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Charles S. Peskin - One of the best experts on this subject based on the ideXlab platform.
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Simulating cardiovascular fluid dynamics by the Immersed Boundary Method
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009Co-Authors: Boyce E Griffith, David M Mcqueen, Charles S. PeskinAbstract:The Immersed Boundary Method is both a general mathematical framework and a particular numerical approach to problems of fluid-structure interaction. In this paper, we describe the application of the Immersed Boundary Method to the simulation of cardiovascular fluid dynamics, focusing on the fluid dynamics of the aortic heart valve (the valve which prevents the backflow of blood from the aorta into the left ventricle of the heart) and aortic root (the initial portion of the aorta, which attaches to the heart). The aortic valve and root are modeled as a system of elastic fibers, and the blood is modeled as a viscous incompressible fluid. Three-dimensional simulation results obtained using a parallel and adaptive version of the Immersed Boundary Method are presented. These results demonstrate that it is feasible to perform three-dimensional Immersed Boundary simulations of cardiovascular fluid dynamics in which realistic cardiac output is obtained at realistic pressures.
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SIMULATING THE FLUID DYNAMICS OF NATURAL AND PROSTHETIC HEART VALVES USING THE Immersed Boundary Method
International Journal of Applied Mechanics, 2009Co-Authors: Boyce E Griffith, Leon H. Charney, David M Mcqueen, Xiaoyu Luo, Charles S. PeskinAbstract:The Immersed Boundary Method is both a general mathematical framework and a partic-ular numerical approach to problems of fluid-structure interaction. In the present work, we describe the application of the Immersed Boundary Method to the simulation of the fluid dynamics of heart valves, including a model of a natural aortic valve and a model of a chorded prosthetic mitral valve. Each valve is mounted in a semi-rigid flow chamber. In the case of the mitral valve, the flow chamber is a circular pipe, and in the case of the aortic valve, the flow chamber is a model of the aortic root. The model valves and flow chambers are Immersed in a viscous incompressible fluid, and realistic fluid Boundary conditions are prescribed at the upstream and downstream ends of the chambers. To con-nect the Immersed Boundary models to the boundaries of the fluid domain, we introduce a novel modification of the standard Immersed Boundary scheme. In particular, near the outer boundaries of the fluid domain, we modify the construction of the regularized delta function which mediates fluid-structure coupling in the Immersed Boundary Method, whereas in the interior of the fluid domain, we employ a standard four-point delta func-tion which is frequently used with the Immersed Boundary Method. The standard delta function is used wherever possible, and the modified delta function continuously transi-tions to the standard delta function away from the outer boundaries of the fluid domain. Three-dimensional computational results are presented to demonstrate the capabilities of our Immersed Boundary approach to simulating the fluid dynamics of heart valves.
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On the foundations of the stochastic Immersed Boundary Method
Computer Methods in Applied Mechanics and Engineering, 2007Co-Authors: Peter R. Kramer, Charles S. Peskin, Paul J. AtzbergerAbstract:We explore the theoretical foundations for the inclusion of thermal fluctuations in the Immersed Boundary Method for simulating microscale fluid systems with Immersed flexible structures, as in cellular and subcellular biology. We investigate in particular the physical validity of the thermal forcing scheme with respect to the coupling of fluid and Immersed structural degrees of freedom and non-equilibrium conditions. We discuss also the shortcomings of a natural alternative scheme in which the thermal fluctuations are applied directly to the structural degrees of freedom through Langevin-type dynamics.
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Fluid-Structure Interaction via an Adaptive Finite Element Immersed Boundary Method
2007Co-Authors: Daniele Boffi, Luca Heltai, Lucia Gastaldi, Charles S. PeskinAbstract:The Immersed Boundary (IB) Method is a mathematical formulation for fluid-structure interaction problems, where Immersed incompressible visco-elastic bodies or boundaries interact with an incompressible fluid. The original numerical scheme associated to the IB Method requires a smoothed approximation of the Dirac delta distribution to link the moving Lagrangian domain with the fixed Eulerian one. We present an adaptive version of the finite element Immersed Boundary Method, where the Dirac delta distribution is treated in a variational way, eliminating the need to approximate it. [ DOI : 10.1685 / CSC06022] About DOI
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Stochastic Immersed Boundary Method incorporating thermal fluctuations
PAMM, 2007Co-Authors: Paul J. Atzberger, Peter R. Kramer, Charles S. PeskinAbstract:We give a brief introduction to the stochastic Immersed Boundary Method which allows for simulation of small length-scale physical systems in which elastic structures interact with a fluid flow in the pre sence of thermal fluctuations. The conventional Immersed Boundary Method is extended to account for thermal fluctuatio ns by introducing stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE’s for which standard n umerical approaches perform poorly. We discuss a numerical Method derived using stochastic calculus to overcome the stiff features of the equations. We then discuss results which indicate that the Method captures physical features predicted by statistical mechanics for small length-scale systems. The stochastic Immersed Boundary Method holds promise as a numerical approach in simulating microscopic mechanical systems in which thermal fluctuations play a fundamental role. A more deta iled discussion of this work is given in [1, 2, 3]. Copyright line will be provided by the publisher
Yong Sheng - One of the best experts on this subject based on the ideXlab platform.
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PIBM: Particulate Immersed Boundary Method for fluid–particle interaction problems
Powder Technology, 2015Co-Authors: Hao Zhang, F. Xavier Trias, Yuanqiang Tan, Assensi Oliva, Dongmin Yang, Shi Shu, Yong ShengAbstract:It is well known that the number of particles should be scaled up to enable industrial scale simulation. The calculations are more computationally intensive when the motion of the surrounding fluid is considered. Besides the advances in computer hardware and numerical algorithms, the coupling scheme also plays an important role on the computational efficiency. In this study, a particulate Immersed Boundary Method (PIBM) for simulating the fluid–particle multiphase flow was presented and assessed in both two- and three-dimensional applications. The idea behind PIBM derives from the conventional momentum exchange-based Immersed Boundary Method (IBM) by treating each Lagrangian point as a solid particle. This treatment enables Lattice Boltzmann Method (LBM) to be coupled with fine particles residing within a particular grid cell. Compared with the conventional IBM, dozens of times speedup in two-dimensional simulation and hundreds of times in three-dimensional simulation can be expected under the same particle and mesh number. Numerical simulations of particle sedimentation in Newtonian flows were conducted based on a combined LBM–PIBM–Discrete Element Method (DEM) scheme, showing that the PIBM can capture the feature of particulate flows in fluid and is indeed a promising scheme for the solution of the fluid–particle interaction problems.
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PIBM: Particulate Immersed Boundary Method for fluid-particle interaction problems
arXiv: Computational Physics, 2014Co-Authors: Hao Zhang, F. Xavier Trias, Yuanqiang Tan, Assensi Oliva, Dongmin Yang, Shi Shu, Yong ShengAbstract:It is well known that the number of particles should be scaled up to enable industrial scale simulation. The calculations are more computationally intensive when the motion of the surrounding fluid is considered. Besides the advances in computer hardware and numerical algorithms, the coupling scheme also plays an important role on the computational efficiency. In this study, a particle Immersed Boundary Method (PIBM) for simulating the fluid-particle multiphase flow was presented and assessed in both two- and three-dimensional applications. The idea behind PIBM derives from the conventional momentum exchange-based Immersed Boundary Method (IBM) by treating each Lagrangian point as a solid particle. This treatment enables LBM to be coupled with fine particles residing within a particular grid cell. Compared with the conventional IBM, dozens of times speedup in two-dimensional simulation and hundreds of times in three-dimensional simulation can be expected under the same particle and mesh number. Numerical simulations of particle sedimentation in the Newtonian flows were conducted based on a combined lattice Boltzmann Method - particle Immersed Boundary Method - discrete element Method scheme, showing that the PIBM can capture the feature of particulate flows in fluid and is indeed a promising scheme for the solution of the fluid-particle interaction problems.
Paul J. Atzberger - One of the best experts on this subject based on the ideXlab platform.
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On the foundations of the stochastic Immersed Boundary Method
Computer Methods in Applied Mechanics and Engineering, 2007Co-Authors: Peter R. Kramer, Charles S. Peskin, Paul J. AtzbergerAbstract:We explore the theoretical foundations for the inclusion of thermal fluctuations in the Immersed Boundary Method for simulating microscale fluid systems with Immersed flexible structures, as in cellular and subcellular biology. We investigate in particular the physical validity of the thermal forcing scheme with respect to the coupling of fluid and Immersed structural degrees of freedom and non-equilibrium conditions. We discuss also the shortcomings of a natural alternative scheme in which the thermal fluctuations are applied directly to the structural degrees of freedom through Langevin-type dynamics.
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Stochastic Immersed Boundary Method incorporating thermal fluctuations
PAMM, 2007Co-Authors: Paul J. Atzberger, Peter R. Kramer, Charles S. PeskinAbstract:We give a brief introduction to the stochastic Immersed Boundary Method which allows for simulation of small length-scale physical systems in which elastic structures interact with a fluid flow in the pre sence of thermal fluctuations. The conventional Immersed Boundary Method is extended to account for thermal fluctuatio ns by introducing stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE’s for which standard n umerical approaches perform poorly. We discuss a numerical Method derived using stochastic calculus to overcome the stiff features of the equations. We then discuss results which indicate that the Method captures physical features predicted by statistical mechanics for small length-scale systems. The stochastic Immersed Boundary Method holds promise as a numerical approach in simulating microscopic mechanical systems in which thermal fluctuations play a fundamental role. A more deta iled discussion of this work is given in [1, 2, 3]. Copyright line will be provided by the publisher
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A note on the correspondence of an Immersed Boundary Method incorporating thermal fluctuations with Stokesian-Brownian dynamics
Physica D: Nonlinear Phenomena, 2007Co-Authors: Paul J. AtzbergerAbstract:Abstract In this paper a direct correspondence is made between the effective stochastic dynamics of elastic structures of an Immersed Boundary Method incorporating thermal fluctuations and Stokesian–Brownian Dynamics. The correspondence is made in the limit of small Reynolds number, in which the fluid relaxes rapidly on the time scale of the motion of the Immersed structures, by performing an averaging procedure directly on the stochastic equations of the Immersed Boundary Method. It is found that there is agreement with Stokesian–Brownian Dynamics for the far-field hydrodynamic interactions and that a fluctuation–dissipation relation is satisfied for the stochastic fluctuations of the effective equations.
Huaxiong Huang - One of the best experts on this subject based on the ideXlab platform.
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An Immersed Boundary Method for endocytosis
Journal of Computational Physics, 2014Co-Authors: Yu-hau Tseng, Huaxiong HuangAbstract:Endocytosis is one of the cellular functions for capturing (engulfing) vesicles or microorganisms. Understanding the biophysical mechanisms of this cellular process is essential from a bioengineering point of view since it will provide guidance for developing effective targeted drug delivery therapies. In this paper, we propose an Immersed Boundary (IB) Method that can be used to simulate the dynamical process of this important biological function. In our model, membranes of the vesicle and the cell are treated as Canham-Helfrich Hamiltonian interfaces. The membrane-bound molecules are modeled as insoluble surfactants such that the molecules after binding are regarded as a product of a ''chemical'' reaction. Our numerical examples show that the Immersed Boundary Method is a useful simulation tool for studying endocytosis, where the roles of interfacial energy, fluid flow and viscous dissipation in the success of the endocytosis process can be investigated in detail. A distinct feature of our IB Method is the treatment of the two binding membranes that is different from the merging of fluid-fluid interfaces. Another important feature of our Method is the strict conservation of membrane-borne receptors and ligands, which is important for predicting the dynamics of the endocytosis process.
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Effect of regularized delta function on accuracy of Immersed Boundary Method
Applied Mathematics and Mechanics-english Edition, 2012Co-Authors: Zhao-xin Gong, Chuan-jing Lu, Huaxiong HuangAbstract:The Immersed Boundary Method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. The effect of the regularized delta function on the accuracy is an important subject in the property study. A Method of manufactured solutions is used in the research. The computational code is first verified to be mistake-free by using smooth manufactured solutions. Then, a jump in the manufactured solution for pressure is introduced to study the accuracy of the Immersed Boundary Method. Four kinds of regularized delta functions are used to test the effect on the accuracy analysis. By analyzing the discretization errors, the accuracy of the Immersed Boundary Method is proved to be first-order. The results show that the regularized delta function cannot improve the accuracy, but it can change the discretization errors in the entire computational domain.
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Accuracy analysis of Immersed Boundary Method using Method of manufactured solutions
Applied Mathematics and Mechanics, 2010Co-Authors: Zhao-xin Gong, Huaxiong HuangAbstract:The Immersed Boundary Method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. This paper analyzes the accuracy of the Immersed Boundary Method. The procedure contains two parts, i.e., the code verification and the accuracy analysis. The code verification provides the confidence that the code used is free of mistakes, and the accuracy analysis gives the order of accuracy of the Immersed Boundary Method. The Method of manufactured solutions is taken as a means for both parts. In the first part, the numerical code employs a second-order discretization scheme, i.e., it has second-order accuracy in theory. It matches the calculated order of accuracy obtained in the numerical calculation for all variables. This means that the code contains no mistake, which is a premise of the subsequent work. The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations. By analyzing the discretization errors, the accuracy of the Immersed Boundary Method is proven to be first order even though the discretization scheme is second order. It has been found that the coarser mesh may not be sensitive enough to capture the influence of the Immersed Boundary, and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.
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Stability Analysis for the Immersed Boundary Method
New Trends in Fluid Mechanics Research, 2007Co-Authors: Zhao-xin Gong, Huaxiong HuangAbstract:In this paper, we analyse the stablity of the Immersed Boundary Method applied to a membrane-fluid system with a plasma membrane Immersed in an incompressible viscous fluid. For small deformations, the Immersed Boundary Method, using a standard regularization technique for the singular force, is shown to be linearly stable.
Boyce E Griffith - One of the best experts on this subject based on the ideXlab platform.
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Hybrid finite difference/finite element Immersed Boundary Method.
International journal for numerical methods in biomedical engineering, 2017Co-Authors: Boyce E Griffith, Xiaoyu LuoAbstract:The Immersed Boundary Method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original Immersed Boundary Methods described Immersed elastic structures using systems of flexible fibers, and even now, most Immersed Boundary Methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the Immersed Boundary Method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this Method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the Immersed Boundary Method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.
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hybrid finite difference finite element Immersed Boundary Method
arXiv: Numerical Analysis, 2016Co-Authors: Boyce E Griffith, Xiaoyu LuoAbstract:The Immersed Boundary Method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original Immersed Boundary Methods described Immersed elastic structures using systems of flexible fibers, and even now, most Immersed Boundary Methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the Immersed Boundary Method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach employs a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this Method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the Immersed Boundary Method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.
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Simulating cardiovascular fluid dynamics by the Immersed Boundary Method
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009Co-Authors: Boyce E Griffith, David M Mcqueen, Charles S. PeskinAbstract:The Immersed Boundary Method is both a general mathematical framework and a particular numerical approach to problems of fluid-structure interaction. In this paper, we describe the application of the Immersed Boundary Method to the simulation of cardiovascular fluid dynamics, focusing on the fluid dynamics of the aortic heart valve (the valve which prevents the backflow of blood from the aorta into the left ventricle of the heart) and aortic root (the initial portion of the aorta, which attaches to the heart). The aortic valve and root are modeled as a system of elastic fibers, and the blood is modeled as a viscous incompressible fluid. Three-dimensional simulation results obtained using a parallel and adaptive version of the Immersed Boundary Method are presented. These results demonstrate that it is feasible to perform three-dimensional Immersed Boundary simulations of cardiovascular fluid dynamics in which realistic cardiac output is obtained at realistic pressures.
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SIMULATING THE FLUID DYNAMICS OF NATURAL AND PROSTHETIC HEART VALVES USING THE Immersed Boundary Method
International Journal of Applied Mechanics, 2009Co-Authors: Boyce E Griffith, Leon H. Charney, David M Mcqueen, Xiaoyu Luo, Charles S. PeskinAbstract:The Immersed Boundary Method is both a general mathematical framework and a partic-ular numerical approach to problems of fluid-structure interaction. In the present work, we describe the application of the Immersed Boundary Method to the simulation of the fluid dynamics of heart valves, including a model of a natural aortic valve and a model of a chorded prosthetic mitral valve. Each valve is mounted in a semi-rigid flow chamber. In the case of the mitral valve, the flow chamber is a circular pipe, and in the case of the aortic valve, the flow chamber is a model of the aortic root. The model valves and flow chambers are Immersed in a viscous incompressible fluid, and realistic fluid Boundary conditions are prescribed at the upstream and downstream ends of the chambers. To con-nect the Immersed Boundary models to the boundaries of the fluid domain, we introduce a novel modification of the standard Immersed Boundary scheme. In particular, near the outer boundaries of the fluid domain, we modify the construction of the regularized delta function which mediates fluid-structure coupling in the Immersed Boundary Method, whereas in the interior of the fluid domain, we employ a standard four-point delta func-tion which is frequently used with the Immersed Boundary Method. The standard delta function is used wherever possible, and the modified delta function continuously transi-tions to the standard delta function away from the outer boundaries of the fluid domain. Three-dimensional computational results are presented to demonstrate the capabilities of our Immersed Boundary approach to simulating the fluid dynamics of heart valves.
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An adaptive, formally second order accurate version of the Immersed Boundary Method
Journal of Computational Physics, 2007Co-Authors: Boyce E Griffith, David M Mcqueen, Richard D. Hornung, Charles S. PeskinAbstract:Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The Immersed Boundary Method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the Immersed Boundary Method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive Immersed Boundary Method of Roma et al. [A multilevel self adaptive version of the Immersed Boundary Method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the Immersed Boundary Method, J. Comput. Phys. 153 (2) (1999) 509–534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the Immersed Boundary Method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the Immersed Boundary Method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75–105]. Actual second order convergence rates are obtained for both the uniform and adaptive Methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this Methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive Method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the Method, but the flow in the vicinity of the model heart valves indicates that the new Methodology provides enhanced Boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.
