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19
On the Volume Conservation of the Immersed Boundary Method
, 2012
"... Abstract. The immersed boundary (IB) method is an approach to problems of fluid-struc-ture interaction in which an elastic structure is immersed in a viscous incompress-ible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the f ..."
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Cited by 12 (4 self)
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Abstract. The immersed boundary (IB) method is an approach to problems of fluid-struc-ture interaction in which an elastic structure is immersed in a viscous incompress-ible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some ver-sions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic bench-mark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests con-sidered herein, the staggered-grid IB scheme generally yields at least a modest im-provement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of
STOCHASTIC EULERIAN LAGRANGIAN METHODS FOR FLUID-STRUCTURE INTERACTIONS WITH THERMAL FLUCTUATIONS
, 2010
"... Abstract. We present approaches for the study of fluid-structure interactions subject to thermal fluctuations. A mixed mechanical description is utilized combining Eulerian and Lagrangian reference frames. We establish general conditions for operators coupling these descriptions. Stochastic driving ..."
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Cited by 10 (1 self)
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Abstract. We present approaches for the study of fluid-structure interactions subject to thermal fluctuations. A mixed mechanical description is utilized combining Eulerian and Lagrangian reference frames. We establish general conditions for operators coupling these descriptions. Stochastic driving fields for the formalism are derived using principles from statistical mechanics. The stochastic differential equations of the formalism are found to exhibit significant stiffness in some physical regimes. To cope with this issue, we derive reduced stochastic differential equations for several physical regimes. We also present stochastic numerical methods for each regime to approximate the fluid-structure dynamics and to generate efficiently the required stochastic driving fields. To validate the methodology in each regime, we perform analysis of the invariant probability distribution of the stochastic dynamics of the fluid-structure formalism. We compare this analysis with results from statistical mechanics. To further demonstrate the applicability of the methodology, we perform computational studies for spherical particles having translational and rotational degrees of freedom. We compare these studies with results from fluid mechanics. The presented approach provides for fluidstructure systems a set of rather general computational methods for treating consistently structure mechanics, hydrodynamic coupling, and thermal fluctuations.
STOCHASTIC REDUCTIONS FOR INERTIAL FLUID-STRUCTURE INTERACTIONS SUBJECT TO THERMAL FLUCTUATIONS
"... Abstract. We investigate the dynamics of elastic microstructures that interact with a fluid flow when subject to thermal fluctuations. We perform analysis to obtain systematically simplified descriptions of the mechanics in the limiting regimes when (i) the coupling forces that transfer momentum bet ..."
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Cited by 4 (1 self)
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Abstract. We investigate the dynamics of elastic microstructures that interact with a fluid flow when subject to thermal fluctuations. We perform analysis to obtain systematically simplified descriptions of the mechanics in the limiting regimes when (i) the coupling forces that transfer momentum between the fluid and microstructures is strong, (ii) the mass of the microstructures is small relative to the displaced mass of the fluid, and (iii) the response to stresses results in hydrodynamics that relax rapidly to a quasi-steady-state relative to the motions of the microstructure. We derive effective equations using a singular perturbation analysis of the Backward Kolmogorov equations of the stochastic process. Our continuum mechanics description is based on the Stochastic Eulerian Lagrangian Method (SELM) which provides a framework for approximation of the fluid-structure interactions when subject to thermal fluctuations. We perform a dimension analysis of the SELM equations to identify key non-dimensional groups and to characterize precisely each of the limiting physical regimes. The reduced equations offer insights into the physical accuracy of SELM descriptions in comparison with classical results. The reduced equations also elimintate rapid time-scales from the dynamics and provide possible approaches for the development of more efficient computational methods for simulations of fluid-structure interactions when subject to thermal fluctuations.
Spatially adaptive stochastic methods for fluidstructure interactions subject to thermal fluctuations in domains with complex geometries
- Journal of Computational Physics
, 2014
"... Abstract. We develop stochastic mixed finite element methods for spatially adaptive simula-tions of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compati-ble stochas ..."
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Cited by 3 (1 self)
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Abstract. We develop stochastic mixed finite element methods for spatially adaptive simula-tions of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compati-ble stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs-Boltzmann dis-tribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with O(N) computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles de-pends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications.
An Efficient Parallel Immersed Boundary Algorithm using a Pseudo-Compressible Fluid Solver
, 2013
"... We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev t ..."
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Cited by 3 (2 self)
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We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible Navier-Stokes equations, thereby reducing the linear systems to a series of one-dimensional tridiagonal systems. We perform numerical sim-ulations of several fluid-structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. For these prob-lems, we compare the proposed algorithm against other second-order projection-based fluid solvers. Lastly, the strong and weak scaling properties of the proposed algorithm are investigated.
IMMERSED BOUNDARY METHOD FOR VARIABLE VISCOSITY AND VARIABLE DENSITY PROBLEMS USING FAST CONSTANT-COEFFICIENT LINEAR SOLVERS I: NUMERICAL METHOD AND RESULTS
"... Abstract. We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered m ..."
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Cited by 2 (2 self)
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Abstract. We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered material properties and are defined by a dynamically updated tesselation. Empirical convergence rates are reported for a test problem of a two-dimensional viscoelastic shell with spatially varying material properties. The reduction to first-order accuracy in the variable density case can be avoided by using an iterative scheme, although this approach may not be efficient enough for practical use. In our timestepping scheme, both the inertial and viscous terms are split into two parts: a constant-coefficient part that is treated implicitly, and a variable-coefficient part that is treated explicitly. This splitting allows the resulting equations to be solved efficiently using fast constant-coefficient linear solvers, and in this work, we use solvers based on the Fast Fourier transform (FFT). As an application of this method, we perform fully three-dimensional, two-phase simulations of red blood cells accounting for variable viscosity and variable density. We study the behavior of red cells during shear flow and during capillary flow. Key words. immersed boundary method, variable viscosity, red blood cells AMS subject classifications. 65M06, 76D05, 76Z05
19b. TELEPHONE NUMBER
"... a. REPORT 16. SECURITY CLASSIFICATION OF: To address the computational challenges associated with contact between moving solid surfaces, such as those in cardiovascular fluid--structure interaction (FSI), parachute FSI, and flapping-wing aerodynamics, we introduce a space--time (ST) interface-tracki ..."
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a. REPORT 16. SECURITY CLASSIFICATION OF: To address the computational challenges associated with contact between moving solid surfaces, such as those in cardiovascular fluid--structure interaction (FSI), parachute FSI, and flapping-wing aerodynamics, we introduce a space--time (ST) interface-tracking method that can deal with topology change (TC). In cardiovascular FSI, our primary target is heart valves. The method is a new version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method, and we call it ST-TC. It includes a master--slave system that maintains the connectivity of the ``parent' ' mesh when there is contact between the moving interfaces. It is an efficient, practical alternative to using
unknown title
, 2014
"... doi:10.1093/imamat/hxu029 Dynamic finite-strain modelling of the human left ventricle in health and disease using an immersed boundary-finite element method ..."
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doi:10.1093/imamat/hxu029 Dynamic finite-strain modelling of the human left ventricle in health and disease using an immersed boundary-finite element method
Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGKModel
, 2014
"... Abstract. This paper aims to study the numerical features of a coupling scheme be-tween the immersed boundary (IB) method and the lattice Boltzmann BGK (LBGK) model by four typical test problems: the relaxation of a circular membrane, the shear-ing flow induced by a moving fiber in the middle of a c ..."
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Abstract. This paper aims to study the numerical features of a coupling scheme be-tween the immersed boundary (IB) method and the lattice Boltzmann BGK (LBGK) model by four typical test problems: the relaxation of a circular membrane, the shear-ing flow induced by a moving fiber in the middle of a channel, the shearing flow near a non-slip rigid wall, and the circular Couette flow between two inversely rotating cylinders. The accuracy and robustness of the IB-LBGK coupling scheme, the perfor-mances of different discrete Dirac delta functions, the effect of iteration on the cou-pling scheme, the importance of the external forcing term treatment, the sensitivity of the coupling scheme to flow and boundary parameters, the velocity slip near non-slip rigid wall, and the origination of numerical instabilities are investigated in detail via the four test cases. It is found that the iteration in the coupling cycle can effectively improve stability, the introduction of a second-order forcing term in LBGK model is crucial, the discrete fiber segment length and the orientation of the fiber boundary ob-viously affect accuracy and stability, and the emergence of both temporal and spatial fluctuations of boundary parameters seems to be the indication of numerical instabil-
