three dimensional volume conserving immersed boundary model for two phase fluid flows abstract we present a volume preserving scheme for two phase immiscible incompressible flows using an immersed boundary method ibm in a three dimensional space the two phase ibm employs a mixture of eulerian and lagrangian variables where the fluid interface is represented by discrete lagrangian markers exerting surface tension forces to the eulerian fluid domain and the markers are advected by the fluid velocity the interactions between the lagrangian markers and the fluid variables are linked by the discretized dirac delta function the present study extends the previous two dimensional research li et al volume preserving immersed boundary methods for two phase fluid flows int j numer meth fluids 69 2012 842 858 to the three dimensional space the key idea of the proposed method is relocating surface points along the normal directions to conserve the total volume we perform a number of numerical experiments to show the efficiency and accuracy of the proposed method

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

an implicit immersed boundary method for three dimensional fluid membrane interactions we present an implicit immersed boundary method for the incompressible navier stokes equations capable of handling three dimensional membrane fluid flow interactions the goal of our approach is to greatly improve the time step by using the jacobian free newton krylov method jfnk to advance the location of the elastic membrane implicitly the most attractive feature of this jacobian free approach is newton like nonlinear convergence without the cost of forming and storing the true jacobian the generalized minimal residual method gmres which is a widely used krylov subspace iterative method is used to update the search direction required for each newton iteration each gmres iteration only requires the action of the jacobian in the form of matrix vector products and therefore avoids the need of forming and storing the jacobian matrix explicitly once the location of the boundary is obtained the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model we then use the immersed boundary method to calculate the hydrodynamic effects and fluid structure interaction effects such as membrane deformation the present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force

eulerian lagrangian treatment of nondilute two phase gels we develop a three dimensional eulerian lagrangian two phase mixture model for a gel in which fluid is represented in eulerian coordinates and network in lagrangian coordinates the equations of motion and boundary conditions in their respective frames are derived using a principle of virtual power the volume fractions of each material at each point of eulerian space are calculated using the position and volume density of network particles network and fluid have separate velocity fields and the model includes viscous elastic and applied forces as well as those due to chemical potentials pressure and intermaterial drag we also develop a second order accurate numerical method for solving the model equations in one dimension illustrations of model behavior under a variety of conditions are shown

approximate projection methods part i inviscid analysis the use of approximate projection methods for modeling low mach number flows avoids many of the numerical complications associated with exact projection methods but introduces additional design choices in developing a robust algorithm in this paper we first explore these design choices in the setting of inviscid incompressible flow using several computational examples we then develop a framework for analyzing the behavior of the different design variations and use that analysis to explain the features observed in the computations as part of this work we introduce a new variation of the approximate projection algorithm that combines the advantages of several of the previous versions

a boundary condition capturing immersed interface method for 3d rigid objects in a flow to simulate the flow around an object we can replace the object with the fluid enclosed by a singular force we can then simulate the flow on a fixed domain with a fluid fluid interface supporting the singular force in this paper we present a boundary condition capturing approach to determine the singular force for a 3d rigid object we apply a discontinuous body force to enforce the rigid motion of the fluid replacing the object and compute the singular force based on the kinematics of the object due to the singular force and the body force the flow is not smooth across the interface we solve the flow using the immersed interface method our boundary condition capturing immersed interface method is very efficient and stable and its accuracy based on the infinity norm is near second order for the velocity and above first order for the pressure

the finite element immersed boundary method with distributed lagrange multiplier we introduce a new formulation for the finite element immersed boundary method which makes use of a distributed lagrange multiplier we prove that a full discretization of our model based on a semi implicit time advancing scheme is unconditionally stable with respect to the time step size

implicit second order immersed boundary methods with boundary mass abstract the immersed boundary method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures immersed boundary computations typically evaluate the elastic forces explicitly in the configuration of the immersed elastic structure in many applications this results in a severe restriction on the time step we present a semi implicit and a fully implicit second order accurate immersed boundary method the methods provide a natural way to handle mass on the immersed elastic structures we demonstrate their performance for a prototypical fluid structure interaction problem the methods are shown to possess superior stability properties that significantly alleviate the typically severe time step restriction of explicit computations

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

an immersed interface method for incompressible navier stokes equations the method developed in this paper is motivated by peskin s immersed boundary ib method and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed cartesian grid the ib method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points our method instead incorporates part of this force into jump conditions for the pressure avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary this has been implemented for the two dimensional incompressible navier stokes equations using a high resolution finite volume method for the advective terms and a projection method to enforce incompressibility in the projection step the correct jump in pressure is imposed in the course of solving the poisson problem this gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional ib method comparisons between this method and the ib method are presented for several test problems numerical studies of the convergence and order of accuracy are included

a moving control volume approach to computing hydrodynamic forces and torques on immersed bodies abstract we present a moving control volume cv approach to computing hydrodynamic forces and torques on complex geometries the method requires surface and volumetric integrals over a simple and regular cartesian box that moves with an arbitrary velocity to enclose the body at all times the moving box is aligned with cartesian grid faces which makes the integral evaluation straightforward in an immersed boundary ib framework discontinuous and noisy derivatives of velocity and pressure at the fluid structure interface are avoided and far field smooth velocity and pressure information is used we re visit the approach to compute hydrodynamic forces and torques through force torque balance equations in a lagrangian frame that some of us took in a prior work bhalla et al 2013 13 we prove the equivalence of the two approaches for ib methods thanks to the use of peskin s delta functions both approaches are able to suppress spurious force oscillations and are in excellent agreement as expected theoretically test cases ranging from stokes to high reynolds number regimes are considered we discuss regridding issues for the moving cv method in an adaptive mesh refinement amr context the proposed moving cv method is not limited to a specific ib method and can also be used for example with embedded boundary methods

properties of the solutions of rational matrix difference equations abstract we prove a comparison theorem for the solutions of a rational matrix difference equation generalizing the riccati difference equation and existence and convergence results for the solutions of this equation moreover we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution

the immersed boundary method for advection electrodiffusion with implicit timestepping and local mesh refinement we describe an immersed boundary method for problems of fluid solute structure interaction the numerical scheme employs linearly implicit timestepping allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method and local mesh refinement making it feasible to resolve the steep gradients associated with the space charge layers as well as the chemical potential which is used in our formulation to control the permeability of the membrane to the possibly charged solute low reynolds number fluid dynamics are described by the time dependent incompressible stokes equations which are solved by a cell centered approximate projection method the dynamics of the chemical species are governed by the advection electrodiffusion equations and our semi implicit treatment of these equations results in a linear system which we solve by gmres preconditioned via a fast adaptive composite grid fac solver numerical examples demonstrate the capabilities of this methodology as well as its convergence properties

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

a new implicit fictitious domain method for the simulation of flow in complex geometries with heat transfer a numerical algorithm for the simulation of flow past immersed objects with heat transfer is proposed and validated which conforms with the ideas of the fictitious domain method a momentum source term is added to account for the presence of the object and a heat source term is proposed to impose the dirichlet boundary condition on the surface of the objects the algorithm is an implicit fictitious domain based method where the entire fluid immersed object domain assumed to be an incompressible fluid the flow domain is constrained to be divergence free whereas a rigidity constraint is imposed on the body domain heat transfer is similarly considered by assuming that the object domain is filled with a fluid with different thermal properties the simple algorithm with a collocated grid arrangement is used for pressure velocity coupling which is unconditionally stable the algorithm is validated by considering stationary forced motion and freely moving objects with both isothermal and freely variable temperature inside the object good agreement with previous numerical and experimental studies for all the test cases is observed

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

a cartesian based embedded geometry technique with adaptive high order finite differences for compressible flow around complex geometries an immersed boundary methodology to solve the compressible navier stokes equations around complex geometries in cartesian fluid dynamics solvers is described the objective of the new approach is to enable smooth reconstruction of pressure and viscous stresses around the embedded objects without spurious numerical artifacts a standard level set represents the boundary of the object and defines a fictitious domain into which the flow fields are smoothly extended boundary conditions on the surface are enforced by an approach inspired by analytic continuation each fluid field is extended independently constrained only by the boundary condition associated with that field unlike most existing methods no jump conditions or explicit derivation of them from the boundary conditions are required in this approach numerical stiffness that arises when the fluid solid interface is close to grid points of the mesh is addressed by preconditioning in addition the embedded geometry technique is coupled with a stable high order adaptive discretization that is enabled around the object boundary to enhance resolution the stencils used to transition the order of accuracy of the discretization are derived using the summation by parts technique that ensures stability applications to shock reflections shock ramp interactions and supersonic and low mach number flows over two and three dimensional geometries are presented

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

an adaptive newton continuation strategy for the fully implicit finite element immersed boundary method the immersed boundary method ib is known as a powerful technique for the numerical solution of fluid structure interaction problems as for instance the motion and deformation of viscoelastic bodies immersed in an external flow it is based on the treatment of the flow equations within an eulerian framework and of the equations of motion of the immersed bodies with respect to a lagrangian coordinate system including interaction equations providing the transfer between both frames the classical ib uses finite differences but the ibm can be set up within a finite element approach in the spatial variables as well fe ib the discretization in time usually relies on the backward euler be method for the semidiscretized flow equations and the forward euler fe method for the equations of motion of the immersed bodies the be fe fe ib is subject to a cfl type condition whereas the fully implicit be be fe ib is unconditionally stable the latter one can be solved numerically by newton type methods whose convergence properties are dictated by an appropriate choice of the time step size in particular if one is faced with sudden changes in the total energy of the system in this paper taking advantage of the well developed affine covariant convergence theory for newton type methods we study a predictor corrector continuation strategy in time with an adaptive choice of the continuation steplength the feasibility of the approach and its superiority to be fe fe ib is illustrated by two representative numerical examples

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations the immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization in this paper we propose several efficient semi implicit schemes to remove this stiffness from the immersed boundary method for the two dimensional stokes flow first we obtain a novel unconditionally stable semi implicit discretization for the immersed boundary problem using this unconditionally stable discretization as a building block we derive several efficient semi implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization our stability analysis and extensive numerical experiments show that our semi implicit schemes offer much better stability property than the explicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit schemes can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme but with a much better stability property

an adaptive version of the immersed boundary method a computational setting for the immersed boundary method employing an adaptive mesh refinement is presented enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested progressively finer rectangular grid patches which dynamically follow the immersed boundary motion the set of equations describing the interaction between a non stationary viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method specially designed for locally refined meshes to an implicit formulation of the immersed boundary method the main contributions of this work concern the formulation and the implementation of a multilevel self adaptive version of the immersed boundary method on locally refined meshes this approach is tested for a particular two dimensional model problem for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary and on the associated uniform mesh built with the resolution of the finest level

tether force constraints in stokes flow by the immersed boundary method on a periodic domain the immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid the immersed elastic material is represented in the fluid equations by a system or field of applied forces the particular case of stokes flow with applied forces on a periodic domain involves two related mathematical complications one of these is that an arbitrary constant vector may be added to the fluid velocity and the other is the constraint that the integral of the applied force must be zero typically forces defined on a freely floating elastic immersed boundary or body satisfy this constraint but there are many important classes of forces that do not for example the so called tether forces that are used to prescribe the simulated configuration of an immersed boundary possibly in a time dependent manner typically do not sum to zero another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system we present a method for periodic stokes flow that when used with tether points admits the use of all forces irrespective of their integral over the domain a byproduct of this method is that the additive constant velocity associated with periodic stokes flow is uniquely determined indeed the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied

truncated newton methods and the modeling of complex immersed elastic structures truncated newton minimization methods are combined with peskin s immersed boundary method to facilitate investigation of the dynamic interaction between a viscous incompressible fluid and immersed elastic objects of complex structure applications to aquatic animal locomotion and platelet aggregation during blood clotting are presented 1993 john wiley sons inc

a fast robust and non stiff immersed boundary method we propose a fast and non stiff approach for the solutions of the immersed boundary method for newtonian incompressible flows in two or three dimensions the proposed methodology is built on a robust semi implicit discretization introduced by peskin in the late 70s which is solved efficiently through the novel use of a fast treecode strategy to compute flow structure interactions optimal multipole type expansions are performed numerically by solving a least squares problem with a new fast iterative algorithm the new immersed boundary method is particularly well suited for three dimensional applications and or for problems where the number of immersed boundary points is large we demonstrate the efficacy and superiority of the method over existing approaches with two simple but illustrative examples in 3d

a spatio temporal design problem for a damped wave equation we analyze in this work a spatio temporal optimal design problem governed by a linear damped one dimensional wave equation the problem consists of simultaneously seeking the spatio temporal layout of two isotropic materials and the static position of the damping set in order to minimize a functional depending quadratically on the gradient of the state the lack of classical solutions for this kind of nonlinear problem is well known we examine a well posed relaxation by using the representation of a two dimensional divergence free vector as a rotated gradient we transform the original optimal design problem into a nonconvex vector variational problem by means of gradient young measures we compute an explicit form of the constrained quasi convexification of the cost density moreover this quasi convexification is recovered by first order laminates which give the optimal distribution of materials and damping set at every point finally we analyze the relaxed problem and some numerical experiments ar

a poroelastic immersed boundary method with applications to cell biology the immersed boundary method is a widely used mixed eulerian lagrangian framework for simulating the motion of elastic structures immersed in viscous fluids in the traditional immersed boundary method the fluid and structure move with the same velocity field in this work a model based on the immersed boundary method is presented for simulating poroelastic media in which the fluid permeates a porous elastic structure of small volume fraction that moves with its own velocity field two distinct methods for calculating elastic stresses are presented and compared the methods are validated on a radially symmetric test problem by comparing with a finite difference solution of the classical equations of poroelasticity finally two applications of the modeling framework to cell biology are provided cellular blebbing and cell crawling it is shown that in both examples poroelastic effects are necessary to explain the relevant mechanics

on computational issues of immersed finite element methods the objective of this paper is to provide a review of recent finite element formulations for immersed methods in these finite element formulations independent lagrangian solid meshes are introduced to move on top of a background eulerian fluid mesh this key feature allows the handling without excessive fluid mesh adaptation multiple deformable solids immersed in viscous fluid in particular pros and cons of both explicit and implicit approaches are illustrated along with subtle differences between incompressible and slightly compressible models

nitsche xfem for the coupling of an incompressible fluid with immersed thin walled structures abstract in this paper we introduce a nitsche xfem method for fluid structure interaction problems involving a thin walled elastic structure lagrangian formalism immersed in an incompressible viscous fluid eulerian formalism the fluid domain is discretized with an unstructured mesh not fitted to the solid mid surface mesh weak and strong discontinuities across the interface are allowed for the velocity and pressure respectively the fluid solid coupling is enforced consistently using a variant of nitsche s method with cut elements robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization several coupling schemes with different degrees of fluid solid time splitting implicit semi implicit and explicit are investigated a series of numerical test in 2d involving static and moving interfaces illustrates the performance of the different methods proposed

a spectral element formulation of the immersed boundary method for newtonian fluids a spectral element formulation of the immersed boundary method ibm is presented the spectral element formulation se ibm is a generalisation of the finite element immersed boundary method fe ibm based on high order approximations of the fluid variables several schemes for tracking the movement of the immersed boundary are considered and a semi implicit euler scheme is shown to offer advantages in terms of accuracy and efficiency high order spectral element approximations provide improved area conservation properties of the ibm due to the incompressibility constraint being more accurately satisfied superior orders of convergence are obtained for se ibm compared with fe ibm in both l2 and h1 norms the area conservation and convergence properties of the scheme are demonstrated on a series of benchmark problems

unconditionally stable discretizations of the immersed boundary equations the immersed boundary ib method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method many implicit methods have been proposed to try to mitigate this timestep restriction but none are known to be unconditionally stable and the observed instability of even some of the fully implicit methods is not well understood in this paper we prove that particular backward euler and crank nicolson like discretizations of the nonlinear immersed boundary terms of the ib equations in conjunction with unsteady stokes flow can yield unconditionally stable methods we also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space we further demonstrate through computational tests that approximate projection methods which do not provide a discretely divergence free velocity field appear to have a stabilizing influence for these problems and that the implicit methods of this paper when used with the full navier stokes equations are no longer subject to such a strict timestep restriction and can be run up to the cfl constraint of the advection terms

the blob projection method for immersed boundary problems a new finite difference numerical method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two dimensional domain is presented the method differs from existing methods in the way the forces exerted by the membranes on the fluid are modeled these are described by a collection of regularized point forces and the velocity field they induce is computed directly on a regular cartesian grid via a smoothed dipole potential comparisons between this method and the immersed boundary method of peskin and mcqueen are presented the results show that the method proposed here preserves volumes better and has a higher order of convergence

efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice boltzmann finite element method the deformation of an initially spherical capsule freely suspended in simple shear flow can be computed analytically in the limit of small deformations d barthes biesel j m rallison the time dependent deformation of a capsule freely suspended in a linear shear flow j fluid mech 113 1981 251 267 those analytic approximations are used to study the influence of the mesh tessellation method the spatial resolution and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice boltzmann finite element method for the description of the capsule membrane a finite element method and the skalak constitutive model r skalak a tozeren r p zarda s chien strain energy function of red blood cell membranes biophys j 13 1973 245 264 have been employed our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid we come to the conclusion that details of the membrane mesh as tessellation method and resolution play only a minor role the hydrodynamic resolution i e the width of the discrete delta function can significantly influence the accuracy of the simulations the discretization of the delta function introduces an artificial length scale which effectively changes the radius and the deformability of the capsule we discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

efficient solutions to robust semi implicit discretizations of the immersed boundary method the immersed boundary method is a versatile tool for the investigation of flow structure interaction in a large number of applications the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well known severe time step restriction for explicit discretizations this excessive stability constraint can be removed with fully implicit or suitable semi implicit schemes but at a seemingly prohibitive computational cost while economical alternatives have been proposed recently for some special cases there is a practical need for a computationally efficient approach that can be applied more broadly in this context we revisit a robust semi implicit discretization introduced by peskin in the late 1970s which has received renewed attention recently this discretization in which the spreading and interpolation operators are lagged leads to a linear system of equations for the interface configuration at the future time when the interfacial force is linear however this linear system is large and dense and thus it is challenging to streamline its solution moreover while the same linear system or one of similar structure could potentially be used in newton type iterations nonlinear and highly stiff immersed structures pose additional challenges to iterative methods in this work we address these problems and propose cost effective computational strategies for solving peskin s lagged operators type of discretization we do this by first constructing a sufficiently accurate approximation to the system s matrix and we obtain a rigorous estimate for this approximation this matrix is expeditiously computed by using a combination of pre calculated values and interpolation the availability of a matrix allows for more efficient matrix vector products and facilitates the design of effective iterative schemes we propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points one of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force we demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method in addition to considering the standard elliptical drop test case we show both the robustness and efficacy of the proposed methodology with a 2d model of a heart valve

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

partially implicit motion of a sharp interface in navier stokes flow we develop a numerical method for the coupled motion of navier stokes flow with an elastic interface of zero thickness which exerts tension and bending forces on the fluid the interface motion is made partially implicit by approximating a backward euler step in the high wavenumbers as in the small scale decomposition method of hou lowengrub and shelley this modified step is combined with the method of beale and layton j t beale a t layton a velocity decomposition approach for moving interfaces in viscous fluids j comput phys 228 2009 3358 67 the fluid velocity is found by computing the stokes velocity and a more regular remainder the resulting scheme is second order in space and first order in time it can be made second order in time by extrapolation the discontinuities in the pressure and velocity gradient are preserved the partially implicit method allows much larger time steps than an explicit method with negligible added effort the formulas in the fourier transform for the implicit approximation in high wavenumbers are similar to those derived in hou and shi t y hou z shi an efficient semi implicit immersed boundary method for the navier stokes equations j comput phys 227 2008 9138 69 in a different context

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

a pre conditioned implicit direct forcing based immersed boundary method for incompressible viscous flows a novel immersed boundary ib method based on an implicit direct forcing idf scheme is developed for incompressible viscous flows the key idea for the present idf method is to use a block lu decomposition technique in momentum equations with taylor series expansion to construct the implicit ib forcing in a recurrence form which imposes more accurate no slip boundary conditions on the ib surface to accelerate the ib forcing convergence during the iterative procedure a pre conditioner matrix is introduced in the recurrence formulation of the ib forcing a jacobi type parameter is determined in the pre conditioner matrix by minimizing the frobenius norm of the matrix function representing the difference between the ib forcing solution matrix and the pre conditioner matrix in addition the pre conditioning parameter is restricted due to the numerical stability in the recurrence formulation consequently the present pre conditioned idf pidf enables accurate calculation of the ib forcing within a few iterations we perform numerical simulations of two dimensional flows around a circular cylinder and three dimensional flows around a sphere for low and moderate reynolds numbers the result shows that pidf yields a better imposition of no slip boundary conditions on the ib surfaces for low reynolds number with a fairly larger time step than ib methods with different direct forcing schemes due to the implicit treatment of the diffusion term for determining the ib forcing finally we demonstrate the robustness of the present pidf scheme by numerical simulations of flow around a circular array of cylinders flows around a falling sphere and two sedimenting spheres in gravity

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

an efficient parallel immersed boundary algorithm using a pseudo compressible fluid solver we propose an efficient algorithm for the immersed boundary method on distributed memory architectures that has the computational complexity of a completely explicit method and also has 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 simulations of several fluid structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm we also compare the proposed algorithm with other second order projection based fluid solvers lastly the execution time and scaling properties of the proposed algorithm are investigated and compared to alternate approaches

a multirate time integrator for regularized stokeslets the method of regularized stokeslets is a numerical approach to approximating solutions of fluid structure interaction problems in the stokes regime regularized stokeslets are fundamental solutions to the stokes equations with a regularized point force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid due to the linearity of the stokes equations the velocity at any point in the fluid can be computed by summing the contributions of regularized stokeslets and the time evolution of positions can be computed using standard methods for ordinary differential equations rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid for problems with boundaries modeled by springs with large spring constants the resulting ordinary differential equations become stiff and hence the time step for explicit time integration methods is severely constrained unfortunately the use of standard implicit time integration methods for the method of regularized stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems here an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method the multirate methods are higher order methods that treat different portions of the ode explicitly with different time steps depending on the stiffness of each component numerical examples on two nontrivial three dimensional problems demonstrate the increased efficiency of the multi explicit approach with no significant increase in numerical error