generalized fictitious methods for fluid structure interactions analysis and simulations we present a new fictitious pressure method for fluid structure interaction fsi problems in incompressible flow by generalizing the fictitious mass and damping methods we published previously in 1 the fictitious pressure method involves modification of the fluid solver whereas the fictitious mass and damping methods modify the structure solver we analyze all fictitious methods for simplified problems and obtain explicit expressions for the optimal reduction factor convergence rate index at the fsi interface 2 this analysis also demonstrates an apparent similarity of fictitious methods to the fsi approach based on robin boundary conditions which have been found to be very effective in fsi problems we implement all methods including the semi implicit robin based coupling method in the context of spectral element discretization which is more sensitive to temporal instabilities than low order methods however the methods we present here are simple and general and hence applicable to fsi based on any other spatial discretization in numerical tests we verify the selection of optimal values for the fictitious parameters for simplified problems and for vortex induced vibrations viv even at zero mass ratio for ever resonance we also develop an empirical a posteriori analysis for complex geometries and apply it to 3d patient specific flexible brain arteries with aneurysms for very large deformations we demonstrate that the fictitious pressure method enhances stability and convergence and is comparable or better in most cases to the robin approach or the other fictitious methods

conservative derivations and dissipative laplacians let 1 d be a finite sequence of derivations on a c algebra u such that the laplacian k 1d k2 is densely defined we prove that if is dissipative then 1 d are conservative on the domain of this has consequences for integrability of lie algebras of derivations the result does not extend to general banach algebras

a semi local spectral hp element solver for linear elasticity problems summary we develop an efficient semi local method for speeding up the solution of linear systems arising in spectral hp element discretization of the linear elasticity equations the main idea is to approximate the element wise residual distribution with a localization operator we introduce in this paper and subsequently solve the local linear system additionally we decouple the three directions of displacement in the localization operator hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver this approach is effective for both nodal and modal bases in the spectral hp element method but here we focus on the modal hierarchical basis in numerical tests we verify that there is no loss of accuracy in the semi local method and we obtain good parallel scalability and substantial speed up compared to the original formulation in particular our tests include both structure only and fluid structure interaction problems with the latter modeling a 3d patient specific brain aneurysm copyright 2014 john wiley sons ltd

curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies the sharp interface curvib approach of ge and sotiropoulos l ge f sotiropoulos a numerical method for solving the 3d unsteady incompressible navier stokes equations in curvilinear domains with complex immersed boundaries journal of computational physics 225 2007 1782 1809 is extended to simulate fluid structure interaction fsi problems involving complex 3d rigid bodies undergoing large structural displacements the fsi solver adopts the partitioned fsi solution approach and both loose and strong coupling strategies are implemented the interfaces between immersed bodies and the fluid are discretized with a lagrangian grid and tracked with an explicit front tracking approach an efficient ray tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies numerical experiments are carried out for two fsi problems vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions for both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements the numerical experiments suggest that both the properties of the structure mass geometry and the local flow conditions can play an important role in determining the stability of the fsi algorithm under certain conditions the fsi algorithm is unconditionally unstable even when strong coupling fsi is employed for such cases however combining the strong coupling iteration with under relaxation in conjunction with the aitken s acceleration technique is shown to effectively resolve the stability problems a theoretical analysis is presented to explain the findings of the numerical experiments it is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the fsi algorithm the stabilizing role of under relaxation is also clarified and the upper bound of the under relaxation coefficient required for stability is derived

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

stabilization of explicit coupling in fluid structure interaction involving fluid incompressibility abstract in this work we propose a stabilized explicit coupling scheme for fluid structure interaction problems involving a viscous incompressible fluid the coupled discrete formulation is based on nitsche s method with a time penalty term giving l 2 control on the fluid pressure variations at the interface for a linear model problem we prove that the scheme is stable in the energy norm irrespectively of the so called added mass effect namely the fluid structure density ratio and the geometry of the domain numerical experiments in the linear and non linear case show that optimal time accuracy can be obtained by performing one defect correction iteration

an analysis of a new stable partitioned algorithm for fsi problems part ii incompressible flow and structural shells stable partitioned algorithms for fluid structure interaction fsi problems are developed and analyzed in this two part paper part i describes an algorithm for incompressible flow coupled with compressible elastic solids while part ii discusses an algorithm for incompressible flow coupled with structural shells the numerical approach described here for structural shells uses robin mixed interface conditions for the pressure and velocity in the fluid which are derived directly from the governing equations the resulting added mass partitioned amp algorithm is stable even for very light structures requires no sub iterations per time step and is second order accurate the stability and accuracy of the amp algorithm is evaluated for linearized fsi model problems a normal mode analysis is performed to show that the new amp algorithm is stable even for the case of very light structures when added mass effects are large exact traveling wave solutions are derived for the fsi model problems and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the amp algorithm for the cases of light medium and heavy structures a summary comparison of the amp algorithm developed here and the one in part i is provided

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

covariance and fisher information in quantum mechanics variance and fisher information are ingredients of the cramer rao inequality we regard fisher information as a riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and fisher information in this approach we show that there is a kind of dual one to one correspondence between the candidates of the two concepts we emphasize that fisher information is obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

inexact accurate partitioned algorithms for fluid structure interaction problems with finite elasticity in haemodynamics in this paper we consider the numerical solution of the three dimensional fluid structure interaction problem in haemodynamics in the case of real geometries physiological data and finite elasticity vessel deformations we study some new inexact schemes obtained from semi implicit approximations which treat exactly the physical interface conditions while performing just one or few iterations for the management of the interface position and of the fluid and structure non linearities we show that such schemes allow to improve the efficiency while preserving the accuracy of the related exact implicit scheme to do this we consider both a simple analytical test case and two real cases of clinical interest in haemodynamics we also provide an error analysis for a simple differential model problem when a bdf method is considered for the time discretization and only few newton iterations are performed at each temporal instant

a convergence study of a new partitioned fluid structure interaction algorithm based on fictitious mass and damping we develop analyze and validate a new method for simulating fluid structure interactions fsis which is based on fictitious mass and fictitious damping in the structure equation we employ a partitioned method for the fluid and structure motions in conjunction with sub iteration and aitken relaxation in particular the use of such fictitious parameters requires sub iterations in order to reduce the induced error in addition to the local temporal truncation error to this end proper levels of tolerance for terminating the sub iteration procedure have been obtained in order to recover the formal order of temporal accuracy for the coupled fsi problem these fictitious terms have a significant effect leading to better convergence rate and hence substantially smaller number of sub iterations through analysis we identify the proper range of these parameters which we then verify by corresponding numerical tests we implement the method in the context of spectral element discretization which is more sensitive than low order methods to numerical instabilities arising in the explicit fsi coupling however the method we present here is simple and general and hence applicable to fsi based on any other discretization we demonstrate the effectiveness of the method in applications involving 2d vortex induced vibrations viv and in 3d flexible arteries with structural density close to blood density we also present 3d results for a patient specific aneurysmal flow under pulsatile flow conditions examining in particular the sensitivity of the results on different values of the fictitious parameters

fluid structure partitioned procedures based on robin transmission conditions in this article we design new partitioned procedures for fluid structure interaction problems based on robin type transmission conditions the choice of the coefficient in the robin conditions is justified via simplified models the strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane as in hemodynamics applications we analyze theoretically the new iterative procedures on a model problem which represents a simplified blood vessel system in particular the robin neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures the theoretical results are checked using numerical experimentation

analysis of the block gauss seidel solution procedure for a strongly coupled model problem with reference to fluid structure interaction the block gauss seidel procedure is widely used for the resolution of the strong coupling in the computer simulation of fluid structure interaction based on a simple model problem this work presents a detailed analysis of the convergence behaviour of the method in particular the model problem is used to highlight some aspects that arise in the context of the application of the block gauss seidel method to fsi problems thus the effects of the time integration schemes chosen of relaxation techniques of physical constraints and non linearities on the convergence of the iterations are investigated

mixed spectral hp element formulation for nonlinear elasticity we present a mixed formulation for high order spectral hp element methods and investigate its stability in large strain analysis of nearly incompressible solids specifically we employ the displacement pressure formulation and use a pressure projection method along with general jacobi polynomial bases to derive the discrete equations we first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements subsequently we investigate numerically the validity of ellipticity and inf sup conditions for various test problems including cases with large deformation we find that the inf sup condition is satisfied even when the pressure interpolation order op is equal to the displacement order ou but to resolve the locking free behavior it is required that op ou 1 and this choice also gives the best convergence rate in large deformation analysis surprisingly the results are more stable for ou op ou k because the ellipticity in kernel is better satisfied with higher k as ou increases compared to the displacement only formulation the mixed formulation seems to have superior accuracy and it is overall more efficient especially for problems with high bulk modulus

computer modeling of cardiovascular fluid structure interactions with the deforming spatial domain stabilized space time formulation hemodynamic factors such as the wall shear stress are believed to affect a number of cardiovascular diseases including atherosclerosis and aneurysm since resolving phenomena in a living human body is currently beyond the capabilities of in vivo measurement techniques computer modeling is expected to play an important role in gaining a better understanding of the relationship between the cardiovascular diseases and the hemodynamic factors we have developed a computer modeling technique for cardiovascular hemodynamic simulations with this modeling technique patient specific 3d geometry of an artery can be analyzed we take into account some of the important factors in human body for the purpose of demonstrating in vivo situations in a virtual world the interaction between the blood flow and the deformation of the arterial walls is a factor that we are specifically focusing on for such fluid structure interactions we have developed a computer modeling tool based on the deforming spatial domain stabilized space time dsd sst formulation this simulation tool is applied to a patient specific model under pulsatile blood flow conditions the simulations show that the flow behavior with compliant arterial walls is different from what we see with rigid arterial walls consequently the distribution of the wall shear stress on the compliant arterial walls is significantly different from that on the rigid arterial walls we deduce that the compliance of the arterial walls needs to be taken into account in cardiovascular hemodynamic simulations and the computer modeling tool we have developed can be effective in investigation of cardiovascular diseases

analysis and optimization of robin robin partitioned procedures in fluid structure interaction problems in the solution of fluid structure interaction fsi problems partitioned procedures are modular algorithms that involve separate fluid and structure solvers that interact in an iterative framework through the exchange of suitable transmission conditions at the fs interface in this work we study using fourier analysis the convergence of partitioned algorithms based on robin transmission conditions we derive for different models of the fluid and the structure a frequency dependent reduction factor at each iteration of the partitioned algorithm which is minimized by choosing optimal values of the coefficients in the robin transmission conditions two dimensional numerical results are also reported which highlight the effectiveness of the optimization procedure

sub iteration leads to accuracy and stability enhancements of semi implicit schemes for the navier stokes equations we present an iterative semi implicit scheme for the incompressible navier stokes equations which is stable at cfl numbers well above the nominal limit we have implemented this scheme in conjunction with spectral discretizations which suffer from serious time step limitations at very high resolution however the approach we present is general and can be adopted with finite element and finite difference discretizations as well specifically at each time level the nonlinear convective term and the pressure boundary condition both of which are treated explicitly in time are updated using fixed point iteration and aitken relaxation eigenvalue analysis shows that this scheme is unconditionally stable for stokes flows while numerical results suggest that the same is true for steady navier stokes flows as well this finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters in unsteady flows second and third order temporal accuracy is obtained for the velocity field at cfl number 5 14 using analytical solutions systematic accuracy stability and cost comparisons are presented against the standard semi implicit method and a recently proposed fully implicit scheme that does not require newton s iterations in addition to its enhanced accuracy and stability the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully implicit schemes

fluid structure interaction problems with strong added mass effect in this paper the so called added mass effect is investigated from a different point of view of previous publications the monolithic fluid structure problem is partitioned using a static condensation of the velocity terms following this procedure the classical stabilized projection method for incompressible fluid flows is introduced the procedure allows obtaining a new pressure segregated scheme for fluid structure interaction problems which has good convergent characteristics even for biomechanical application where the added mass effect is strong the procedure reveals its power when it is shown that the same projection technique must be implemented in staggered fsi methods copyright 2009 john wiley sons ltd

artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows this note investigates the so called arti cial added mass e ect which is responsible for devastating instabilities within sequentially staggered fluid structure interaction fsi simulations where incompressible fluids are considered a discrete representation of the added mass operatorma is given and instability conditions are evaluated for different temporal discretisation schemes it is proven that for every sequentially staggered scheme and given spatial discretisation of a problem a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable the analysis is quite general and does not depend upon the particular spatial discretisation schemes used however here special attention is put on stabilised finite elements employed on the fluid partition numerical investigations further highlight the results

truly monolithic algebraic multigrid for fluid structure interaction the coupling of flexible structures to incompressible fluids draws a lot attention during the last decade many different solution schemes have been proposed in this contribution we concentrate on strong coupling fluid structure interaction by means of monolithic solution schemes therein a newton krylov method is applied to the monolithic set of nonlinear equations such schemes require good preconditioning to be efficient we propose two preconditioners that apply algebraic multigrid techniques to the entire fluid structure interaction system of equations the first is based on a standard block gauss seidel approach where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem the second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields the resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance

robin based semi implicit coupling in fluid structure interaction stability analysis and numerics in this paper we propose a semi implicit coupling scheme for the numerical simulation of fluid structure interaction systems involving a viscous incompressible fluid the scheme is stable irrespective of the so called added mass effect and allows for conservative time stepping within the structure the efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical convective nonlinearities through the use of the chorin temam projection scheme within the fluid stability comes from the implicit pressure solid coupling and a specific robin treatment of the explicit viscous solid coupling derived from nitsche s method

stable loosely coupled type algorithm for fluid structure interaction in blood flow we introduce a novel loosely coupled type algorithm for fluid structure interaction between blood flow and thin vascular walls this algorithm successfully deals with the difficulties associated with the added mass effect which is known to be the cause of numerical instabilities in fluid structure interaction problems involving fluid and structure of comparable densities our algorithm is based on a time discretization via operator splitting which is applied in a novel way to separate the fluid sub problem from the structure elastodynamics sub problem in contrast with traditional loosely coupled schemes no iterations are necessary between the fluid and structure sub problems this is due to the fact that our novel splitting strategy uses the added mass effect to stabilize rather than to destabilize the numerical algorithm this stabilizing effect is obtained by employing the kinematic lateral boundary condition to establish a tight link between the velocities of the fluid and of the structure in each sub problem the stability of the scheme is discussed on a simplified benchmark problem and we use energy arguments to show that the proposed scheme is unconditionally stable due to the crucial role played by the kinematic lateral boundary condition the proposed algorithm is named the kinematically coupled scheme

short note on the similarity between dirichlet neumann with interface artificial compressibility and robin neumann schemes for the solution of fluid structure interaction problems in this note the similarity between two implicit partitioned solution techniques for fluidstructure interaction fsi problems is analyzed using finite volume discretization of the flow equations fluid structure interaction refers to the mutual interaction between a fluid flow and a flexible structure partitioned solution techniques solve the flow equations and the structural equations separately these techniques are classified as implicit or strongly coupled if they satisfy the interaction conditions on the fluid structure interface in each time step and as explicit or loosely coupled if they do not both techniques analyzed in this note use block gauss seidel gs iterations meaning that the flow equations and the structural equations are solved consecutively within a time step until some convergence tolerance is reached as the flow and structural equations are solved separately the interaction conditions on the fluid structure interface have to be converted into boundary conditions on the common boundary of the fluid and structure subdomains several types of boundary conditions can be applied resulting in different decompositions in the case of dirichlet neumann dn decomposition the flow equations are solved with a dirichlet boundary condition given velocity on the fluid structure interface while the structural equations are solved with a neumann boundary condition given stress on the interface conversely robinneumann rn decomposition introduced in 1 denotes a robin boundary condition on the fluid side of the interface and a neumann boundary condition on the structure side the first technique in this comparison is block gauss seidel iterations applied to the monolithic system previously multiplied by a suitable permutation matrix leading to a robin neumann decomposition gs rn this first technique includes a simplified version of the structural model in the flow equations by means of a robin boundary condition to accelerate the convergence of the gs iterations 1 2 the second technique is block gauss seidel iterations with dirichletneumann decomposition and interface artificial compressibility gs dn iac this second

robin robin preconditioned krylov methods for fluid structure interaction problems in this work we propose a robin robin preconditioner combined with krylov iterations for the solution of the interface system arising in fluid structure interaction fsi problems it can be seen as a partitioned fsi procedure and in this respect it generalizes the ideas introduced in s badia f nobile c vergara j comput phys 227 2008 7027 7051 we analyze the convergence of gmres iterations with the robin robin preconditioner on a model problem and compare its efficiency with some existing algorithms the method is shown to be very efficient for many challenging fluid structure interaction problems such as those characterized by a large added mass effect or by enclosed fluids in particular the possibility to solve balloon type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches c 2009 elsevier b v all rights reserved

an analysis of a new stable partitioned algorithm for fsi problems part i incompressible flow and elastic solids stable partitioned algorithms for fluid structure interaction fsi problems are developed and analyzed in this two part paper part i describes an algorithm for incompressible flow coupled with compressible elastic solids while part ii discusses an algorithm for incompressible flow coupled with structural shells importantly these new added mass partitioned amp schemes are stable and retain full accuracy with no sub iterations per time step even in the presence of strong added mass effects e g for light solids the numerical approach described here for bulk compressible solids extends the scheme of banks et al 1 2 for inviscid compressible flow and uses robin mixed boundary conditions with the fluid and solid solvers at the interface the basic amp robin conditions involving a linear combination of velocity and stress are determined from the outgoing solid characteristic relation normal to the fluid solid interface combined with the matching conditions on the velocity and traction two alternative forms of the amp conditions are then derived depending on whether the fluid equations are advanced with a fractional step method or not the stability and accuracy of the amp algorithm is evaluated for linearized fsi model problems the full nonlinear case being left for future consideration a normal mode analysis is performed to show that the new amp algorithm is stable for any ratio of the solid and fluid densities including the case of very light solids when added mass effects are large in contrast it is shown that a traditional partitioned algorithm involving a dirichlet neumann coupling for the same fsi problem is formally unconditionally unstable for any ratio of densities exact traveling wave solutions are derived for the fsi model problems and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the amp algorithm for the cases of light medium and heavy solids