Immersed Boundary Finite Element Research Articles

An eigenvalue stabilization technique for immersed boundary finite element methods in explicit dynamics

The application of immersed boundary methods in static analyses is often impeded by poorly cut elements (small cut elements problem), leading to ill-conditioned linear systems of equations and stability problems. While these concerns may not be paramount in explicit dynamics, a substantial reduction in the critical time step size based on the smallest volume fraction χ of a cut element is observed. This reduction can be so drastic that it renders explicit time integration schemes impractical. To tackle this challenge, we propose the use of a dedicated eigenvalue stabilization (EVS) technique.The EVS-technique serves a dual purpose. Beyond merely improving the condition number of system matrices, it plays a pivotal role in extending the critical time increment, effectively broadening the stability region in explicit dynamics. As a result, our approach enables robust and efficient analyses of high-frequency transient problems using immersed boundary methods. A key advantage of the stabilization method lies in the fact that only element-level operations are required.This is accomplished by computing all eigenvalues of the element matrices and subsequently introducing a stabilization term that mitigates the adverse effects of cutting. Notably, the stabilization of the mass matrix Mc of cut elements – especially for high polynomial orders p of the shape functions – leads to a significant raise in the critical time step size Δtcr.To demonstrate the efficiency of our technique, we present two specifically selected dynamic benchmark examples related to wave propagation analysis, where an explicit time integration scheme must be employed to leverage the increase in the critical time step size.

Impact of Multi-Grade Localized Calcifications on Aortic Valve Dynamics under Helical Inflow: A Comparative Hemodynamic Study

This study investigates the hemodynamic impacts of localized aortic valve calcification, utilizing immersed boundary-finite element (IBFE) method simulations with realistic inflow patterns of uniform and helical blood flow from the left ventricular outflow tract (LVOT). We modeled the aortic valve leaflets with varying grades of calcification, assessing their influence on valve performance, including transvalvular hemodynamics, wall shear stress (WSS) indices, and vortical structures. The findings highlighted that calcification significantly restricts leaflet motion, diminishes the orifice area, disrupts flow efficiency, and consequently increases the left ventricular workload. Advanced calcification resulted in elevated WSS, especially at the leaflet tips, which indicates a heightened risk of endothelial damage and further calcification. Asymmetrical calcifications redirect flow towards the ascending aorta wall, potentially inducing structural damage and increased stress on the remaining healthy leaflets. Calcification was also found to alter the naturally occurring helical blood flow patterns, affecting the system’s fluid transport efficiency and possibly contributing to cardiovascular disease progression. The study revealed a significant alteration in vortex formation, with calcification causing distorted and complex vortex structures, which may influence the dynamics of blood flow and valve function. These insights into the hemodynamic changes induced by calcification contribute to a better understanding of the progression of aortic valve diseases and could inform more effective diagnostic and treatment strategies.

An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces

We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. That is, integration is performed over a fixed fictitious domain. Source terms are included to account for interface conditions and extend the boundary data into the complement of the embedded domain. The calculation of these extra terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation of data back to the quadrature points is performed using Taylor expansions. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.

Immersed boundary finite element method for blood flow simulation

We present an efficient and accurate immersed boundary (IB) finite element (FE) method for numerically solving the incompressible Navier–Stokes equations. Particular emphasis is given to internal flows with complex geometries (blood flow in the vasculature system). IB methods are computationally costly for internal flows, mainly due to the large percentage of grid points that lie outside the flow domain. In this study, we apply a local refinement strategy, along with a domain reduction approach, in order to reduce the grid that covers the flow domain and increase the percentage of grid nodes that fall inside the flow domain. The proposed method utilizes an efficient and accurate FE solver with the incremental pressure correction scheme (IPCS), along with the boundary condition enforced IB method to numerically solve the transient, incompressible Navier–Stokes flow equations. We verify the accuracy of the numerical method using the analytical solution for Poiseuille flow in a cylinder. We further examine the accuracy and applicability of the proposed method by considering flow within complex geometries, such as blood flow in aneurysmal vessels and the aorta, flow configurations which would otherwise be extremely difficult to solve by most IB methods. Our method offers high accuracy, as demonstrated by the verification examples, and high efficiency, as demonstrated through the solution of blood flow within complex geometry on an off-the-shelf laptop computer.

The Comparison of Different Constitutive Laws and Fiber Architectures for the Aortic Valve on Fluid-Structure Interaction Simulation.

Built on the hybrid immersed boundary/finite element (IB/FE) method, fluid–structure interaction (FSI) simulations of aortic valve (AV) dynamics are performed with three different constitutive laws and two different fiber architectures for the AV leaflets. An idealized AV model is used and mounted in a straight tube, and a three-element Windkessel model is further attached to the aorta. After obtaining ex vivo biaxial tensile testing of porcine AV leaflets, we first determine the constitutive parameters of the selected three constitutive laws by matching the analytical stretch–stress relations derived from constitutive laws to the experimentally measured data. Both the average error and relevant R-squared value reveal that the anisotropic non-linear constitutive law with exponential terms for both the fiber and cross-fiber directions could be more suitable for characterizing the mechanical behaviors of the AV leaflets. We then thoroughly compare the simulation results from both structural mechanics and hemodynamics. Compared to the other two constitutive laws, the anisotropic non-linear constitutive law with exponential terms for both the fiber and cross-fiber directions shows the larger leaflet displacements at the opened state, the largest forward jet flow, the smaller regurgitant flow. We further analyze hemodynamic parameters of the six different cases, including the regurgitant fraction, the mean transvalvular pressure gradient, the effective orifice area, and the energy loss of the left ventricle. We find that the fiber architecture with body-fitted orientation shows better dynamic behaviors in the leaflets, especially with the constitutive law using exponential terms for both the fiber and cross-fiber directions. In conclusion, both constitutive laws and fiber architectures can affect AV dynamics. Our results further suggest that the strain energy function with exponential terms for both the fiber and cross-fiber directions could be more suitable for describing the AV leaflet mechanical behaviors. Future experimental studies are needed to identify competent constitutive laws for the AV leaflets and their associated fiber orientations with controlled experiments. Although limitations exist in the present AV model, our results provide important information for selecting appropriate constitutive laws and fiber architectures when modeling AV dynamics.

Adaptive mesh refinement immersed boundary method for simulations of laminar flows past a moving thin elastic structure

One of the critical issues in numerical simulation of fluid-structure interaction problems is inaccuracy of the solutions, especially for flows past a stationary thin elastic structure where large deformations occur. High resolution is required to capture the flow characteristics near the fluid-structure interface to enhance accuracy of the solutions within proximity of the thin deformable body. Hence, in this work, an algorithm is developed to simulate fluid-structure interactions of moving deformable structures with very thin thicknesses. In this algorithm, adaptive mesh refinement (AMR) is integrated with immersed boundary finite element method (IBFEM) with two-stage pressure-velocity corrections. Despite successive interpolation of the flow field by IBM, the governing equations were solved using a fixed structured mesh, which significantly reduces the computational time associated with mesh reconstruction. The cut-cell IBM is used to predict the body forces while FEM is used to predict deformation of the thin elastic structure in order to integrate the motions of the fluid and solid at the interface. AMR is used to discretize the governing equations and obtain solutions that efficiently capture the thin boundary layer at the fluid-solid interface. The AMR-IBFEM algorithm is first verified by comparing the drag coefficient, lift coefficient, and Strouhal number for a benchmark case (laminar flow past a circular cylinder at Re = 100) and the results showed good agreement with those of other researchers. The algorithm is then used to simulate 2-D laminar flows past stationary and moving thin structures positioned perpendicular to the freestream direction. The results also showed good agreement with those obtained from the arbitrary Lagrangian-Eulerian (ALE) algorithm for elastic thin boundaries. It is concluded that the AMR-IBFEM algorithm is capable of predicting the characteristics of laminar flow past an elastic structure with acceptable accuracy (error of ∼0.02%) with only ∼1% of the computational time for simulations with full mesh refinement.

A sharp interface method for an immersed viscoelastic solid

The immersed boundary–finite element method (IBFE) is an approach to describing the dynamics of an elastic structure immersed in an incompressible viscous fluid. In this formulation, there are generally discontinuities in the pressure and viscous stress at fluid–structure interfaces. The standard immersed boundary approach, which connects the Lagrangian and Eulerian variables via integral transforms with regularized Dirac delta function kernels, smooths out these discontinuities, which generally leads to low order accuracy. This paper describes an approach to accurately resolve pressure discontinuities for these types of formulations, in which the solid may undergo large deformations. Our strategy is to decompose the physical pressure field into a sum of two pressure–like fields, one defined on the entire computational domain, which includes both the fluid and solid subregions, and one defined only on the solid subregion. Each of these fields is continuous on its domain of definition, which enables high accuracy via standard discretization methods without sacrificing sharp resolution of the pressure discontinuity. Numerical tests demonstrate that this method improves rates of convergence for displacements, velocities, stresses, and pressures as compared to the conventional IBFE method. Further, it produces much smaller errors at reasonable numbers of degrees of freedom. The performance of this method is tested on several cases with analytic solutions, a nontrivial benchmark problem of incompressible solid mechanics, and an example involving a thick, actively contracting torus.

Some Effects of Different Constitutive Laws on FSI Simulation for the Mitral Valve

In this paper, three different constitutive laws for mitral leaflets and two laws for chordae tendineae are selected to study their effects on mitral valve dynamics with fluid-structure interaction. We first fit these three mitral leaflet constitutive laws and two chordae tendineae laws with experimental data. The fluid-structure interaction is implemented in an immersed boundary framework with finite element extension for solid, that is the hybrid immersed boundary/finite element(IB/FE) method. We specifically compare the fluid-structure results of different constitutive laws since fluid-structure interaction is the physiological loading environment. This allows us to look at the peak jet velocity, the closure regurgitation volume, and the orifice area. Our numerical results show that different constitutive laws can affect mitral valve dynamics, such as the transvalvular flow rate, closure regurgitation and the orifice area, while the differences in fiber strain and stress are insignificant because all leaflet constitutive laws are fitted to the same set of experimental data. In addition, when an exponential constitutive law of chordae tendineae is used, a lower closure regurgitation flow is observed compared to that of a linear material model. In conclusion, combining numerical dynamic simulations and static experimental tests, we are able to identify suitable constitutive laws for dynamic behaviour of mitral leaflets and chordae under physiological conditions.

A Shear-Slip Mesh Update – Immersed Boundary Finite Element model for computational simulations of material transport in EPB tunnel boring machines

Earth Pressure Balance (EPB) machines are widely used in mechanized tunneling operations in soft soils in a variety of soil conditions. In contrast to slurry shield machines, the pressure distribution at the tunnel face is not well defined. This paper proposes a computational model for the numerical simulation of the material transport inside the pressure chamber of EPB machines. The governing equations of the soil flow are discretized by using the Finite Element (FE) method and the Fractional Step (FS) scheme. A pressure-dependent viscoplastic fluid is adopted to model the behavior of the soil paste, which is assumed to be a compressible homogeneous mixture. The flow behavior along the boundaries of the chamber wall and the cutterhead rotators is described by the Navier slip law. The moving boundaries associated with the rotators and the mixing arms inside the pressure chamber are imposed in the numerical model by means of the Shear-Slip Mesh Update (SSMU) method. The description of the rotating screw in the conveyor system is enabled by using the Immersed Boundary (IB) method. The combination of the two methods enables to exploit the advantages and to mutually compensate for the disadvantages associated with each of the two methods. The numerical components in the model are presented and verified by several numerical benchmark examples. The setup for the chamber model is presented in detail. Selected results from the simulation of a pressure chamber of a tunnel boring machine are shown and some important observations and remarks are discussed.

Flow-Structure Interaction Simulations of the Aortic Heart Valve at Physiologic Conditions: The Role of Tissue Constitutive Model.

The blood flow patterns in the region around the aortic valve depend on the geometry of the aorta and on the complex flow-structure interaction between the pulsatile flow and the valve leaflets. Consequently, the flow depends strongly on the constitutive properties of the tissue, which can be expected to vary between healthy and diseased heart valves or native and prosthetic valves. The main goal of this work is to qualitatively demonstrate that the choice of the constitutive model of the aortic valve is critical in analysis of heart hemodynamics. To accomplish that two different constitutive models were used in curvilinear immersed boundary-finite element-fluid-structure interaction (CURVIB-FE-FSI) method developed by Gilmanov et al. (2015, "A Numerical Approach for Simulating Fluid Structure Interaction of Flexible Thin Shells Undergoing Arbitrarily Large Deformations in Complex Domains," J. Comput. Phys., 300, pp. 814-843.) to simulate an aortic valve in an anatomic aorta at physiologic conditions. The two constitutive models are: (1) the Saint-Venant (StV) model and (2) the modified May-Newman&Yin (MNY) model. The MNY model is more general and includes nonlinear, anisotropic effects. It is appropriate to model the behavior of both prosthetic and biological tissue including native valves. Both models are employed to carry out FSI simulations of the same valve in the same aorta anatomy. The computed results reveal dramatic differences in both the vorticity dynamics in the aortic sinus and the wall shear-stress patterns on the aortic valve leaflets and underscore the importance of tissue constitutive models for clinically relevant simulations of aortic valves.

A stabilised immersed boundary method on hierarchical b-spline grids

In this work, an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b-spline grid and employs the non-symmetric and penalty-free version of Nitsche’s method to enforce the boundary conditions. The strategy allows for h- and p-refinement and employs a so-called ghost penalty term to stabilise the cut cells. An effective procedure based on hierarchical subdivision and sub-cell merging, which avoids excessive numbers of quadrature points, is used for the integration of the cut cells. A basic Laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty-free Nitsche method as well as their impact on accuracy. The methodology is also applied to the incompressible Navier–Stokes equations, where the SUPG/PSPG stabilisation is employed. Simulations of the lid-driven cavity flow and the flow around a cylinder at low Reynolds number show the good performance of the methodology. Excessive ill-conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field.

17 A case-control study with computational modelling of acute left ventricular dysfunction

Introduction The pathophysiology of acute left ventricular dysfunction after acute ST-elevation myocardial infarction (STEMI) is incompletely understood. In this study, we carried out a case-control study of left ventricular (LV) biomechanical behaviour in vivo using an immersed boundary-finite element (IB/FE) method. Methods Cardiac magnetic resonance images were acquired from six healthy volunteers (the healthy group) and six patients with acute LV dysfunction post-acute STEMI (the MI group) with no-reflow, defined as an acute reduction in myocardial blood flow despite a patent epicardial coronary artery. The two groups were age matched. The passive and active myocardial parameters were determined from the measured in vivo data (LV cavity volume and strain) to match the LV dynamics in diastole and systole. Results Student’s t-test was used for significance test. Our results show that even though the average measured peak cuff-pressure in the MI group is much lower than the healthy group (122 ± 21 vs. 150 ± 20mmHg, p Conclusions In summary, we have shown for the first time that, compared with controls, patients with recent ST-elevation myocardial infarction exhibit increased active tension generation in combination with predictable changes in end-diastolic volume and ejection fraction. Active tension generation is a novel biomechanical index, and its clinical and prognostic significance merits further study. Funding British Heart Foundation (PG/14/64/31043)

Study of cardiovascular function using a coupled left ventricle and systemic circulation model

To gain insight into cardio-arterial interactions, a coupled left ventricle-systemic artery (LV–SA) model is developed that incorporates a three-dimensional finite-strain left ventricle (LV), and a physiologically-based one-dimensional model for the systemic arteries (SA). The coupling of the LV model and the SA model is achieved by matching the pressure and the flow rate at the aortic root, i.e. the SA model feeds back the pressure as a boundary condition to the LV model, and the aortic flow rate from the LV model is used as the input for the SA model. The governing equations of the coupled system are solved using a combined immersed-boundary finite-element (IB/FE) method and a Lax–Wendroff scheme. A baseline case using physiological measurements of healthy subjects, and four exemplar cases based on different physiological and pathological scenarios are studied using the LV–SA model. The results of the baseline case agree well with published experimental data. The four exemplar cases predict varied pathological responses of the cardiovascular system, which are also supported by clinical observations. The new model can be used to gain insight into cardio-arterial interactions across a range of clinical applications.

An immersed structure approach for fluid-vegetation interaction

We present an immersed structure approach for modeling the interaction between surface flows and vegetation. Fluid flow and rigid and flexible vegetative obstacles are coupled through a local drag relation that conserves momentum. In the presented method, separate meshes are used for the fluid domain and vegetative obstacles. Taking techniques from immersed boundary finite element methods, the effects of the fluid on the vegetative structures and vice versa are calculated using integral transforms. Using a simple elastic structure model we incorporate bending and moving vegetative obstacles. We model flexible vegetation as thin, elastic, inextensible cantilever beams. Using the immersed structure approach, a fully coupled fluid-vegetation interaction model is developed assuming dynamic fluid flow and quasi-static bending. This relatively computationally inexpensive model allows for thousands of vegetative obstacles to be included in a simulation without requiring an extremely refined fluid mesh. The method is validated with comparisons to mean velocity profiles and bent vegetation heights from experiments that are reproduced computationally. We test the method on several channel flow setups. We calculate the bulk drag coefficient in these flow scenarios and analyze their trends with changing model parameters including stem population density and flow Reynolds number. Bulk drag models are the primary method of incorporating small-scale drag from individual plants into a value that can be used in larger-scale models. Upscaled bulk drag quantities from this method may be utilized in larger-scale simulations of flow through vegetation regions.

Immersed Boundary Solution of Natural Convection in a Square Cavity with an Enclosed Rosette-Shaped Hot Cylinder

Numerical solutions are shown for the natural convection flow inside a cold square enclosure with a hot rosette-shaped cylinder inside. Solutions are obtained using an immersed-boundary finite element method. The study investigates the effect of the rosette-shape factor on the heat transfer and fluid flow for Rayleigh numbers between 103 and 107. Results indicate the formation of recirculation cells of number, size, and location depending strongly on the shape of the cylinder and on the Rayleigh number. All solutions reach a steady state, except when the rosette-shape factor is 0.6 at Ra = 106, for which a periodic transient solution is obtained.

Three-dimensional Numerical Study of the Mixing Behaviour of Twin-screw Elements

Abstract In this work the numerical modeling of the flow inside co-rotating twin-screw extruders is performed and solutions are analyzed to determine the mixing behavior of two screw elements: conveying and mixing elements. The flow around intermeshing screws is computed using an immersed boundary finite element method capable of dealing with complex moving solid boundaries. The flow is considered isothermal and the material behaves as a generalized non-Newtonian fluid. Because the viscosity depends on the shear rate, solutions will be shown for various rotation velocities of the screw. The 3D solutions are then analyzed in order to determine various parameters characterizing the flow mixing such as the residence time and the linear stretch. Residence time distribution inside the twin-screw extruder is first computed by using a particle tracking algorithm based on a fourth order Runge-Kutta method. A large number of particles are tracked inside the extruder and the resulting particle data is used to determine the distribution of the residence time and of the linear stretch. The spatial distribution of the residence time is also computed by solving a transport equation tracking the injection time of the polymer melt. The methodology shows important differences in the mixing behavior of the screw elements considered.

Three-dimensional Finite Element Solution of the Flow in Single and Twin-Screw Extruders

Abstract This work is aimed at the numerical modeling of the flow inside single and twin-screw extruders. Numerical solutions are obtained using a recently developed immersed boundary finite element method capable of solving the flow in the presence of complex non-stationary solid boundaries. The method is first validated against the solution obtained on a body-conforming grid for a single screw extruder and then applied to a twin-screw mixer. The time dependent single screw problem can also be solved in a rotating reference frame for which a steady state solution can be obtained. This allows the evaluation of time integration errors of the moving immersed interface algorithm. For instance the flow is considered isothermal and the material behaves as a Generalized Newtonian fluid. Because the viscosity depends on the shear rate, solutions will be shown for various rotation velocities of the screw and compared with solutions obtained on body-conforming grids. The method is shown to give very accurate results and can be used for a more in-depth investigation of the material behavior in extruders.