a continuum model accounting for defect and mass densities in solids with inelastic material behaviour in this paper the analysis of structures with inelastic material behaviour is considered taking into account the evolution of defects and changes in mass density the underlying kinematical concept of an oriented continuum is general enough to describe the micro and macrobehaviour of material bodies appropriately based on the logical and consistent variational arguments for a lagrangian functional the dynamic balance laws boundary and transversality conditions all related to the evolution of defect density and mass changes are derived for macro and microstresses of deformational as well as of configurational type the adopted procedure which formally leaves the balance laws unaltered leads to the additional balance law for changes in defect density and additional boundary conditions for the changes in mass and defect densities driving forces or affinities associated with the evolution of defect and mass densities and a generalization of the j integral representing the thermodynamic forces on defects are obtained a nonlocal constitutive model accounting for changes in the defect density is presented

chaos control in economical model by time delayed feedback method abstract a two dimensional map describing chaotic behaviour of an economic model has been stabilized on various periodic orbits by the use of pyragas time delayed feedback control the method avoids fancy data processing used in the ott grebogi yorke approach and is based solely on the plain measurement and time lag of a scalar signal which in our case is a value of sales of a firm following an active investment strategy behrens feichtinger model we show that the application of this control method is very straightforward and one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the system s economic properties

a class of multistep methods based on a super future points technique for solving ivps in this article the details of new methods based on a super future points technique for improving stability regions compared with those of the extended bdf ebdf and modified ebdf mebdf have been presented for solving initial value problems ivps numerical results related to five test problems show that our new methods have good performance in saving cpu time compared with the corresponding mebdf method

existence for coupled pseudomonotone strongly monotone systems and application to a cahn hilliard model with elasticity abstract a system of two operator equations is considered one of pseudomonotone type and the other of strongly monotone type both being strongly coupled conditions are given that allow to reduce the solvability of this system to a single operator equation of a pseudomonotone mapping this result is applied to a coupled system consisting of a parabolic equation of fourth order in space of cahn hilliard type and a nonlinear elliptic equation of second order modeling a quasi steady mechanical equilibrium using an appropriate notion of weak solutions and a framework for evolution equations developed by groger 2001 the system is reduced to a single parabolic operator equation and the existence of solutions is shown under restrictions on the strength of the coupling

a level set approach to the computation of twinning and phase transition dynamics a computational method is proposed for the dynamics of solids capable of twinning and phase transitions in a two dimensional sharp interface model of twinning the stored energy function is a nonconvex potential with multiple wells the evolution of twin interfaces is governed by field equations and jump conditions of momentum balance and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation a regularized version of the model is constructed based on the level set method a level set function which changes signs across the interface is introduced the evolution of this function is described by a hamilton jacobi equation whose velocity coefficient is determined by the kinetic relation jump conditions are thereby eliminated allowing finite difference discretization numerical simulations exhibit complex evolution of the interface including cusp formation needle growth spontaneous tip splitting and topological changes that result in microstructure refinement the results capture experimentally observed phenomena in martensitic crystals

spn graphs when copositive spn abstract a real symmetric matrix a is copositive if x t a x 0 for every nonnegative vector x a matrix is spn if it is a sum of a real positive semidefinite matrix and a nonnegative one every spn matrix is copositive but the converse does not hold for matrices of order greater than 4 a graph g is an spn graph if every copositive matrix whose graph is g is spn in this paper we present sufficient conditions for a graph to be spn in terms of its possible blocks and necessary conditions for a graph to be spn in terms of forbidden subgraphs we also discuss the remaining gap between these two sets of conditions and make a conjecture regarding the complete characterization of spn graphs

irreducible elements of the copositive cone we call an element a of the n n copositive cone cn irreducible with respect to the nonnegative cone nn if it cannot be written as a nontrivial sum a c n of a copositive matrix c and an elementwise nonnegative matrix n note that our concept of irreducibility differs from the standard one normally studied in matrix theory this property was studied by baumert 1965 2 who gave a characterisation of irreducible matrices we demonstrate here that baumert s characterisation is incorrect and give a correct version of his theorem which establishes a necessary and sufficient condition for a copositive matrix to be irreducible for the case of 5 5 copositive matrices we give a complete characterisation of all irreducible matrices we show that those irreducible matrices in c5 which are not positive semidefinite can be parameterized in a semi trigonometric way finally we prove that every 5 5 copositive matrix which is not the sum of a nonnegative and a semidefinite matrix can be expressed as the sum of a nonnegative matrix with zero diagonal and a single irreducible matrix

dynamic solid solid transitions with phase characterized by an order parameter abstract we develop a thermodynamically consistent continuum theory for the study of solid solid phase transitions where deformation dominates heat and mass transfer we use an order parameter to characterize the notion of phase and identify phase interfaces with thin transition zones within which the strain and order parameter exhibit large gradients to model the growth of one phase at the expense of another we introduce configurational forces subject to their own balance that work against changes in the order parameter by studying traveling waves and the asymptotics of a transition layer we establish connections between our theory and more standard approaches that identify phase interfaces with surfaces of strain discontinuity

thin interface asymptotics for an energy entropy approach to phase field models with unequal conductivities karma and rappel phys rev e 57 1998 4342 recently developed a new sharp interface asymptotic analysis of the phase field equations that is especially appropriate for modeling dendritic growth at low undercoolings their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis and has the added advantage that interfacial kinetic effects can also be eliminated however their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases when applied to a standard phase field model with unequal conductivities anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp interface solidification models as discussed further by almgren siam j appl math 59 1999 2086 in this paper we apply their asymptotic methodology to a generalized phase field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities the additional degrees of freedom associated with the generalized phase field equations can be used to eliminate the anomalous terms that arise for unequal conductivities

summation by parts in time we develop a new high order accurate time integration technique for initial value problems we focus on problems that originate from a space approximation using high order finite difference methods on summation by parts form with weak boundary conditions and extend that technique to the time domain the new time integration method is global high order accurate unconditionally stable and together with the approximation in space it generates optimally sharp fully discrete energy estimates in particular it is shown how stable fully discrete high order accurate approximations of the maxwells equations the elastic wave equations and the linearized euler and navier stokes equations can obtained even though we focus on finite difference approximations we stress that the methodology is completely general and suitable for all semi discrete energy stable approximations numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

l0 stable split linear multistep formulas for parabolic pdes abstract new types of split linear multistep methods slmms are applied to the numerical solution of parabolic differential equations the formulas derived achieve up to fourth order in time and unlike the crank nicolson method are l0 stable

homogeneous nucleation of a solid solid dilatational phase transformation abstract the problem of homogeneous coherent nucleation of a spherical particle in an infinite matrix is considered a landau ginzburg type potential is used to describe the material response the underlying landau potential is taken to be of the 2 3 4 type and the gradient energy contribution is taken to be quadratic the nucleation problem is posed as an energy extremum problem and the finite element method in conjunction with a perturbed lagrangian algorithm is used to obtain solutions with nucleus structure the present nonlinear model spans the range from classical to nonclassical nucleation and exhibits many of the physical phenomena associated with nonclassical nucleation including divergence in radius and interface thickness of the critical nucleus and vanishing of the nucleation energy as the instability temperature is approached

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

criteria for copositive matrices of order four abstract it is proved that the copositivity of a symmetric matrix of order n is equivalent to the copositivity of two symmetric matrices of order n 1 if the matrix has a row whose off diagonal elements are all nonpositive this result is used for deriving criteria for copositive matrices of order 3 and 4

modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in odes and daes abstract for many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well known backward differentiation formulae bdf more recently however new classes of formulae which can offer some important advantages over bdf have emerged in particular some recent large scale independent comparisons have indicated that modified extended backward differentiation formulae mebdf are particularly efficient for general stiff initial value problems and for linearly implicit daes with index 3 in the present paper we survey some of the more important theory associated with these formulae discuss some of the practical applications where they are particularly effective e g in the solution of damped highly oscillatory problems and describe some significant recent extensions to the applicability of mebdf codes

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

on the effect of temporal error in high order simulations of unsteady flows abstract the present paper focuses on the effect that temporal error has on unsteady solutions modeled with a high spatial order finite element streamline upwind petrov galerkin solver six l stable first through fourth order time integration methods are discussed and exercised on two canonical test cases the convecting isentropic vortex and the two dimensional circular cylinder in crossflow all results are shown on grids composed of p2 third order elements the isentropic vortex case is used to verify temporal order of accuracy assess the relative split between dissipative and dispersive errors for each method with various time steps and quantify the relative computational cost of each method the results from the two dimensional circular cylinder in crossflow simulations at various reynolds numbers are compared to experimental results and accepted numerical results for vortex shedding frequency strouhal number mean drag coefficient and rms lift and drag coefficient the convergence of those parameters with respect to time step size is shown these results highlight the benefits of high order greater than second order time integration methods and the need for time step convergence studies on unsteady simulations

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

time domain analog circuit simulation recent developments of new methods for simulating electric circuits are described emphasis is put on methods that fit existing datastructures for backward differentiation formulae methods these methods can be modified to apply to hierarchically organized datastructures which allows for efficient simulation of large designs of circuits in the electronics industry

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

evolution of phase boundaries by configurational forces an initial boundary value problem modeling the evolution of phase interfaces in materials showing martensitic transformations is studied the model which is derived rigorously from a sharp interface model with phase interfaces driven by configurational forces and which generalizes that model consists of the equations of linear elasticity coupled with a nonlinear partial differential equation of hyperbolic character governing the evolution of the order parameter it is proved that in one space dimension global solutions exist for which the order parameter belongs to the space of functions of bounded variation other models for interface motion by martensitic transformations and by interface diffusion are suggested

a test for copositive matrices abstract the paper presents necessary and sufficient conditions that a symmetric matrix be copositive or strictly copositive the conditions are given in terms of the eigenvalues and eigenvectors of the principal submatrices of the given matrix

elastic phase transitions a new model abstract in this note we consider a new model for one dimensional clastic phase transitions the model is described by two hyperbolic systems one which propagates information at speeds c and the other at speeds where c 0 the model supports slow moving under compressive shocks which connect separate phases of the material and microstructural equilibrium solutions similar to those predicted by viscosity capillarity theories for such materials our principal results are contained in section 3 and describe the long term behavior of the system section 4 presents numerical simulations for our model for a variety of initial and boundary conditions these results confirm the analysis of sections 2 and 3

optimization of high order diagonally implicit runge kutta methods abstract this article presents constrained numerical optimization of high order linearly and algebraically stable diagonally implicit runge kutta methods after satisfying the desired order conditions undetermined coefficients are optimized with respect to objective functions which consider accuracy stability and computational cost constraints are applied during the optimization to enforce stability properties to ensure a well conditioned method and to limit the domain of the abscissa two promising third order methods are derived using this approach labelled sdirk 3 1 2 2 3 l 14 and sdirk 3 1 4 l sa 5 both optimized schemes have a good balance of properties the relative error norm of the latter the l 2 norm scaled by a function of the number of implicit stages is a factor of two smaller than comparable methods found in the literature variations on these methods are discussed relative to trade offs in their accuracy and stability properties a novel fifth order scheme sdirk 5 1 5 l 02 is derived with a significantly lower relative error norm than the comparable fifth order a stable reference method in addition the optimized scheme is l stable the accuracy and relative efficiency of the runge kutta methods are verified through numerical simulation of van der pol s equation as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder and in the turbulent wake of a naca 0012 airfoil these results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high order runge kutta methods with a balance between competing objectives

efficient time integrators in the numerical method of lines the numerical method of lines has long been acknowledged as a very powerful approach to the numerical solution of time dependent partial differential equations this method in its original form involved making a simple approximation to the space derivatives and by so doing reducing the problem to that of solving a system of initial value ordinary differential equations and then using a black box package as the time integrator however in the past twenty years or so moving mesh algorithms in space have been developed and this allows much more challenging problems for example those with moving fronts to be solved efficiently and reliably regridding of the space variables poses special problems for the time integrator since sufficient back information to allow multistep formulae to run at high order is not available immediately after the regridding has been performed in this paper we survey some of the options available for the time integration when using a moving grid method of lines code in particular we derive runge kutta starters for use after grid adaptation has been carried out and we show how a considerable saving in computational effort can be made if just a few spatial points are moved during each regridding

bifurcation in model reference adaptive control systems abstract we study the effect of unmodeled dynamics on the performance of a simple model reference adaptive control mechanism exact stability boundaries are computed and different routes to chaotic behavior and global instability are described typically we observed a region of period doubling and hopf bifurcations similar to those observed in the ziegler nichols approach to pid control design the analysis applies to higher order and certain nonlinear systems as well

corrigendum to spn graphs when copositive spn linear algebra appl 509 2016 82 113 abstract in this corrigendum an error in the proof of theorem 6 4 in shaked monderer 2016 2 is pointed out this theorem states that every graph t n consisting of n 2 triangles sharing a common base is spn we have an alternative proof for the case n 5 but for all n 5 it remains open whether t n is spn as a result the question whether k 2 n n 4 is spn also remains open

solving time dependent pdes via an improved modified extended bdf scheme in a previous article the known mebdf scheme had been extended to solve odes that arise in the method of lines solution of time dependent pdes in that article it was shown that mebdf is superior to the widely used bdf approach in certain important classes of problems in this paper by following a different logic we were able to improve the stability characteristics of mebdf appropriate modifications to the code have also been made consequently we investigate whether the current approach is better to the one previously presented it is indeed shown that the instigated improvements have had a positive effect on both accuracy and efficiency numerical results are given to support this claim

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

exponential fitting bdf algorithms explicit and implicit 0 stable methods we present bdf type formulas capable of the exact integration with only round off errors of differential equations the solutions of which belong to the space generated by the linear combinations of 1 eax xeax plots of their 0 stability regions in terms of are provided we will see that the explicit method is 0 stable with many parameters h plots of their regions absolute stability that include all the negative real axis are provided numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems with the explicit method

liquid vapor transformations with surfactants phase field model and isogeometric analysis surfactants are compounds that find energetically favorable to be located at the boundaries between fluids they are able to modify the properties of those interfaces for example reducing surface tension here we propose a new model for liquid vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid the aforementioned model is derived from a free energy functional by using a coleman noll approach the proposed theory emanates from the isothermal navier stokes korteweg equations which describe single component two phase flow and naturally allow for phase transformations we believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes from a numerical point of view the proposed model poses significant challenges to existing discretization methods including stiffness in space and time internal and boundary layers as well as higher order partial differential operators to overcome these challenges we propose algorithms based on isogeometric analysis which permit an accurate and efficient discretization finally we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

extrapolated stabilized explicit rungekutta methods extrapolated stabilized explicit rungekutta methods eserk are proposed to solve multi dimensional nonlinear partial differential equations pdes in such methods it is necessary to evaluate the function nt times per step but the stability region is o nt2 hence the computational cost is o nt times lower than for a traditional explicit algorithm in that way stiff problems can be integrated by the use of simple explicit evaluations in which case implicit methods usually had to be used therefore they are especially well suited for the method of lines mol discretizations of parabolic nonlinear multi dimensional pdes in this work first s stages first order methods with extended stability along the negative real axis are obtained they have slightly shorter stability regions than other traditional first order stabilized explicit rungekutta algorithms also called rungekuttachebyshev codes later they are used to derive nt stages second and fourth order schemes using richardson extrapolation the stability regions of these fourth order codes include the interval 0 01nt2 0 nt being the number of total functions evaluations which are shorter than stability regions of rock4 methods for example however the new algorithms neither suffer from propagation of errors as other rungekuttachebyshev codes as rock4 or dumka nor internal instabilities additionally many other types of higher order and also lower order methods can be obtained easily in a similar way these methods also allow adaptation of the length step with no extra cost hence the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way i e with minimal number of additional stages we compare the new techniques with other well known algorithms with good results in very stiff diffusion or reactiondiffusion multi dimensional nonlinear equations

a positivity preserving and conservative variational scheme for phase field modeling of two phase flows abstract we present a positivity preserving variational scheme for the phase field modeling of incompressible two phase flows with high density ratio the variational finite element technique relies on the allen cahn phase field equation for capturing the phase interface on a fixed eulerian mesh with mass conservative and energy stable discretization the mass conservation is achieved by enforcing a lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase field equation to make the scheme energy stable in a variational sense we discretize the spatial part of the lagrange multiplier in the phase field equation by the mid point approximation the proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second order accuracy of both spatial and temporal discretizations we integrate the allen cahn phase field equation with the incompressible navier stokes equations for modeling a broad range of two phase flow and fluid fluid interface problems the coupling of the implicit discretizations corresponding to the phase field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase indicator field we perform a standalone test to verify the accuracy and stability of the allen cahn two phase solver we examine the convergence and accuracy properties of the coupled phase field solver through the standard benchmarks of the laplace young law and a sloshing tank problem two and three dimensional dam break problems are simulated to assess the capability of the phase field solver for complex air water interfaces involving topological changes on unstructured meshes finally we demonstrate the phase field solver for a practical offshore engineering application of wave structure interaction

a 17th order radau iia method for package radau applications in mechanical systems a new radau iia code with an order of 17 is programmed for the well known implicit runge kutta program package radau several numerical examples are reported in this article to demonstrate the performance and efficiency of the new algorithm they are real stiff problems obtained from the simulation of mechanical systems

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

extended backward differentiation methods in the numerical solution of the neutral volterra integro differential equations the numerical solution of first order volterra integro differential equations of neutral type with continuous kernel is considered preliminary results from the application of extended backward differentiation methods are given and comparisons are made with adams bashforth adams moulton predictor corrector methods and collocation methods

recent advances in methods for numerical solution of o d e initial value problems abstract this is a review paper which describes recent advances in numerical methods and computer codes for solving initial value problems of ordinary differential equations particular emphasis is placed upon stiff systems

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

drift instabilities and chaos in forecasting and adaptive decision theory abstract bifurcation theory shows that policy adaptation and the rational expectations hypothesis of macro economics can be used to explain unpredictability rapid changes in solution structure and chaos in decision problems with uncertainty structural errors lead to catastrophic instability and forecasts become irrelevant short evaluation horizon and the application of measures designed to give quick response give multiperiodicity and chaos finally wrong interpretations of the context lead to global bifurcations laminar drift complacency and chaotic bursting the discrete map representing these dynamics is of interest in its own right it is non invertible and displays bifurcation behaviour not commonly seen in systems derived from physical considerations

spatially periodic solutions in a 1d model of phase transitions with order parameter abstract a model of phase transitions with convex strain energy is investigated within the limits of 1d nonlinear bar theory the model is a special case of a coupled field theory using an order parameter that has been developed by fried and gurtin to study the nucleation and propagation of phase boundaries the system of governing equations studied here consists of a wave equation coupled to a nonlinear reaction diffusion equation using phase plane methods the equilibria of the system have been constructed in order to obtain the macroscopic response of the bar it is demonstrated that a large number of coexisting spatially periodic inhomogeneous solutions can occur with the number of these solutions being inversely proportional to the diffusion coefficient in the reaction diffusion subsystem a stability analysis of the equilibria is presented and the bifurcation diagram is discussed in the context of quasistatic loading it is shown that though solutions with more than one interface are unstable they are only weakly so and can thus persist for a long time the nucleation and propagation of phase boundaries are illustrated via a numerical study which shows how nucleation relates to the loss of stability of the homogeneous equilibria

small amplitude chaos and ergodicity in adaptive control abstract the theory for continuous iterated maps on the real line is used to study the dynamics of a continuous family of one parameter adaptive systems we establish sufficient and necessary conditions for uniform asymptotic stability and ultimate boundedness exact bifurcation boundaries and sufficient conditions for the existence of chaotic attractors and ergodicity of the estimated parameter it is argued that the chaotic dynamics are structurally stable that they persist in higher dimensional adaptive control systems and that they do not necessarily lead to poor performance when the controller is modified to deal with the infinite drift and small divisor problems a novel modification involving a sign change in the adaptive law is proposed

parameter range reduction for ode models using cumulative backward differentiation formulas we consider fitting an ode model to time series data of the system variables we assume that the parameters of the model have some initial range of possible values and the goal is to reduce these ranges to produce a smaller parameter region from which to start a global nonlinear optimization algorithm we introduce the class of cumulative backward differentiation formulas cbdfs and show that they inherit the accuracy and stability properties of their generating backward differentiation formulas bdfs discretizing the system with these cbdfs and applying consistency conditions results in reductions of the parameter ranges we show that these reductions are better than can be obtained simply using bdfs in addition cbdfs inherit any monotonicity properties with respect to the parameters that the vector field possesses and we exploit these properties to make the consistency checking more efficient we illustrate with several examples analyze some of the behavior of our range reduction method and discuss how the method could be extended and improved

a multirate w method for electrical networks in state space formulation subunits of coupled technical systems typically behave on differing time scales which are often separated by several orders of magnitude an ordinary integration scheme is limited by the fastest changing component whereas so called multirate methods employ an inherent step size for each subsystem to exploit these settings however the realization of the coupling terms is crucial for any convergence thus the approach to return to one step methods within the multirate concept is promising this paper introduces the multirate w method for ordinary differential equations and gives a theoretical discussion in the context of partitioned rosenbrock wanner methods finally the matlab implementation of an embedded scheme of order 3 2 is tested for a multirate version of prothero robinson s equation and the inverter chain benchmark

on the notion of a vector and a stress tensor for a general class of anisotropic diffuse interface models we show that under equilibrium conditions an anisotropic phase field model of a diffuse interface leads to the notion of a divergence free tensor whose components are negligible away from the interface near the interface it plays the role of a stress tensor and involves the phase field generalization of the cahn hoffman vector we show that this tensor may be used to derive the equilibrium conditions for edges in interfaces we then extend these ideas from the phase field model to a general class of multiple order parameter models in this broader context we demonstrate that it is also possible to construct a vector and we use it to derive the three dimensional form of the gibbs thomson equation in the sharp interface limit we derive the associated stress tensor for the multiple order parameter case and show that it also leads to the appropriate equilibrium conditions for edges in interfaces and multiple junctions in the sharp interface limit this approach shows that both multiple order parameter models and less physically based phase field models share a common framework the former providing a generalization of the notion of the vector for phase field models as well as providing a connection to the classical vector theory for a sharp interface for the special case of a phase field model with non convex surface energy we show that edges can be represented by weak shocks in which the spatial derivatives of the phase field are not continuous and we derive the associated jump conditions in all the situations considered the notion of the vector and the stress tensor play a central role

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

finite element approximation of a phase field model for surface diffusion of voids in a stressed solid we consider a fully practical finite element approximation of the degenerate cahn hilliard equation with elasticity find the conserved order parameter theta x t in 1 1 and the displacement field underline u x t in bbb r 2 such that gamma frac partial theta partial t nabla cdot b theta nabla gamma delta theta gamma 1 psi theta tfrac 12 c theta cal c underline underline cal e underline u underline underline cal e underline u quad nabla cdot c theta cal c underline underline cal e underline u underline 0 subject to an initial condition theta 0 cdot in 1 1 on theta and boundary conditions on both equations here gamma in bbb r 0 is the interfacial parameter psi is a nonsmooth double well potential underline underline cal e is the symmetric strain tensor cal c is the possibly anisotropic elasticity tensor c s c 0 frac 12 1 c 0 1 s with c 0 gamma in bbb r 0 and b s 1 s 2 is the degenerate diffusion mobility in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions finally some numerical experiments are presented

how floquet theory applies to index 1 differential algebraic equations local stability of periodic solutions is established by means of a floquet theory for index 1 differential algebraic equations linear differential algebraic equations with periodic coefficients are considered in detail and a natural notion of the monodromy matrix is obtained that generalizes the well known theory for regular ordinary differential equations

optimal high order diagonally implicit runge kutta schemes for nonlinear diffusive systems on atmospheric boundary layer abstract nonlinear diffusion equations are extensively applicable in diverse fields of science and engineering numerical stability is a common concern in this class of equations in the present study a three stage third order diagonally implicit runge kutta dirk scheme is introduced by optimizing the error and linear stability analysis for a commonly used nonlinear diffusive system in atmospheric boundary layer the proposed scheme is stable for a wide range of time steps and able to resolve different diffusive systems with diagnostic turbulence closures or prognostic ones with a diagnostic length scale with enhanced accuracy and stability compared to current schemes it maintains a stability which makes it appropriate for the solution of stiff problems the procedure implemented in this study is quite general and can be used in other diffusive systems as well

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

stability concepts in the numerical solution of difference and differential equations abstract several algorithms have been proposed for the stable numerical computation of non dominant solutions of linear difference equations the unifying approach of these algorithms is that they replace the task of solving the given initial value problem by that of solving a boundary value problem which is in some sense equivalent a similar task is involved in the numerical solution of initial value problems for ordinary differential equations using linear multistep methods however in this case it is always assumed that the numerical solution will be found by forward recurrence and this assumption imposes severe restrictions on the numerical methods that can be used by adapting some of the ideas that have been proposed for linear recurrence relations new algorithms are derived for the numerical solution of initial value problems and it is shown that these new methods can be competitive with the best methods currently available

an adaptive variational procedure for the conservative and positivity preserving allen cahn phase field model abstract in this paper we present an adaptive variational procedure for unstructured meshes to capture fluid fluid interfaces in two phase flows the two phases are modeled by the phase field finite element formulation which involves the conservative allen cahn equation coupled with the incompressible navier stokes equations the positivity preserving variational formulation is designed to maintain the bounded and stable solution of the allen cahn equation for the adaptivity procedure we consider the residual based error estimates for the underlying differential equations of the two phase system in particular the adaptive refinement coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the allen cahn equation the coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh thus providing the ease of numerical implementation furthermore the proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations we investigate the adaptive phase field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume conserved interface motion driven by the mean curvature flow for two circles in a square domain we then assess the accuracy and efficiency of the proposed procedure by modeling the free surface motion in a sloshing tank in contrast to the non adaptive eulerian grid counterpart we demonstrate that the mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy the mass loss in the allen cahn equation via adaptivity process is also reduced by nearly three times compared to the non adaptive mesh finally we apply the adaptive numerical framework to solve the application of a dam breaking problem with topological changes

an mebdf package for the numerical solution of large sparse systems of stiff initial value problems abstract an efficient algorithm for the numerical integration of large sparse systems of stiff initial value ordinary differential equations odes and differential algebraic equations daes is described the algorithm is constructed by embedding a standard sparse linear algebraic equation solver into a suitably modified mebdf code an important practical application of this algorithm is in the numerical solution of time dependent partial differential equations pdes particularly in two or more space dimensions using the method of lines mol a code based on this algorithm is illustrated by application to several problems of practical interest and its performance is compared to that of the standard code lsodes

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

spn completable graphs abstract an spn matrix is a matrix which is the sum of a real positive semidefinite matrix and a symmetric nonnegative one we solve the spn completion problem we show that the spn completable graphs are the graphs in which every odd cycle induces a complete subgraph

a diffuse interface model for alloys with multiple components and phases a nonisothermal phase field model for alloys with multiple phases and components is derived the model allows for arbitrary phase diagrams we relate the model to classical sharp interface models by formally matched asymptotic expansions in addition we discuss several examples and relate our model to the ones already existing

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems

a simulation based evolutionary approach to lna circuit design optimization abstract in this paper we propose a simulation based evolutionary approach for designing low noise amplifier lna integrated circuits ics based on a genetic algorithm ga the levenberg marquardt lm method and a circuit simulator the simulation based evolutionary approach is developed for design optimization of lna circuits for a given lna circuit the simulation based evolutionary approach simultaneously optimizes the electrical specifications such as s 11 s 12 s 21 s 22 k factor the noise figure and the input third order intercept point in the process between simulation and optimization first of all the necessary parameters of the lna circuit for circuit simulation are loaded by solving a set of nonlinear ordinary differential equations the circuit simulator will then be performed for the circuit simulation and specification evaluation once the specification meets the aforementioned seven constraints we output the optimized parameters otherwise we activate the ga for the global optimization in the meanwhile the lm method searches the local optima according to the results of the ga we then call circuit simulator to compute and evaluate newer results until the specification is matched in numerical experiment 10 parameters of the lna circuit including device configuration and biasing condition are optimized with respect to the constraints the design of lna circuit is with 0 18 m metal oxide silicon filed effect transistors benchmark results also computationally confirm the robustness and efficiency of the proposed method this simulation based evolutionary approach in general can be applied to optimal design of other analog and radio frequency circuits we believe that this systematical approach will help ic design optimization and benefit computer aided design of wireless communication system on a chip