Prediction of Flow Features in Centrifugal Blood Pumps

Transcription

1 ECCM-2001 European Conference on Computational Mechanics June 26-29, 2001 Cracow, Poland Prediction of Flow Features in Centrifugal Blood Pumps Marek Behr and Dhruv Arora Department of Mechanical Engineering and Materials Science Rice University, Houston, TX, USA e mail: behr@rice.edu Sebastian Schulte-Eistrup Heart Center NRW Clinic for Thoraic and Cardiovascular Surgery, Bad Oeynhausen, Germany e mail: schulte-eistrup@gmx.de Key words: Computational Hemodynamics, Finite Element Method, Centrifugal Blood Pumps Abstract. We present preliminary results from numerical simulations of blood flow in a prototype of a centrifugal pump. The simulations involve incompressible Navier-Stokes governing equations, stabilized space-time finite element formulations, and parallel implementation on distributed memory architectures. The examples illustrate our approach of custom hybrid structured-unstructured tetrahedral meshes, which accommodate the rotation of the pump impeller with respect to the stationary housing, without assuming axisymmetry of either part.

2 Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup 1 Introduction The development of implantable left ventricular assist devices (LVADs), in the form of continuous-flow axial and centrifugal pumps, offers hope for many heart attack victims waiting for donor hearts. These autonomous devices are intended as a medium-term bridge to transplant, or, if enough progress is made, even as a permanent solution. Some of the engineering challenges in the design of blood pumps that need to be addressed are: Efficiency. Implanted blood pumps should have minimal power requirements while providing the necessary blood flow volume. Another consideration is the size, which must make implantation, in children as well as in adults, possible. Shear-stress levels. Excessive levels of shear-stress in the blood flow through the pump lead to platelet, and then white cell, damage. Gradual and smooth acceleration of the fluid throughout the flow path is an important design goal. Stagnation and clotting. Areas of persistent stagnation in the flow field may result in thrombus formation and catastrophic failure of the device. Pump components must be shaped to avoid recirculation and large stagnant flow zones. We are extending our CFD methodology in order to speed up and automate the analysis of blood flow in centrifugal pumps. Our approach is based on a space-time weak form of the incompressible Navier-Stokes equations, and finite element discretization using unstructured tetrahedral meshes [1]. We are employing a mixed velocity-pressure formulation, using equalorder interpolations for the velocity and pressure fields. This convenient choice of interpolation is made possible by consistent stabilization techniques, in particular the Galerkin Least-Squares (GLS) approach. Another GLS benefit is the inclusion of SUPG-like [2] terms in the formulation, which provide optimal stability for high element-level Reynolds numbers. The governing equations are recalled in Section 2, and the weighted-residual formulation is presented in Section 3. In the presence of large mesh deformations, conventional mesh moving techniques often lead to unacceptably stretched elements, and a breakdown of accuracy. This necessitates remeshing, defined as creating new inter-element connectivity, and sometimes a new set of nodes. Every remeshing requires a projection of the solution from the old mesh to the new one, and therefore introduces additional inaccuracies to the computation. In the case of e.g. blood pump impeller, it is clear that such remeshing would have to be done rather frequently, many times per revolution. To circumvent this difficulty, we are employing the Shear-Slip Mesh Update Method (SSMUM) [3], which involves generating rigid unstructured meshes attached to individual boundaries of the domain, and connecting them together via a regular layer of elements. The objective is to restrict remeshing to that small layer, and control projection errors by limiting remeshing to changes in inter-element connectivity only. With this approach, the cost of remeshing becomes negligible, and consequently, such remeshing can be carried out frequently, even at every time step. As an alternative to slipping mesh approach and Chimera methods, SSMUM does not entail accuracy losses due to frequent interpolation, nor does it have the restrictions on the time step size which are inherent in the slipping mesh technique. The method is further elucidated in Section 4, and examples based on actual pump geometries are given in 2

3 ECCM-2001, Cracow, Poland Section 5. Throughout this research, we are employing scalable, portable and efficient implementations of the methods involved, utilizing iterative solution update techniques and standard messagepassing libraries. The simulations are carried out on distributed-memory parallel computers, including CRAY T3E-1200 and IBM SP [4]. 2 Governing Equations We consider blood as a viscous, incompressible fluid occupying at an instant t (0,T) a bounded region Ω t R n sd, with boundary Γt,wheren sd is the number of space dimensions. In the DSD/SST formulation, the spatial domain may change with respect to time, and the subscript t indicates such time-dependence. The symbols u(x,t) and p(x,t) represent the velocity and pressure. The external forces (e.g., the gravity) are represented by f(x,t). The momentum and mass balance equations can be written as follows: ( ) u ρ t + u u f σ = 0 on Ω t, (1) u =0 on Ω t, (2) where the density ρ is assumed to be constant. We consider only the Newtonian fluid model, for which the deviatoric component of the stress tensor σ is related linearly to the strain rate tensor: σ(u,p)= pi +2µε(u), ε(u) = 1 ( ) u +( u) T, (3) 2 where µ is the dynamic viscosity. The Dirichlet and Neumann-type boundary conditions are: u = g on (Γ t ) g, (4) n σ = h on (Γ t ) h, (5) where (Γ t ) g and (Γ t ) h are complementary subsets of the boundary Γ t. Statement of the fluid flow problem is completed by imposing suitable divergence-free initial condition for the velocity field. Remark 1 Blood can be modeled as a Newtonian fluid only if shear-stress levels in the flow are moderate, and the clearances involved are large when compared with blood cell diameter. For higherfidelity simulations, a non-newtonian constitutive model may be adopted. For a review of the body of knowledge in this field see [5]. Another recent work thoroughly examines the applicability of continuum models, including viscoelastic ones, to blood as a liquid [6]. We also anticipate that microstructure-based models [7] will play an importantrole in blood flow simulations. Because the typical meshes are not able to resolve the flow features well enough to fully capture turbulence effects at high Reynolds numbers, introduction of a turbulence model becomes 3

4 Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup necessary. In the present study, a simple Smagorinsky turbulence model is employed. In this model, the kinematic viscosity ν is augmented by an eddy viscosity ν t whichisdefinedas ν t =(Ch) 2 (2ε(u):ε(u)) 1 2, (6) where C =0.15 and h is the element length defined here as the maximum of the edge lengths for the element. 3 WeakForm To construct the finite element function spaces for the space-time method, we partition the time interval (0,T) into subintervals I n =(t n,t n+1 ),wheret n and t n+1 belong to an ordered series of time levels 0=t 0 <t 1 < <t N = T.LetΩ n =Ω tn and Γ n =Γ tn.wewilldefinethe space-time slab Q n as the domain enclosed by the surfaces Ω n, Ω n+1,andp n,wherep n is the surface described by the boundary Γ t as t traverses I n.asitisthecasewithγ t, surface P n is decomposed into (P n ) g and (P n ) h with respect to the type of boundary condition (Dirichlet and Neumann) being applied. After introducing suitable trial solution spaces for the velocity and pressure [1], (Su) h n and (Sp h ) n, and test function spaces, (Vu) h n and (Vp h ) n, the stabilized space-time formulation of Equations (1) and(2) is written as follows: given (u h ) n,finduh (Su h) n and p h (Sp h) n such that w h (Vu h) n and q h (Vp h) n: ( ) u w h h ρ Q n t + uh u h f dq + ε(w h ):σ(u h,p h )dq Q n + q h u h dq + (w h ) + n ρ ( ) (u h ) + n (u h ) n dω Q n Ω n + + (n el ) n e=1 [ ρ (n el ) n e=1 Q e n [ ( 1 w h τ MOM ρ ρ t ) ] + u h w h σ(w h,q h ) ( ) ] u h t + uh u h f σ(u h,p h ) dq τ CONT w h ρ u h dq = w h h h dp. (7) (P n) h Q e n The following notation is being used in Equation (7): (u h ) ± n = lim u(t n ± ε),...dq=...dωdt, ε 0 Q n I n Ω h t...dp = P n I n Γ h t...dγdt. (8) In most applications of our method, both velocity and pressure fields are represented using piecewise linear continuous interpolation functions. The stabilization parameters τ MOM and τ CONT follow definitions given in [1]. 4

5 ECCM-2001, Cracow, Poland Remark 2 Both viscoelastic continuum, e.g. Oldroyd-B, and microstructure-based non-newtonian models require significant modifications to the weak form given above. Either the stress, or the conformation tensor, join the velocity and pressure fields as independent unknowns. For a review of finite element approaches to the modeling of viscoelastic fluids, including the popular Elastic- Viscous Stress Splitting (EVSS) approach, see [8]. A stress-velocity-pressure GLS extension of the formulation (7) for Oldroyd-type fluids has been investigated [9]. The nonlinear system given by Equation (7) is solved with the Newton-Raphson method with an analytic Jacobian. At each nonlinear iteration, a sparse linear system is solved using the GMRES iterative mesh update technique [10]. 4 Deforming Domain Issues The mesh update methods developed in conjunction with the DSD/SST formulation and applied to complex flow problems include special-purpose and automatic mesh moving techniques. In most cases, the automatic mesh moving needs to be combined with remeshing. Remeshing, defined as creating new inter-element connectivity and sometimes new set of nodes, could be a bottleneck, depending on factors such as how frequently and how locally remeshing is required and whether an automatic mesh generator needs to be called at each remesh. Limiting the frequency of remeshing would limit the applicability to problems with modest changes in the spatial domain. Generous use of remeshing would lift such limitations, but would result in excessive costs in terms of automatic mesh generation and projection of the solution from the old mesh to the new one. Furthermore, these projections introduce additional inaccuracies to the computation, which, depending on the frequency and scope of remeshing, could be excessive. The Shear-Slip Mesh Update Method (SSMUM) [11, 3] was designed to accommodate some specific classes of problems which could be handled with the DSD/SST formulation, without limitations on the frequency of remeshing. These problems are characterized by large, but linear or rotational, motion of some parts of the boundary with respect to other parts. This situation occurs when one object forming the computational boundary undergoes translation or rotation with respect to other boundaries. In the SSMUM, we introduce a regular, typically hand-made, thin layer of elements; at each time step let the elements in this layer to undergo shear deformation; restrict remeshing to that layer as changes in inter-element connectivity; and thus control projection errors and cost of remeshing. Because the remeshing in the thin layer is accomplished by simply changing the inter-element connectivity, no nodal projection is needed. As a special case of this approach, the nodal positions for the old (deformed) and new (good-quality) meshes can be matched, so that the new element shapes are very comparable to the shapes prior to the shear deformation. A typical sequence of events is shown in Figure 1. This approach is related to the one used previously in the context of finite-difference turbomachinery simulations [12]. Applications of the SSMUM are based on special-purpose mesh designs which combine regions of rigid, non-deforming elements with layers of the shear-absorbing elements. A trans- 5

6 Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup Deformation Reconnect Figure 1: Shear-slip layer concept in 2D (top) and 3D (bottom): the sequence of deformation followed by reconnecting. lating object may be embedded in a strip (in 2D) or a tube (in 3D) of rigid elements that move glued to that object. Similarly, a rotating object may be embedded in a disk of rigid elements which rotate glued to that object. These non-deforming regions are immersed in another set of non-deforming elements spanning the exterior boundaries, and connected with the previously described shear-slip layer. This concept, which was discussed in more detail in [3], is illustrated for the 2D case in Figure 2. Figure 2: Shear-slip layer concept: special-purpose meshes. Regions of deforming elements are showningrey,andregionsofrigidelementsareshowninwhite. 5 Numerical Examples We now concentrate our attention on a specific family of geometries, representing centrifugal blood pumps. A typical configuration is schematically shown in Figure 3. It includes an inflow tube positioned above the impeller, and an outflow tube on the cylindrical part of the housing. 6

7 ECCM-2001, Cracow, Poland The impeller and its shaft are held loosely in place by two fluid-filled hemispherical bearings. The impeller has a small number of large primary top vanes, typically six, and also a lesser number of secondary bottom vanes, typically two. The main purpose of the lower vanes is not the transfer of the momentum to the fluid, but washing out the fluid from the low-clearance area under the impeller. bearing top vanes (6) bottom vanes (2) Figure 3: Centrifugal blood pumps: typical configuration. For the purpose of flow simulation, only the internal surfaces of the pump, such as the ones showninfigure3, need to be represented. The actual pump is slightly larger, containing motor and control components directly beneath the housing; the impeller rotation is driven through magnetic couplings. We have developed meshes for two models of such centrifugal pumps, and performed preliminary flow simulations using one of these models. 5.1 KP305 Centrifugal Pump The KP305 is a compact centrifugal pump design [13], with a chamber diameter of 40 mm, inflow tube diameter of 9 mm, and outflow tube diameter of 11.9 mm. This pump is designed to deliver the flow volume of 5 l/min at approximately 3000 rpm. The external CAD geometry, as well as the finite element representation of the internal surfaces are shown in Figure 4. Also visible is the portion of the SSMUM layer, which surrounds the impeller. Due to typically low clearances on the lower side of the impeller, the most natural layer design is an axisymmetric inverted pot with its main axis coinciding with the axis of rotation. The pot intersects the lower 7

8 Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup Figure 4: KP305 centrifugal blood pump: CAD model and the computational mesh. surface of the housing and the upper tip of the impeller shaft. The outer diameter of the SSMUM layer is 37.5 mm and its maximum thickness is 0.5 mm. The mesh shown in Figure 4 is generated in a semi-automatic process. At first, the internal surface of the chamber and the external surface of the impeller is extracted from the CAD model and made watertight. The model surface is then cut at locations where the SSMUM will intersect it; this produces two disjoint sections of the model surface, one of which will be stationary and another will be spinning. The layer surface elements (inner and outer) are generated and attached to the two sections, forming two watertight surfaces. An automatic mesh generator [14] is then used to generate two unstructured meshes, which are then combined with a structured layer mesh to form the final mesh. In the example presented here, the total number of elements is 488,689, and the number of space-time nodes is 172,152. The SSMUM layer has 96 segments in circumferential direction, and 45 segments in the remaining direction. 1 The initial simulation involved 768 time steps, with a time step size of seconds, and 4 time 4 96 steps elapsing between each SSMUM reconnect. Each time step involved 4 nonlinear iterations, and a Krylov space size of 100 in the GMRES solver. While the flow field is still developing, at the end of the 768 time steps we were able to observe the pressure distribution on the impeller surface shown in Figure 5, as well as the potential persistent areas of slow-moving fluid, which present a clotting risk. These are in evidence in Figure 6, which shows the isosurface of the velocity magnitude close to the stagnation value, in a stationary reference frame (at 5 in/s), as well as in a rotating reference frame attached to the impeller (at 10 in/s relative to the impeller). Each bubble which brings the isosurface away from the solid boundary is a stagnation area. While the stagnation areas in the outflow tube are observed to be merely transient, the areas where the inflow impinges on the upper portion of the shaft, and around the bottom portion of 8

9 ECCM-2001, Cracow, Poland Figure 5: KP305 centrifugal blood pump: Pressure distribution on the impeller surface the shaft, remain so for the duration of the simulation, and therefore are a cause for concern. Figure 6: KP305 centrifugal blood pump: Stagnation areas in a stationary frame of reference (left) and in a rotating frame of reference (right). 5.2 PI710 Centrifugal Pump The PI710 is a larger centrifugal pump design, with a chamber diameter of 2.24 in, and a uniform inflow and outflow tube diameter of in. The external CAD geometry, and the finite element representation of the internal surfaces are shown in Figure 7. Also visible is the portion of the SSMUM layer, surrounding the impeller. Because of the significant slant of the upper impeller surface and the housing lid, the top portion of the SSMUM pot is now a cone, becoming flat only in the vicinity of the shaft. The outer diameter of the SSMUM layer is 2.12 in and its maximum thickness is 0.04 in. As the development of the KP305 pump has been stopped, our simulations now focus on the 9

10 Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup Figure 7: PI710 centrifugal blood pump: CAD model and the computational mesh. PI710 model. In addition to the stagnation area detection, the levels of shear-stress present in the flow field are also being investigated. 6 Conclusions We have presented aspects of a computational approach used to perform simulations of unsteady, incompressible blood flow in a centrifugal pump. The method is based on the Deformable-Spatial-Domain/ Stabilized Space-Time (DSD/SST) finite element formulation of the Navier-Stokes equations. A numerical example involved a 3D computation of a complex unsteady flow in the KP305 centrifugal blood pump. 7 Acknowledgment This research was supported in part by National Science Foundation through computing resources provided by the National Partnership for Advanced Computational Infrastructure. References [1] M. Behr and T.E. Tezduyar. Finite element solution strategies for large-scale flow simulations. Computer Methods in Applied Mechanics and Engineering, 112, 3 24, (1994). 10

11 ECCM-2001, Cracow, Poland [2] A.N. Brooks and T.J.R. Hughes. Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32, , (1982). [3] M. Behr and T.E. Tezduyar. Shear-Slip Mesh Update Method. Computer Methods in Applied Mechanics and Engineering, 174, , (1999). [4] M. Behr, D.M. Pressel, and W.B. Sturek. Comments on CFD code performance on scalable architectures. Computer Methods in Applied Mechanics and Engineering, 190, , (2000). [5] A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular fluid dynamics: Problems, models and methods. Technical Report EPFL/DMA 11.98, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, (2000). to appear in Computing and Visualization in Science. [6] K.K. Yeleswarapu. Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood. PhD thesis, Department of Mechanical Engineering, University of Pittsburgh, (1996). [7] M. Pasquali. Polymer Molecules in Free Surface Coating Flows. PhD thesis, Department of Chemical Engineering, University of Minnesota, (1999). [8] F.P.T. Baaijens. An iterative solver for the DEVSS/DG method with application to smooth and non-smooth flows of the upper convected Maxwell fluid. Journal of Non-Newtonian Fluid Mechanics, 75, , (1998). [9] M. Behr. Stabilized Finite Element Methods for Incompressible Flows with Emphasis on Moving Boundaries and Interfaces. PhD thesis, Department of Aerospace Engineering and Mechanics, University of Minnesota, (1992). [10] Y. Saad and M. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal of Scientific and Statistical Computing, 7, , (1986). [11] M. Behr and T.E. Tezduyar. A note on Shear-Slip Mesh Update Method. In Lecture Notes of the Workshop on Parallel Computing in Applied Fluid Mechanics. Associazione Amici Scuola Normale Superiore, Pisa, Italy, (1997). [12] J.M. Janus. Advanced 3-D CFD Algorithm for Turbomachinery. PhD thesis, Department of Aerospace Engineering, Mississippi State University, (1989). [13] S. Schulte-Eistrup, M. Behr, T. Takano, K. Nonaka, T. Maeda, J. Linneweber, S. Kawahito, T. Motomura, S. Ichikawa, M. Mikami, K. Minami, R. Koerfer, T. Tezduyar, and Y. Nosé. Computational fluid dynamic evaluation of Baylor miniature gyro centrifugal blood pump. In Proceedings of the 8th Congress of the International Society for Rotary Blood Pumps, Aachen, Germany. (2000). 11

12 Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup [14] A. Johnson. Mesh Generation and Update Strategies for Parallel Computation of Flow Problems with Moving Boundaries and Interfaces. PhD thesis, Department of Aerospace Engineering and Mechanics, University of Minnesota, (1995). 12