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1448
Coupled Eulerian-Lagrangian simulations using a level set method Sean Mauch ^, Dan Meiron ^'*, Raul Radovitzky ^, Ravi Samtaney ^ '^ Applied and Computational Mathematics, Caltech, Pasadena, CA 91125, USA ^Aeronautics and Astronautics, MIT, Cambridge, MA 02139, USA ^ Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0541, USA
Abstract We present an algorithm for the coupled solution of high velocity fluid-structure interaction problems in which the simulation of the fluid is accomplished using Eulerian methods whereas the simulation of the solid utilizes a Lagrangian approach. The algorithm is a variant of the Ghost Fluid Method (Fedkiw [1]) coupled with a new linear time approach to computing the distance function. Several example applications are presented. Keywords: Fluid-structure interaction; Level set; Eulerian-Lagrangian coupling; AMR; Ghost fluid method
1. Introduction In the simulation of high velocity fluid and solid mechanics (such as arise for example in penetration and impact problems) it is sometimes advantageous to utilize a Lagrangian description for the solid and an Eulerian description for the fluid. The Lagrangian description is a natural approach to describing phenomena such as plastic deformation of the solid while the Eulerian approach deals well with instances where there exist regions of large strain rate but where strength effects are unimportant. There exist several approaches in the literature that adopt this approach. For example one can track the evolution of the solid boundary and compute, at every time step, a conforming fluid mesh. Alternatively one can use Arbitrary Lagrangian Eulerian methods (ALE) to allow the coordinate system to transition smoothly from pure Lagrangian in the solid to pure Eulerian in the fluid. In this paper we present an alternative approach wherein the solid boundary is captured using level set based techniques. The level set information is used to inform the fluid mechanics solver of the location, velocities and accelerations of the solid and in turn can inform the solid solver of the load induced by the fluid. An advantage of this approach is that there is no need to generate dynamically conforming meshes for the fluid mechanics. While the current version of this algorithm has first order accuracy, it has the advantage that it allows one to easily couple di-
verse solid and fluid mechanics approaches even on parallel architectures. One potential difficulty with this approach is the need to repeatedly generate a distance function in the fluid domain from the solid boundary. In three dimensions, conventional algorithms make such a procedure potentially more costly than the solution of the fluid or solid mechanics problems which can effectively be performed in a time that is linearly proportional to the number of mesh points in the fluid or elements in the solid. To overcome this difficulty we also present a new linear time algorithm for generation of the level set from a triangular surface mesh. We describe in this paper the major aspects of the solvers and coupling algorithm. We then provide some examples of this approach implemented on parallel architectures.
2. Eulerian fluid mechanics We begin by describing the Eulerian fluid mechanics algorithm We solve the Euler equations, written below in strong conservation form, for inviscid compressible flow:
u,+^,(U)+GyiV)+n,(V) = 5(U), where
(1)
I] = {p,pu,pv,pw,E,pX}^
(2)
^(U) = {pu,pu^ -h /7, puv, puw,(E + p)u, pXuY ^(U) = {pv,puv,pv^ -\-p,pvw,{E -\- p)v,pXv}^
* Corresponding author. Tel.: -hi (626) 395-8157; Fax: +\ (626) 568-9102; E-mail:
[email protected]
H{\]) = {pw,puw,pvw,pw^
-\- p,{E + p)w,pXwY.
© 2003 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics 2003 K.J. Bathe (Editor)
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(3)
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S. Mauch et al. /Second MIT Conference on Computational Fluid and Solid Mechanics The above equations are closed by a general equation of state (EoS) expressed functionally as: xiP^^^P^^) = 0. In order to deal with more general EoS we have extended the work of Glaister [2] so that the solver can deal with multiple species. Given left and right states (ULMR) at a cell interface / + 1 / 2 between cells / and / - h i , the Roe [3] flux is 1
F(ULMR)=-(F(UL)
1
+
F(UR))--S-'\A\S{UR-UL).
(4) Here S is the matrix of left eigenvectors of the Jacobian matrix A = ( § ) , 1
1
0 0
1
0
u—c
u
0 0
u+c
0
V
V
1 0
V
0
w
w
0
1
w
0
ho- - p c '/Pe
V
w ho-^u c -Px/Pe
S= 0+ U C
X
X
0 0
k
^ = {^;^ J / = 1... L, n = 1... A^, m = 1... M} ^Imn = {[XiJ,k.Xi+\,i,k^
X {yi,j,k,yi,j + \,k\ X U/J,/t,Z/J,^+l]},
/ = !...//, j = \...Jm.
8Q = {Sp\p = l...Pl
1
5~^ are the right eigenvectors of Jacobian A, and A is the diagonal matrix of the eigenvalues of A
k=\...Kn.
(8)
Sp = {A,l
q = \...Qp,
(9)
i.e., the Lagrangian boundary comprises of P triangulated surfaces, Sp. The pressure, velocity and position of the nodes on the Lagrangian boundary are exchanged. Given the Lagrangian boundary, a signed distance level-set function is computed using the demonstrably optimal Closest Point Transform (CPT) algorithm (10)
0
0
0
0
(t>{Xi,j,k J) = C,mm[d(Xij,k,Sp),p = 1... P]
where Cs = -\-l in the solid and C^ = — 1 in the fluid. Note that the level set (j){xij^k^t) = 0 defines 8Q. Thermodynamic variables are then extrapolated by advection in pseudotime. We define xfr = (p, p) if the perfect gas EoS is used, or T/T = (^,p,A,i,...,A„) for the general EoS solver. Then,
0
u
0
0
0
0
0
0
u
0
0
0
0
0
0
u
0
0
0
0
0
0
u-\-c
0
0
0
0
0
0
u
A =
(6)
(11)
The sound speed c is given by the following general thermodynamic relation: .-{^P\
coupling is explicit. The two solvers exchange information at the beginning of every time step. A key idea of this coupling algorithm is that the zero mass flux boundary condition for the Euler equations at the interface between the fluid and the solid be strictly enforced. The traction boundary condition is applied via imposition of the fluid pressure forces on the Lagrangian boundary of the solid. The Eulerian fluid domain, ^ , is decomposed as follows:
where, in a parallel implementation, the fluid subdomains Q^imn reside on a logical Cartesian mesh of L,M,N processors along the x,y,z directions, respectively. The Lagrangian boundary 8Q which, in general, is parametrized as a triangular surface mesh, is broadcast to all processors
(5)
u—c 0
1449
P^
+ Pp'
(7)
Second order accuracy is achieved via linear reconstruction with Van Leer type slope limiting applied to projections in characteristic state space.
3. Fluid-solid coupling in 3D It is important to emphasize that the "solid solver" in this approach can be as simple as a purely rigid solid (in which case one is simply computing flow in a possibly complex geometry) to something as complex as a fully Lagrangian solver based on finite elements. The underlying assumption in the development of the algorithm is that the
n(Xij,k,t)
=
V0 |V0|
(12)
where n is the normal to the level set 0. The above extrapolation is solved in a band of ghost cells, i.e., 0 < (pixij^kJ) < ^' Typically we choose O to be four to five times the mesh spacing. In the same band of ghost cells we reconstruct the velocity field to enforce the zero mass flux boundary condition by extrapolation and reflection of the normal velocity component in a local frame attached to 5Q. Let \l/ = (U'i,U' j,U-k). Then, \l/r-\-n'V\l/ = 0
(13)
u{Xij^kJ) = (—w 'n-\-2Vs -h^U'UU'h)
(14)
The final step in the coupling algorithm is the trilinear interpolation of pressure p{Xijj;,t) on to nodes on Sp.
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4. Adaptive mesh refinement
form of grid functions. In addition, GrACE utilizes a variety of domain decomposition algorithms to enable loadbalanced computation on distributed architectures. An issue which has to be addressed is that of time synchronization between the fluid and the solid solvers. With moving boundaries, and multiple meshes care must be taken in addressing this issue. We have taken the simplest approach here in which the finest fluid mesh negotiates the time step with the solid solver and all fluid levels march the solution forward by this time step. This implies that there is no adaptivity in time. In future work we will explore the use of a more sophisticated algorithms.
One potential difficulty that arises with the approach proposed above is the possibility that small scale features on the solid boundary will be totally smeared out in the fluid due to inadequate resolution of the boundary by the fluid solver. In order to overcome this we use Adaptive Mesh Refinement (AMR) in order to capture small scale features of the fluid flow as well as those of the fluidsolid boundary. Since the fluid-solid boundary is expressed in terms of a level set, the distance function becomes an additional field used to determine which parts of the fluid mesh are chosen for refinement The AMR algorithm is essentially of the classic BergerColella [4] type and is implemented using the GrACE C++ class library which provides a suitable framework for explicit solutions of hyperbolic conservation laws [5]. GrACE provides a hierarchy of scalable distributed dynamic grids and data structures and storage for field quantities in the
400
5. The coupling algorithm The algorithm itself is of splitting type in which each solver, fluid and solid, transfer information relevant to the coupling, coordinate a time step and then proceed
(a)
(b)
(c)
(d)
E
•J/b
T • 350 L - • : • \ - r 325 s. • "
2.69 2.54 2.40 2.26 2.12 1.98 1.84 1.70 1.55 1.41 1.27 1.13
300 275 2b0 ??5 »200 175
%
-•
L ! - - ' - ' • •
150
fe?'---
1?5
1- -\;
100
l'::U:
75 50 25
-; -
a99
M a85 0.70 0.56
i - ;:t r •-..;--5* •
0 P t i i t l n t ii t l l M l t 50 100 150
200
11 > I ..n i I I i 11 M I 11 I 250 300 350 400
Fig. 1. Shock propagation of over a complex two-dimensional contour at two different times. The right colunm shows the mesh corresponding to the results shown in the left column. Base mesh: 50 x 50 with 4 AMR levels.
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S. Mauch et al. /Second MIT Conference on Computational Fluid and Solid Mechanics
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Y
1451 Y
(a)
(b)
Fig. 2. Detonation propagation in a cylinder with radial springs. Base mesh: 64 x 32 x 32 with 3 AMR levels for an effective unigrid simulation at 256 x 128 x 128 resolution. The simulation was performed on the Los Alamos National Laboratory Nirvana SGI Origin on 128 processors. 160 140
Ir (a)
I (b)
pfeAx!)
p(g/c!c) 2.87 2.77 2.68
120 2.63 2.56 2.49 2,43 2.36 2.29 2.23 2.16 2.09 I 2.03
100 1 80 60
:
• >•
40 20
\ .
i
50
}
(C)
.
.
