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We introduce a new paradigm for immersed finite element and isogeometric methods based on interpolating function spaces from an unfitted background mesh into Lagrange finite element spaces defined on a foreground mesh that captures the domain geometry but is otherwise subject to minimal constraints on element quality or connectivity. This is a generalization of the concept of Lagrange extraction from the isogeometric analysis literature and also related to certain variants of the finite cell and material point methods. Crucially, the interpolation may be approximate without sacrificing high-order convergence rates, which distinguishes the present method from existing finite cell, CutFEM, and immersogeometric approaches. The interpolation paradigm also permits non-invasive reuse of existing finite element software for immersed analysis. We analyze the properties of the interpolation-based immersed paradigm for a model problem and implement it on top of the open-source FEniCS finite element software, to apply it to a variety of problems in fluid, solid, and structural mechanics where we demonstrate high-order accuracy and applicability to practical geometries like trimmed spline patches.

This work develops, discretizes, and validates a continuum model of a molybdenum disulfide (MoS$_2$) monolayer interacting with a periodic holey silicon nitride substrate via van der Waals (vdW) forces. The MoS$_2$ layer is modeled as a geometrically nonlinear Kirchhoff-Love shell, and vdW forces are modeled by a Lennard-Jones potential, simplified using approximations for a smooth substrate topography. The material parameters of the shell model are calibrated by comparing small-strain tensile and bending tests with atomistic simulations. This model is efficiently discretized using isogeometric analysis (IGA) for the shell structure and a pseudo-time continuation method for energy minimization. The IGA shell model is validated against fully-atomistic calculations for several benchmark problems with different substrate geometries. The continuum simulations reproduce deflections, strains and curvatures predicted by atomistic simulations, which are known to strongly affect the electronic properties of MoS$_2$, with deviations well below the modeling errors suggested by differences between the widely-used reactive empirical bond order and Stillinger-Weber interatomic potentials. Agreement with atomistic results depends on geometric nonlinearity in some cases, but a simple isotropic St. Venant-Kirchhoff model is found to be sufficient to represent material behavior. We find that the IGA discretization of the continuum model has a much lower computational cost than atomistic simulations, and expect that it will enable efficient design space exploration in strain engineering applications. This is demonstrated by studying the dependence of strain and curvature in MoS$_2$ over a holey substrate as a function of the hole spacing on scales inaccessible to atomistic calculations. The results show an unexpected qualitative change in the deformation pattern below a critical hole separation.

This paper reviews several existing peridynamic models for frictional contact (previously documented only in the gray literature), and extends them to remedy various shortcomings. In particular, we introduce a state-based nonlocal friction formulation that corrects for loss of angular momentum balance and objectivity in a widely used frictional extension of short-range contact forces. We demonstrate the properties of various peridynamic contact models by applying them in finite element and meshfree peridynamic analyses of benchmark problems and an impact/penetration test.

We extend a recently-developed framework for isogeometric analysis of composite laminates to drive material damage evolution with a smoothed strain field. This builds on ideas from gradient-enhanced continuum damage modeling, and is intended to limit the dependence of damage predictions on the choice of discrete mesh. The resulting enhanced framework models each lamina of a composite shell structure as a Kirchhoff–Love thin shell. To account for the anisotropic damage modes of laminae, we smooth a tensor-valued strain by solving an elliptic partial differential equation (PDE) system on each lamina. This strain-smoothing PDE system is formulated to be independent of the choice of coordinates and is applicable to general manifold shell geometries. Numerical examples illustrate the enhanced damage model’s validity, mesh-independence, and applicability to complex industrial geometries.

We propose a new way to include strong velocity discontinuities in the material point method (MPM). We use the recent idea of damage gradient partitioning to dynamically determine whether or not deformed locations of material points are on the same side of a discontinuity as each node of a finite element discretization of the Eulerian velocity field. Velocity basis functions associated with the finite element nodes are then altered to ensure that their contributions to discontinuous components of the velocity are supported only at points assigned to the same side of the discontinuity. The partitioning of material points relative to each finite element node eliminates the need for indexing partitions, and accommodates discontinuities that do not partition material into disjoint bodies (e.g., propagating cracks). By building velocity discontinuities directly into the approximation space, sliding contact, separation, and multi-body interaction are possible within a single velocity field. The methodology avoids the need to track contact surfaces, construct and evolve explicit cracks, or use multiple velocity fields. Several numerical examples are computed to support these claims. The examples are also compared to the conventional MPM, demonstrating the method’s superiority in dealing with sliding contact and separation.

This paper introduces tIGAr , a library for using the open-source finite element (FE) automation software FEniCS to perform isogeometric analysis (IGA). The library uses a global variant of Bezier extraction to avoid modifying the finite element assembly procedures of FEniCS . This makes much of the convenient functionality of FEniCS directly available for IGA. General rational splines are implemented through an abstraction that sees only an extracted representation of the IGA function space. Through this abstraction, an enormous variety of spline spaces can be used for analysis, so long as a corresponding preprocessor is developed for each one, implementing a simple interface. As examples, we discuss preprocessors for B-splines specified analytically, non-uniform rational B-splines (NURBS) designed using the open-source software igakit , and T-splines designed using commercial software. We then demonstrate the implementation of solvers for several non-trivial partial differential equations that benefit from IGA. We also evaluate the parallel performance of tIGAr on a distributed-memory supercomputer . Finally, we outline possibilities for further development of IGA in FEniCS . Source code for tIGAr is continuously updated online at https://github.com/david-kamensky/tIGAr .

Part I of this series introduced a formulation of phase field fracture in which the inclusion of micro-scale inertia leads to a hyperbolic system of partial differential equations governing the coupled problems of nonlinear elasticity and material damage. This paper applies that formulation to a coupled isogeometric–meshfree discretization of compressible fluid–structure interaction that discretizes the damage field’s governing equation using the reproducing-kernel particle method and the fluid–structure momentum balance equation using isogeometric analysis. The hyperbolic character of the coupled system permits efficient fully-explicit time integration of the entire system (not just the momentum equation, as in previous semi-implicit procedures for phase field analysis of dynamic fracture). Results for dynamic fracture problems match those computed in Part I using implicit time integration and standard finite elements. However, the isogeometric–meshfree discretization of this paper provides greater flexibility in representing structural fragmentation from blast loading. The blast loading computations at the end of this paper demonstrate the model’s ability to represent extreme events involving air-blast–structure interaction.

We propose a new phase field model of dynamic brittle fracture, in which material damage evolves according to a hyperbolic partial differential equation. This model can be stably discretized using explicit time integration, without imposing crippling time step restrictions with refinement in space. The model is derived from microforce balance by including effects of microscopic inertia. Quantitative predictions of the proposed model differ from those of parabolic and elliptic phase field models in that there is a mild rate-toughening effect, but major qualitative solution features are essentially the same. We compute finite element approximations of solutions to several dynamic fracture scenarios to support these claims. Part II of this series will incorporate the proposed model into a hybrid isogeometric–meshfree framework for air-blast–structure interaction and provide further demonstrations of the model’s physical and numerical properties.

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

The functional complexity of native and replacement aortic heart valves are well known, incorporating such physical phenomenons as time-varying non-linear anisotropic soft tissue mechanical behavior, geometric non-linearity, complex multi-surface time varying contact, and fluid-structure interactions to name a few. It is thus clear that computational simulations are critical in understanding AV function and for the rational basis for design of their replacements. However, such approaches continued to be limited by ad-hoc approaches for incorporating tissue fibrous structure, high-fidelity material models, and valve geometry. To this end, we developed an integrated tri-leaflet valve pipeline built upon an isogeometric analysis (IGA) framework. A high-order structural tensor (HOST) based method was developed for efficient storage and mapping the two-dimensional fiber structural data onto the valvular 3D geometry. We then developed a neural network (NN) material model that learned the responses of a detailed mesostructural model for exogenously cross-linked planar soft tissues. The NN material model not only reproduced the full anisotropic mechanical responses but also demonstrated a considerable efficiency improvement, as it was trained over a range of realizable fibrous structures. Results of parametric simulations were then performed, as well as population based bicuspid aortic heart valve fiber structure, that demonstrated the efficiency and robustness of the present approach. In summary, the present approach that integrates HOST and NN material model provides an efficient computational analysis framework with increased physical and functional realism for the simulation of native and replacement tri-leaflet heart valves. This article is protected by copyright. All rights reserved.

In this two-part paper we begin the development of a new class of methods for modeling fluid–structure interaction (FSI) phenomena for air blast. We aim to develop accurate, robust, and practical computational methodology, which is capable of modeling the dynamics of air blast coupled with the structure response, where the latter involves large, inelastic deformations and disintegration into fragments. An immersed approach is adopted, which leads to an a-priori monolithic FSI formulation with intrinsic contact detection between solid objects, and without formal restrictions on the solid motions. In Part I of this paper, the core air-blast FSI methodology suitable for a variety of discretizations is presented and tested using standard finite elements. Part II of this paper focuses on a particular instantiation of the proposed framework, which couples isogeometric analysis (IGA) based on non-uniform rational B-splines and a reproducing-kernel particle method (RKPM), which is a meshfree technique. The combination of IGA and RKPM is felt to be particularly attractive for the problem class of interest due to the higher-order accuracy and smoothness of both discretizations, and relative simplicity of RKPM in handling fragmentation scenarios. A collection of mostly 2D numerical examples is presented in each of the parts to illustrate the good performance of the proposed air-blast FSI framework.

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. © 2018. This is the authors’ accepted and refereed manuscript to the article. Locked until 12.4.2019 due to copyright restrictions. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

This work formulates frictionless contact between solid bodies in terms of a repulsive potential energy term and illustrates how numerical integration of the resulting forces is computationally similar to the "pinball algorithm" proposed and studied by Belytschko and collaborators in the 1990s. We thereby arrive at a numerical approach that has both the theoretical advantages of a potential-based formulation and the algorithmic simplicity, computational efficiency, and geometrical versatility of pinball contact. The singular nature of the contact potential requires a specialized nonlinear solver and an adaptive time stepping scheme to ensure reliable convergence of implicit dynamic calculations. We illustrate the effectiveness of this numerical method by simulating several benchmark problems and the structural mechanics of the right atrioventricular (tricuspid) heart valve. Atrioventricular valve closure involves contact between every combination of shell surfaces, edges of shells, and cables, but our formulation handles all contact scenarios in a unified manner. We take advantage of this versatility to demonstrate the effects of chordal rupture on tricuspid valve coaptation behavior.

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.

This work describes a fluid–structure interaction (FSI) design optimization framework and applies it to improving the structural performance of a water brake used to stop aircraft landing on short runways. Inside the water brake, a dissipative torque is exerted on a rotor through interactions between rotor blades and a surrounding fluid. We seek to optimize blade shape over a parameterized design space, to prevent potentially-damaging stress concentrations without compromising performance. To avoid excessive numbers of costly simulations while exploring the design space, we use a surrogate management framework that combines derivative-free pattern search optimization with automated construction of a low-fidelity surrogate model, requiring only a handful of high-fidelity FSI simulations. We avoid the difficult problem of generating fluid and structure meshes at new points in the design space by using immersogeometric FSI analysis. The structure is analyzed isogeometrically: its design geometry also serves as a computational mesh. This geometry is then immersed in an unfitted fluid mesh that does not depend on the structure’s design parameters. We use this framework to make significant improvements to a baseline design found in the literature. Specifically, there is a 35% reduction of von Mises stress variance and a 25% reduction of maximum of stress, while the resisting torque and mass of the optimized blades remain uncompromised.

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 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.

We introduce a residual-based stabilized formulation for incompressible Navier–Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf–sup stable spaces with H 1 -conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier–Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.

In this paper we design and analyze an explicit partitioned procedure for a 2D dynamic local-to-nonlocal (LtN) coupling problem, based on a new nonlocal Robin-type transmission condition. The nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter δ characterizing the range of nonlocal interactions, and the local subproblem is described by the classical heat equation. We consider a heterogeneous system where the local and nonlocal subproblems present different physical properties, and employ no overlapping region between the two subdomains. We first propose a new generalization of classical local Neumann-type condition by converting the local flux to a correction term in the nonlocal model, and show that the proposed Neumann-type boundary formulation recovers the local case as O ( δ 2 ) in the L ∞ norm. We then extend the nonlocal Neumann-type boundary condition to a Robin-type boundary condition, and develop a local-to-nonlocal coupling formulation with Robin–Dirichlet transmission conditions. To stabilize the explicit coupling procedure and to achieve asymptotic compatibility, the choice of the coefficient in the Robin condition is obtained via amplification factor analysis for the discretized system with coarse grids. Employing a high-order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver, the selection of optimal Robin coefficients are verified with numerical experiments on heterogeneous and complicated domains. With the developed optimal coupling strategy, we numerically demonstrate the coupling framework’s asymptotic convergence to the local limit with an O ( δ ) = O ( h ) rate, when there is a fixed ratio between the horizon size δ and the spatial discretization size h .

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 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.

The use of replacement heart valves continues to grow due to the increased prevalence of valvular heart disease resulting from an ageing population. Since bioprosthetic heart valves (BHVs) continue to be the preferred replacement valve, there continues to be a strong need to develop better and more reliable BHVs through and improved the general understanding of BHV failure mechanisms. The major technological hurdle for the lifespan of the BHV implant continues to be the durability of the constituent leaflet biomaterials, which if improved can lead to substantial clinical impact. In order to develop improved solutions for BHV biomaterials, it is critical to have a better understanding of the inherent biomechanical behaviors of the leaflet biomaterials, including chemical treatment technologies, the impact of repetitive mechanical loading, and the inherent failure modes. This review seeks to provide a comprehensive overview of these issues, with a focus on developing insight on the mechanisms of BHV function and failure. Additionally, this review provides a detailed summary of the computational biomechanical simulations that have been used to inform and develop a higher level of understanding of BHV tissues and their failure modes. Collectively, this information should serve as a tool not only to infer reliable and dependable prosthesis function, but also to instigate and facilitate the design of future bioprosthetic valves and clinically impact cardiology.

This paper adapts the concept of residual-based shock-capturing viscosity to the setting of solid mechanics. To evaluate the residual of the momentum balance equation, one requires the divergence of the Cauchy stress. Solid constitutive models used in simulations of extreme events involving shocks typically specify the Cauchy stress in terms of a local rate equation rather than an explicit formula involving the current deformation gradient and/or strain rate; unlike in widely-used models for fluid mechanics, there is usually no closed-form expression for spatial derivatives of the solid Cauchy stress. We therefore investigate the evolution of stress gradients given rate-form models involving various objective rates of Cauchy stress. The rate equations we derive for the stress gradient are then integrated numerically to furnish the stress divergence needed to define the shock viscosity. The properties of this shock viscosity are demonstrated in benchmark problems using Lagrangian isogeometric analysis and an immersed isogeometric–meshfree simulation framework.

In the mid-2010s, we began applying a combination of isogeometric analysis and immersed boundary methods to the problem of bioprosthetic heart valve (BHV) fluid–structure interaction (FSI). This chapter reviews how our research on BHV FSI (1) crystallized the emerging concept of immersogeometric analysis, (2) introduced a new semi-implicit numerical method for weakly enforcing constraints in time dependent problems, which we refer to as the dynamic augmented Lagrangian approach, and (3) clarified the important role of mass conservation in immersed FSI analysis. We illustrate these contributions with selected numerical results and discuss future improvements to, and applications of, the computational FSI techniques we have developed.

The purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

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.

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.