an extended pressure finite element space for two phase incompressible flows with surface tension we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set or vof method is applied then the interface which is implicitly given by the zero level of the level set function is in general not aligned with the triangulation that is used in the discretization of the flow problem this non alignment causes severe difficulties w r t the discretization of the localized surface tension force and the discretization of the flow variables in cases with large surface tension forces the pressure has a large jump across the interface in standard finite element spaces due to the non alignment the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure in many simulations these effects cause large oscillations of the velocity close to the interface so called spurious velocities in this paper for a simplified model problem we give an analysis that explains why known standard methods for discretization of the localized force term and for discretization of the pressure variable often yield large spurious velocities in the paper s grosz a reusken finite element discretization error analysis of a surface tension force in two phase incompressible flows preprint 262 igpm rwth aachen siam j numer anal accepted for publication we introduce a new and accurate method for approximation of the surface tension force in the present paper we use the extended finite element space xfem presented in n moes j dolbow t belytschko a finite element method for crack growth without remeshing int j numer meth eng 46 1999 131 150 t belytschko n moes s usui c parimi arbitrary discontinuities in finite elements int j numer meth eng 50 2001 993 1013 for the discretization of the pressure we show that the size of spurious velocities is reduced substantially provided we use both the new treatment of the surface tension force and the extended pressure finite element space

galerkin finite element methods with symmetric pressure stabilization for the transient stokes equations stability and convergence analysis we consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient stokes equation the analysis is valid for a class of symmetric pressure stabilization operators but also for standard inf sup stable velocity pressure spaces without stabilization provided the initial data are chosen as a specific method dependent ritz projection we get unconditional stability and optimal convergence for both pressure and velocity approximations in natural norms for arbitrary interpolations of the initial data a condition between the space and time discretization parameters has to be verified in order to guarantee pressure stability

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

interior penalty variational multiscale method for the incompressible navier stokes equation monitoring artificial dissipation in this paper we apply the interior penalty method analyzed in e burman and m a fernndez numer math 107 2007 no 1 39 77 mr2317827 2008c 65317 to flows at high reynolds number as a possible measure of solution quality we propose to monitor the ratio between the artificial dissipation induced by the numerical method and the computed physical dissipation for smooth flows we prove that for our method the artificial dissipation serves as an a posteriori error estimator and also that it vanishes at an optimal rate in the case of flows with multiscale features we discuss a heuristic approach relating the decay of the artificial dissipation to the decay rate of the power spectrum some numerical results in two space dimensions are presented examining the relation between the numerical dissipation and solution quality

a face oriented stabilized nitsche type extended variational multiscale method for incompressible two phase flow a novel extended variational multiscale method for incompressible two phase flow is proposed in this approach the level set method which allows for accurately representing complex interface evolutions is combined with an extended finite element method for the fluid field sharp representation of the discontinuities at the interface related to surface tension effects and large material parameter ratios are enabled by this approach to capture the discontinuities jump enrichments are applied for both velocity and pressure field nitsche s method is then used to weakly impose the continuity of the velocity field for a stable formulation on the entire domain residual based variational multiscale terms are supported by appropriate face oriented ghost penalty and fluid stabilization terms in the region of enriched elements both face oriented terms as well as interfacial terms related to nitsche s method are introduced such that it is accounted for viscous and convection dominated transient flows as a result stability and wellconditioned systems are guaranteed independent of the interface location the proposed method is applied to four numerical examples of increasing complexity two dimensional rayleigh taylor instabilities a twodimensional collapsing water column three dimensional rising bubbles as well as a three dimensional bubble coalescence excellent agreement with either analytical solutions or numerical and experimental reference data as well as robustness for all flow regimes is demonstrated for all examples

large eddy simulation in complex geometric configurations using boundary body forces anumericalmethodispresentedthatallowslargeeddysimulation les ofturbulente owsincomplexgeometric cone gurations with moving boundaries and that retains the advantages of solving the navier stokes equations on e xed orthogonal grids the boundary conditions are applied independently of the grid by assigning body forces over surfaces that need not coincide with coordinate lines the use of orthogonal nondeforming grids simplie es grid generation facilitates theimplementation of high order nondissipativediscretization schemes andminimizes the spatial and temporal variations in e lter width that complicate unstructured deforming grid les dynamic subgrid scaleturbulence models areparticularly appealing in combination with the body forceprocedure because the dynamic model accounts automatically for the presence of solid walls without requiring damping functions the method is validated by simulations of the turbulent e ow in a motored axisymmetric piston cylinder assembly for which detailed experimental measurements are available computed mean and rms velocity proe les show very good agreement with measured ensemble averages thepresent numerical code runs on small personal computerlike workstations for a comparable level of accuracy computational requirements memory and cpu time are at least a factor of 10 lower compared to published simulations for the same cone guration obtained using an unstructured boundary e tted deforming grid approach

hardy spaces for the conjugated beltrami equation in a doubly connected domain abstract we consider hardy spaces associated to the conjugated beltrami equation on doubly connected planar domains there are two main differences with previous studies baratchart et al 2010 2 first while the simple connectivity plays an important role in baratchart et al 2010 2 the multiple connectivity of the domain leads to unexpected difficulties in particular we make strong use of a suitable parametrization of an analytic function in a ring by its real part on one part of the boundary and by its imaginary part on the other then we allow the coefficient in the conjugated beltrami equation to belong to w 1 q for some q 2 while it was supposed to be lipschitz in baratchart et al 2010 2 we define hardy spaces associated with the conjugated beltrami equation and solve the corresponding dirichlet problem the same problems for generalized analytic function are also solved

a conservative immersed interface method for large eddy simulation of incompressible flows we propose a conservative second order accurate immersed interface method for representing incompressible fluid flows over complex three dimensional solid obstacles on a staggered cartesian grid the method is based on a finite volume discretization of the incompressible navier stokes equations which is modified locally in cells that are cut by the interface in such a way that accuracy and conservativity are maintained a level set technique is used for description and tracking of the interface geometry so that an extension of the method to moving boundaries and flexible walls is straightforward numerical stability is ensured for small cells by a conservative mixing procedure discrete conservation and sharp representation of the fluid solid interface render the method particularly suitable for large eddy simulations of high reynolds number flows accuracy second order grid convergence and robustness of the method is demonstrated for several test cases inclined channel flow at re 20 flow over a square cylinder at re 100 flow over a circular cylinder at re 40 re 100 and re 3900 as well as turbulent channel flow with periodic constrictions at re 10 595

nonlinear riemann hilbert problems with lipschitz continuous boundary data doubly connected domains here we investigate nonlinear riemann hilbert problems for hyperbolic functions w z u z i v z in a given doubly connected domain g 2 where the nonlinear boundary data i e the family of curves are implicitly given by f u v 0 for g 2 in the w plane with a lipshitz continuous function f and these curves for every are non closed but going to infinity in order to prove the solvability of the nonlinear problem we first reduce the problem to a system of nonlinear cauchysingular integral equations on the boundary g 2 employing the topological degree of quasilinear fredholm mappings together with approximation arguments montel theorem we prove the existence of at least one solution

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 unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

generalized navier boundary condition and geometric conservation law for surface tension we consider two fluid flow problems in an arbitrary lagrangian eulerian ale framework the purpose of this work is twofold first we address the problem of the moving contact line namely the line common to the two fluids and the wall second we perform a stability analysis in the energy norm for various numerical schemes taking into account the gravity and surface tension effects the problem of the moving contact line is treated with the so called generalized navier boundary conditions owing to these boundary conditions it is possible to circumvent the incompatibility between the classical no slip boundary condition and the fact that the contact line of the interface on the wall is actually moving the energy stability analysis is based in particular on an extension of the geometry conservation law gcl concept to the case of moving surfaces this extension is useful to study the contribution of the surface tension the theoretical and computational results presented in this paper allow us to propose a strategy which offers a good compromise between efficiency stability and artificial diffusion

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 boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

derivation and validation of a novel implicit second order accurate immersed boundary method a novel implicit second order accurate immersed boundary method ibm for simulating the flow around arbitrary stationary bodies is developed implemented and validated in this paper the ibm is used to efficiently take into account the existence of bodies within the fluid domain the flow domain consist of simple cartesian cells whereas the body can be arbitrary at the triangulated interface of the body and the fluid the immersed boundary the coefficients obtained from discretizing the navier stokes equations are closed with a second order accurate interpolation arising from the immersed boundary condition employed at the interface two different conditions are developed in this paper and it is shown that for the mirroring method the resulting coefficients lead to a well posed and diagonally dominant system which can be efficiently solved with a preconditioned krylov sub space solver the immersed boundary condition generates a fictitious reversed velocity field inside the immersed boundary which is excluded from the continuity equation to account for the presence of the ib in the pressure correction equation resulting in no mass flux over the ib the force acting on the object from the fluid is determined by integrating the pressure and the viscous forces over the object the method is validated by simulating the flow around a sphere for a range of re numbers it is shown that the drag is very well in agreement with experimental data accuracy and convergence studies are employed proving the second order accuracy of the method and showing the superiority in convergence rate compared to other ibm finally the drag force of a cluster of non spherical particles is employed to show the generality and potential of the method

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

a new face oriented stabilized xfem approach for 2d and 3d incompressible navier stokes equations abstract we develop a new stabilized xfem based fixed grid approach for the transient incompressible navier stokes equations using cut elements our framework is based on a new flexible strategy to handle several degrees of freedom dofs at nodes required for complex shaped embedded meshes as they occur for fixed grid fluid structure interaction problems boundary conditions on embedded boundaries are imposed weakly using a nitsche type approach ghost penalty terms for velocity and pressure are added for stability reasons and to improve the conditioning of the system matrix the idea of ghost penalties previously developed for stokes problems is extended to the incompressible navier stokes equations by using face oriented fluid stabilizations for both viscous and convective dominated flows furthermore the need for additional terms to enforce boundary conditions for non viscous flows is addressed a detailed numerical study of our stabilized fluid formulation including studies of nitsche s parameter and the ghost penalty parameter shows optimal error convergence and a good system conditioning in the viscous and the convective dominated cases compared to other fixed grid fluid formulations our method is much more accurate and less sensitive to the location of the interface regarding the accuracy and the sensitivity with respect to the interface position a clear improvement could be observed for the viscous and pressure fluxes as well as for the imposition of the boundary condition results of a transient two dimensional high reynolds number flow over a cylinder and the flow over a thick plate in 3d confirm the applicability of the proposed method

an observation on korn s inequality for nonconforming finite element methods by utilizing a simple observation on traces of rigid motions we are able to strengthen a result of brenner 2004 on korn s inequality for nonconforming finite element methods the approach here is tightly connected to the theory developed in brenner s work our motivation for the analysis was the desire to show that a robust darcy stokes element satisfies korn s inequality and to achieve this the stronger result seems necessary

slip with friction and penetration with resistance boundary conditions for the navier stokes equations numerical tests and aspects of the implementation we consider slip with friction and penetration with resistance boundary conditions in the steady state navier stokes equations this paper describes some aspects of the implementation of these boundary conditions for finite element discretizations numerical tests on two and three dimensional channel flows across a step using the slip with friction boundary condition study the influence of the friction parameter on the position of the reattachment point and the reattachment line of the recirculating vortex respectively

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

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 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

a nitsche cut finite element method for the oseen problem with general navier boundary conditions abstract in this work a nitsche based imposition of generalized navier conditions on cut meshes for the oseen problem is presented other methods from literature dealing with the generalized navier condition impose this condition by means of substituting the tangential robin condition in a classical galerkin way these methods work fine for a large slip length coefficient but lead to conditioning and stability issues when it approaches zero we introduce a novel method for the weak imposition of the generalized navier condition which remains well posed and stable for arbitrary choice of slip length including zero the method proposed here builds on the formulation done by juntunen and stenberg 2009 they impose a robin condition for the poisson problem by means of nitsche s method for an arbitrary combination of the dirichlet and neumann parts of the condition the analysis conducted for the proposed method is done in a similar fashion as in massing et al 2018 but is done here for a more general type of boundary condition the analysis proves stability for all flow regimes and all choices of slip lengths also an l 2 optimal estimate for the velocity error is shown which was not conducted in the previously mentioned work a numerical example is carried out for varying slip lengths to verify the robustness and stability of the method with respect to the choice of slip length even though proofs and formulations are presented for the more general case of an unfitted grid method they can easily be reduced to the simpler case of a boundary fitted grid with the removal of the ghost penalty stabilization terms

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

continuous interior penalty finite element method for oseen s equations in this paper we present an extension of the continuous interior penalty method of douglas and dupont interior penalty procedures for elliptic and parabolic galerkin methods in computing methods in applied sciences lecture notes in phys 58 springer verlag berlin 1976 pp 207 216 to oseen s equations the method consists of a stabilized galerkin formulation using equal order interpolation for pressure and velocity to counter instabilities due to the pressure velocity coupling or due to a high local reynolds number we add a stabilization term giving l2 control of the jump of the gradient over element faces edges in two dimensions to the standard galerkin formulation boundary conditions are imposed in a weak sense using a consistent penalty formulation due to nitsche we prove energy type a priori error estimates independent of the local reynolds number and give some numerical examples recovering the theoretical results

accurate immersed boundary reconstructions for viscous flow simulations cells crossed by the immersed boundary is treated a novel interpolation method based on momentum balance and massconservationbasedon finitevolumemethodareintroducedtocorrectlypredictthevelocityandpressure fieldin the vicinity of the boundaries the assessment of the solution quality is based on calculations of taylor decaying vortices examples of laminar and turbulent flow calculations are also reported

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

a ghost cell immersed boundary method for flow in complex geometry an efficient ghost cell immersed boundary method gcibm for simulating turbulent flows in complex geometries is presented a boundary condition is enforced through a ghost cell method the reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second order accuracy of the base solver both dirichlet and neumann boundary conditions can be treated the current ghost cell treatment is both suitable for staggered and non staggered cartesian grids the accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface numerical results are compared with experimental data and boundary fitted grid results the method is further extended to an existing ocean model mitgcm to simulate geophysical flow over a three dimensional bump the method is easily implemented as evidenced by our use of several existing codes

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

nitsche s method for general boundary conditions we introduce a method for treating general boundary conditions in the finite element method generalizing an approach due to nitsche 1971 for approximating dirichlet boundary conditions we use poisson s equations as a model problem and prove a priori and a posteriori error estimates the method is also compared with the traditional galerkin method the theoretical results are verified numerically

a stabilized nitsche cut finite element method for the oseen problem abstract we provide the numerical analysis for a nitsche based cut finite element formulation for the oseen problem which has been originally presented for the incompressible navier stokes equations by schott and wall 2014 and allows the boundary of the domain to cut through the elements of an easy to generate background mesh the formulation is based on the continuous interior penalty cip method of burman et al 2006 which penalizes jumps of velocity and pressure gradients over inter element faces to counteract instabilities arising for high local reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure since the mesh does not fit the boundary dirichlet boundary conditions are imposed weakly by a stabilized nitsche type approach the addition of cip like ghost penalties in the boundary zone allows to prove that our method is inf supstable and to derive optimal order a priori error estimates in an energy type norm irrespective of how the boundary cuts the underlying mesh all applied stabilization techniques are developed with particular emphasis on low and high reynolds numbers two and three dimensional numerical examples corroborate the theoretical findings finally the proposed method is applied to solve the transient incompressible navier stokes equations on a complex geometry

hardy spaces of the conjugate beltrami equation we study hardy spaces of solutions to the conjugate beltrami equation with lipschitz coefficient on dini smooth simply connected planar domains in the range of exponents 1 infty we analyse their boundary behaviour and certain density properties of their traces we derive on the way an analog of the fatou theorem for the dirichlet and neumann problems associated with the equation mbox div sigma nabla u 0 with l p boundary data

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

weak imposition of the slip boundary condition on curved boundaries for stokes flow we study the finite element approximation of two methods to weakly impose a slip boundary condition for incompressible fluid flows the lagrange multiplier method and nitsche s method for each method we can distinguish several formulations depending on the values of some real parameters in the case of a spatial domain with a polygonal or polyhedral boundary we prove convergence results of their finite element approximations extending previous results of verfurth 33 and we show numerical results confirming them in the case of a spatial domain with a smooth curved boundary numerical results show that approximations computed on polygonal domains approximating the original domain may not converge to the exact solution depending on the values of the aforementioned parameters and on the finite element discretization these negative results seem to highlight babuska s like paradox due to the approximation of the boundary by polygonal ones in particular they seem to contradict some of verfurth s theoretical convergence results

a stabilized cut finite element method for the three field stokes problem we propose a nitsche based fictitious domain method for the three field stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh the dependent variables of velocity pressure and extra stress tensor are discretized on the background mesh using linear finite elements this equal order approximation is stabilized using a continuous interior penalty cip method on the unfitted domain boundary dirichlet boundary conditions are weakly enforced using nitsche s method we add cip like ghost penalties in the boundary region and prove that our scheme is inf sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh additionally we demonstrate that the condition number of the system matrix is bounded independently of the boundary location we corroborate our theoretical findings numerically

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

a penalty free nonsymmetric nitsche type method for the weak imposition of boundary conditions in this paper we show that the nonsymmetric version of nitsche s method for the weak imposition of boundary conditions is stable without penalty term for nonconforming elements we prove the same result for the symmetric formulation as well we prove optimal h 1 error estimates and l 2 error estimates that are suboptimal with half an order in h both the pure diffusion and the convection diffusion problems are discussed

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

a nitsche based cut finite element method for the coupling of incompressible fluid flow with poroelasticity abstract the focus of this contribution is the numerical treatment of interface coupled problems concerning the interaction of incompressible fluid flow and permeable elastic structures the main emphasis is on extending the range of applicability of available formulations on especially three aspects these aspects are the incorporation of a more general poroelasticity formulation the use of the cut finite element method cutfem to allow for large interface motion and topological changes of the fluid domain and the application of a novel nitsche based approach to incorporate the beavers joseph saffmann tangential interface condition this last aspect allows one to extend the practicable range of applicability of the proposed formulation down to very low porosities and permeabilities which is important in several examples in application different aspects of the presented formulation are analyzed in a numerical example including spatial convergence the sensitivity of the solution to the nitsche penalty parameters varying porosities and permeabilities and a varying beavers joseph interface model constant finally a numerical example analyzing the fluid induced bending of a poroelastic beam provides evidence of the general applicability of the presented approach

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

an accurate robust and easy to implement method for integration over arbitrary polyhedra application to embedded interface methods we present an accurate method for the numerical integration of polynomials over arbitrary polyhedra using the divergence theorem the method transforms the domain integral into integrals evaluated over the facets of the polyhedra the necessity of performing symbolic computation during such transformation is eliminated by using one dimensional gauss quadrature rule the facet integrals are computed with the help of quadratures available for triangles and quadrilaterals numerical examples in which the proposed method is used to integrate the weak form of the navier stokes equations in an embedded interface method eim are presented the results show that our method is as accurate and generalized as the most widely used volume decomposition based methods moreover since the method involves neither volume decomposition nor symbolic computations it is much easier for computer implementation also the present method is more efficient than other available integration methods based on the divergence theorem efficiency of the method is also compared with the volume decomposition based methods and moment fitting methods to our knowledge this is the first article that compares both accuracy and computational efficiency of methods relying on volume decomposition and those based on the divergence theorem

an eulerian lagrangian method for fluid structure interaction based on level sets abstract we describe a method for fluid structure interaction with the interface and free surfaces defined by level sets the fluid is treated by an eulerian mesh whereas the solid or structure by a lagrangian mesh the level set method provides a compact framework for writing the weak form and for developing and computing the discretized equations lagrange multiplier methods and penalty methods are used to couple the fluid and structure results are presented for a few compressible fluid problems representative of high pressure waves

a second order projection method for the incompressible navier stokes equations in arbitrary domains we present a projection method for the numerical solution of the incompressible navier stokes equations in an arbitrary domain that is second order accurate in both space and time the original projection method was developed by chorin in which an intermediate velocity field is calculated from the momentum equations which is then projected onto the space of divergence free vector fields our method is based on the projection method developed by bell and co workers which is designed for problems in regular domains we use the continuity equation to derive a pressure equation to compute the gradient part of the vector field an integral form of the continuity equation is used to give us a natural way to define the discrete divergence operator for cells near the boundary which ensures the diagonal dominance of the resulting pressure equation we then use the restarted version of the gmres method to solve the pressure equation

a representation of curved boundaries for the solution of the navier stokes equations on a staggered three dimensional cartesian grid a method is presented for representing curved boundaries for the solution of the navier stokes equations on a non uniform staggered three dimensional cartesian grid the approach involves truncating the cartesian cells at the boundary surface to create new cells which conform to the shape of the surface we discuss in some detail the problems unique to the development of a cut cell method on a staggered grid methods for calculating the fluxes through the boundary cell faces for representing pressure forces and for calculating the wall shear stress are derived and it is verified that the new scheme retains second order accuracy in space in addition a novel cell linking method is developed which overcomes problems associated with the creation of small cells while avoiding the complexities involved with other cell merging approaches techniques are presented for generating the geometric information required for the scheme based on the representation of the boundaries as quadric surfaces the new method is tested for flow through a channel placed oblique to the grid and flow past a cylinder at re 40 and is shown to give significant improvement over a staircase boundary formulation finally it is used to calculate unsteady flow past a hemispheric protuberance on a plate at a reynolds number of 800 good agreement is obtained with experimental results for this flow