scholarly journals Numerical Analysis of the Immersed Boundary Method for Cell-Based Simulation

2016 ◽  
Author(s):  
Fergus R. Cooper ◽  
Ruth E. Baker ◽  
Alexander G. Fletcher
Keyword(s):  
Open Source ◽  
Immersed Boundary Method ◽  
Interaction Strength ◽  
Cell Interaction ◽  
Biological Tissues ◽  
Immersed Boundary ◽  
Boundary Node ◽  
Time Step ◽  
Boundary Method ◽  
Mesh Spacing

AbstractMathematical modelling provides a useful framework within which to investigate the organization of biological tissues. With advances in experimental biology leading to increasingly detailed descriptions of cellular behaviour, models that consider cells as individual objects are becoming a common tool to study how processes at the single-cell level affect collective dynamics and determine tissue size, shape and function. However, there often remains no comprehensive account of these models, their method of solution, computational implementation or analysis of parameter scaling, hindering our ability to utilise and accurately compare different models. Here we present an effcient, open-source implementation of the immersed boundary method (IBM), tailored to simulate the dynamics of cell populations. This approach considers the dynamics of elastic membranes, representing cell boundaries, immersed in a viscous Newtonian fluid. The IBM enables complex and emergent cell shape dynamics, spatially heterogeneous cell properties and precise control of growth mechanisms. We solve the model numerically using an established algorithm, based on the fast Fourier transform, providing full details of all technical aspects of our implementation. The implementation is undertaken within Chaste, an open-source C++ library that allows one to easily change constitutive assumptions. Our implementation scales linearly with time step, and subquadratically with mesh spacing and immersed boundary node spacing. We identify the relationship between the immersed boundary node spacing and fluid mesh spacing required to ensure fluid volume conservation within immersed boundaries, and the scaling of cell membrane stiffness and cell-cell interaction strength required when refining the immersed boundary discretization. This study provides a recipe allowing consistent parametrisation of IBM models.

FINITE ELEMENT APPROACH TO IMMERSED BOUNDARY METHOD WITH DIFFERENT FLUID AND SOLID DENSITIES

2011 ◽  
Vol 21 (12) ◽  
pp. 2523-2550 ◽  
Author(s):  
DANIELE BOFFI ◽  
NICOLA CAVALLINI ◽  
LUCIA GASTALDI
Keyword(s):  
Finite Element ◽  
Immersed Boundary Method ◽  
Stokes Equations ◽  
Numerical Procedure ◽  
Biological Tissues ◽  
Immersed Boundary ◽  
Fluid Structure ◽  
Finite Element Approach ◽  
Boundary Method ◽  
Element Approach

The Immersed Boundary Method (IBM) has been designed by Peskin for the modeling and the numerical approximation of fluid-structure interaction problems, where flexible structures are immersed in a fluid. In this approach, the Navier–Stokes equations are considered everywhere and the presence of the structure is taken into account by means of a source term which depends on the unknown position of the structure. These equations are coupled with the condition that the structure moves at the same velocity of the underlying fluid. Recently, a finite element version of the IBM has been developed, which offers interesting features for both the analysis of the problem under consideration and the robustness and flexibility of the numerical scheme. Initially, we considered structure and fluid with the same density, as it often happens when dealing with biological tissues. Here we study the case of a structure which can have a density higher than that of the fluid. The higher density of the structure is taken into account as an excess of Lagrangian mass located along the structure, and can be dealt with in a variational way in the finite element approach. The numerical procedure to compute the solution is based on a semi-implicit scheme. In fluid-structure simulations, nonimplicit schemes often produce instabilities when the density of the structure is close to that of the fluid. This is not the case for the IBM approach. In fact, we show that the scheme enjoys the same stability properties as in the case of equal densities.

A Discrete-Forcing Immersed Boundary Method for the Fluid-Structure Interaction of an Elastic Slender Body

2011 ◽  
Author(s):  
Injae Lee ◽  
Haecheon Choi
Keyword(s):  
Immersed Boundary Method ◽  
Present Method ◽  
Stokes Equations ◽  
Slender Body ◽  
Body Motion ◽  
Computational Time ◽  
Immersed Boundary ◽  
Flexible Beam ◽  
Time Step ◽  
Boundary Method

In the present study, a new immersed boundary method for the simulation of flow around an elastic slender body is suggested. The present method is based on the discrete-forcing immersed boundary method by Kim et al. (J. Comput. Phys., 2001) and is fully coupled with the elastic slender body motion. The incompressible Navier-Stokes equations are solved in an Eulerian coordinate and the elastic slender body motion is described in a Lagrangian coordinate, respectively. The elastic slender body is modeled as a thin flexible beam and is segmented by finite number of blocks. Each block is then moved by the external and internal forces such as the hydrodynamic, tension, bending, and buoyancy forces. With the proposed method, we simulate several flow problems including flows over a flexible filament, an oscillating insect wing, and a flapping flag. We show that the present method does not impose any severe limitation on the size of computational time step. The results obtained agree very well with those from previous studies.

scholarly journals Immersed boundary method combined with proper generalized decomposition for simulation of a flexible filament in a viscous incompressible flow

2017 ◽  
Vol 39 (2) ◽  
pp. 109-119
Author(s):  
Cuong Q. Le ◽  
H. Phan-Duc ◽  
Son H. Nguyen
Keyword(s):  
Immersed Boundary Method ◽  
Stokes Equations ◽  
Fluid Pressure ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Proper Generalized Decomposition ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Boundary Method ◽  
Fluid Filament

In this paper, a combination of the Proper Generalized  Decomposition (PGD) with the Immersed Boundary method (IBM) for solving  fluid-filament interaction problem is proposed. In this combination, a  forcing term constructed by the IBM is introduced to Navier-Stokes equations  to handle the influence of the filament on the fluid flow. The PGD is  applied to solve the Poission's equation to find the fluid pressure  distribution for each time step. The numerical results are compared with  those by previous publications to illustrate the robustness and  effectiveness of the proposed method.

Benchmark simulations of flow past rigid bodies using an open-source, sharp interface immersed boundary method

2017 ◽  
Vol 1 (1) ◽  
pp. 1 ◽  
Author(s):  
Clarence W. Rowley ◽  
Alexander J. Smits ◽  
Nicoleta Herzog ◽  
Hrvoje Jasak ◽  
Daniel Brunner ◽  
...  
Keyword(s):  
Open Source ◽  
Immersed Boundary Method ◽  
Rigid Bodies ◽  
Sharp Interface ◽  
Immersed Boundary ◽  
Flow Past ◽  
Boundary Method

scholarly journals An Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

2013 ◽  
Vol 3 (3) ◽  
pp. 247-262 ◽  
Author(s):  
Wei-Fan Hu ◽  
Ming-Chih Lai
Keyword(s):  
Immersed Boundary Method ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Boundary Method ◽  
Stretching Factor ◽  
Vesicle Dynamics ◽  
Energy Stable

AbstractWe develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.

Application of SPH-FE method for fluid-structure interaction using immersed boundary method

2018 ◽  
Vol 35 (8) ◽  
pp. 2802-2824 ◽  
Author(s):  
Farhoud Kalateh ◽  
Ali Koosheh
Keyword(s):  
Immersed Boundary Method ◽  
Fluid Structure Interaction ◽  
Kernel Functions ◽  
Immersed Boundary ◽  
Fe Method ◽  
Time Step ◽  
Fluid Structure ◽  
Content Type ◽  
Structure Interaction ◽  
Boundary Method

Purpose This paper aims to propose a new smoothed particle hydrodynamics (SPH)-finite element (FE) algorithm to study fluid–structure interaction (FSI) problems. Design/methodology/approach The fluid domain is discretized based on the theory of SPH), and solid part is solved through FE method, similar to other SPH-FE methods in the previous studies. Instead of master-slave technique, the interpolating (kernel) functions of immersed boundary method are implemented to couple fluid and solid domains. The procedure of modeling completely follows the classic IB framework where forces and velocities are transferred between interacting parts. Three benchmark FSI problems are simulated and the results are compared with those of similar numerical and experimental works. Findings The proposed SPH-FE algorithm with promising and acceptable results can be utilized as a reliable method to simulate FSI problems. Originality/value Contrary to most SPH-FE algorithms, the calculation of contact force is not required at interacting boundaries and no iterative process is proposed to calculate forces, velocities and positions at new time step.

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