multi solution theory of signal processing based on finite element method any function can be approached by finite element to any accuracy because space of finite element of rough division is included in the space of fine division one can get multi solution theory by finite element method the multi solution theory bases on the usually finite element theory will result in a lot oft calculation representation theory of group is used to translate the bases of finite element to orthogonal bases and by this method a multi solution theory is developed bases on finite element method the theory can be applied to two dimension signal compact

a fundamental mean square convergence theorem for sdes with locally lipschitz coefficients and its applications a version of the fundamental mean square convergence theorem is proved for stochastic differential equations sdes in which coefficients are allowed to grow polynomially at infinity and which satisfy a one sided lipschitz condition the theorem is illustrated on a number of particular numerical methods including a special balanced scheme and fully implicit methods the proposed special balanced scheme is explicit and its mean square order of convergence is 1 2 some numerical tests are presented

computational methods for inverse finite elastostatics abstract in the inverse motion problem in finite hyper elasticity the classical formulation relies on conservation laws based on eshelby s energy momentum tensor this formulation is shown to be lacking in several regards for a particular class of inverse motion problems where the deformed configuration and cauchy traction are given and the undeformed configuration must be calculated it is shown that for finite element calculations a simple re examination of the equilibrium equations provides a more suitable finite element formulation this formulation is also shown to involve only minor changes to existing elements designed for forward motion calculations examples illustrating the method in simple and complex situations involving a neo hookean material are presented

a return mapping algorithm for plastic damage models 3 d and plane stress formulation three dimensional and plane stress formulations of the return mapping algorithm for a class of plastic damage models are derived using the spectral decomposition form of the stress an efficient plane stress computation scheme based on the spectral return mapping algorithm is developed the consistent algorithmic tangent stiffness for the present algorithm is formulated the validation and performance of the present return mapping algorithm is demonstrated by numerical examples copyright 2001 john wiley sons ltd

bifurcation with a two dimensional kernel abstract we present a theorem ensuring the existence of local solution branches for one parameter bifurcation problems in which the linearization at the trivial solution possesses a two dimensional kernel in particular we provide a straightforward test that is sufficient for the existence of local solution continua we demonstrate our abstract theorem with several concrete examples for second order systems of elliptic partial differential equations with symmetry

a group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell abstract the accurate prediction of the buckling load of thin shell structures is an important yet elusive goal it is particularly important in the aerospace industry where thin shell members are commonly used as structural elements due to a lack of adequate analytical results current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis see d bushnell computerized buckling analysis of shells martinus nijhoff dordrecht 1985 this paper focuses on recent results of the group theoretic approach to a numerical global postbuckling analysis of a perfect axially compressed cylindrical shell with built in end conditions the built in end conditions obviate the existence of a trivial membrane solution branch the example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells the primary axisymmetric solution branch is riddled with closely spaced symmetry breaking bifurcation points in a numerical arc length continuation scheme the close proximity of the bifurcation points on the primary path manifests itself in severe ill conditioning of the tangent stiffness matrix group theory helps one systematically find an optimal set of basis vectors or symmetry modes which reflect the symmetry of a given solution path the immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block diagonalizes and the numerical ill conditioning is avoided thus an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry breaking bifurcations is made relatively simple understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification

on random imperfections for structures of regular polygonal symmetry a theoretical study is made on the effect of random initial imperfections on the load bearing capacities of structures of regular polygonal symmetry the group theoretic method is employed to implement the symmetry in the formulation in view of the concept of critical initial imperfection the explicit form of probability density function of the load bearing capacity of structures is derived for random initial imperfections in particular tight bounds on the range of load bearing capacity are presented for various types of simple and double critical points the theoretical and empirical probability distribution functions are compared for simple truss structures to show the validity and effectiveness of the present method

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

group theoretic exploitations of symmetry in computational solid and structural mechanics the use of group theory in simplifying the study of problems involving symmetry is a well established approach in various branches of physics and chemistry and major applications in these areas date back more than 70 years within the engineering disciplines the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group theoretic methods over the past 40 years this paper reviews the advances made in the application of group theory in areas such as bifurcation analysis vibration analysis and finite element analysis and summarizes the various implementation procedures currently available illustrative examples of typical solution procedures are drawn from recent work of the author it is shown how the group theoretic approach through the characteristic vector space decomposition enables considerable simplifications and reductions in computational effort to be achieved in many cases group theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained before any actual calculations are carried out copyright 2009 john wiley sons ltd

multiple equilibria for unlinked and weakly linked cellular structural forms abstract it is demonstrated that a structural form comprising n unlinked or weakly linked separate identical cells can by its nature suffer an explosion of unstable post buckling states associated with an n fold compound critical point examples of rigid link models and atomic matrix cellular models employing a lennard jones potential are included attention is paid to criteria for the appearance of homogeneous localized but distributed and thoroughly localized solutions for the atomic models generalized coordinates that exploit local and global symmetries of unlinked cells are introduced in a block diagonalization context to clarify the bifurcation structure a discretized lagrangian formulation is introduced to untangle the paths for weakly linked cells

a return mapping algorithm for a class of cyclic plasticity models a return mapping algorithm for a class of cyclic plasticity models is presented the constitutive model includes a quadratic yield criterion and multi component formulations of non linear isotropic and kinematic hardening thus both initially isotropic and anisotropic materials can be described and a proper description of the hysteresis loops and the cyclic hardening response of materials under cyclic loading is obtained the return mapping algorithm employs the closest point projection scheme in combination with a decomposition method to obtain an efficient and robust integration of the constitutive model the consistent tangent operator is derived in closed form and is found to be unsymmetric due to the non linear evolution of the kinematic hardening terms the accuracy and robustness of the algorithm are assessed through numerical examples

mixed formulation and locking in path following nonlinear analysis abstract the arc length riks strategy has rapidly become a standard tool for path following analysis of nonlinear structures due to its theoretical ability to surpass limit points the aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle locking effect intrinsic to the nonlinear nature of the problem as a consequence its sanitization has to be pursued within a reformulation of the structural model the use of a mixed stress displacement variant of the algorithm in particular appears very promising in this respect the topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in 39 it is shown that with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm

an orthogonal residual procedure for non linear finite element equations a general and robust solution procedure for non linear finite element equations with limit points is developed at each equilibrium iteration the magnitude of the load is adjusted such that the residual force is orthogonal to the current displacement increment from the last equilibrium state the method implements the physical condition that the orthogonal residual force will neither increase nor decrease the magnitude of the current displacement increment vector the orthogonality condition is formulated directly in terms of conjugate variables and therefore does not contain any scaling parameters passage of load and displacement limit points is discussed as well as the relation to line search minimum residual and arc length methods the method is illustrated by two examples

symmetric failures in symmetric control systems abstract this paper discusses the fault tolerance of symmetric systems with respect to controllability which is a fundamental characteristic of control systems in particular we reveal the underlying mathematical mechanism of the loss of controllability for symmetric systems induced by failures based on the decomposition of the symmetric systems into subsystems under the symmetry the controllability of the entire system can be discussed by checking that of each subsystem the analysis of the fault tolerance in this paper is an extension of this idea with the aid of the chain adapted transformation matrix for the decomposition the result is shown as a necessary condition for symmetric systems to retain the controllability despite some symmetric failures we also discuss sufficient conditions

statistics of normally distributed initial imperfections abstract this paper offers a theoretical study on the probabilistic nature of critical loads buckling loads of structures subject to normally distributed initial imperfections explicit form of probability density function of critical loads are derived for various types of critical points double bifurcation points of structures with regular polygonal symmetry are dealt with by means of the group theoretic bifurcation theory the distribution of minimum values of the critical loads is investigated to present a statistical design index the theoretical and empirical probability density functions for simple structures are compared to show the validity and effectiveness of this method the method is quite efficient when it is directly applicable otherwise the explicit forms at least can greatly supplement the inefficiency of the conventional random method

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

post buckling analysis of shallow shells by the field boundary element method the non linear field boundary element technique is applied to the analysis of snap through phenomena in thin shallow shells the equilibrium path is traced by using the arc length method and the solution strategy is discussed in detail the results show that as compared to the approaches based on the popular symmetric variational galerkin finite element formulation the current approach based on an unsymmetric variational petrov galerkin field boundary element formulation gives a faster convergence while using fewer degrees of freedom the illustrative numerical examples deal with post buckling responses of several shallow shells with different geometries

structural stability of multi folding structures with contact problem this paper presents the theoretical basis for both static and dynamic numerical approaches to the elastic stability of a folding multi layered truss both analyses are based on bifurcation theory and include geometrical nonlinearity the dynamic analysis includes an allowance for contact between nodes comparisons are made between published experimental folding patterns and the patterns obtained from both numerical methods in which bifurcations are demonstrated as elastic unstable snap through behaviour several folding patterns are identified during the elastic instability where the folding behaviour of the truss is shown to be a function of the initial geometry and velocity of the dynamic loading the authors suggest that the understanding of the behaviour will be very useful for the development of light weight structures subject to dynamic loading based on the bifurcation static analysis and dynamic analysis using both the displacement control method and the load control method

implementation of cyclic plasticity models based on a general form of kinematic hardening this paper deals with implementation of cyclic plastic constitutive models in which a general form of strain hardening and dynamic recovery is employed to represent the multilinear as well as non linear evolution of back stress first in order to incorporate such a general form of kinematic hardening in finite element methods successive substitution and its convergence are discussed for implicitly integrating stress moreover a new expression of consistent tangent modulus is derived by introducing a set of fourth rank constitutive parameters into discretized kinematic hardening then the constitutive parameters introduced are specified in three cases of the general form of kinematic hardening the three cases have distinct capabilities of simulating ratcheting and cyclic stress relaxation numerical examples are given to verify the convergence in successive substitution and the new expression of consistent tangent stiffness error maps for implicitly integrating stress under non proportional as well as proportional loading are also given to show that the multilinear case of the general form provides high accuracy even if strain increment is very large copyright 2002 john wiley sons ltd

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

structural localization phenomena and the dynamical phase space analogy the localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints a from a modal perspective b from a formulation which allows the amplitude to modulate in an asymptotically defined slow space and c from a dynamical analogy in phase space suggested by the form of the underlying differential equation a simple strut on an asymmetric nonlinear foundation provides a typical illustrative example the three approaches emphasize different features of the localization phenomenon the modal view illustrates the distinctive effects of boundary conditions the modulated approach generates a convenient second order differential equation in the amplitude function and the dynamical phase space analogy suggests a useful interpretation of localization as a homoclinic connection comparisons are also made with nonlinear numerical solutions as the strut length approaches infinity it is shown that the fully localized solution represents the unstable post buckled state with the lowest energy allowing evaluation of the minimum energy barrier relevant to dynamical impact studies attention is drawn to the possibility of spatial chaos becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

finite element modelling of spatially chaotic structures a simple non linear mechanical system comprising a pin jointed string of finite length links supported by elastic springs at the pins and compressed by an axial load is viewed from two perspectives when seen as an initial value problem equilibrium equations provide an iterative non linear mapping when seen as a boundary value problem it becomes a simple finite element model at loads less than the critical buckling load a preferred buckling configuration is found that is localized along the length in the limit of infinite length this is described as a homoclinic connection in phase space joining the flat equilibrium state to itself the infinite sequence of homoclinic points thus defined embeds within the complex topological structure of a homoclinic tangle within which also appear periodic quasi periodic and chaotic spatial solutions implications in the finite element setting are discussed 1997 john wiley sons ltd

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

on nonisothermal elastoplastic analysis of shell components employing realistic hardening responses abstract in the present paper efficient numerical algorithms for elastoplastic analysis of shell like structural components will be proposed employing nonisothermal realistic highly nonlinear hardening responses the closest point projection integration algorithm is presented using a reissner mindlin type kinematic shell model completely formulated in tensor notation further a consistent elastoplastic tangent modulus is derived which ensures high convergence rates in the global iteration approach the integration algorithm has been implemented into a layered assumed strain isoparametric finite element which also enables geometrical nonlinearities including finite rotations the nonisothermal elastoplastic response of a circular cylindrical shell and a box column under axial compression is analysed under the assumption of an adiabatic process the increase in temperature is computed during elastoplastic deformation robustness and numerical stability of the proposed algorithms are demonstrated

bifurcation hierarchy of a rectangular plate abstract the mechanism of the complex recursive bifurcation behavior of a four sides simply supported rectangular plate is investigated such complex behavior is due to the hidden symmetry of the plate associated with the periodic nature of the solutions the group theoretic bifurcation theory is employed to arrive at a lattice of subgroups which expresses the rule for the behavior this rule is shown to suffer degeneration due to the restriction by the boundaries compared with that of geometrical symmetry the governing equation of this plate is discretized by means of galerkin s method with the use of the double fourier series as shape functions the tangential stiffness matrix of the plate is shown to be block diagonalized by appropriately permuting the order of the fourier series following the rule presented the bifurcation analysis of the plate is carried out to assess the validity of the rule and to demonstrate the merit of the block diagonalization as a result of this the vital role of the bifurcation rule in the proper understanding and successful analysis of the complex bifurcation behavior has been demonstrated

finite symmetric systems and their analysis abstract symmetry of linear mechanical systems permits one to substantially reduce the effort in their analysis classification and structural analysis of the finite linear symmetric systems models are studied in this paper there are two variants of the symmetry approach mechanical and algebraic in accordance with the former a symmetric system is replaced by one or several small nonsymmetric subsystems which are subjected to special loads obtained from the initial set the total response of the original symmetric system is found by special superposition of partial responses of these subsystems the algebraic approach is based on the explicit block diagonal decomposition of the matrix equation corresponding to a symmetric system while both approaches have the same efficiency the latter is easier to implement and describe it is presented here the condtions under which the symmetry technique may be utilized do not include symmetry of the applied loads specifically symmetry of those loads which form the right side of the associated equations nevertheless if the loads are symmetric the efficiency of the symmetry approach substantially increases group theory which is widely used in this paper is the mathematical tool for the study of symmetry and all necessary notions are introduced

solution method for nonlinear problems with multiple critical points a generalized displacement control method has been proposed for geometrically nonlinear analysis and has been proven to be bery effective by several problems with multiple limit points and snap back points this method can be implemented easily in a general purpose finite element program

a semi analytical approach to the non linear dynamic response problem of beams at large vibration amplitudes part ii multimode approach to the steady state forced periodic response abstract the semi analytical approach to the non linear dynamic response of beams based on multimode analysis has been presented in part i of this series of papers azrar et al 1999 journal of sound and vibration 224 183 207 1 the mathematical formulation of the problem and single mode analysis have been studied the objective of this paper is to take advantage of applying this semi analytical approach to the large amplitude forced vibrations of beams various types of excitation forces such as harmonic distributed and concentrated loads are considered the governing equation of motion is obtained and can be considered as a multi dimensional form of the duffing equation using the harmonic balance method the equation of motion is converted into non linear algebraic form techniques of solution based on iterative incremental procedures are presented the non linear frequency and the non linear modes are determined at large amplitudes of vibration the basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated the percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution also the percentage contributions of the higher modes to the bending moment near to the clamps are given in order to determine accurately the error introduced in the non linear bending stress estimated by different approximations solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency the non linear deflection shapes in various regions of the solution the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes the numerical results obtained here for the non linear forced response are compared with those from the linear theory with available non linear results based on various approaches and with the single mode analysis

an arc length control procedure to solve parabolic problems abstract an arc length method is applied as a time step selector for a class of stiff parabolic problems related to heat conduction and radiation the resulting system of nonlinear equations is solved by iteration not only on the temperature vector but also on the current time step the procedure is linked with an a posteriori estimate of the time integration error used as input to the arc length selection algorithm in addition a quasi newton method us used to obtain approximate factorizations of the jacobian matrix during the iterative process

loss of regularity for kolmogorov equations the celebrated hormander condition is a sufficient and nearly necessary condition for a second order linear kolmogorov partial differential equation pde with smooth coefficients to be hypoelliptic as a consequence the solutions of kolmogorov pdes are smooth at all positive times if the coefficients of the pde are smooth and satisfy hormander s condition even if the initial function is only continuous but not differentiable first order linear kolmogorov pdes with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth in this article we consider the intermediate regime of nonhypoelliptic second order kolmogorov pdes with smooth coefficients the main observation of this article is that there exist counterexamples to regularity preservation in that case more precisely we give an example of a second order linear kolmogorov pde with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the pde is not even locally holder continuous from the perspective of probability theory the existence of this example pde has the consequence that there exists a stochastic differential equation sde with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally holder continuous in other words degenerate noise can have a roughening effect a further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the sde more precisely we prove for an example sde with globally bounded and smooth coefficients that the standard euler approximations converge to the exact solution of the sde in the strong and numerically weak sense but at a rate that is slower then any power law

block diagonalization analysis of symmetric plates abstract this paper presents a block diagonalization method to solve stiffness equations of isotropic symmetric plates by means of a suitable local coordinate transformation chosen based on group theory the stiffness matrix is decomposed into a block diagonal form the stiffness equation in the local coordinate is solved block by block thus realizing numerical efficiency and greatly reducing the requisite amount of computer memory the efficiency has been further upgraded with the aid of the concept of augmented orbit this method is applied to a square isotropic plate subject to asymmetric loads to show its usefulness

improvement of the scaled corrector method for bifurcation analysis using symmetry exploiting block diagonalization in the nonlinear bifurcation analysis for large scaled structures the standard eigenanalysis of the tangent stiffness matrix yields important information but at the same time demands a large amount of computational cost the scaled corrector method was developed as a numerically efficient eigenanalysis free bifurcation analysis strategy which exploits byproducts of the numerical iteration for path tracing this method however has a problem in its accuracy especially when eigenvalues are nearly or exactly coincidental as a remedy for this we propose a new bifurcation analysis method through the implementation of bifurcation mechanism of a symmetric structure into the scaled corrector method the bifurcation mode is accurately approximated by decomposing a scaled corrector vector into a number of vectors by means of block diagonalization method in group theoretic bifurcation theory and in turn by choosing the predominant one among these vectors in order to demonstrate the usefulness of this method it is applied to the bifurcation analysis of reticulated regular hexagonal truss domes to compute accurately the locations of double bifurcation points and nearly coincidental bifurcation points and associated bifurcation modes

tracing post limit point paths with reduced basis technique abstract a reduced basis technique and a problem adaptive computational algorithm are presented for predicting the post limit point paths of structures in the proposed approach the structure is discretized by using displacement finite element models the nodal displacement vector is expressed as a linear combination of a small number of vectors and a rayleigh ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations to circumvent the difficulties associated with the singularity of the stiffness matrix at limit points a constraint equation defining a generalized arc length in the solution space is added to the system of nonlinear algebraic equations and the rayleigh ritz approximation functions or basis vectors are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc length the potential of the proposed approach and its advantages over the reduced basis load control technique are outlined the effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap through and snap back phenomena

lower order symmetric mechanism modes and bifurcation behavior of deployable bar structures with cyclic symmetry abstract kinematic bifurcation is generally unavoidable for deployable structures stating at one critical point along the compatibility path a deployable structure could exhibit unexpected motions it can either keep transforming along the primary path or get into one of the bifurcation paths in this study a group theoretic approach which is applicable for bifurcation analyses on symmetric deployable bar structures is developed with the aid of symmetry groups and their irreducible representations the key issue is to decompose the compatibility matrix of a deployable structure into block matrices in the symmetry adapted coordinate system and then extract new mechanism modes with lower order symmetries these symmetric mechanisms are associated with independent bifurcation paths and can identify the feasibility of the paths further using the prediction correction algorithm the lower order symmetric mechanisms are utilized to lead the structure smoothly transform into expected bifurcation paths to verify the effectiveness of the proposed approach several 2d linkages and 3d deployable structures are analyzed which possess inherent cyclic symmetries numerical results indicate that the proposed approach can effectively trace the complex and independent bifurcation paths

symmetry preserving return mapping algorithms and incrementally extremal paths a unification of concepts in this work we seek to characterize the conditions under which an elastic plastic stress update algorithm preserves the symmetries inherent to the material response from a numerical standpoint the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality for the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving for hardening plasticity symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations we show that these restrictions amount to the existence of a pseudo internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli in view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and hence in symmetric tangents we address the question of whether any connections exist between the two approaches we show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence

studies in anisotropic plasticity with reference to the hill criterion algorithms based upon the notion of return mapping have been developed for the hill yield function of anisotropic plasticity the relative accuracy of two algorithms is assessed by means of iso error maps the choice of the algorithm turns out to be much more critical for the orthotropic hill criterion than for the underlying isotropic von mises plasticity model a tangent operator that is consistent with the developed integration algorithm is formulated and its efficiency is assessed compared with the classical continuum tangent operator the model has been applied to three shell plate structures

some computational aspects of a group theoretic finite element approach to the buckling and postbuckling analyses of plates and shells of revolution abstract progress in the application of group theoretic methods to the nonlinear bifurcation analysis of symmetric structures has provided new hope in understanding the global postbuckling behavior of symmetric thin shell structures however what is still lacking in the computational mechanics community is a general purpose fe code with a group theoretic capability designed to take optimal advantage of existing symmetry this paper addresses some of the important issues associated with developing such a code the long term goal being to provide structural analysts with new numerical tools for doing careful global analyses of imperfection sensitive structures which may eventually lead to more accurate predictions of the maximum load carrying capacity of thin imperfection sensitive shell structures

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

an adaptive sub incremental strategy for the solution of homogenization based multi scale problems abstract in this contribution several schemes for the solution of homogenization based multi scale constitutive problems undergoing finite strains with inelastic material behavior are investigated these schemes are aimed at improving the robustness and efficiency of the newton raphson method in the multilevel finite element ml fem framework an adaptive sub incremental strategy is proposed for the discrete representative volume element rve boundary value problem the procedure is able to ensure the convergence of the solution algorithm in the presence of several sources of non linearity and obtains improved initial guesses for the newton raphson scheme in the ml fem framework the enlargement of the convergence bowl of the newton raphson procedure at the micro scale allows larger macroscopic deformation gradients to be prescribed and significantly reduces the overall computational cost of ml fem analyses the proposed strategy preserves the quadratic rates of asymptotic convergence that characterize the newton raphson scheme at the macroscopic level numerical examples of both micro scale and two scale finite element simulations are presented to demonstrate the improved robustness and efficiency of the solution procedures proposed

exact block diagonalization of large eigenvalue problems for structures with symmetry we consider large eigenvalue problems for skeletal structures with symmetry we present an algorithm based upon a novel combination of group theoretic ideas and substructuring techniques that block diagonalizes such systems exactly and efficiently the procedure requires only the structural matrices of a repeating substructure together with the symmetry modes which are obtained from symmetry considerations alone we first present a simple paradigmatic example and then follow with several non trivial applications involving large lattice structures

nonlinear analysis based on homogeneous incremental systems abstract the performance of path following techniques depends on so called constraint equations which are used in the augmented incremental systems it is common practice in the literature that these equations are explicitly defined following the arbitrary geometrical or physical ideas in the present paper their role is discussed in order to establish a reliable method of augmentation it is shown that the explicitly defined constraint equation is not necessary the implicit formulation is proposed which is based on the rank analysis performed on the rectangular matrix of the homogeneous incremental system the proposed method allows to solve problems where the well known continuation schemes fail the theoretical considerations are illustrated by the numerical examples

on stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions in a recent article jentzen et al 2016 commun math sci 14 1477 1500 doi 10 4310 cms 2016 v14 n6 a1 it has been established that for every arbitrarily slow convergence speed and every nat

sensitivity analysis of bifurcation load of finite dimensional symmetric systems three methods are presented for sensitivity analysis of bifurcation load factor of finite dimensional conservative symmetric systems subjected to a set of symmetric proportional loads in the first method a conventional method with diagonalization is utilized to derive an explicit formula of sensitivity coefficients corresponding to a minor imperfection next a new concept is introduced to find the sensitivity coefficients of the load factor displacements and the eigenmodes under fixed lowest eigenvalue of the tangent stiffness matrix based on this concept a method is presented for finding approximate sensitivity coefficients of the buckling load factor finally a direct method is presented to find the accurate sensitivity coefficients of the bifurcation load factor displacements at buckling and the buckling mode of a symmetric system note that different formula should be used for sensitivity analysis of a limit point load factor in the examples the proposed three methods are compared in view of accuracy of the results and simplicity in coding

exploitation of symmetry in graphs with applications to finite and boundary elements analysis summary we present explicit and parametric forms of transformation matrices for three well known and widely used symmetry groups s2 c2v and c4v group representation theory is the most powerful method for exploiting symmetry we propose an efficient algorithm for systematic generation of reducible representations that can be combined linearly to obtain the projection operators the exact column spaces of these projection operators are calculated and integrated through special orderings leading to exact explicit and parametric forms of transformation matrices the transformation matrices could be used directly for block diagonalization of single variable scalar field problems another algorithm is proposed to extend the application of the method to nonscalar and multivariable field problems finally the generality and efficiency of the proposed method in relation to computation times and the accuracy of results are illustrated through examples from spectral decomposition free vibration buckling of fems and boundary element analysis of a symmetric field copyright 2012 john wiley sons ltd

bifurcation hierarchy of symmetric structures abstract a group theoretic method for the analysis of bifurcation behavior of regular polygonal symmetric structures is described possible bifurcation paths and points of these structures are categorized in terms of dihedral and cyclic groups which express the symmetry in particular we offer a complete description of those double bifurcation points which occur due to group symmetry the type the number and the stability of bifurcation paths branching at these points are determined by deriving bifurcation equations the existence of a potential function plays a substantial role for the existence of bifurcation paths as a result of these all possible bifurcation process can be known a priori as a natural consequence of bifurcation hierarchy before actual numerical analysis an implementation of this method in numerical analysis shows its validity and usability

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

echelon modes in uniform materials abstract the mechanism of the echelon mode a series of parallel short wrinkles of soils that looks like a flight of stairs or wild geese arranged in formation observed in the triaxial compression test is revealed by means of the group theoretic bifurcation theory the study of the direct bifurcation of uniform materials in cylindrical domains with periodic boundaries has revealed the existence of various kinds of deformation patterns in particular a classical diamond checkerboard pattern solution and a pair of oblique stripe pattern solutions simultaneously branch at a bifurcation point of multiplicity four with the use of the periodic boundaries these oblique stripe solutions which cannot be directly obtained by the customary bifurcation analysis with non periodic boundaries have thus been made accessible the secondary bifurcation from these stripe solutions realizes the echelon mode the triaxial compression tests on soil specimens have demonstrated that the pattern of wrinkles shear bands and echelon modes of the specimens do indeed follow the present theory and can be understood as a natural consequence of pattern selection or in other words symmetry breaking bifurcation

a note on strong approximation of sdes with smooth coefficients that have at most linearly growing derivatives abstract in the literature on strong approximation of sdes polynomial error rate results for numerical schemes are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition in the present article we show that the latter condition is far from being sufficient for polynomial error rates we construct an sde with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can at best be approximated with a logarithmic rate whatever numerical method based on observations of the driving brownian motion at finitely many fixed times is used most interestingly it furthermore turns out that using a method that properly adjusts the number of evaluations of the driving brownian motion to its actual path the latter sde can in fact be approximated with polynomial rate 1 in terms of the average number of evaluations that are used to the best of our knowledge this is only the second example in the literature of an sde for which there exist adaptive methods that perform superior to non adaptive ones with respect to the convergence rate

a picture of the global bifurcation diagram in ecological interacting and diffusing systems abstract this paper concerns global phenomena of pattern formation in stationary reaction diffusion equations possessing turing s diffusion induced instability which appear typically in mathematical biology global bifurcation diagrams with respect to two diffusion parameters are presented by integrating two complementary approaches analytical and numerical simple and double bifurcation analysis using group theoretic methods for the compact lie group d x results from singular perturbations when one diffusion constant is sufficiently small and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large are the main analytical results a new numerical method is used to trace all bifurcating branches we examine interrelations between those local and semi global results under the light of global pictures obtained by numerical studies a variety of interesting and new diagrams near double eigenvalues are observed coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions

non linear b stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward euler return maps as a particular case more importantly it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary helmholz free energy function b stability since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm the requirement of b stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity the analysis that follows depends critically on the assumption of convexity in particular the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation the proposed algorithms obey the discrete counterpart of this classical principle

modified stiffness iteration to pinpoint multiple bifurcation points abstract a modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented the iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones two different transformation matrices are employed to modify the stiffness matrix one is a non orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix the other is an orthogonal one which is chosen with reference to the symmetry of the system under consideration the use of the non orthogonal transformation matrix is a key idea termed stiffness amplification method in this paper whereas the orthogonal matrix is based on block diagonalization which is becoming popular in group theoretic analysis of symmetric structures extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

a return mapping algorithm for cyclic viscoplastic constitutive models abstract this paper presents a return mapping algorithm for cyclic viscoplastic constitutive models that include material memory effects the constitutive model is based on multi component forms of kinematic and isotropic hardening variables in conjunction with von mises yield criterion armstrong frederick a f type rules are used to describe the nonlinear evolution of each of the multi component kinematic hardening variables a saturation type exponential rule is used to describe the nonlinear evolution of each of the isotropic hardening variables the concept of memory surface is used to describe the strain range dependent material memory effects that are induced by the prior strain histories in this paper the above class of cyclic viscoplastic constitutive models is formulated within a consistent thermodynamic framework that encompasses the standard generalized materials framework furthermore a complete algorithmic treatment of the above rate dependent constitutive model is also presented for any desired stress or strain constrained configuration subspace a generalized midpoint algorithm is used to integrate the rate constitutive equations the consistent tangent operator is obtained by linearizing the return mapping algorithm and is found to be unsymmetric due to the presence of nonlinear evolution rules for the kinematic hardening variables several numerical examples representing the cyclic hardening and softening behavior transient and stabilized hysteresis behavior and the non fading memory effects of the material are presented these examples demonstrate the accuracy and robustness of the present algorithmic framework for modeling the cyclic viscoplastic behavior of the material

bending instabilities of elastic tubes the present paper examines instabilities of long thin elastic tubes both initially straight and initially bent tubes are analyzed under in plane bending tube response a combination of ovalization instability and bifurcation instability buckling is investigated using a nonlinear finite element fe technique which employs polynomial functions in the longitudinal tube direction and trigonometric functions to describe cross sectional deformation it is demonstrated that the interaction between the two instability modes depends on the value and the sign of the initial tube curvature the ovalization of initially bent tubes is examined in detail and in particular the case of opening moments furthermore the paper emphasizes on bifurcation instability it is shown that buckling may occur prior to or beyond the ovalization limit point depending on the value of the initial curvature using the nonlinear fe formulation the location of bifurcation on the primary path is detected post buckling equilibrium paths are traced and the corresponding wavelengths of the buckled configurations are calculated moreover results over a wide range of initial curvature values are presented extending the findings of previous works finally several analytical approaches introduced in previous research works are also employed to estimate the moments causing ovalization and bifurcation instability these approaches are based on nonlinear flexible shell theory or simplified ring analysis the efficiency and accuracy of those analytical methods with respect to the nonlinear fe formulation are examined

a new approach in non linear mechanics the large time increment method a new algorithm family taking into account the entire loading process in a single large time increment is proposed to compute structures with physical non linearities and is tested on some examples in elastoplasticity the method considerably reduces the number of transfers between local and global levels hence the numerical cost of calculation is also diminished