partitioning strategies for the interaction of a fluid with a poroelastic material based on a nitsche s coupling approach abstract we develop a computational model to study the interaction of a fluid with a poroelastic material the coupling of stokes and biot equations represents a prototype problem for these phenomena which feature multiple facets on one hand it shares common traits with fluid structure interaction on the other hand it resembles the stokes darcy coupling for these reasons the numerical simulation of the stokes biot coupled system is a challenging task the need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem such as the finite element method for spatial discretization and finite differences for time stepping the application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others at each time step in this work we develop and thoroughly analyze a loosely coupled scheme for stokes biot equations the scheme is based on nitsche s method for enforcing interface conditions once the interface operators corresponding to the interface conditions have been defined time lagging allows us to build up a loosely coupled scheme with good stability properties the stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem the error of the resulting method is also analyzed showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes in order to restore good approximation properties while maintaining the computational efficiency of the loosely coupled approach we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

local critical perturbations of unimodal maps we introduce a new complete metric in the space mathcal v 2 of unimodal c2 maps of the interval with two maps close if they are close in the c2 metric and differ only on a small interval containing their critical points we identify all structurally stable maps in the sense of this metric they are maps for which either 1 the trajectory of the critical point is attracted to a topologically attracting at least from one side periodic orbit but never falls into this orbit or 2 the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same minimal period we verify the generalized fatou conjecture for mathcal v 2 and show that structurally stable maps form an open dense subset of mathcal v 2

getting rid of the negative schwarzian derivative condition in this paper we will show that the assumption on the negative schwarzian derivative is redundant in the case of c 3 unimodal maps with a non at critical point the following theorem will be proved for any c 3 unimodal map of an interval with a non at critical point there exists an interval around the critical value such that the flrst entry map to this interval has negative schwarzian derivative another theorem proved in the paper provides useful cross ratio estimates thus all theorems proved only for unimodal maps with negative schwarzian derivative can be easily generalized

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

incremental displacement correction schemes for incompressible fluid structure interaction in this paper we introduce a class of incremental displacement correction schemes for the explicit coupling of a thin structure with an incompressible fluid these methods enforce a specific robin neumann explicit treatment of the interface coupling we provide a general stability and convergence analysis that covers both the incremental and the non incremental variants their stability properties are independent of the added mass effect the superior accuracy of the incremental schemes with respect to the original non incremental variant is highlighted by the error estimates and then confirmed in a benchmark by numerical experiments

a strongly conservative finite element method for the coupled stokes biot model abstract in this paper based on interior penalty discontinuous galerkin method and mixed finite element method we consider a strongly conservative discretization for the rearranged stokes biot model theoretical analysis and numerical modeling are carried out from the view of mathematics we firstly present the existence and uniqueness of solution of the numerical scheme then the analysis of stability and priori error estimates are derived from the point of numerical simulation we report the numerical examples under uniform meshes which well validate the analysis of convergence and the strong mass conservation

an efficient preconditioner for monolithically coupled large displacement fluid structure interaction problems with pseudo solid mesh updates we present a block preconditioner for the efficient solution of the linear systems that arise when employing newton s method to solve monolithically coupled large displacement fluid structure interaction problems in which the update of the moving fluid mesh is performed by the equations of large displacement elasticity following a theoretical analysis of the preconditioner we propose an efficient implementation that yields a solver with near optimal computational cost in the sense that the time for the solution of the linear systems scales approximately linearly with the number of unknowns we evaluate the performance of the preconditioner in selected two and three dimensional test problems

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

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

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

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

metric attractors for smooth unimodal maps we classify the measure theoretic attractors of general c 3 unimodal maps with quadratic critical points the main ingredient is the decay of geometry

fluid structure interaction with large structural displacements the calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain this paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium to be studied in a fixed reference configuration the resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions this approach treats the structure in a fully lagrangian way and uses an associated arbitrary lagrangian eulerian ale formulation for the fluid such a strategy has three advantages which are detailed in the present paper it can be discretised in time by implicit stable energy conserving time integration schemes and solved by simple iterative uncoupled algorithms the fluid and the structural problems can be approximated by independent finite element spaces the most recent geometrically exact nonlinear shell models can be used for the structures the full strategy has been implemented into an industrial cfd code results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper

explicit robin neumann schemes for the coupling of incompressible fluids with thin walled structures abstract we introduce a class of explicit coupling schemes for the numerical solution of fluid structure interaction problems involving a viscous incompressible fluid and a general thin walled structure e g including damping and non linear behavior the fundamental ingredient in these methods is a parameter free explicit robin interface condition for the fluid which enables the fluid solid splitting through appropriate extrapolations of the solid velocity and fluid stress on the interface the resulting solution procedures are genuinely partitioned stability and error estimates are provided for all the variants depending on the extrapolations using energy arguments within a representative linear setting in particular we show that one of them simultaneously yields added mass free unconditional stability and optimal first order time accuracy a comprehensive numerical study involving different examples from the literature supports the theory

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

an asymptotically compatible formulation for local to nonlocal coupling problems without overlapping regions abstract in this paper we design and analyze an explicit partitioned procedure for a 2d dynamic local to nonlocal ltn coupling problem based on a new nonlocal robin type transmission condition the nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter characterizing the range of nonlocal interactions and the local subproblem is described by the classical heat equation we consider a heterogeneous system where the local and nonlocal subproblems present different physical properties and employ no overlapping region between the two subdomains we first propose a new generalization of classical local neumann type condition by converting the local flux to a correction term in the nonlocal model and show that the proposed neumann type boundary formulation recovers the local case as o 2 in the l norm we then extend the nonlocal neumann type boundary condition to a robin type boundary condition and develop a local to nonlocal coupling formulation with robin dirichlet transmission conditions to stabilize the explicit coupling procedure and to achieve asymptotic compatibility the choice of the coefficient in the robin condition is obtained via amplification factor analysis for the discretized system with coarse grids employing a high order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver the selection of optimal robin coefficients are verified with numerical experiments on heterogeneous and complicated domains with the developed optimal coupling strategy we numerically demonstrate the coupling framework s asymptotic convergence to the local limit with an o o h rate when there is a fixed ratio between the horizon size and the spatial discretization size h

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

explicit strategies for incompressible fluid structure interaction problems nitsche type mortaring versus robin robin coupling we discuss explicit coupling schemes for fluid structure interaction problems where the added mass effect is important in this paper we show the close relation between coupling schemes using nitsche s method and a robin robin type coupling in the latter case the method may be implemented either using boundary integrals of the stresses or the more conventional discrete lifting operators recalling the explicit method proposed in comput methods appl mech engrg 198 5 8 766 784 2009 we make the observation that this scheme is stable under a hyperbolic type cfl condition but that optimal accuracy imposes a parabolic type cfl conditions due to the splitting error two strategies to enhance the accuracy of the coupling scheme under the hyperbolic cfl condition are suggested one using extrapolation and defect correction and one using a penalty free non symmetric nitsche method finally we illustrate the performance of the proposed schemes on some numerical examples in two and three space dimensions

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

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

decay of geometry for unimodal maps negative schwarzian case we show that decay of geometry holds for unimodal maps of the interval which have negative schwarzian derivative sufficient finite smoothness and a nondegenerate critical point the proof is based on pseudo analytic extensions of order at least 2 they allow us to modify sullivan s principle that rescaled high iterates of one dimensional maps tend to analytic limits in such a way that no passage to a limit is actually needed but the maps are shown to approach the analytic class in a well defined sense as a technical improvement this method yields a uniform estimate in the case of renormalizable maps

hausdorff dimension and conformal measures of feigenbaum julia sets 1 1 statement of the results one of the first questions usually asked about a fractal subset of r is whether it has the maximal possible hausdorff dimension n it certainly happens if the set has positive lebesgue measure on the other hand it is easy to construct fractal sets of zero measure but of dimension n moreover this phenomenon is often observable for fractal sets produced by conformal dynamical systems iterated rational functions or kleinian groups in particular the analogy with kleinian groups suggested that the julia sets of feigenbaum maps should have hausdorff dimension two in this paper we will show that this is not always the case a quadratic polynomial or more generally a quadratic like map is called feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds see 2 for a precise definition feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years by analogy with kleinian groups it was anticipated that their julia sets have dimension 2 and this problem has been around for a while however in this paper we will show that this is not always the case

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

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

a stabilised immersed boundary method on hierarchical b spline grids abstract in this work an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b spline grid and employs the non symmetric and penalty free version of nitsche s method to enforce the boundary conditions the strategy allows for h and p refinement and employs a so called ghost penalty term to stabilise the cut cells an effective procedure based on hierarchical subdivision and sub cell merging which avoids excessive numbers of quadrature points is used for the integration of the cut cells a basic laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty free nitsche method as well as their impact on accuracy the methodology is also applied to the incompressible navier stokes equations where the supg pspg stabilisation is employed simulations of the lid driven cavity flow and the flow around a cylinder at low reynolds number show the good performance of the methodology excessive ill conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field

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

a lagrange multiplier method for a stokes biot fluid poroelastic structure interaction model we study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium the free fluid is governed by the stokes equations while the flow in the poroelastic medium is modeled using the biot poroelasticity system equilibrium and kinematic conditions are imposed on the interface a mixed darcy formulation is employed resulting in continuity of flux condition of essential type a lagrange multiplier method is employed to impose weakly this condition a stability and error analysis is performed for the semi discrete continuous in time and the fully discrete formulations a series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters

decay of geometry for unimodal maps an elementary proof we prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry by an elementary and purely real argument this completes a real approach to milnor s attractor problem for smooth unimodal maps with critical order not greater than 2

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

a projection semi implicit scheme for the coupling of an elastic structure with an incompressible fluid we address the numerical simulation of fluid structure systems involving an incompressible viscous fluid this issue is particularly difficult to face when the fluid added mass acting on the structure is strong as it happens in hemodynamics for example indeed several works have shown that in such situations implicit coupling seems to be necessary in order to avoid numerical instabilities although significant improvements have been achieved during the last years solving implicit coupling often exhibits a prohibitive computational cost in this work we introduce a semi implicit coupling scheme which remains stable for a reasonable range of the discretization parameters the first idea consists in treating implicitly the added mass effect whereas the other contributions geometrical non linearities viscous and convective effects are treated explicitly the second idea relies on the fact that this kind of explicit implicit splitting can be naturally performed using a chorin temam projection scheme in the fluid we prove conditional stability of the scheme for a fully discrete formulation several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches

explicit coupling schemes for a fluid fluid interaction problem arising in hemodynamics in this work we propose a new approach to the loosely coupled time marching of fluid fluid interaction problems involving the incompressible navier stokes equations the methods combine a specific explicit robin robin treatment of the interface coupling with a weakly consistent interface pressure stabilization in time a priori energy estimates guaranteeing stability of the splitting are obtained for a total pressure formulation of the coupled problem the performance of the proposed schemes is illustrated on several numerical experiments related to simulation of aortic blood flow

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

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

generalized robin neumann explicit coupling schemes for incompressible fluid structure interaction stability analysis and numerics we introduce a new class of explicit coupling schemes for the numerical solution of fluid structure interaction problems involving a viscous incompressible fluid and an elastic structure these methods generalize the arguments reported in 13 10 to the case of the coupling with thick walled structures the basic idea lies in the derivation of an intrinsic interface robin consistency at the space semi discrete level using a lumped mass approximation in the structure the fluid solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface based on these methods a new parameter free robin neumann iterative procedure is also proposed for the partitioned solution of implicit coupling a priori energy estimates guaranteeing the added mass free stability of the schemes and the convergence of the iterative procedure are established the accuracy and robustness of the methods are illustrated in several numerical examples

fully decoupled time marching schemes for incompressible fluid thin walled structure interaction in this paper we introduce a class of fully decoupled time marching schemes velocity pressure displacement splitting for the coupling of an incompressible fluid with a thin walled elastic or viscoelastic structure the time splitting combines a projection method in the fluid with a specific robin neumann treatment of the interface coupling a priori energy estimates guaranteeing unconditional stability are established for some of the schemes the accuracy and performance of the methods proposed are illustrated by a thorough numerical study

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

pairing interaction and its q deformed versions we investigate the behaviour of the pairing model within the context of quantum algebras the pairing hamiltonian is diagonalized exactly for different values of the deformation parameter q in systems with 8 and 50 particles the same hamiltonian is solved with the help of the coherent state variational method we find that the variational method gives reliable results when compared with the exact ones we also discuss the differences between the two deformation procedures