Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three- dimensional finite element simulations of blood flow a b c d b* Hyun Jin Kim , C. A. Figueroa , T. J. R. Hughes , K. E. Jansen , C. A. Taylor a b Department of Mechanical Engineering, Departments of Bioengineering and Surgery, Stanford University, E350 Clark Center, 318 Campus Drive, Stanford, CA 94305, USA c Institute for Computational Engineering and Sciences, The University of Texas at Austin, ACES 5.430A (C0200), 201 East 24th Street, Austin, TX 78712, USA d Scientific Computation Research Center and the Department of Mechanical, Aeronautical & Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA * Corresponding author: Charles A. Taylor, E350 Clark Center, 318 Campus Drive, Stanford, CA 94305-5431, USA. Telephone: 1 (650) 725 6128, fax: 1 (650) 725 9082, email: [email protected] Abstract In three-dimensional simulations of blood flow in the vascular system, velocity and pressure fields in the computational domain are highly affected by outflow boundary conditions. This fact has motivated the development of novel methods to couple three- dimensional numerical models with one-dimensional numerical models or, alternatively, with zero-dimensional or one-dimensional analytic models. In all such methods described to date, whether they are explicit or implicit, the relationship between flow and pressure at the outlet boundary is enforced weakly. This coupling does not include any constraints on the shape of velocity profiles nor on the distribution of pressure at the interface. As a result, there remain some classes of problems that are, at best, difficult to solve, and at worst, intractable, with current numerical methods for modeling blood flow. These include problems with significant flow reversal during part of the cardiac cycle or 1 geometric complexity in the proximity of the outlet of the numerical domain. We have implemented a novel method to resolve these challenging problems whereby the augmented Lagrangian method is used to enforce a constraint on the shape of the velocity profile at the interface between the upstream numerical domain and the downstream analytic domain. This constraint on the velocity profile is added to the Coupled Multidomain method for implicitly coupling the computational domain with downstream analytic models. In this study, an axisymmetric profile is imposed after ensuring that each constrained outlet boundary is circular. We demonstrate herein that including a constraint on the shape of the velocity profile does not affect the pressure and velocity fields except in the immediate vicinity of the constrained outlet boundaries. Furthermore, this new method enables the solution of problems which diverged with an unconstrained method. Keywords: Blood flow; Boundary conditions; Augmented Lagrangian method; Coupled Multidomain Method Introduction Computational simulations have evolved as a powerful tool for quantifying blood flow and pressure in the cardiovascular system [1] for studying the hemodynamics of healthy and diseased blood vessels [2-8], designing and evaluating medical devices [9- 11] and predicting the outcomes of surgeries [12-15]. As computing power and numerical methods advance, simulation-based methods are expected to be even more extensively used in studying the cardiovascular system. Much progress has been made in simulating 2 blood flow and pressure accurately including the development of novel outlet boundary conditions. Conventional outlet boundary conditions employed include constant or time- varying prescribed pressure, zero traction or prescribed velocity [2-5, 8-11, 13-17]. These boundary conditions are severely limiting in that they are not appropriate models of the downstream vasculature that is best described using models that replicate vascular impedance. In an effort to model the interactions between the computational domain and the downstream vasculature, new outlet boundary conditions have been developed to couple the computational domain with simpler models such as a resistance, impedance, lumped parameter model or one-dimensional models [12, 18-24]. With this coupling, outlet boundary conditions are derived naturally through the interactions between the computational domain and downstream analytic models enabling highly realistic velocity and pressure fields. In particular, we previously described a highly versatile method, the Coupled Multidomain Method [21] to couple a three-dimensional finite element model of the cardiovascular system to a variety of downstream analytic models such as a resistance, impedance or Windkessel models and have demonstrated physiologically realistic velocity and pressure fields. However, we have encountered challenges in solving certain classes of problems including those where complex flow structures propagate to the outlet boundary due to vessel curvature or branches immediately upstream of the boundary. These problems can often be resolved by artificially extending the vessel with a long, straight segment. Yet, the result is that a significant part of the computational domain can reside in regions that are not of interest. Furthermore, some parts of the cardiovascular system such as the 3 branches off the arch of the aorta, the infrarenal segment of the aorta or the lower extremity vessels have significant retrograde flow during part of the cardiac cycle [25]. As flow is forced back into the computational domain from the downstream analytic models, the velocity distribution at the interface becomes irregular and, in most cases, results in divergence of the simulation. Outlet boundary conditions which involve coupling between three-dimensional and zero-dimensional or one-dimensional models including the Coupled Multidomain Method generally enforce weak relationships between flow and pressure at each interface. This coupling does not include any constraints on the shape of velocity profiles nor on the pressure distribution at the interface. However, zero-dimensional or one-dimensional models are derived based on an assumed shape of the velocity profile and the assumption of uniform pressure over the cross section [26-28]. In this paper, we extend the Coupled Multidomain Method to include a constraint on the shape of velocity profiles. In finite element methods, constraints are generally enforced using penalty methods or Lagrange multiplier methods [29]. Both methods are used in a variety of applications; to enforce incompressibility in a computational domain or mass flux through the boundary [22, 30-35]. The augmented Lagrangian method enforces constraints using both penalty and Lagrange multiplier methods and can be used to achieve faster convergence and enforce constraints more strongly [36-40]. However, penalty, Lagrange multiplier or augmented Lagrangian methods have not previously been applied in computational fluid dynamic studies to enforce constraints on the shape of velocity profiles. 4 In this paper, we present an augmented Lagrangian method to weakly enforce the shape of velocity profiles at outlet boundaries. By constraining the shape of velocity profiles, a three-dimensional computational domain is consistently coupled to zero- dimensional or one-dimensional models. This paper is organized as follows. First, we present the derivation of the augmented Lagrangian method for constraining the shape of velocity profiles of outlet boundaries and describe constraint and profile functions. We then demonstrate this method by applying it to simulated blood flow in a circular vessel and an abdominal aorta model to demonstrate that this new method only affects velocity and pressure in the immediate vicinity of the outflow boundary. We next use an idealized aortic bifurcation model to demonstrate that this method can be used to truncate branch vessels very close to a main vessel of interest. Finally, we demonstrate the utility of this method by applying it to solve for pulsatile flow in a subject-specific thoracic aorta model where there is significant retrograde flow during early diastole. Methods 1. Governing equations (strong form) Blood flow in the large vessels of the cardiovascular system can be approximated nsd as the flow of an incompressible Newtonian fluid in a domain Ω∈ℜ where nsd is the number of spatial dimensions [16]. The boundary Γ of the spatial domain Ω is split into a Dirichlet partition Γg and a Neumann partition Γh such that Γ =∂Ω=Γg ∪ Γh and ΓΓ=∅gh∩ . The three-dimensional equations of an incompressible Newtonian fluid consist of the three momentum balance equations and the continuity equation subject to suitable boundary and initial conditions. 5 ρρvvvpdivf,t +⋅∇=−∇+()τ + div() v = 0 (1) 1 where τμ==2DDv with (∇+∇vT ) 2 vxt( , )= gxt ( , ) x∈Γg (2) vx( ,0)= v0 ( x ) x∈Ω (3) The unknowns are the fluid velocity vvvv= (,,)x yz and pressure p . The density ρ and the viscosity μ of the fluid are assumed to be constant, and f is the external body force. Furthermore, constraint functions are defined to enforce a shape of the velocity profile on a part of the Neumann partition Γ . For each constrained face Γ , the h hk following constraint functions are imposed 2 ckk1( vxt , , )=⋅−Φ=() vxt ( , ) n ( vxt ( , ), xt , ) ds 0 for k=1, ..., nc x∈Γh ∫Γ k hk 2 cvxt(, ,)=⋅= vxttds (,) 0 (4) k 22∫Γ () hk 2 cvxt(, ,)=⋅= vxtt (,) ds 0 k33∫Γ () hk Here, nc is the number of faces where the constraints on the shape of the velocity profile are enforced and Φk (vxt (