Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2005 M. Papadrakakis, E. O˜ nate and B. Schrefler (Eds) c
CIMNE, Barcelona, 2005
SIMULATION OF BIDIRECTIONAL FLUID-STRUCTURE INTERACTION BASED ON EXPLICIT COUPLING APPROACHES OF LATTICE BOLTZMANN AND P-FEM SOLVERS S. Geller∗ , J. T¨ olke∗ , M. Krafczyk∗ , D. Scholz† , A. D¨ uster† and E. Rank† ∗ Institut
f¨ ur Computeranwendungen im Bauingenieurwesen Technische Universit¨at Braunschweig 38106 Braunschweig, Germany e-mail: geller, toelke,
[email protected], web page: http://www.cab.bau.tu-bs.de/ † Lehrstuhl
f¨ ur Bauinformatik Technische Universit¨at M¨ unchen 80290 M¨ unchen, Germany e-mail: d.scholz, duester,
[email protected], web page: http://www.inf.bv.tum.de/
Key words: Fluid-Structure Interaction, Lattice Boltzmann Method, p-FEM, Octreebased grid Abstract. In the last few years the Lattice Boltzmann method which has been derived from kinetic theory, has matured as an efficient approach to solve the Navier-Stokes equations in the limit of small Mach numbers. In our approach we couple a LB-solver for fluid flow based on unstructured quadtree/octree Eulerian grids with a p-FEM-solver for structure mechanics based on a Lagrangian description to predict bidirectional fluid-structure interaction. After a brief introduction of the fundamental concepts of the Lattice Boltzmann method and the p-Version of the Finite Element Method, the coupling algorithm is presented. Due to the underlying Eulerian LB-grid, problems with regard to the activation/deactivation of fluid nodes in case of moving structures are discussed and some solution strategies are indicated.
1
INTRODUCTION
Nowadays efficient solvers both for fluid and structure mechanics problems are available. Recent research and development has focused on the coupling of the two fields. The most common approach is the Arbitrary Lagrangian Eulerian (ALE) formulation. In this case, an ALE-based FV/FE discretization for the Navier-Stokes equations is combined with a FE-solver for the structure. There are partioned as well as monolitic approaches [29]. The advantage of the ALE method is the mass conservation and the prevention of
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S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
the activation/deactivation of fluid nodes due to the moving mesh. The most important disadvantage of this approach is the remeshing procedure if large displacements occur. We present a coupling between a LB solver based on adaptive quad- and octree grids and a p-FEM structure solver. The coupling is achieved by a moving interface mesh described in section 4. The adaption of the quad-/octrees requires only local operations and therefore a time consuming remeshing procedure can be omitted. The paper is organized as follows. In section 2 a short introduction into the Lattice Boltzmann method is given and the implementation of boundary conditions for moving walls and the evaluation of forces on the boundary is discussed. In section 3 a p-FEM structure solver able to calculate thin and shell elements is introduced. In section 4 the coupling algorithm will be discussed. 2
The Lattice-Boltzmann method
2.1
Basics
The lattice Boltzmann equation fi (t + ∆t, ~x + ~ci ∆t) − fi (t, ~x) = Ωi ,
i = 0, . . . , b − 1,
(1)
is a finite difference discretization of the discrete velocity model [12]. fi are the particle distribution functions, ∆t is the time step, Ω the collision operator and {~ci , i = 0, . . . , b−1} the b microscopic velocities of the model. The left hand side of equation (1) is the propagation step and the right hand side the collision step. Here we use the d3q19 model [24] with the following speeds creating a space-filling lattice with nodal unit distance ∆x = c∆t, ( ) {~ci } =
0 c −c 0 0 0 0 c −c c −c c −c c −c 0 0 0 0 0 0 0 c −c 0 0 c −c −c c 0 0 0 0 c −c c −c 0 0 0 0 0 c −c 0 0 0 0 c −c −c c c −c −c c
(2)
where c is a constant velocity determining the speed of sound c2s = c2 /3. The collision operator, which acts only on the nodal values of the distribution functions, is given with Ωi = M−1 Si [(Mf ) − M eq ] .
(3)
M transforms the particle distributions functions in moment space and vice versa, m = Mf
f = M−1 m
(4)
The moments m = Mf are labeled as m = (p/c2s , e, , ρ0 ux , qx , ρ0 uy , qy , ρ0 uz , qz , 3pxx , 3πxx , pww , πww , pxy , pyz , pxz , mx , my , mz )
(5)
where p is the pressure, ux , uy , uz is the velocity vector, e, pxx , pww , pxy , pyz , pxz are related to the stress tensor and the other quantities are irrelevant for the simulation of incompressible flows. ρ0 is a constant reference density. M eq is the vector composed of the 2
S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
equilibrium moments and S is a diagonal collision matrix. The exact definitions are given in [14, 17]. It can be shown by a multi-scale expansion [10], that p and ~u are solutions of the incompressible Navier-Stokes equation. The error introduced by the finite Mach number Ma ≡ ucs0 is of O(Ma2 ). The order of convergence with respect to the corresponding Navier-Stokes solution is first order in time and second order in space [15, 16]. 2.2
Grid refinement
The use of locally refined meshes for the direct discretization of the Navier-Stokes equations is mandatory for an efficient solution of CFD problems. The LB method is based on quadratic elements in 2D and cubic elements in 3D respectively. Therefore we use recursively refined quadtrees in 2D and octrees in 3D [4]. Due to the coupling of space and time discretization ∆x = c∆t, this leads to a nested time stepping scheme (see Figure 1), i.e. for each grid level l we have ∆xl = c∆tl . Figure 1 shows the staggered time step computation scheme for one coarse cell time step. The level index is linked with the grid cell size, Level 0 is equal to ∆x, whereas Level l corresponds to 2−l ∆x. One time step in level 0 means two time steps in the level 1 and so on. For an absolute number k of refinement levels we have 2k internal iterations. The analysis of the LB equations on different grid levels reveals, that it is not sufficient to glue two Cartesian grids of different solution together by interpolating the distributions during the propagation from the coarse to the fine grid. It was first shown in [7], that in order to obtain smooth transitions for pressure, velocity and stresses, one has to rescale the distributions in an appropriate manner. A detailed description of the mesh refinement procedure can be found in [3, 4, 34, 35].
Level 0
Propagation
Level 1
Propagation
Propagation
Level 2
Level 3
Propagation
Propagation
Propagation
Prop .
Prop. Prop.
Prop.
Prop.
t
n
t
1 4
n+
t
Propagation
Prop.
Prop.
1 2
n+
t
3 4
n+
Prop.
t
n+1
normal propagated distributions time interpolated propageted distributions
Figure 1: staggered time step calculation.
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S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
2.3
Boundary conditions
The macroscopic flow quantities can only be set implicitly via the boundary nodes’ particle distribution functions. A well known and simple way to introduce no-slip walls is the so-called bounce back scheme which allows spatial second order accuracy if the boundary is aligned with one of the lattice vectors ~ei = ~c/∆t and first order otherwise. As we have arbitrarily shaped moving objects, we use the modified bounce back scheme developed in [1, 18] for velocity boundary conditions, which is second order accurate for arbitrarily shaped boundaries.
Figure 2: Interpolations for second order bounce back scheme.
Here we identify two cases: (a) the wall has a distance less then 0.5 ∆x from the node and (b) the wall has a distance between 0.5 and 1.0 ∆x from the node. ei uw t+1 t t fIA (6) = 2q · fiF + (1 − 2q) · fiA − 2ρwi 2 , 0.0 < q < 0.5 cs 1 ei uw 2q − 1 t t+1 t fIA · fIA + · fiA − ρwi 2 , 0.5 ≤ q ≤ 1.0 (7) = 2q 2q qcs Thus we obtain second order accurate results in space even for curved geometries [8]. For a detailed discussion of LBE boundary conditions we refer to [9]. In contrast to the simple bounce-back scheme the use of these interpolation based no-slip boundary conditions result in a notable mass loss across the no-slip lines. Yet, the results obtained with bounce-back were inferior which highlights the importance of a proper geometric resolution of the flow domain. Pressure boundary conditions are implemented by setting the incoming distributions to [30] fI = −fi + fIeq (P0 , ~u) + fieq (P0 , ~u) where P0 is the given pressure, ~eI = −~ei and ~u is obtained from the local velocity.
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(8)
S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
2.4
Force evaluation at the boundary
There are two possibilities to evaluate forces on boundaries using the LB method. The first one is the pressure/stress integration based method where one has to extrapolate the pressure and stresses from the fluid nodes to the boundary, the second is the momentumexchange based method. A detailed description and comparison of both methods can be found in [20]. As the first method is more complex to implement and seems to be inferior to the second one, we use the second one. The force on each link to a boundary (see Figure 3) results from the momentum-exchange between the particle distributions hitting the moving object and the moving object itself [22]: F =
∆x3 (~ci − ~uB )(fi + fI ) ∆t
(9)
where ~uB is the velocity of the moving boundary. Figure 3 shows the links of the LB grid which contribute to the force on the structure. The forces are integrated by a linear interpolation scheme to obtain the forces acting on the interface points. The weight for the force integration for point A is the normalized distance between the intersection point S and the structure interface point A. All fluid nodes with a link to the boundary contribute to the force transmitted to the structure interface points. fluid nodes
fluid nodes with link to the boundary
solid nodes structure interface points
fluid
A
S
B A
B
solid
Figure 3: Interpolation scheme to compute forces acting on the structure
2.5
Initialization of fluid nodes
In Figure 4 the extrapolation scheme for the computation of the distribution functions for new fluid nodes is shown. We compute the moments at a new node by one-dimensional linear extrapolations from the surrounding fluid nodes followed by averaging. To obtain the distributions, the moments are transformed by equation (4). Another problem is the change from fluid to solid node. In both cases we see localized oscillations in the flow field, methods for reducing them are under investigation. Using a refined mesh at the fluid-structure interface the oscillations are substantially reduced.
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S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
new fluid node to be initialized node with extrapolation weight 2.0 node with extrapolation weight −1.0 direction of motion (solid)
Figure 4: Interpolation scheme for new fluid nodes.
3
Structure simulation using Three-Dimensional, high-order Finite Elements
The problem usually arising when using a strictly three-dimensional approach for thinwalled parts of structures is that standard low-order finite elements are very sensitive to large aspect ratios, and locking effects are very likely to occur. Therefore, one would have to use a large number of small isotropic elements. In contrast, high-order elements can cope with high aspect ratios [27, 28], provided that the polynomial degree is sufficiently high. In [5, 6, 25] the use of high-order hexahedral elements for thin-walled structures is described. Curved structures (see Figure 5) can be taken into account by applying appropriate mapping techniques [6, 11]. A very important feature in the context of high-order Q N5 E9 N6
F2 E10
F3
E5 x1
E6 E1 F1 N2
E2
N1
E12 F6
N8
N5
x33
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N7 x3 F4 x2
x11 E8
N4 N7
z N4
x22
N2
E4 E7
N1
x
E3
y
N3
N3
Figure 5: Discretization of thin-walled structures with hexahedral elements.
elements are anisotropic Ansatz spaces. Using the approach presented in [5] one can define different polynomial degrees in the different local directions of the hexahedral element. For a shell-like structure as presented in Figure 5 we apply a high polynomial degree for the in-plane direction, whereas in thickness direction a lower polynomial degree can be used in order to reduce the computational effort. In our implementation, it is not only possible to define different polynomial degrees for the different local directions, but also 6
S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
for the different components of the displacement field. Applying this approach, the ’model error’ related to every plate or shell theory turns into the discretization error of the three-dimensional approximation. The main advantage is that this error can be controlled by varying the polynomial degree over the thickness in a sequence of computations. It should be mentioned that this error could not be controlled when using fixed kinematic assumptions. For the spatial discretization of the structural problem, high-order elements are used, whereas the time domain is discretized using the generalized-α-method, which is secondorder accurate and has favorable numerical damping properties [2]. 4
Coupling
Since we have very different grids for the structure (coarse and curved p-elements) and the fluid (very small cubes, locally refined octree), we introduce an interface mesh to couple both meshes. The interface mesh is a moving surface mesh built of flat triangles. On each node the values for the velocity, the forces and other physical quantities required for the exchange are stored. The mesh is constructed as follows. The nodes are defined by the Gaussian integration points of the p-FEM solver, then a triangulation to obtain the triangles is done. The interface mesh can be adapted by the p-FEM solver as well as by the LB solver. An interface mesh and the coupling is shown in Figure 6. For problems in two dimensions the interface mesh is reduced to a polyline. Structure FEM−Mesh
Interface Mesh
displacements forces
Fluid LBM−Grid displacements forces
Figure 6: Coupling through interface mesh
The fluid solver rescales the system in such a way, that numerical errors are reduced (avoiding very small and/or large numerical values) while dimensionless quantities are conserved. Therefore we have to scale the forces and the time in the fluid solver system to 7
S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
the real system. By setting the dimensionless drag coefficient equal in both systems, we obtain the following scaling from the lattice Boltzmann system (f) to the original system (s): Ff ρs · u2s Fs = · (10) Hs Hf ρf · u2f with reference height H, reference density ρ and reference velocity u. By equating the H Reynolds number in both systems and using u = ∆T we obtain the following time scaling: ∆Ts = ∆Tf ·
νf · Hs2 νs · Hf2
(11)
The explicit time coupling is sketched in Figure 7 and can be summarized as follows: (1) The fluid solver computes the load vector on the interface mesh points. (2) Exchange of loads through interface mesh. (3) The structure solver computes the new displacements due to the loads. (4) Exchange of the new displacements through interface mesh. (5) Fluid solver: Time interpolation of the position of the interface surface geometry and dynamic adaption of the octree. (6) Repeat step 5 for all internal iterations (nested time stepping). (7) Repeat step 5 and 6 for the number n of fluid time steps.
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S. Geller, J. T¨ olke, M. Krafczyk, D. Scholz, A. D¨ uster and E. Rank
LB--Fluid--Solver
p--FEM--Structure--Solver forces
fluid step = 0
structure step = 0
internal step = 1
interpolated surface geometry
internal step = 2level n∆T fluid
fluid step = 1
∆T structure
internal step = 1
interpolated surface geometry
internal step = 2level
fluid step = n
displacements new surface geometry
structure step = 0
Figure 7: Coupling algorithm.
5
CONCLUSIONS
An approach to couple a p-FEM structure solver with a LB fluid solver was presented. General problems as the moving interface mesh and the adaption of the octree result in time consuming intensive programming efforts. As a first example we coupled the fluid solver with a rigid circular cylinder attached to two springs and obtained a quantitative agreement to the results of Mittal and Kumar [21]. Present work is focussing on simulation of different coupled problems, which will be presented in near future. The explicit coupling algorithm will be extended by a predictor-corrector scheme. REFERENCES [1] M. Bouzidi, M. Firdaouss, P. Lallemand. Momentum transfer of a Boltzmann-Lattice fluid with boundaries, Physics of Fluids 13(11), 3452-3459, (2001). [2] J. Chung, G. Hulbert. A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-α-Method. J. of Applied Mechanics, vol. 60, pp. 1562-1566, (1993). [3] B. Crouse, E. Rank, M. Krafczyk, J. T¨olke. A LB-based approach for adaptive flow simulations, Int. J. of Modern Physics B 17, 109-112, (2002). 9 2
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[31] J. T¨olke, M. Krafczyk, E. Rank, R, Berrioz. Discretization of the Boltzmann equation in velocity space using a Galerkin approach, Comp. Phys. Comm. 129, 91-99, (2000). [32] J. T¨olke, M. Krafczyk, M. Schulz, E. Rank. Implicit discretization and non-uniform mesh refinement approaches for FD discretizations of LBGK Models, Int. J. of Mod. Phys. C 9(8), 1143-1157, (1998). [33] J. T¨olke, M. Krafczyk, E. Rank. A Mutligrid-Solver for the Discrete BoltzmannEquation, Journal of Statistical Physics 107, 573-591, (2002). [34] D. Yu. Viscous Flow Computations with the Lattice Boltzmann equation method, PhD thesis, Univ. of Florida, (2002). [35] D. Yu, R. Mei, W. Shyy. A multi-block lattice Boltzmann method for viscous fluid flows, Int. J. Numer. Methods Fluids 39(2), 99-120, (2002).
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