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Yildiran IN, Beratlis N, Capuano F, Loke YH, Squires K, Balaras E. Pressure boundary conditions for immersed-boundary methods. JOURNAL OF COMPUTATIONAL PHYSICS 2024; 510:113057. [PMID: 38912295 PMCID: PMC11192505 DOI: 10.1016/j.jcp.2024.113057] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/25/2024]
Abstract
Immersed boundary methods have seen an enormous increase in popularity over the past two decades, especially for problems involving complex moving/deforming boundaries. In most cases, the boundary conditions on the immersed body are enforced via forcing functions in the momentum equations, which in the case of fractional step methods may be problematic due to: i) creation of slip-errors resulting from the lack of explicitly enforcing boundary conditions on the (pseudo-)pressure on the immersed body; ii) coupling of the solution in the fluid and solid domains via the Poisson equation. Examples of fractional-step formulations that simultaneously enforce velocity and pressure boundary conditions have also been developed, but in most cases the standard Poisson equation is replaced by a more complex system which requires expensive iterative solvers. In this work we propose a new formulation to enforce appropriate boundary conditions on the pseudo-pressure as part of a fractional-step approach. The overall treatment is inspired by the ghost-fluid method typically utilized in two-phase flows. The main advantage of the algorithm is that a standard Poisson equation is solved, with all the modifications needed to enforce the boundary conditions being incorporated within the right-hand side. As a result, fast solvers based on trigonometric transformations can be utilized. We demonstrate the accuracy and robustness of the formulation for a series of problems with increasing complexity.
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Affiliation(s)
- Ibrahim Nasuh Yildiran
- Department of Mechanical and Aerospace Engineering, The George Washington University, 800 22nd Street NW, Washington DC, 20052, USA
| | - Nikolaos Beratlis
- School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, 85287, USA
| | - Francesco Capuano
- Department of Fluid Mechanics, Universitat Politècnica de Catalunya – BarcelonaTech, 08019 Barcelona, Spain
| | - Yue-Hin Loke
- Division of Cardiology, Children’s National Health System, Washington, DC 20010, USA
| | - Kyle Squires
- School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, 85287, USA
| | - Elias Balaras
- Department of Mechanical and Aerospace Engineering, The George Washington University, 800 22nd Street NW, Washington DC, 20052, USA
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Lee JH, Griffith BE. On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method. JOURNAL OF COMPUTATIONAL PHYSICS 2022; 457:111042. [PMID: 35300097 PMCID: PMC8923617 DOI: 10.1016/j.jcp.2022.111042] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method's performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretization combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta functions). Earlier work with this methodology suggested that such coarse structural meshes can yield improved accuracy for shear-dominated cases and, further, found that accuracy improves when the structural mesh spacing is increased. However, these results were limited to simple test cases that did not include substantial pressure loading on the structure. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations in a broader range of tests. Our results indicate that kernels satisfying a commonly imposed even-odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy this condition. We also find that narrower kernels are more robust, in the sense that they yield results that are less sensitive to relative changes in the Eulerian and Lagrangian mesh spacings, and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.
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Affiliation(s)
- Jae H Lee
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
- Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD, USA
- Institute for Computational Medicine, Johns Hopkins University, Baltimore, MD, USA
| | - Boyce E Griffith
- Departments of Mathematics, Applied Physical Sciences, and Biomedical Engineering, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill, NC, USA
- Computational Medicine Program, University of North Carolina School of Medicine, Chapel Hill, NC, USA
- McAllister Heart Institute, University of North Carolina School of Medicine, Chapel Hill, NC, USA
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Acharya S, Halder S, Kou W, Kahrilas PJ, Pandolfino JE, Patankar NA. A fully resolved multiphysics model of gastric peristalsis and bolus emptying in the upper gastrointestinal tract. Comput Biol Med 2022; 143:104948. [PMID: 35091365 PMCID: PMC9014465 DOI: 10.1016/j.compbiomed.2021.104948] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Revised: 10/09/2021] [Accepted: 10/13/2021] [Indexed: 11/22/2022]
Abstract
Over the past few decades, in silico modeling of organ systems has significantly furthered our understanding of their physiology and biomechanical function. In spite of the relative importance of the digestive system in normal functioning of the human body, there is a scarcity of high-fidelity models for the upper gastrointestinal tract including the esophagus and the stomach. In this work, we present a detailed numerical model of the upper gastrointestinal tract that not only accounts for the fiber architecture of the muscle walls, but also the multiphasic components they help transport during normal digestive function. Construction details for 3D models of representative stomach geometry are presented along with a simple strategy for assigning circular and longitudinal muscle fiber orientations for each layer. We developed a fully resolved model of the stomach to simulate gastric peristalsis by systematically activating muscle fibers embedded in the stomach. Following this, for the first time, we simulate gravity-driven bolus emptying into the stomach due to density differences between ingested contents and fluid contents of the stomach. Finally, we present a case of retrograde flow of fluid from the stomach into the esophagus, resembling the phenomenon of acid reflux. This detailed computational model of the upper gastrointestinal tract provides a foundation for future models to investigate the biomechanics of acid reflux and probe various strategies for gastric bypass surgeries to address the growing problem of obesity.
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Affiliation(s)
- Shashank Acharya
- Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA
| | - Sourav Halder
- Theoretical and Applied Mechanics Program, Northwestern University, Evanston, IL, 60208, USA
| | - Wenjun Kou
- Division of Gastroenterology and Hepatology, Feinberg School of Medicine, Northwestern University, Chicago, IL, 60611, USA
| | - Peter J Kahrilas
- Division of Gastroenterology and Hepatology, Feinberg School of Medicine, Northwestern University, Chicago, IL, 60611, USA
| | - John E Pandolfino
- Division of Gastroenterology and Hepatology, Feinberg School of Medicine, Northwestern University, Chicago, IL, 60611, USA
| | - Neelesh A Patankar
- Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA; Theoretical and Applied Mechanics Program, Northwestern University, Evanston, IL, 60208, USA.
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Kingora K, Sadat-Hosseini H. A novel interpolation-free sharp-interface immersed boundary method. JOURNAL OF COMPUTATIONAL PHYSICS 2022; 453:110933. [PMID: 35250049 PMCID: PMC8896509 DOI: 10.1016/j.jcp.2021.110933] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
This paper describes a novel 2nd order direct forcing immersed boundary method designed for simulation of 2D and 3D incompressible flow problems with complex immersed boundaries. In this formulation, each cell cut by the immersed boundary (IB) is reshaped to conform to the shape of the IB. IBs are modeled as a series of 2D planes in 3D space that connect seamlessly at the edges of the cut cells, in a way that mimics conformal grid. IBs are represented in a continuous and consistent fashion from one cell to another, thus eliminating spatial pressure oscillations originating from inconsistent description of the IB as well as the traditional stair-step problem, leading to a more accurate resolution of the boundary layer. Boundary conditions are enforced at the exact location of the IB devoid of interpolation, which guarantees sound simulations even on grids with high aspect ratio, and enables simulations of flow packed with multiple IBs in close proximity. Boundary conditions for each phase across the IB are enforced independently, yielding a unique capability to solve flows with zero-thickness IBs. Simulations of a large number of 2D and 3D test cases confirm the prowess of the devised immersed boundary method in solving flows over multiple loosely/closely-packed IBs; stationary, moving and highly morphing IBs; as well as IBs with zero-thickness. Results show that predictions from the proposed scheme agree remarkably well with theoretical models and experimental data for both integral variables and local flow fields and they are often with less than 1% deviation from solutions obtained by conformal grid of similar resolution.
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Luo Y, Li X, Hao W. Projection-based model reduction for the immersed boundary method. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2022; 38:e3558. [PMID: 34865313 DOI: 10.1002/cnm.3558] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2021] [Revised: 09/30/2021] [Accepted: 11/27/2021] [Indexed: 06/13/2023]
Abstract
Fluid-structure interactions are central to many biomolecular processes, and they impose a great challenge for computational and modeling methods. In this paper, we consider the immersed boundary method (IBM) for biofluid systems, and to alleviate the computational cost, we apply reduced-order techniques to eliminate the degrees of freedom associated with the large number of fluid variables. We show how reduced models can be derived using Petrov-Galerkin projection and subspaces that maintain the incompressibility condition. More importantly, the reduced-order model (ROM) is shown to preserve the Lyapunov stability. We also address the practical issue of computing coefficient matrices in the ROM using an interpolation technique. The efficiency and robustness of the proposed formulation are examined with test examples from various applications.
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Affiliation(s)
- Yushuang Luo
- Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania, USA
| | - Xiantao Li
- Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania, USA
| | - Wenrui Hao
- Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania, USA
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Vadala-Roth B, Acharya S, Patankar NA, Rossi S, Griffith BE. Stabilization approaches for the hyperelastic immersed boundary method for problems of large-deformation incompressible elasticity. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2020; 365:112978. [PMID: 32483394 PMCID: PMC7263477 DOI: 10.1016/j.cma.2020.112978] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses Lagrangian coordinates to describe the structural deformations, stresses, and resultant forces. Integral transforms with Dirac delta function kernels connect the Eulerian and Lagrangian frames. The fluid and the structure are both typically treated as incompressible materials. Upon discretization, however, the incompressibility of the structure is only maintained approximately. To obtain an immersed method for incompressible hyperelastic structures that is robust under large structural deformations, we introduce a volumetric energy in the solid region that stabilizes the formulation and improves the accuracy of the numerical scheme. This formulation augments the discrete Lagrange multiplier for the incompressibility constraint, thereby improving the original method's accuracy. This volumetric energy is incorporated by decomposing the strain energy into isochoric and dilatational components, as in standard solid mechanics formulations of nearly incompressible elasticity. We study the performance of the stabilized method using several quasi-static solid mechanics benchmarks, a dynamic fluid-structure interaction benchmark, and a detailed three-dimensional model of esophageal transport. The accuracy achieved by the stabilized immersed formulation is comparable to that of a stabilized finite element method for incompressible elasticity using similar numbers of structural degrees of freedom.
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Affiliation(s)
- Ben Vadala-Roth
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
| | - Shashank Acharya
- Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA
| | - Neelesh A. Patankar
- Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA
| | - Simone Rossi
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill, NC, USA
| | - Boyce E. Griffith
- Departments of Mathematics, Applied Physical Sciences, and Biomedical Engineering, University of North Carolina, Chapel Hill, NC, USA
- Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill, NC, USA
- Computational Medicine Program, University of North Carolina, Chapel Hill, NC, USA
- McAllister Heart Institute, University of North Carolina, Chapel Hill, NC, USA
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Kaiser AD, McQueen DM, Peskin CS. Modeling the mitral valve. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2019; 35:e3240. [PMID: 31330567 DOI: 10.1002/cnm.3240] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2019] [Revised: 06/18/2019] [Accepted: 07/14/2019] [Indexed: 06/10/2023]
Abstract
This work is concerned with modeling and simulation of the mitral valve, one of the four valves in the human heart. The valve is composed of leaflets, the free edges of which are supported by a system of chordae, which themselves are anchored to the papillary muscles inside the left ventricle. First, we examine valve anatomy and present the results of original dissections. These display the gross anatomy and information on fiber structure of the mitral valve. Next, we build a model valve following a design-based methodology, meaning that we derive the model geometry and the forces that are needed to support a given load and construct the model accordingly. We incorporate information from the dissections to specify the fiber topology of this model. We assume the valve achieves mechanical equilibrium while supporting a static pressure load. The solution to the resulting differential equations determines the pressurized configuration of the valve model. To complete the model, we then specify a constitutive law based on a stress-strain relation consistent with experimental data that achieves the necessary forces computed in previous steps. Finally, using the immersed boundary method, we simulate the model valve in fluid in a computer test chamber. The model opens easily and closes without leak when driven by physiological pressures over multiple beats. Further, its closure is robust to driving pressures that lack atrial systole or are much lower or higher than normal.
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Affiliation(s)
- Alexander D Kaiser
- Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, New York
| | - David M McQueen
- Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, New York
| | - Charles S Peskin
- Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, New York
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Leveraging collective effects in externally driven colloidal suspensions: experiments and simulations. Curr Opin Colloid Interface Sci 2019. [DOI: 10.1016/j.cocis.2018.10.002] [Citation(s) in RCA: 37] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
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