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

a priori error estimates of regularized elliptic problems approximations of the dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth generalized functions via convolution in this work we show a priori rates of convergence of this approximation process in standard sobolev norms with minimal regularity assumptions on the approximation of the dirac delta distribution the application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp h 1 and l 2 error estimates for the corresponding regularized problem as an application we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non regularized counterpart provided that the support of the dirac delta approximation is set to a multiple of the mesh size at a fraction of the implementation complexity numerical experiments are provided to support our theories

extended immersed boundary method using fem and rkpm abstract the proposed extended immersed boundary method eibm has several distinct features in comparison with the immersed boundary ib method in the ib method the interaction between the immersed elastic boundary and the surrounding viscous fluid is replaced with an equivalent body force distribution within the fluid domain the key ingredient of the ib method is the enforcement of the energy input from the equivalent body force into the fluid domain to be the same at all time as that of the elastic forces within the ib in eibm instead of the volumeless immersed elastic boundary we consider the submerged elastic solid which occupies a finite volume within the fluid domain in particular we replace the kinematic and dynamic matching of the fluid solid interface and the effect of the submerged solid with a judiciously chosen collection of equivalent nodal forces calculated in the context of finite element formulations the employment of the finite element representation enables a realistic stress analysis for the submerged solid subject to large deformations finally in eibm we also replace the discretized delta function commonly used in the ib method with the kernel functions in meshless reproducing kernel particle method rkpm with this improvement using the same finite support domain we could improve the discretized delta function from c 1 to c n continuity where n is chosen according to the required smoothness and resolution of such functions and more importantly enables the use of non uniform meshing in the fluid domain these extensions may eventually open doors to the modelling of complex biological fluid involving molecular cellular and flexible vessel flow interactions

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

stability analysis for the immersed fiber problem a linear stability analysis is performed on a two dimensional version of the immersed fiber problem formulated by c peskin to model the flow of fluid in the presence of a mesh of moving elastic fibers the purpose of the analysis is to isolate the modes in the solution which are associated with the fiber and thereby determine the effect of the presence of a fiber on the fluid the results are used not only to make conclusions about the stability of the problem but also to suggest guidelines for developing numerical methods for flows with immersed fibers

an unfitted nitsche method for incompressible fluid structure interaction using overlapping meshes we consider the extension of the nitsche method to the case of fluid structure interaction problems on unfitted meshes we give a stability analysis for the space semi discretized problem and show how this estimate may be used to derive optimal error estimates for smooth solutions irrespectively of the mesh interface intersection we also discuss different strategies for the time discretization using either fully implicit or explicit coupling loosely coupled schemes some numerical examples illustrate the theoretical discussion

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

a fictitious domain approach with lagrange multiplier for fluid structure interactions we study a recently introduced formulation for fluid structure interaction problems which makes use of a distributed lagrange multiplier in the spirit of the fictitious domain method the time discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided the finite element space discretization is discussed and optimal convergence estimates are proved

benchmarking the immersed finite element method for fluid structure interaction problems we present an implementation of a fully variational formulation of an immersed method for fluid structure interaction problems based on the finite element method while typical implementation of immersed methods are characterized by the use of approximate dirac delta distributions fully variational formulations of the method do not require the use of said distributions in our implementation the immersed solid is general in the sense that it is not required to have the same mass density and the same viscous response as the surrounding fluid we assume that the immersed solid can be either viscoelastic of differential type or hyperelastic here we focus on the validation of the method via various benchmarks for fluid structure interaction numerical schemes this is the first time that the interaction of purely elastic compressible solids and an incompressible fluid is approached via an immersed method allowing a direct comparison with established benchmarks

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

error estimates in weighted sobolev norms for finite element immersed interface methods abstract when solving elliptic partial differential equations in a region containing immersed interfaces possibly evolving in time it is often desirable to approximate the problem using an independent background discretisation not aligned with the interface itself optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface at the cost of a higher complexity in the implementation a much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions as done in immersed boundary methods these methods are known to have inferior convergence properties depending on the global regularity of the solution across the interface when compared to enriched methods in this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon restricted to a small neighbourhood of the interface in particular we show that optimal approximations can be constructed in a natural and inexpensive way simply by reformulating the problem in a distributionally consistent way and by resorting to weighted norms when computing the global error of the approximation

the disturbance decoupling problem for implicit linear discrete time systems the disturbance decoupling problem for implicit linear discrete time systems is studied in detail necessary and sufficient conditions for the problem to be solved are given the results are obtained with the aid of the concepts of almost invariant sliding and coasting subspaces for implicit systems

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

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

a natural framework for isogeometric fluid structure interaction based on bem shell coupling abstract the interaction between thin structures and incompressible newtonian fluids is ubiquitous both in nature and in industrial applications in this paper we present an isogeometric formulation of such problems which exploits a boundary integral formulation of stokes equations to model the surrounding flow and a non linear kirchhoff love shell theory to model the elastic behavior of the structure we propose three different coupling strategies a monolithic fully implicit coupling a staggered elasticity driven coupling and a novel semi implicit coupling where the effect of the surrounding flow is incorporated in the non linear terms of the solid solver through its damping characteristics the novel semi implicit approach is then used to demonstrate the power and robustness of our method which fits ideally in the isogeometric paradigm by exploiting only the boundary representation b rep of the thin structure middle surface

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

invariant measures for a two species asymmetric process we consider a process of two classes of particles jumping on a one dimensional lattice the marginal system of the first class of particles is the one dimensional totally asymmetric simple exclusion process when classes are disregarded the process is also the totally asymmetric simple exclusion process the existence of a unique invariant measure with product marginals with density and for the first and first plus second class particles respectively was shown by ferrari kipnis and saada recently derrida janowsky lebowitz and speer have computed this invariant measure for finite boxes and performed the infinite volume limit based on this computation we give a complete description of the measure and derive some of its properties in particular we show that the invariant measure for the simple exclusion process as seen from a second class particle with asymptotic densities and is equivalent to the product measure with densities to the left of the origin and to the right of the origin

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

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

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

mixed formulation for interface problems with distributed lagrange multiplier we study a mixed formulation for elliptic interface problems which has been recently introduced when dealing with a test problem arising from fluid structure interaction applications the formulation which involves a lagrange multiplier defined in the solid domain can be approximated by standard finite elements on meshes which do not need to fit with the interface in this paper we discuss a modification of the original formulation involving a different approach for the analysis and the numerical implementation of the lagrange multiplier new two dimensional numerical results confirm the good performances of the proposed schemes

an unfitted formulation for the interaction of an incompressible fluid with a thick structure via an xfem dg approach a numerical procedure that combines an extended finite element formulation and a discontinuous galerkin technique is presented with the final aim of providing an effective tool for the simulation of three dimensional 3d fluid structure interaction problems in this work we consider a thick structure immersed in a fluid we describe the numerical models and discuss the specific implementation issues arising in three dimensions finally 3d numerical results are provided to show the effectiveness of the approach

on the hyper elastic formulation of the immersed boundary method abstract the immersed boundary ib method is both a mathematical formulation and a numerical method for fluid structure interaction problems in which immersed incompressible visco elastic bodies or boundaries interact with an incompressible fluid previous formulations of the ib method were not able to treat appropriately immersed materials of finite nonzero thickness modeled by general hyper elastic constitutive laws because of the lack of appropriate transmission conditions between the immersed body and the surrounding fluid in the case of a nonzero jump in normal stress at the solid fluid interface such a jump does not arise when the solid is comprised of fibers that run parallel to the interface but typically does arise in other cases e g when the solid contains elastic fibers that terminate at the solid fluid interface we present a derivation of the ib method that takes into account in an appropriate way the missing term the derivation presented in this paper starts from a separation of the stress that appears in the principle of virtual work the stress is divided into its fluid like and solid like components and each of these two terms is treated in its natural framework i e the eulerian framework for the fluid like stress and the lagrangian framework for the solid like stress we describe how the ib method can be used in conjunction with standard formulations of continuum mechanics models for immersed incompressible elastic materials and present some illustrative numerical experiments

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 unified mathematical framework and an adaptive numerical method for fluid structure interaction with rigid deforming and elastic bodies many problems of interest in biological fluid mechanics involve interactions between fluids and solids that require the coupled solution of momentum equations for both the fluid and the solid in this work we develop a mathematical framework and an adaptive numerical method for such fluid structure interaction fsi problems in which the structure may be rigid deforming or elastic we employ an immersed boundary ib formulation of the problem that permits us to avoid body conforming discretizations and to use fast cartesian grid solvers rigidity and deformational kinematic constraints are imposed using a formulation based on distributed lagrange multipliers and a conventional ib method is used to describe the elasticity of the immersed body we use cartesian grid adaptive mesh refinement amr to discretize the equations of motion and thereby obtain a solution methodology that efficiently captures thin boundary layers at fluid solid interfaces as well as flow structures shed from such interfaces this adaptive methodology is validated for several benchmark problems in two and three spatial dimensions in addition we use this scheme to simulate free swimming including the maneuvering of a two dimensional model eel and a three dimensional model of the weakly electric black ghost knifefish

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

a study on unfitted 1d finite element methods in the present paper we consider a 1d poisson model characterized by the presence of an interface where a transmission condition arises due to jumps of the coefficients we aim at studying finite element methods with meshes not fitting such an interface it is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions we focus on so called embedded approaches frequently adopted to treat fluid structure interaction problems with the aim of recovering higher order of approximation also in presence of non fitting meshes we implement several methods inspired by the immersed boundary method the fictitious domain method and the extended finite element method in particular we present four formulations in a comprehensive and unified format proposing several numerical tests and discussing their performance moreover we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions