methods in Applied Mechanicsand Engineering , 194 (2005) 4135-4195.


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Figure 2: Idealized aneurysm. Inow and outow waveforms. Notice the time lag attributableto the distensibility of the wall.

[2] J.A. Cottrell, A. Reali, Y. Bazilevs, and T.J.R. Hughes. Isogeometric analysis of structuralvibrations. Computer Methods in Applied Mechanics and Engineering , 195 (2006) 5257-5296.

[3] Y. Bazilevs, L. Beirao da Veiga, J.A. Cottrell, T.J.R. Hughes, and G. Sangalli. Isogeometricanalysis: Approximation, stability and error estimates for h-rened meshes. Mathematical Methods and Models in Applied Sciences, 16 (2006) 1031-1090.

[4] Y. Bazilevs, V.M. Calo, Y. Zhang, and T.J.R. Hughes. Isogeometric uid-structure in-teraction analysis with applications to arterial blood ow. Computational Mechanics , 38(2006) 310-322.

[5] A.-V. Salsac, M.A. Fernandez, J.-M. Chomaz, and P. Le Tallec. Effects of the exibility of the arterial wall on the wall shear stress and wall tension in abdominal aortic aneurysms.

Proceedings of the 58th Annual Meeting of the Division of Fluid Dynamics , Chicago, IL,November 2005.


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Figure 3: Patient-specic abdominal aorta. (a) Geometrical model; (b) Control mesh; (c)NURBS mesh; (d) Contours of arterial wall velocity magnitude plotted on a current congura-tion during early systole.


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Modeling Co m plexities in Turbulent SprayCo m bustion

Venkat RamanDept. of Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin, USA.

[email protected]

Abstract

Turbulent spray combustion is a common process found in aircraft and automobileengines as well as chemical reactors. Spray combustion is a multilscale multiphysics prob-lem involving the nonlinear interaction of spray evolution and evaporation, gas-phaseturbulent mixing, and combustion. A detailed description of this complex process re-quires models for each of the physical processes involved. In this study, we focus on thedevelopment of the multiphase combustion models. Currently, spray combustion mod-els are directly extended from corresponding single-phase combustion models. However,combustion in the presence of evaporating droplets has many unique features that renderinvalid many of the assumptions underlying the single-phase models. The objective of thiswork is to introduce a novel probability density function (PDF) based approach, whichaddresses many of the multiphase combustion modeling challenges. Numerical algorithmsand preliminary results are presented here.

Keywords: Spray combustion , large eddy simulation, probability density function, com-bustion regime.

Liquid fuel based combustion is widely encountered in aircraft and automobile engines aswell as chemical reactors. In typical combustors, the liquid fuel is sprayed using an atomizerthat produces a ne mist of droplets. The droplets evolve in the background turbulent gas-phase ow and evaporate. This fuel vapor then mixes with the gas-phase oxidizer and reactsin a high-temperature environment. The proper dispersion of fuel inside the combustor iscritical in maintaining stability and in reducing emissions. The complete description of thespray combustion system requires models for tracking the spray droplets, gas-phase turbulentow, and combustion. Since the focus of this work is the modeling of the combustion process,standard and state-of-the-art approaches for the other two components will be used. The spraydroplets will be evolved using a Lagrangian approach. The gas-phase turbulent ow will bedescribed using the large eddy simulation (LES) method.


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Single-phase combustion models are derived based on the nature of the combustion regime.For instance, the fuel and oxidizer could be molecularly mixed before entering the combustionchamber leading to premixed combustion. If the fuel and oxidizer mix insider the combustor,a non-premixed combustion process is supposed to exist. The controlling parameters andthe ame structure are vastly different in the two cases giving rise to very different combustionmodels. Since the combustion regime is determined by the boundary conditions, the combustionmodel can be chosen apriori . In spray combustion, the fuel is released in vapor form throughdroplet evaporation, which in itself depends on the physical dispersion of droplets and therate of liquid evaporation. Both these processes are dependent on the gas-phase turbulence

and combustion. Consequently, the combustion regime can change within the reactor due toa number of reasons including the relative rates of evaporation and mixing, effect of dropletinertia on the gas-phase turbulence, and the spatial distribution of the reaction zone [1]. Fig. 1shows the ame structure for two different droplet structure in a coowing oxidizer stream.When the droplets are close enough, the ame is pushed outside and a ame front typicalof premixed combustion is formed. If the droplets are dispersed far enough, the ame frontadvances into the inter-droplet spacing leading to a partially-premixed combustion process.While these cases demonstrate the extreme effects, droplet structure and its evolution play acritical role in determining the combustion process. For this reason, a combustion model thatpresumes the combustion regime will not be fully valid in spray combustion.

In this work, a novel method termed the probability density function (PDF) approach isused [2]. The main advantage in this approach is that the chemical source terms appear closedand do not require modeling. This, in turn, implies that the combustion regime is not xed ina simulation. While this is certainly advantageous, mixing of scalars has to be modeled andposes a tremendous challenge. In the lecture, models recently proposed by the author to addressthis issue will be discussed. In the PDF approach, the transport equation for the joint-PDFof all chemical species and other scalars that dene the thermochemical state of the systemis solved directly. This transport equation is high dimensional and cannot be solved usingconventional nite-volume or nite-difference discretization scheme. Typically, a LagrangianMonte-Carlo method is used. In practical simulations, this PDF solver is coupled with the sprayand turbulence models and evolved in a temporally accurate manner. This coupled solver was

used to simulate canonical ows in an effort to verify and validate this computational tool(Fig. 2). The nature of information exchange between the solvers is very important in thecontext of spray combustion. It will be demonstrated that the current techniques for simulatingsuch ows change the underlying physics of the combustion process.


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References

[1] V. Raman and H. Koo, Effect of Droplet Inertia on Evaporation and Spray Combustion,Submitted to the The 32nd International Combustion Symposium , 2008

[2] V. Raman and H. Koo, LES/Filtered-Density Function Approach for Turbulent SprayCombustion, Submitted to Combustion and Flame , 2008

Figure 1: Multiple droplet combustion with detailed resolution of the droplet interface. Dropletsare dispersed closely (right) and (right) far apart.

Figure 2: Spray droplet population superimposed on the instantaneous contours of gas-phasetemperature. The inset shows more details about the reaction zone.


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plenary lectures merge b5

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