bifurcation analysis in van der pol s oscillator with delayed feedback in this paper a classical van der pol s equation with generally delayed feedback is considered it is shown that there are bogdanov takens bifurcation triple zero and hopf zero singularities by analyzing the distribution of the roots of the associated characteristic equation in the situation that the zero is as a simple eigenvalue the normal forms of the reduced equations are obtained by the center manifold theory and normal form method for functional differential equation and hence the stability of the fixed point is determined and transcritical and pitchfork bifurcations are found

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

a partitioned strongly coupled fluid structure interaction method to model heart valve dynamics a new method for the calculation of fluid structure interaction fsi of highly flexible bodies is presented this innovative algorithm demonstrates the strong coupling of a commercial computational fluid dynamics code with an in house coded structural solver the strong response of the pressure distribution to the displacement can be approximated by a reduced order model for the fluid solver the jacobian of this reduced order model is then used in the structural solver to obtain a stable and full implicit iteration scheme the method is demonstrated on a 2d model of a flexible aortic valve during the cardiac cycle furthermore the model is able to calculate shear stresses on the leaflet

anyonic realization of the quantum affine lie superalgebras we give a realization of the quantum affine lie superalgebras in terms of anyons defined on a one or two dimensional lattice the deformation parameter q being related to the statistical parameter of the anyons by the construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators as a byproduct a realization that is deformed in any sector of the quantum superalgebras is obtained

comment on the q analogues of the harmonic oscillator the heisenberg relation for the q analogues of the quantum harmonic oscillator introduced independently by macfarlane and biedenharn 1989 for the quantum group su 2 q is derived by a method analogous to that used by schwinger 1968 for the su 2 case the author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics

van der pol s oscillator under delayed feedback the effect of delayed feedback on oscillatory behaviour is investigated for the classical van der pol oscillator it is shown how the presence of delay can change the amplitude of limit cycle oscillations or suppress them altogether the result is compared to the conventional proportional and derivative feedback the derivative like effect of delay is also demonstrated in a modified equation where a delayed term provides the damping

fluid structure interaction and multi body contact application to aortic valves in this article we present a partitioned procedure for fluid structure interaction problems in which contacts among different deformable bodies can occur a typical situation is the movement of a thin valve e g the aortic valve immersed in an incompressible viscous fluid e g the blood in the proposed strategy the fluid and structure solvers are considered as independent black boxes that exchange forces and displacements the structure solvers are moreover not supposed to manage contact by themselves the hypothesis of non penetration among solid objects defines a non convex optimization problem to solve the latter we use an internal approximation algorithm that is able to directly handle the cases of thin structures and self contacts a numerical simulation on an idealized aortic valve is finally realized with the aim of illustrating the proposed scheme

a lbm dlm fd method for 3d fluid structure interactions the previously developed lbm dlm fd method derived from the lattice boltzmann method and the distributed lagrange multiplier fictitious domain method is extended to deal with 3d fluid structure interactions in our current algorithm the fluid motion is solved by lbm the deformation of the solid body is solved by the finite element method and the lagrange multiplier is solved on the low order mesh three numerical examples are employed to validate the lbm dlm fd method and reveal the potential of the method to deal with the fluid elastic body interaction problems

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

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

oscillations in a van der pol equation with delayed argument we consider the van der pol equation d2dt2u t eddtu t eu2 t r ddtu t r u t 0 with the delayed time r where r and e are real parameters with e 0 small and 0 r 2 and prove the existence and stability of a periodic solution with period near 2 and amplitude near 2 cosr with these properties this solution is unique up to translations in time

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

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

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

validation of an immersed thick boundary method for simulating fluid structure interactions of deformable membranes this paper constitutes an extension of the work of mendez et al 2014 36 for three dimensional simulations of deformable membranes under flow an immersed thick boundary method is used combining the immersed boundary method with a three dimensional modeling of the structural part the immersed boundary method is adapted to unstructured grids for the fluid resolution using the reproducing kernel particle method an unstructured finite volume flow solver for the incompressible navier stokes equations is coupled with a finite element solver for the structure the validation process relying on a number of test cases proves the efficiency of the method and its robustness is illustrated when computing the dynamics of a tri leaflet aortic valve the proposed immersed thick boundary method is able to tackle applications involving both thin and thick membranes closed and open membranes in significantly high reynolds number flows and highly complex geometries

stability and bifurcation analysis in van der pol s oscillator with delayed feedback abstract the classical van der pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered linear stability of the equations is investigated by analyzing the associated characteristic equations it is found that there exist the stability switches when delay varies and the hopf bifurcation occurs when the delay passes through a sequence of critical values the bifurcation diagram is drawn in e k plane and the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem

periodic solutions for van der pol equation with time delay for van der pol equation with distributed delay when the distributed delay kernel is the general gamma distributed delay kernel the existence and the stability of periodic solution are obtained by using the linear chain trick and geometric singular perturbation theory on the other hand for van der pol equation with discrete delay the existence and the stability are proved by employing a technique which is based on inertial manifold finally numerical simulations are also given to illustrate the results

deformation of the embedding the cartan chevalley generators of g being a maximal singular sub algebra of are written in terms of the generators of using a q boson realization then a deformation scheme for is presented starting from i e in a basis which manifestly exhibits for q 1 the content of the singular subalgebra g

hopf pitchfork bifurcation in van der pol s oscillator with nonlinear delayed feedback abstract first we identify the critical values for hopf pitchfork bifurcation second we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system furthermore we find some interesting phenomena such as the coexistence of two asymptotically stable states two stable periodic orbits and two attractive quasi periodic motions which are verified both theoretically and numerically

zero hopf bifurcation for van der pol s oscillator with delayed feedback in this paper we study the dynamical behaviors of the following van der pol oscillator with delay x e x 2 1 x x eg x t t in the case that its associated characteristic equation has a simple zero root and a pair of purely imaginary roots zero hopf singularity the normal form is obtained by performing a center manifold reduction and by using the normal form theory developed by faria and magalhaes a critical value e 0 of e in 0 2 is obtained to predict the bifurcation diagrams from which saddle node bifurcation pitchfork bifurcation hopf bifurcation the existence and stability of the periodic solutions and heteroclinic bifurcation are determined some examples are given to confirm the theoretical results

the construction of a poisson structure out of a symmetry and a conservation law of a dynamical system a method to construct hamiltonian theories for systems of both ordinary and partial differential equations is presented the knowledge of a lagrangian is not at all necessary to achieve the result the only ingredients required for the construction are one solution of the symmetry perturbation equation and one constant of motion of the original system it turns out that the poisson bracket structure for the dynamical variables is far from becoming uniquely determined by the differential equations of motion examples in classical mechanics as well as in field theory are presented

a dlm fd method for fluid flexible body interactions in this study we extended the distributed lagrange multiplier fictitious domain dlm fd formulation of glowinski et al int j multiphase flow 25 1999 755 for the fluid rigid body interactions to deal with the fluid flexible body interactions by replacing newton s equations of motion for the rigid body with the continuum equations for the general solid material similar to the rigid body case where the dlm is introduced as a pseudo body force to enforce the constraint of rigid body motion of the fictitious fluid in the solid domain the lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid for our computational scheme a first order accurate fractional step scheme is employed to decouple the entire system into three sub systems a fluid problem a solid problem and a lagrange multiplier problem the flow problem is solved with the projection method on half staggered grids the solid problem is solved with the lagrangian finite element method and the newton iterative method and the incompressibility of the material is implemented with the penalty function method the proposed method is applied to two typical fluid structure interaction problems the flow driven oscillation of a flexible plate along the flow direction and the self sustained oscillation across the flow direction both results compare favorably with previously reported numerical and experimental results and show that our method is suited to the simulation of the motion of an incompressible non linear elastic material in a fluid

distributed lagrange multiplier fictitious domain method in the framework of lattice boltzmann method for fluid structure interactions a new implementation of the lattice boltzmann method lbm for fluid structure interactions is presented the idea of the distributed lagrange multiplier fictitious domain method dlm fd is introduced in the framework of the lattice boltzmann algorithm this implementation employs a fixed mesh for the solution of the fluid problem and a lagrangian formulation for the solid problem the main advantage of the method is that the re meshing procedure normally required in the ale method is circumvented numerical examples are provided to verify the algorithm and illustrate the capacity of the method to deal with the fluid elastic solid interactions

a fluid structure interaction method with solid rigid contact for heart valve dynamics a computational method is proposed for problems where fluid structure interaction is combined with solid rigid contact this combination is particularly important for the dynamics and impact of heart valves the navier stokes equation in an eulerian setting is coupled to a neo hookean solid model using a lagrangian description a fictitious domain method extended with a local mesh adaptation algorithm provides the required flexibility with respect to the motion and deformation of the valve in addition it ensures the solids ability of sustaining pressures present in the fluid making use of the fact that the fluid and solid mesh are not required to be connected conformingly it is shown that the model can be extended with a contact algorithm without introducing meshing complications near the contact surfaces

coupled fluid structure interaction hemodynamics in a zero pressure state corrected arterial geometry abstract hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the distensible arterial wall a numerical procedure for solving the fluid structure interaction problem encountered in cardiovascular flows is presented we consider a patient specific carotid bifurcation geometry obtained from 3d reconstruction of in vivo acquired tomography images which yields a geometrical representation of the artery corresponding to its pressurized state to recover the geometry of the artery in its zero pressure state which is required for a fluid structure interaction simulation we utilize inverse finite elastostatics time dependent flow simulations with in vivo measured inflow volume flow rate in the 3d undeformed artery are performed through the finite element method the coupled momentum method for fluid structure interaction is adopted to incorporate the influence of wall compliance in the numerical computation of the time varying flow domain to demonstrate the importance in recovering the zero pressure state of the artery in hemodynamic simulations we compute the time varying flow field with compliant walls for the original and the zero pressure state corrected geometric configurations of the carotid bifurcation the most important resulting effects in the hemodynamic environment are evaluated our results show a significant change in the wall shear stress distribution and the spatiotemporal extent of the recirculation regions

a partitioned fluid structure algorithm for elastic thin valves with contact we present a numerical method to simulate the movements of a thin valve immersed in an incompressible viscous fluid the fluid and structure meshes are not matching the kinematic continuity is imposed using lagrange multipliers the method therefore belongs to the fictitious domain fd family this approach allows very large displacements we propose a partitioned fluid structure algorithm which keeps the fluid and structure solvers independent and which is able to manage contact without assuming that the structure solvers include contact capabilities various numerical tests are proposed in particular when the displacement is moderate we show that the results are in good agreement with arbitrary lagrangian eulerian ale simulations we then present results with contact on a rigid wall finally we show that the proposed approach can be mixed with ale when the wall is elastic

eulerian finite element method for the numerical modeling of fluid dynamics of natural and pathological aortic valves we present a finite element methodology tailored for the simulation of pulsatile flow in the full aorta and sinus of valsalva interacting with highly deformable thin leaflets we describe an extension of the so called resistive immersed surface method to circumvent stability issues resulting from the bad conditioning of the linear system especially when flow and geometry become complex after the inclusion of the aorta we use a lagrange multiplier technique that couples the dynamics of valve and flow a banded level set variant allows to address the singularity of the resulting linear system while featuring in addition to the parallel implementation higher accuracy and an affordable computational burden high fidelity computational geometries are built and simulations are performed under physiological conditions several numerical experiments illustrate the ability of the model to capture the basic fluidic phenomena in both healthy and pathological configurations we finally examine numerically the flow dynamics in the sinus of valsalva after transcatheter aortic valve implantation we show numerically that flow may be subject to stagnation in the lower part of the sinuses we highlight the far reaching implications of this phenomenon and we hope inciting adequate studies to further investigate its potential clinical consequences display omitted hemodynamics in full aorta and aortic sinuses coupled with highly deformable valve an exact lagrange multiplier technique couples the dynamics of valve and flow a damped newton strategy allows more stability for relatively large reynolds numbers numerical examples in 2d and 3d with healthy and pathological valves numerical investigations pinpoint a risk of blood stagnation after tavi

quantum algebras in classical mechanics it is shown that the equations of motion do not determine their hamiltonian descriptions uniquely in particular different poisson bracket relations are admissible for a given physical problem the free symmetric top and a charged particle moving on a homogeneous and constant magnetic field for instance may be described by a classical hamiltonian theory with an su 2 q poisson bracket structure

quantization of non standard hamiltonian systems the quantization of classical theories that admit more than one hamiltonian description is considered this is done from a geometrical viewpoint both at the quantization level geometric quantization and at the level of the dynamics of the quantum theory a spin system is taken as an example in which all the steps can be completed it is shown that the geometry of the quantum theory imposes restrictions on the physically allowed non standard quantum theories

the construction of the q analogues of the harmonic oscillator operators from ordinary oscillator operators the q analogues of boson operators are constructed from ordinary boson operators by an embedding approach accordingly the q deformed quantum algebras su n q can be obtained straightforwardly from corresponding universal enveloping algebras of lie algebra su n

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects this paper presents the development and validation of a new parallel unstructured multi grid preconditioned implicit method for the simulation of three dimensional 3d unsteady compressible flows using a so called immersed membrane method imm g h xia y zhao j h yeo an immersed membrane method for simulation of flui structure interaction in bio fluid flows in 1st international bioengineering conference ibec 2004 08 10 september 2004 the novel feature of the method presented is that it employs a unique combination of a parallel multi grid scheme with low mach number preconditioning and the imm so that 3d unsteady low mach number flows with arbitrarily moving objects can be efficiently simulated

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

q oscillators realization of fq 4 and gq 3 a realization of the quantum superalgebras corresponding to f 4 and g 3 is given in terms of q deformed bosonic and fermionic oscillators

bogdanov takens singularity in van der pol s oscillator with delayed feedback abstract in this paper we study the classical van der pol s equation with delayed feedback we investigate the bogdanov takens singularity and bifurcation occurring in the system with the variation of the original parameters by using the normal form method and predict the suitable choice in the feedback control in order to obtain the asymptotic stable state

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

bi hamiltonian structure in frenet serret frame abstract we reduce the problem of constructing bi hamiltonian structure in three dimensions to the solutions of a riccati equation in moving coordinates of frenet serret frame all explicitly constructed examples in the literature are exhausted by constant solutions we conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi hamiltonian

realizability of the normal form for the triple zero nilpotency in a class of delayed nonlinear oscillators abstract the effects of delayed feedback terms on nonlinear oscillators have been extensively studied and there are important applications in many areas of science and engineering we study a particular class of second order delay differential equations near a point of triple zero nilpotent bifurcation using center manifold and normal form reduction we show that the three dimensional nonlinear normal form for the triple zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay differential equation

a three dimensional computational analysis of fluid structure interaction in the aortic valve numerical analysis of the aortic valve has mainly been focused on the closing behaviour during the diastolic phase rather than the kinematic opening and closing behaviour during the systolic phase of the cardiac cycle moreover the fluid structure interaction in the aortic valve system is most frequently ignored in numerical modelling the effect of this interaction on the valve s behaviour during systolic functioning is investigated the large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood valve interaction modelling this has impeded numerical analyses of valves operating under physiological conditions a numerical method known as the lagrange multiplier based fictitious domain method is used to describe the large leaflet motion within the computational fluid domain this method is applied to a three dimensional finite element model of a stented aortic valve the model provides both the mechanical behaviour of the valve and the blood flow through it results show that during systole the leaflets of the stented valve appear to be moving with the fluid in an essentially kinematical process governed by the fluid motion

a variational framework for fluid solid interaction problems based on immersed domains theoretical bases abstract in this work the theoretical bases to handle the fluid solid interaction problem based on the conception of immersed domains are developed starting from a well known governing variational principle from the continuum mechanics the purpose of the present work is twofold firstly a variational principle to model the interaction between a fluid and a thin structure characterized as a shell for which its volume is geometrically neglected is devised by incorporating suitable kinematical assumptions into the original problem this leads to a rather general formulation for the fluid shell interaction problem that embraces methodologies such as the immersed boundary method secondly no longer is the volume of the solid disregarded and the variational principle is manipulated in order to attain a formulation that comprises the problem of an augmented fluid which is extended as a fictitious fluid to the whole domain of analysis interacting with an immersed solid in some parts the formulation resembles the immersed finite element method as well as the recently presented immersed continuous method due to the appearance of a volume force defined over the fictitious fluid domain one of the main differences lies in the fact that in the present approach the volume force is the lagrange multiplier that imposes the continuity of the velocity field along the whole overlapped domain the other noteworthy difference dwells in the handling of the mass conservation for the artificial fluid three aspects of this setting must be pointed out as original contributions of the work i a consistent generalization of the theoretical continuous setting of immersed boundary methods is carried out ii the variational basis for the finite element approximations is clearly posed written and easy to understand and iii when dealing with an immersed solid a more general formulation valid for all classes of combinations fluid solid is developed eliminating the restriction to compressible fluid compressible solid of the immersed continuous method and the incompressible fluid incompressible solid of the immersed finite element method

patient specific initial wall stress in abdominal aortic aneurysms with a backward incremental method patient specific wall stress simulations on abdominal aortic aneurysms may provide a better criterion for surgical intervention than the currently used maximum transverse diameter in these simulations it is common practice to compute the peak wall stress by applying the full systolic pressure directly on the aneurysm geometry as it appears in medical images since this approach does not account for the fact that the measured geometry is already experiencing a substantial load it may lead to an incorrect systolic aneurysm shape we have developed an approach to compute the wall stress on the true diastolic geometry at a given pressure with a backward incremental method the method has been evaluated with a neo hookean material law for several simple test problems the results show that the method can predict an unloaded configuration if the loaded geometry and the load applied are known the effect of incorporating the initial diastolic stress has been assessed by using three patient specific geometries acquired with cardiac triggered mr the comparison shows that the commonly used approach leads to an unrealistically smooth systolic geometry and therefore provides an underestimation for the peak wall stress our backward incremental modelling approach overcomes these issues and provides a more plausible estimate for the systolic aneurysm volume and a significantly different estimate for the peak wall stress when the approach is applied with a more complex material law which has been proposed specifically for abdominal aortic aneurysm similar effects are observed and the same conclusion can be drawn