David Kamensky

Immersogeometric cardiovascular fluid–structure interaction analysis with divergence-conforming B-splines

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.

Statistical debugging with elastic predicates

Traditional debugging and fault localization methods have addressed localization of sources of software failures. While these methods are effective in general, they are not tailored to an important class of software, including simulations and computational models, which employ floating-point computations and continuous stochastic distributions to represent, or support evaluation of, an underlying model. To address this shortcoming, we introduce elastic predicates, a novel approach to predicate-based statistical debugging. Elastic predicates introduce profiling of values assigned to variables within a failing program. These elastic predicates are better predictors of software failure than the static and uniform predicates used in existing techniques such as Cooperative Bug Isolation (CBI). We present experimental results for established fault localization benchmarks and widely used simulations that show improved effectiveness.

Statistical Debugging for Simulations

Predictions from simulations have entered the mainstream of public policy and decision-making practices. Unfortunately, methods for gaining insight into faulty simulations outputs have not kept pace. Ideally, an insight gathering method would automatically identify the cause of a faulty output and explain to the simulation developer how to correct it. In the field of software engineering, this challenge has been addressed for general-purpose software through statistical debuggers. We present two research contributions, elastic predicates and many-valued labeling functions, that enable debuggers designed for general-purpose software to become more effective for simulations employing random variates and continuous numbers. Elastic predicates address deficiencies of existing debuggers related to continuous numbers, whereas many-valued labeling functions support the use of random variates. When used in combinations, these contributions allow a simulation developer tasked with localizing the program statement causing the faulty simulation output to examine 40% fewer statements than the leading alternatives. Our evaluation shows that elastic predicates and many-valued labeling functions maintain their ability to reduce the number of program statements that need to be examined under the imperfect conditions that developers experience in practice.

A framework for designing patient-specific bioprosthetic heart valves using immersogeometric fluid-structure interaction 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.

An anisotropic constitutive model for immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves

This paper considers an anisotropic hyperelastic soft tissue model, originally proposed for native valve tissue and referred to herein as the Lee-Sacks model, in an isogeometric thin shell analysis framework that can be readily combined with immersogeometric fluid-structure interaction (FSI) analysis for high-fidelity simulations of bioprosthetic heart valves (BHVs) interacting with blood flow. We find that the Lee-Sacks model is well-suited to reproduce the anisotropic stress-strain behavior of the cross-linked bovine pericardial tissues that are commonly used in BHVs. An automated procedure for parameter selection leads to an instance of the Lee-Sacks model that matches biaxial stress-strain data from the literature more closely, over a wider range of strains, than other soft tissue models. The relative simplicity of the Lee-Sacks model is attractive for computationally-demanding applications such as FSI analysis and we use the model to demonstrate how the presence and direction of material anisotropy affect the FSI dynamics of BHV leaflets.

Projection-based stabilization of interface Lagrange multipliers in immersogeometric fluid–thin structure interaction analysis, with application to heart valve modeling

This paper discusses a method of stabilizing Lagrange multiplier fields used to couple thin immersed shell structures and surrounding fluids. The method retains essential conservation properties by stabilizing only the portion of the constraint orthogonal to a coarse multiplier space. This stabilization can easily be applied within iterative methods or semi-implicit time integrators that avoid directly solving a saddle point problem for the Lagrange multiplier field. Heart valve simulations demonstrate applicability of the proposed method to 3D unsteady simulations. An appendix sketches the relation between the proposed method and a high-order-accurate approach for simpler model problems.

Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves

We present a methodology for immersing spline-based representations of thin flexible structures into stabilized discretizations of unsteady viscous incompressible flows [1]. The fluid and structure subproblems are coupled through a Lagrange multiplier field, augmented with penalization of the interface velocity difference. The fully discrete system is advanced in time using a semi-implicit algorithm. The stability of this algorithm may be analyzed by relating it to fully implicit integration of a surrogate problem, which penalizes the time integral of interface velocity difference. We apply this methodology to the analysis of bioprosthetic heart valves, where mass conservation in the fluid sub-problem is essential to obtaining useful solutions. The Lagrange multiplier and penalty forces acting on the fluid sub-problem are concentrated on a surface of co-dimension one to the fluid domain, which can produce unacceptable violations of mass conservation in stabilized fluid discretizations. We find that a simple modification of stabilization parameters within an order-h neighborhood of the structure can greatly reduce this error, even when the fluid is discretized using continuous equal-order pressure and velocity spaces, defined over a quasi-uniform mesh.

An Immersogeometric Method for the Simulation of Turbulent Flow Around Complex Geometries

In this chapter we summarize a recently proposed immersogeometric method for the simulation of incompressible flow around geometrically complex objects. The method immerses the objects into unfitted tetrahedral finite elements meshes and weakly enforces Dirichlet boundary conditions on the surfaces of the objects. Adaptively refined quadrature rules are used to faithfully capture the flow domain geometry in the discrete problem without modifying the unfitted finite element mesh. A variational multiscale formulation which provides accuracy and robustness in both laminar and turbulent flow conditions is employed. 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 flow quantities of interest are in very good agreement with reference values obtained from standard boundary-fitted approaches. Our results also 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 a full-scale agricultural tractor.