In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm. The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid–structure interface traction, arriving at Nitsche’s method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance. To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

This paper builds on a recently developed immersogeometric fluid–structure interaction (FSI) methodology for bioprosthetic heart valve (BHV) modeling and simulation. It enhances the proposed framework in the areas of geometry design and constitutive modeling. With these enhancements, BHV FSI simulations may be performed with greater levels of automation, robustness and physical realism. In addition, the paper presents a comparison between FSI analysis and standalone structural dynamics simulation driven by prescribed transvalvular pressure, the latter being a more common modeling choice for this class of problems. The FSI computation achieved better physiological realism in predicting the valve leaflet deformation than its standalone structural dynamics counterpart.

We propose a framework that combines variational immersed-boundary and arbitrary Lagrangian–Eulerian (ALE) methods for fluid–structure interaction (FSI) simulation of a bioprosthetic heart valve implanted in an artery that is allowed to deform in the model. We find that the variational immersed-boundary method for FSI remains robust and effective for heart valve analysis when the background fluid mesh undergoes deformations corresponding to the expansion and contraction of the elastic artery. Furthermore, the computations presented in this work show that the arterial wall deformation contributes significantly to the realism of the simulation results, leading to flow rates and valve motions that more closely resemble those observed in practice.

We present a tetrahedral finite cell method for the simulation of incompressible flow around geometrically complex objects. The method immerses such objects into non-boundary-fitted meshes of tetrahedral finite elements and weakly enforces Dirichlet boundary conditions on the objects' surfaces. Adaptively-refined quadrature rules faithfully capture the flow domain geometry in the discrete problem without modifying the non-boundary-fitted finite element mesh. A variational multiscale formulation provides accuracy and robustness in both laminar and turbulent flow conditions. We assess the accuracy of the method by analyzing the flow around an immersed sphere for a wide range of Reynolds numbers. We show that quantities of interest such as the drag coefficient, Strouhal number and pressure distribution over the sphere are in very good agreement with reference values obtained from standard boundary-fitted approaches. We place particular emphasis on studying the importance of the geometry resolution in intersected elements. Aligning with the immersogeometric concept, our results show that the faithful representation of the geometry in intersected elements is critical for accurate flow analysis. We demonstrate the potential of our proposed method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor. KeywordsImmersed method, Complex geometry, Immersogeometric finite elements, Geometric accuracy in intersected elements, Weakly enforced boundary conditions, Tetrahedral finite cell method AbstractWe present a tetrahedral finite cell method for the simulation of incompressible flow around geometrically complex objects. The method immerses such objects into non-boundary-fitted meshes of tetrahedral finite elements and weakly enforces Dirichlet boundary conditions on the objects' surfaces. Adaptively-refined quadrature rules faithfully capture the flow domain geometry in the discrete problem without modifying the non-boundary-fitted finite element mesh. A variational multiscale formulation provides accuracy and robustness in both laminar and turbulent flow conditions. We assess the accuracy of the method by analyzing the flow around an immersed sphere for a wide range of Reynolds numbers. We show that quantities of interest such as the drag coefficient, Strouhal number and pressure distribution over the sphere are in very good agreement with reference values obtained from standard boundary-fitted approaches. We place particular emphasis on studying the importance of the geometry resolution in intersected elements. Aligning with the immersogeometric concept, our results show that the faithful representation of the geometry in intersected elements is critical for accurate flow analysis. We demonstrate the potential of our proposed method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor.

This paper uses a divergence-conforming B-spline fluid discretization to address the long-standing issue of poor mass conservation in immersed methods for computational fluid–structure interaction (FSI) that represent the influence of the structure as a forcing term in the fluid subproblem. We focus, in particular, on the immersogeometric method developed in our earlier work, analyze its convergence for linear model problems, then apply it to FSI analysis of heart valves, using divergence-conforming B-splines to discretize the fluid subproblem. Poor mass conservation can manifest as effective leakage of fluid through thin solid barriers. This leakage disrupts the qualitative behavior of FSI systems such as heart valves, which exist specifically to block flow. Divergence-conforming discretizations can enforce mass conservation exactly, avoiding this problem. To demonstrate the practical utility of immersogeometric FSI analysis with divergence-conforming B-splines, we use the methods described in this paper to construct and evaluate a computational model of an in vitro experiment that pumps water through an artificial valve.

We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L2" role="presentation"> error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest. KeywordsImmersed finite element methods, Weak boundary and coupling conditions, Non-symmetric Nitsche method AbstractWe explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L 2 error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.

A novel stabilized formulation for 3D compressible viscous flows on moving domains is developed. New weak imposition of essential boundary conditions and sliding-interface formulations are also proposed in the context of moving-domain compressible flows. The new formulation is successfully tested on a set of examples spanning a wide range of Reynolds and Mach numbers showing its superior robustness. Experimental validation of the new formulation is also carried out with good success. In addition, the formulation is applied to simulate flow inside a gas turbine stage, illustrating its potential to support design of real engineering systems through high-fidelity aerodynamic analysis.

Numerous studies have suggested that medical image derived computational mechanics models could be developed to reduce mortality and morbidity due to cardiovascular diseases by allowing for patient-specific surgical planning and customized medical device design. In this work, we present a novel framework for designing prosthetic heart valves using a parametric design platform and immersogeometric fluid-structure interaction (FSI) analysis. We parameterize the leaflet geometry using several key design parameters. This allows for generating various perturbations of the leaflet design for the patient-specific aortic root reconstructed from the medical image data. Each design is analyzed using our hybrid arbitrary Lagrangian-Eulerian/immersogeometric FSI methodology, which allows us to efficiently simulate the coupling of the deforming aortic root, the parametrically designed prosthetic valves, and the surrounding blood flow under physiological conditions. A parametric study is performed to investigate the influence of the geometry on heart valve performance, indicated by the effective orifice area and the coaptation area. Finally, the FSI simulation result of a design that balances effective orifice area and coaptation area reasonably well is compared with patient-specific phase contrast magnetic resonance imaging data to demonstrate the qualitative similarity of the flow patterns in the ascending aorta.