1 Introduction

Ventricular Assist Devices (VADs) are devices which provide mechanical circulatory support to a single ventricle of the heart [4, 33]. They are used primarily as a bridge to transplant, extending the life of the patient until a compatible donor can be found. Two device types are available: pulsatile displacement pump designs, and a continuous flow impeller designs. Devices now available to the pediatric community include only pulsatile designs, and the we therefore choose to focus on pulsatile VADs in this work. The pediatric population suffers from increased risk of thromboembolic events (i.e., blood clots) while using VADs, and thrombo-embolic events may occur in up to 40 % [5] of cases. This has made these devices too risky for long-term use, and reliable only as a short-term bridge to transplant. However, in children, particularly those with congenital heart defects or cardiomyopathy, it may be of particular importance to develop a long-term reliable device. Pediatric patients in heart failure due to dilated cardiomyopathies have shown recovery of the native heart tissue when a VAD is used in a bridge to recovery scenario. This is an effect that has not been observed in the adult population [23].

A pulsatile VAD provides mechanical support to a single ventricle of the heart. A patient may receive two VADs if support is required for both ventricles, depending on the underlying disease state. The design of the device is as follows: Two domed chambers are separated by a flexible polyurethane membrane. One chamber is an air compartment, which is driven pneumatically. The other is a blood chamber, which delivers blood from the right atrium/left ventricle to the pulmonary arteries/aorta, for a Right/Left heart VAD, respectively [4]. The flow in the air chamber moves the thin membrane, which causes displacement in the blood chamber and drives the blood through the device.

The low survival rates of VADs may be alleviated if one had a better understanding of how specific flow features in the blood chamber are linked to the formation of blood clots. For this, as a first step, one needs the ability to accurately predict the blood flow itself inside the device. The latter is not possible without fluid–structure interaction (FSI) modeling that involves the interaction of air, blood, and a thin membrane separating the two fluids. For the modeling to be realistic, the relatively complex geometry of the VAD, the large, time-dependent motions of the membrane, and the actual pump operating conditions (i.e., flow rates and pressures) must be taken into account. In this work we present computational FSI methods in which these effects are incorporated, and which may be used for high-fidelity VAD design. While there is extensive literature on patient specific modeling for pediatric applications with and without FSI [31, 52, 53], very little has been done to date for numerical simulation of pulsatile VADs. In [30] an idealized VAD design was analyzed with a linearly elastic shell model for the membrane and non-physiological outlet boundary conditions, and in [50] VAD simulations were performed using imposed membrane motion rather than true FSI. We stress that membrane motion in a VAD is expected to be critical to the device’s performance, and thus accurate prediction of the time-dependent membrane response is crucial to the simulation and design of the device. To the best of the authors’ knowledge this is the first 3D, full-scale, high-fidelity FSI modeling of pulsatile VADs.

The paper is outlined as follows. In Sect. 2, we present the computational methods employed to simulate the VAD–FSI problem. A rotation-free isogeometric Kirchhoff–Love shell formulation is used to model the thin membrane in combination with a moving-domain ALE–VMS finite element formulation for the blood and air flow. The FSI solution strategy involves strong coupling, which is accomplished using a combination of sparse-matrix-based and matrix-free techniques. Strong coupling is essential for convergence of the coupled FSI equation system for this application. In Sect. 3 we provide a detailed description of the VAD problem setup and present a numerical simulation of the device. In the course of the simulation we periodically remesh the fluid mechanics domain to maintain good quality of the finite element discretization. The simulation predicts physiologically realistic blood flow features and membrane deformation patterns. In Sect. 4 we draw conclusions and present future research directions.

2 Numerical methods for VAD–FSI

In this section we briefly discuss the fluid and structural mechanics formulations used in this work, namely ALE–VMS and Isogeometric Analysis (IGA), respectively. We mostly summarize the main features of these methods and provide references where the reader may find the mathematical details of these techniques. One of the main computational challenges of this work is robust FSI coupling, which we present in some detail in this section

2.1 ALE–VMS fluid mechanics formulation

Standard Galerkin methods are not a sufficiently robust technology for advection-dominated flows. For this reason, stabilized methods [22, 37, 39, 42, 43, 78, 80, 85–88] were designed to circumvent this shortcoming of the Galerkin technique. Stabilized methods, which are essentially residual-based modifications of the Galerkin method, exhibit uniform stability and convergence behavior across the full range of advective and diffusive phenomena.

The basic theory of variational multiscale (VMS) methods was developed in [38], wherein stabilized methods were first identified as multiscale methods. Relationship between stabilized methods and subgrid scale modeling was also identified in [38], and now presents an important research direction [41]. Recently, in [7], the authors proposed a residual-based turbulence modeling and computational framework that is based on the VMS theory, named RBVMS. This technique performs well on both laminar and turbulent flows, for a wide range of Reynolds numbers.

The extension of the RBVMS framework to the moving-domain case, where the motion of the fluid mechanics domain is handled using the Arbitrary Lagrangian–Eulerian (ALE) formulation [40], was named ALE–VMS in [14, 69]. The ALE–VMS formulation discretized with linear tetrahedral FEM is used in this work to compute the fluid mechanics part of the VAD problem.

An important additional feature of the ALE–VMS methodology is weak enforcement of essential boundary conditions. Weakly enforced essential boundary conditions were introduced in [15] in order to improve solution accuracy on meshes with insufficient boundary-layer resolution [6, 16, 17, 34]. Although the weak BCs are now routinely used for wind-turbine aerodynamics [13, 35, 36] and ship hydrodynamics [1–3], we do not use them in this work. However, we feel that they will likely be beneficial in cardiovascular blood flow and FSI computations in that they may further improve boundary-layer accuracy and produce more accurate wall quantities such as wall shear stress or oscillating shear index, which are critically important in numerous cardiovascular applications [10, 52, 65, 68–74, 96, 99, 100].

2.2 Rotation-free isogeometric thin shell formulation

The circular membrane separating the blood and air chambers of the device is a very thin structure. The membrane stress-free reference configuration is not flat, but convex. As the membrane undergoes large cyclic deformation, it is almost always in a state of compression, which leads to local buckling and wrinkling. As a result, it is desirable to represent the membrane or thin shell with numerical technology that is efficient and capable of representing the underlying complex structural dynamics without posing significant challenges associated with robustness of the structural mechanics computations and large local deformations of the fluid mechanics domain boundary.

Low-order, bi-linear quadrilateral finite elements, which are widely used and are considered standard shell element technology, exhibit several shortcomings: (1) These elements require the use of displacement and rotation degrees of freedom to describe shell kinematics; (2) One needs a fine mesh to represent shell geometries with high local curvature, and to simultaneously achieve the desired solution accuracy; (3) Ad-hoc element technology is necessary to overcome membrane and shear locking; (4) In the case of implicit time integration employed in this work, the presence of rotational degrees of freedom doubles the size of the solution and right-hand-side residual arrays, quadruples the size of the left-hand-side matrix, and results in an order-of-magnitude increase in linear solver time.

Isogeometric shell analysis was recently proposed in [20] to address the shortcomings of standard shell technology listed above. It was found that higher-order continuity (\(C^1\) and above) of the IGA basis functions significantly improved the per-degree-of-freedom accuracy and robustness of thin shell discretizations as compared to the FEM. Furthermore, the increased continuity of the IGA discretizations enabled the use of shell kinematics without rotational degrees of freedom [19, 21, 49], leading to further computational cost savings. The isogeometric rotation-free Kirchhoff–Love shell formulation for structures composed of multiple structural patches, called the bending strip method, was developed in [48], which enabled the application of the rotation-free IGA technology to real-life structures, such as wind turbine rotors (see [11, 12, 36]). Besides significant savings in computational time, the rotation-free shell discretization makes FSI coupling simpler than the discretization with rotational degrees-of-freedom. Finally, the smooth structural motion computed with IGA gives a smooth fluid mechanics mesh at the fluid–structure boundary, which adds accuracy and robustness to the fluid mechanics computation.

Non-uniform rational B-splines (NURBS) [61] are employed in this work to discretize the structural mechanics equations of the membrane separating the blood and air chambers. T-splines [8, 28], a relative newcomer to IGA currently receiving significant attention, are also well suited for the proposed structural modeling approach. For related rotation-free shell formulations the reader is also referred to [25–27, 57, 59, 60].

2.3 FSI coupling

In order to take advantage of the benefits of IGA for structural mechanics, and to leverage the existing advanced automatic mesh generation tools for the FEM, we choose to couple low-order FEM for the fluid and IGA for structural mechanics. As a result, the FSI coupling assumes a nonmatching fluid–structure interface discretization. Nonmatching interface discretizations in FSI problems necessitate the use of interpolation or projection of kinematic and traction data between the nonmatching surface meshes (see, e.g., [13, 18, 32, 69, 71, 73, 74, 76, 88, 92, 95–98], where [76] is more comprehensive than [74]). A computational procedure, which can simultaneously handle the data transfer for IGA and FEM discretizations, was proposed in [13]. The procedure also includes a robust approach in identifying “closest points” for arbitrary shaped surfaces. While such interface projections are rather straightforward for loosely-coupled FSI algorithms, they require special techniques (such as developed in [18, 71, 72, 75, 76, 88, 89, 93, 95] as well as this paper) for strongly-coupled methods that are monolithic-like and that are necessary for the present application.

A full discretization of the FSI formulation leads to coupled, nonlinear equation systems that need to be solved at every time step.The equation systems can be written as follows:

$$\begin{aligned} {\mathbf{N}_\mathrm{1}}~({\mathbf{d}_\mathrm{1}},{\mathbf{d}_\mathrm{2}},{\mathbf{d}_\mathrm{3}})&= {\mathbf{0}},\end{aligned}$$
(1)
$$\begin{aligned} {\mathbf{N}_\mathrm{2}}~({\mathbf{d}_\mathrm{1}},{\mathbf{d}_\mathrm{2}},{\mathbf{d}_\mathrm{3}})&= {\mathbf{0}}, \end{aligned}$$
(2)
$$\begin{aligned} {\mathbf{N}_\mathrm{3}}~({\mathbf{d}_\mathrm{1}},{\mathbf{d}_\mathrm{2}},{\mathbf{d}_\mathrm{3}})&= {\mathbf{0}}. \end{aligned}$$
(3)

Here \({\mathbf{N}_\mathrm{1}}, {\mathbf{N}_\mathrm{2}}\), and \( {\mathbf{N}_\mathrm{3}}\) are the discrete residual functions, and \({\mathbf{d}_\mathrm{1}}, {\mathbf{d}_\mathrm{2}}\), and \({\mathbf{d}_\mathrm{3}}\) are the vectors of nodal (or control-point in the case of IGA) unknowns, corresponding to the fluid mechanics, structural mechanics, and mesh problems,

In the block-iterative coupling [18, 47, 56, 81, 82, 84, 88, 90, 91, 94], the fluid, structure, and mesh systems are treated as separate blocks, and the nonlinear iterations are carried out sequentially. First, the fluid block is solved, then the structure, and then the mesh. In solving a given block of equations the most current values of the other blocks of unknowns are used. The sequence of solves is repeated until all the equation systems are solved to an a priori set tolerance. This strategy is the easiest to implement, and it performs very well in applications where the structure is heavy relative to the surrounding fluid.

In the present application, the membrane separating the blood and air chambers of the VAD is extremely thin, and its mass is significantly smaller than the mass of the surrounding fluid that is displaced as a result if the membrane motion. Because of the relatively low structural mass, block-iterative FSI is not an appropriate technique for this application. Instead, we employ the quasi-direct coupling technique [18, 88, 90, 91, 94], where the fluid+structure and mesh systems are treated as two separate blocks, and the nonlinear iterations are carried out one block at a time until all the equation systems are solved to an a priori set tolerance. In an iteration step, given the solution at \(i\), the solution \(i+1\) is obtained by solving the following two blocks of equations:

$$\begin{aligned} \left. \frac{\partial {\mathbf{N}_\mathrm{1}}}{\partial {\mathbf{d}_\mathrm{1}}} \right| _i \varDelta {\mathbf{d}_\mathrm{1}^i}+ \left. \frac{\partial {\mathbf{N}_\mathrm{1}}}{\partial {\mathbf{d}_\mathrm{2}}} \right| _i \varDelta {\mathbf{d}_\mathrm{2}^i}&= - {\mathbf{N}_\mathrm{1}}\left( {\mathbf{d}_\mathrm{1}^i},~{\mathbf{d}_\mathrm{2}^i},~{{\mathbf{d}_\mathrm{3}^i}}\right) , \end{aligned}$$
(4)
$$\begin{aligned} \left. \frac{\partial {\mathbf{N}_\mathrm{2}}}{\partial {\mathbf{d}_\mathrm{1}}} \right| _i \varDelta {\mathbf{d}_\mathrm{1}^i}+ \left. \frac{\partial {\mathbf{N}_\mathrm{2}}}{\partial {\mathbf{d}_\mathrm{2}}} \right| _i \varDelta {\mathbf{d}_\mathrm{2}^i}&= - {\mathbf{N}_\mathrm{2}}\left( {\mathbf{d}_\mathrm{1}^i},~{\mathbf{d}_\mathrm{2}^i},~{{\mathbf{d}_\mathrm{3}^i}}\right) , \end{aligned}$$
(5)
$$\begin{aligned} {\mathbf{d}_\mathrm{1}^{i+1}}&= {\mathbf{d}_\mathrm{1}^i}+ \varDelta {\mathbf{d}_\mathrm{1}^i}, \end{aligned}$$
(6)
$$\begin{aligned} {\mathbf{d}_\mathrm{2}^{i+1}}&= {\mathbf{d}_\mathrm{2}^i}+ \varDelta {\mathbf{d}_\mathrm{2}^i}, \end{aligned}$$
(7)
$$\begin{aligned} \left. \frac{\partial {\mathbf{N}_\mathrm{3}}}{\partial {\mathbf{d}_\mathrm{3}}} \right| _i \varDelta {{\mathbf{d}_\mathrm{3}^i}}&= - {\mathbf{N}_\mathrm{3}}\left( {\mathbf{d}_\mathrm{1}^{i+1}},~{\mathbf{d}_\mathrm{2}^{i+1}},~{{\mathbf{d}_\mathrm{3}^i}}\right) , \end{aligned}$$
(8)
$$\begin{aligned} {\mathbf{d}_\mathrm{3}^{i+1}}&= {{\mathbf{d}_\mathrm{3}^i}}+ \varDelta {{\mathbf{d}_\mathrm{3}^i}}. \end{aligned}$$
(9)

The above systems of linear equations are solved using a GMRES technique [63], requiring the computation of matrix-vector products. In this work the matrix-vector products involving \(\left. \frac{\partial {\mathbf{N}_\mathrm{1}}}{\partial {\mathbf{d}_\mathrm{1}}} \right| _i, \left. \frac{\partial {\mathbf{N}_\mathrm{2}}}{\partial {\mathbf{d}_\mathrm{2}}} \right| _i\), and \( \left. \frac{\partial {\mathbf{N}_\mathrm{3}}}{\partial {\mathbf{d}_\mathrm{3}}} \right| _i\) are computed using a sparse-matrix-based approach, where the tangent matrices are derived analytically and assembled into a sparse-matrix data structure in a standard fashion. The remaining matrix-vector products involving \(\left. \frac{\partial {\mathbf{N}_\mathrm{1}}}{\partial {\mathbf{d}_\mathrm{2}}} \right| _i\) and \(\left. \frac{\partial {\mathbf{N}_\mathrm{2}}}{\partial {\mathbf{d}_\mathrm{1}}} \right| _i\) are approximated using a matrix-free approach, namely

$$\begin{aligned} \left. \frac{\partial {\mathbf{N}_\mathrm{1}}}{\partial {\mathbf{d}_\mathrm{2}}} \right| _i \varDelta {\mathbf{d}_\mathrm{2}^i}&= \frac{{\mathbf{N}_\mathrm{1}}\left( {\mathbf{d}_\mathrm{1}^i},~{\mathbf{d}_\mathrm{2}^i}+ \epsilon _1 \varDelta {\mathbf{d}_\mathrm{2}^i},~{{\mathbf{d}_\mathrm{3}^i}}\right) - {\mathbf{N}_\mathrm{1}}\left( {\mathbf{d}_\mathrm{1}^i},~{\mathbf{d}_\mathrm{2}^i},~{{\mathbf{d}_\mathrm{3}^i}}\right) }{\epsilon _1}, \end{aligned}$$
(10)
$$\begin{aligned} \left. \frac{\partial {\mathbf{N}_\mathrm{2}}}{\partial {\mathbf{d}_\mathrm{1}}} \right| _i \varDelta {\mathbf{d}_\mathrm{1}^i}&= \frac{{\mathbf{N}_\mathrm{2}}\left( {\mathbf{d}_\mathrm{1}^i}+ \epsilon _2 \varDelta {\mathbf{d}_\mathrm{1}^i},~{\mathbf{d}_\mathrm{2}^i},~{{\mathbf{d}_\mathrm{3}^i}}\right) - {\mathbf{N}_\mathrm{2}}\left( {\mathbf{d}_\mathrm{1}^i},~{\mathbf{d}_\mathrm{2}^i},~{{\mathbf{d}_\mathrm{3}^i}}\right) }{\epsilon _2}, \end{aligned}$$
(11)

where \(\epsilon _1\) and \(\epsilon _2\) are relatively small real numbers. This mixed analytical/numerical, matrix-based/vector-based approach was proposed for coupled problems in general in [79], and for FSI problems specifically in [88], where the numerical, vector-based computation is applied to the computation with the \(\left. \frac{\partial {\mathbf{N}_\mathrm{1}}}{\partial {\mathbf{d}_\mathrm{3}}} \right| _i\) block.

We feel the proposed approach is well suited for cases that require a relatively larger number of GMRES iterations (\(O\)(100)) for good overall nonlinear convergence. This is because the sparse matrices are formed once every nonlinear iteration (or, possibly, once every time step to further decrease the computational cost), making the associated computational cost independent of the number of GMRES iterations employed. Although matrix-vector products given by Eqs. (10) and (11) need to be performed once per GMRES iteration, the FEM assembly takes place over a narrow band of fluid elements near the fluid–structure interface, which is a lot less expensive than assembling the discrete residuals over the entire fluid mechanics domain. For a comprehensive exposition of sparse-matrix-based and matrix-free approaches see [18].

3 VAD simulation

3.1 Problem setup

Geometry

For the initial study, we use a generic VAD device as our computational domain. Geometric parameters, such as width, height, and angles of the entrance/exit arms are consistent with current designs, and are meant to be a generic representation of current commercially available devices. The chosen design for the initial study has a width of 7.7 cm, and an apex to apex height of 4.5 cm. The incline angle between the arms and the main blood chamber is 30\(^\circ \), with one assigned exclusively as the inlet, and the other as an outlet. The outlet faces are 1.5 cm in diameter. The air chamber has one small inlet/outlet port of diameter 0.8 cm. These are labeled in Fig. 1.

Fig. 1
figure 1

The computational domain, with the blood domain in red, and the air domain in blue. The inlet and outlet face of the blood chamber are labeled 1 and 2, respectively. The air-side inlet/outlet face is labeled 3. (Color figure online)

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A stroke volume of 73 mL was chosen for this device, which yields an ejection fraction of 68 %. A beat frequency of 80 bpm is used, for a pump output of 5.8 L/min. This initial study uses a VAD that is too large to be considered a pediatric model, but all data is within an acceptable physiologic range for adult models.

Boundary conditions

Each pump cycle may be broken up into two components: the fill stage and the ejection stage. We impose the fill period of 0.45 s, and the ejection period of 0.3 s, and we also enforce that each stage must fill or eject the same volume, 73 mL. For simplicity, the flow is assumed to behave sinusoidally during each stage. We can therefore impose the air chamber inflow flow rate \(q\) at a given time \(t\) as

$$\begin{aligned} q = \left\{ \begin{array}{l@{\quad }l} q_{e} \sin ^{\frac{1}{2}}\left( \frac{t}{0.3}\pi \right) &{} \text{ if }\; \,t<0.3 \\ q_{f} \sin ^{\frac{1}{2}}\left( \frac{t-0.3}{0.45}\pi \right) &{}\text{ otherwise } \end{array} \right. , \end{aligned}$$
(12)

where \(q_e\) and \(q_f\) are constants equal to the peak flow rate of each stage. The constants may be obtained from the equations

$$\begin{aligned} \int _0^{0.3} q_e \sin ^{\frac{1}{2}}\left( \frac{t}{0.3}\pi \right) ~\text{ d }t = 73, \end{aligned}$$
(13)

and

$$\begin{aligned} \int _{0.3}^{0.75} q_f \sin ^{\frac{1}{2}}\left( \frac{t-0.3}{0.45}\pi \right) ~\text{ d }t = -73, \end{aligned}$$
(14)

and are equal to \(q_e = 319.02\) cc/s and \(q_f = -212.68\) cc/s.

On the blood side, we alternate boundary conditions at the inlet/outlet between a Neumann condition and a Dirichlet condition as necessary since we do not directly compute the valve motion in the simulation. At the outlet, for example, we have two conditions. If we are in the fill stage, then we impose a zero-velocity (i.e., no flow) boundary condition. During the ejection, however, we impose a resistance boundary condition

$$\begin{aligned} p = C_r q + p_0, \end{aligned}$$

where \(q\) is the volumetric flow rate on the outlet face, \(C_r\) is a prescribed resistance value, \(p_0\) is the distal pressure, and \(p\) is the pressure at the outlet face. For the simulation we choose \(p_0\) to be 65 mmHg, which enforces a minimum pressure of 65 mmHg during the expel. The resistance value is set to \(C_r = 183\) g/(s cm\(^4\)), which gives a maximum systolic pressure of 108 mmHg. The inlet face uses the same boundary conditions, but, obviously, with opposite phase.

The structural membrane is simply supported around the circumference.

Remark

Note that, because the incompressible flow assumption is employed for both blood and air, the flow rate into the air chamber must equal to the flow rate out of the blood chamber, and vice versa. As a result, by controlling the total volume of the air going in and out of the air chamber, we automatically control the total volume of the blood flowing in and out of the blood chamber. The proposed quasi-direct FSI coupling guarantees that at every nonlinear iteration this balance holds. Loosely-coupled FSI approaches, besides being unsuitable for this problem due to the strong added mass effect, cannot guarantee this balance unless special procedures are devised to enforce it (see, e.g., [51]).

Blood, air, and membrane properties

Both air and blood are treated as incompressible, Newtonian fluids. The blood density and dynamic viscosity are set to \(1\) g/cm\(^3\) and \(0.04\) poise, respectively. The air density and dynamic viscosity are set to \(1.205\times 10^{-3}\) g/cm\(^3\) and \(2\times 10^{-4}\) poise, respectively. Given the VAD geometry, fluid properties, and flow rates employed, the peak Reynolds number is about 10,000 in the blood chamber, and 7,000 in the air chamber. These values are based on the inlet/outlet branch diameters and flow speeds. Note that the VAD blood chamber Reynolds number, which is higher than that in the large blood vessels of the human cardiovascular system (e.g., the thoracic aorta), is in the turbulent range.

The membrane is a flexible thin sheet, commonly made of polyurethane. We use membrane material properties consistent with those of the Penn State VAD, the LionHeart [29]. The LionHeart membrane has a thickness of 0.38 mm, density of 1.1 g/cm\(^3\), and Young’s modulus of 550 MPa [29]. In our simulation, we use a thinner membrane of \(0.25\) mm, which is reflective of the smaller device used for the pediatric population, as was provided in a private communication from the authors of [62]. The membrane initial configuration is obtained by taking a circular disc, which is exactly represented using a single NURBS patch with four corner singularities, and displacing the interior control points in the direction normal to the plane of the disc toward the air chamber. The initial shape of the membrane is assumed to be sinusoidal, and the control-point displacement \(d\) is given by the equation

$$\begin{aligned} d = 1.52\cos \left( \frac{r}{3.85}\frac{\pi }{2}\right) , \end{aligned}$$
(15)

where \(r\) is the radial distance of the control point from the center of the disc.

Meshing, mesh moving, and remeshing

The blood and air chamber volumes in the reference configuration are meshed using MeshSim automatic mesh generator (Symmetrix Inc., Clifton Park, NY). The number of elements in the air chamber is 238,322 and in the blood chamber is 497,160. The membrane is discretized using 1,024 \(C^1\)-continuous quadratic NURBS elements. The simulations are run for two time cycles of 0.75 s each, with a time step size of 1.0 ms. Generalized-\(\alpha \) time integration is used for the coupled FSI equation system (see [9, 24, 44]).

As the computation proceeds, the fluid mechanics mesh is moved using equations of elastostatics with Jacobian-based stiffening [18, 45, 77, 83], which better preserves the mesh quality in the simulations than the no-stiffening approach and delays the necessity to remesh. However, due to very large motions of the membrane the mesh eventually becomes highly deformed and a remesh is necessary to preserve the quality of the fluid mechanics discretization. The necessity to remesh is quantified in terms of the change in the element volume as measured by the ratio of the Jacobian determinants of the elements in the current step and the step immediately after the previous remesh. For this simulation, remeshing is performed once the ratio of 72 % for compression or 170 % for expansion is achieved.

During the remesh the surface meshes of the blood and air chamber, including those at the fluid–structure interface, are preserved, and a new tetrahedral mesh is generated on the interior of both subdomains. The solution data at the current step, which includes fluid velocity, acceleration, and pressure, as well as mesh velocity and displacement, is transferred to the new mesh by means of a nodal interpolation procedure that involves the computation of the inverse mapping. To efficiently locate the element in the old mesh containing the nodal point of interest, we use a “point in a polygon” method [58]. Once the data is transferred to the new mesh, the FSI computation continues. No special procedures for transferring the pressure data (e.g., pressure clipping [46, 74, 88]) are employed.

3.2 Simulation Results

The VAD simulation was carried out in a parallel computing environment at the San Diego Supercomputing Center [67]. The simulation was run for two time cycles. All the data presented is gathered from the second time cycle. The time \(t=0\) in all figures refers to the beginning of the second cycle.

Figures 2, 3, and 4 show snapshots of the computed blood flow speed and membrane deformation. The simulation captures a very complex membrane motion, with many folds, clearly seen in Figs. 3 and 4. The deformed membrane surface is notably smooth, with no sharp kinks on the mesh edges, which is due to the underlying smoothness of the NURBS discretization. This buckling motion is smoother than is typically attained using more traditional methods. Since the structural kinematics is used to drive the fluid mechanics mesh deformation, the smoother buckling motion ensures that the fluid mechanics mesh at the fluid–structure interface remains smooth.

Fig. 2
figure 2

Flow speed (cm/s) in the deformed blood chamber configuration at \(t=0.15\) s

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Fig. 3
figure 3

Top view of the membrane deformed configuration at \(t=0.15\) s. Despite the complex deformation pattern, the wrinkles on the membrane surface are smooth

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Fig. 4
figure 4

The membrane deformed configuration at time a \(t=0\) s, b \(t=0.15\) s, c \(t=0.3\) s, and d \(t= 0.525\) s

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During the fill stage, the inlet jet impinges on the chamber wall, and flows along the wall creating a strong vortex. The vortex is destroyed early in the eject phase, as seen in Fig. 5. This strong vortex is a chief source of the wall shear stress and flow stagnation in the center of the device, and may play an important role in thrombus formation. Strong rotating flow during filling was also observed experimentally in [62] and will be of interest in the future validation efforts.

Fig. 5
figure 5

Blood flow speed (cm/s) at 0.5 cm above the plane separating the blood and air chambers. In-plane vectors shown during (a) expel stage (\(t = 0.14\) s) and (b) fill stage (\(t = 0.665\) s)

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Figures 6 and 7 show the time history of volume-averaged pressure and flow speed. The pressure drop across the membrane is small relative to the mean pressure, which is not surprising as it takes little effort to move the membrane. The peak average flow speed in the blood chamber during the fill stage is nearly 20 % greater than during the eject stage. Although the eject stage imposes a flow rate 50 % higher than the fill stage, the corresponding peak average flow speed is lower. This is in large part due to the rotational flow seen in Fig. 5, which is present only during the fill stage.

Fig. 6
figure 6

Time history of the volume-averaged pressure in the blood and air chambers

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Fig. 7
figure 7

Time history of the volume-averaged flow speed in the blood chamber

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4 Conclusions and future work

This paper addressed several computational challenges involved in the FSI modeling of pulsatile VADs. These include large, buckling motions of the membrane, the need for periodic remeshing of the fluid mechanics domain, and the necessity to employ tightly coupled FSI solution strategies due to the very strong added mass effect present in the problem. Structural modeling of the membrane makes use of IGA, which has several accuracy and robustness benefits associated with the smoothness of the underlying discretization. The strong FSI coupling is efficiently implemented using a combination of matrix-free and sparse-matrix-based techniques. The simulations captured the essential blood flow features and structural deformations observed clinically and experimentally in pulsatile VADs. This is the first 3D, full-scale, high-fidelity FSI modeling of pulsatile VADs.

The computational FSI tools developed here provide a foundation for the study of the fluid and structural mechanics inside pulsatile VADs, with clinically relevant implications. We intend to use such simulations in the future to investigate design improvements that will mitigate risk of thrombosis, especially for pediatric populations. Methodically exploring a parameterized design space using computational FSI combined with modern optimization techniques and uncertainty quantification [54, 55, 64, 66, 103] may lead to novel designs that will improve patient outcomes. Thrombus formation involves a complex interplay between hemodynamics and blood chemistry, presenting significant modeling challenges [101, 102]. Future work could incorporate reduced order models of blood chemistry to capture the key features of this process.

A strong validation effort is also planned. Particle Image Velocimetry data is available for the Penn State device [62], which we intend to simulate in the future.