new integrable systems related to the relativistic toda lattice the bi hamiltonian structure of the relativistic toda lattice is exploited to introduce some new integrable lattice systems their integrable discretizations are obtained by means of the general procedure proposed recently by the author backlund transformations between the new systems and the relativistic toda lattice in both the continuous and discrete time formulations are established

the generalized symmetry method for discrete equations the generalized symmetry method is applied to a class of completely discrete equations including the adler bobenko suris list assuming the existence of a generalized symmetry we derive a few integrability conditions suitable for testing and classifying the equations of this class those conditions are used at the end to test for integrability discretizations of some well known hyperbolic equations

optimizing the layout of discrete objects in structures and materials a projection based topology optimization approach abstract this paper uses topology optimization to optimize the layout of discrete objects such as fixed shape void stiffening or functional inclusions in structures and periodic materials the methodology follows a material distribution approach and the heaviside projection method hpm is manipulated to ensure that the designed objects are discrete of fixed size length scale and shape and satisfy minimum spacing requirements these restrictions are achieved implicitly through projection without additional constraints or heuristics the methodology is built in the spirit of free form topology optimization and thus discrete objects are shown to appear vanish or translate across the design domain during the design evolution the technique is demonstrated on problems governed by mechanical stiffness including the design of structures and material microstructures whose elastic stiffness is to be enhanced by optimizing the location of stiff circular inclusions

three point partial difference equations linearizable by local and nonlocal transformations we consider a class of nonlinear partial difference equations defined on three points of a plane lattice we construct conditions for this class of partial difference equations to be linearizable through a point or a cole hopf transformation using these conditions we are able to classify all multilinear linearizable equations belonging to this class

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

symmetries and conservation laws of the abs equations and corresponding differential difference equations of volterra type a sequence of canonical conservation laws for all the adler bobenko suris abs equations is derived and is employed in the construction of a hierarchy of master symmetries for equations h1 h3 q1 q3 for the discrete potential and schwarzian kdv equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless virasoro type algebras in particular for the discrete potential kdv the structure of its symmetry algebra is explicitly given interpreting the hierarchies of symmetries of equations h1 h3 q1 q3 as differential difference equations of yamilov s discretization of krichever novikov equation corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them

a new two dimensional lattice model that is consistent around a cube for two dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions i e that it is consistent around a cube cac as a consequence of cac one can construct a lax pair for the model recently adler bobenko and suris conducted a search based on this principle and certain additional assumptions one of those assumptions was the tetrahedron property which is satisfied by most known equations we present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property its lax pair is also presented and some basic properties are discussed

on the lagrangian formulation of multidimensionally consistent systems multidimensional consistency has emerged as a key integrability property for partial difference equations p es defined on the space time lattice it has led among other major insights to a classification of scalar affine linear quadrilateral p es possessing this property leading to the so called adler bobenko suris abs list recently a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete lagrangian multi forms this description is based on a fundamental and highly non trivial property of lagrangians for those integrable lattice equations namely the fact that on the solutions of the corresponding p e the lagrange forms are closed i e they obey a closure relation here we extend those results to the continuous case it is known that associated with the integrable p es there exist systems of partial differential equations pdes in fact differential equations with regard to the parameters of the lattice as independent variables which equally possess the property of multidimensional consistency in this paper we establish a universal lagrange structure for affine linear quad lattices alongside a universal lagrange multi form structure for the corresponding continuous pdes and we show that the lagrange forms possess the closure property

continuous symmetries of the lattice potential kdv equation in this paper we present a set of results on the integration and on the symmetries of the lattice potential korteweg de vries lpkdv equation using its associated spectral problem we construct the soliton solutions and the lax technique enables us to provide infinite sequences of generalized symmetries finally using a discrete symmetry of the lpkdv equation we construct a large class of non autonomous generalized symmetries

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

a new efficient convergence criterion for reducing computational expense in topology optimization reducible design variable method summary a new efficient convergence criterion named the reducible design variable method rdvm is proposed to save computational expense in topology optimization there are two types of computational costs one is to calculate the governing equations and the other is to update the design variables in conventional topology optimization the number of design variables is usually fixed during the optimization procedure thus the computational expense linearly increases with respect to the iteration number some design variables however quickly converge and some other design variables slowly converge the idea of the proposed method is to adaptively reduce the number of design variables on the basis of the history of each design variable during optimization using the rdvm those design variables that quickly converge are not considered as design variables for the next iterations this means that the number of design variables can be reduced to save the computational costs of updating design variables then the iteration will repeat until the number of design variables becomes 0 in addition the proposed method can lead to faster convergence of the optimization procedure which indeed is a more significant time saving it is also revealed that the rdvm gives identical optimal solutions as those by conventional methods we confirmed the numerical efficiency and solution effectiveness of the rdvm with respect to two types of optimization static linear elastic minimization and linear vibration problems with the first eigenvalue as the objective function for maximization copyright 2012 john wiley sons ltd

achieving minimum length scale in topology optimization using nodal design variables and projection functions a methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology the projection is made via mesh independent functions that are based upon the minimum length scale a simple linear projection scheme and a non linear scheme using a regularized heaviside step function to achieve nearly 0 1 solutions are examined the new approach is demonstrated on the minimum compliance problem and the popular simp method is used to penalize the stiffness of intermediate volume fraction elements solutions are shown to meet user defined length scale criterion without additional constraints penalty functions or sensitivity filters no instances of mesh dependence or checkerboard patterns have been observed copyright 2004 john wiley sons ltd

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

generalized symmetry integrability test for discrete equations on the square lattice we present an integrability test for discrete equations on the square lattice which is based on the existence of a generalized symmetry we apply this test to a number of equations obtained in different recent papers as a result we prove the integrability of seven equations which differ essentially from the qv equation introduced by viallet and thus from the adler bobenko suris list of equations contained therein

evolutionary topology optimization using design space adjustment based on fixed grid design space optimization for topology based on fixed grid is proposed and its superiority to conventional topology optimization is shown in the conventional topology optimization the design domain is fixed it is however desirable to make the design domain evolve into a better one during optimization process by increasing or decreasing the number of design pixels or variables which we call design space optimization a breakthrough in obtaining sensitivities when design space expands has been made recently with necessary mathematical background but due to coupling effect and others sensitivity results have not been satisfactory three innovative implementations are developed in this paper firstly the proper characteristics of artificial material are defined the second one is to decouple neighbouring elements for exact design space sensitivities the previous design space optimization has been tedious because only one layer can be added so the third innovation is a new expansion strategy with multi layers based on design space sensitivities as a result the proposed evolutionary method can get an optimum much faster than ever before especially for large scale problems it is also conjectured that this gives higher probability of getting the global optimum as confirmed by numerical examples

large scale topology optimization using preconditioned krylov subspace methods with recycling summary the computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis we propose fast iterative solvers for large threedimensional topology optimization problems to address this problem since the linear systems in the sequence of optimization steps change slowly from one step to the next we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems in addition we introduce a minres minimum residual method version with recycling and a short term recurrence to make recycling more efficient for symmetric problems furthermore we discuss preconditioning to ensure fast convergence we show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density we demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast pc

generalized symmetry classification of discrete equations of a class depending on twelve parameters we carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square which belong to a twelve parametric class the direct result of this classification is a list of equations containing no new examples however as an indirect result of this work we find a number of integrable examples pretending to be new one of them has a nonstandard symmetry structure the others are analogues of the liouville equation in the sense that they are darboux integrable we also enumerate all equations of the class which are linearizable via a two point first integral and specify the nature of integrability of some known equations

a discrete linearizability test based on multiscale analysis in this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad graph by the multiple scale reduction around their harmonic solution we show that the a1 a2 and a3 linearizability conditions restrain the number of the parameters which enter into the equation a subclass of the equations which pass the a3 c integrability conditions can be linearized by a mobius transformation

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

topology optimization of continuum structures with local stress constraints we introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria we first consider relevant stress criteria for porous composite materials initially by studying the stress states of the so called rank 2 layered materials then on the basis of the theoretical study of the rank 2 microstructures we propose an empirical model that extends the power penalized stiffness model also called simp for solid isotropic microstructure with penalization for inter mediate densities in a second part solution aspects of topology problems are considered to deal with the so called singularity phenomenon of stress constraints in topology design an constraint relaxation of the stress constraints is used we describe the mathematical programming approach that is used to solve the numerical optimization problems and show results for a number of example applications 1998 john wiley sons ltd

a new scheme for imposing a minimum length scale in topology optimization a new scheme for imposing a minimum length scale in topology optimization is presented it guarantees the existence of an optimal design for a large class of topology optimization problems of practical interest it is formulated as one constraint that is computationally cheap and for which sensitivities are also cheap to compute the constraint value is ideally zero but it can be relaxed to a positive value the effect of the method is illustrated in topology optimization for minimum compliance and design of compliant mechanisms notably the method produces compliant mechanisms with distributed flexibility something that has previously been difficult to obtain using topology optimization for the design of compliant mechanisms the term mole method is suggested for the method copyright 2003 john wiley sons ltd

a nodal variable method of structural topology optimization based on shepard interpolant summary a method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated the original discrete valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain with the use of the shepard family of interpolants this density field is mapped onto the design space defined by a finite number of nodal density variables the employed interpolation scheme has an explicit form and satisfies range restricted properties that makes it applicable for physically meaningful density interpolation its ability to resolve more complex spatial distribution of the material density within an individual element as compared with the conventional elementwise design variable approach actually provides certain regularization to the topology optimization problem numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques copyright 2011 john wiley sons ltd

on the classification of darboux integrable chains we study a differential difference equation of the form tx n 1 f t n t n 1 tx n with unknown t t n x depending on x and n the equation is called a darboux integrable if there exist functions f called an x integral and i called an n integral both of a finite number of variables x t n t n 1 t n 2 tx n txx n such that dxf 0 and di i where dx is the operator of total differentiation with respect to x and d is the shift operator dp n p n 1 the darboux integrability property is reformulated in terms of characteristic lie algebras that give an effective tool for classification of integrable equations the complete list of equations of the form above admitting nontrivial x integrals is given in the case when the function f is of the special form f x y z z d x y

multiscale analysis of discrete nonlinear evolution equations the method of multiscale analysis is constructed for discrete systems of evolution equations for which the problem is that of the far behaviour of an input boundary datum discrete slow space variables are introduced in a general setting and the related finite differences are constructed the method is applied to a series of representative examples the toda lattice the nonlinear klein gordon chain the takeno system and a discrete version of the benjamin bona mahoney peregrini equation among the resulting limit models we find a discrete nonlinear schrodinger equation with reversed spacetime a three wave resonant interaction system and a discrete modified volterra model

topology optimization of structures using meshless density variable approximants this paper proposes a new structural topology optimization method using a dual level point wise density approximant and the meshless galerkin weak forms totally based on a set of arbitrarily scattered field nodes to discretize the design domain the moving least squares mls method is used to construct shape functions with compactly supported weight functions to achieve meshless approximations of system state equations the mls shape function with the zero order consistency will degenerate to the well known shepard function while the mls shape function with the first order consistency refers to the widely studied mls shape function the shepard function is then applied to construct a physically meaningful dual level density approximant because of its non negative and range restricted properties first in terms of the original set of nodal density variables this study develops a nonlocal nodal density approximant with enhanced smoothness by incorporating the shepard function into the problem formulation the density at any node can be evaluated according to the density variables located inside the influence domain of the current node second in the numerical implementation we present a point wise density interpolant via the shepard function method the density of any computational point is determined by the surrounding nodal densities within the influence domain of the concerned point according to a set of generic design variables scattered at field nodes an alternative solid isotropic material with penalization model is thus established through the proposed dual level density approximant the lagrangian multiplier method is included to enforce the essential boundary conditions because of the lack of the kronecker delta function property of mls meshless shape functions two benchmark numerical examples are employed to demonstrate the effectiveness of the proposed method in particular its applicability in eliminating numerical instabilities copyright 2012 john wiley sons ltd

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

the non autonomous ydkn equation and generalized symmetries of boll equations in this paper we study the integrability of a class of nonlinear non autonomous quad graph equations compatible around the cube introduced by boll in the framework of the generalized adler bobenko and suris abs classification we show that all these equations possess three point generalized symmetries which are subcases of either the yamilov discretization of the krichever novikov equation or of its non autonomous extension we also prove that all those symmetries are integrable as they pass the algebraic entropy test

improving multiresolution topology optimization via multiple discretizations summary unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization this paper proposes a framework for improving multiresolution topology optimization imtop via multiple distinct discretizations for 1 finite elements 2 design variables and 3 density this approach leads to high fidelity resolution with a relatively low computational cost in addition an adaptive multiresolution topology optimization amtop procedure is introduced which consists of selective adjustment and refinement of design variable and density fields various two dimensional and three dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element based approach copyright 2012 john wiley sons ltd

on discrete 2d integrable equations of higher order we study two dimensional discrete integrable equations of order 1 with respect to one independent variable and m with respect to another one a generalization of the multidimensional consistency property is proposed for this type of equations the examples are related to the backlund darboux transformations for the lattice equations of bogoyavlensky type

integrated topology optimization with embedded movable holes based on combined description by material density and level sets abstract this paper presents a novel topology description model for topology optimization problems with embedded movable holes combining the ability of the level set model for accurate geometrical description of the prescribed hole shapes and the high efficiency of the material density based method by this means any arbitrary hole shapes can be represented accurately and smoothly while the topological changes can be easily handled by the material distribution model an explicit mathematical expression is defined to obtain the actual structural layout by combing both models moreover an effective model for the non overlap constraint is proposed in a unified and systematic manner to avoid the overlaps between the holes and between each hole and the design domain boundary therein the non overlap constraint for all the embedded holes is imposed as a single explicit integral constraint over the design domain thus avoiding the difficulties in the overlap detection of multiple arbitrary shaped geometries such a non overlap constraint is accurate and differentiable facilitating an analytical design sensitivity analysis numerical examples are given to demonstrate the effectiveness and efficiency of the present method

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

lie point symmetries of differential difference equations we present an algorithm for determining the lie point symmetries of differential equations on fixed non transforming lattices i e equations involving both continuous and discrete independent variables the symmetries of a specific integrable discretization of the krichever novikov equation the toda lattice and toda field theory are presented as examples of the general method

construction scheme for discrete miura transformations a direct and elementary scheme for the construction of miura type transformations and discrete differential equations related to them scalar and vector is presented the scheme is illustrated using as examples the volterra and toda models a discrete differential analogue of the calogero degasperis equation is discussed in detail this example is used to show how to construct conservation laws higher symmetries and solutions for an equation obtained with the help of the scheme

non invertible transformations of differential difference equations we discuss aspects of the theory of non invertible transformations of differential difference equations and in particular the notion of miura type transformation we introduce the concept of non miura type linearizable transformation and we present techniques that allow one to construct simple linearizable transformations and might help one to solve classification problems this theory is illustrated by the example of a new integrable differential difference equation depending on five lattice points interesting from the viewpoint of the non invertible transformation which relate it to an itoh narita bogoyavlensky equation

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

conservation laws and symmetries for a nonholonomic deformed volterra equation a nonholonomic deformed volterra equation is studied its lax representation infinitely many of conservation laws and generalized commutation symmetries are given a degenerate recursion operator to generate the generalized symmetries is proposed

master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

differential difference equations associated with the fractional lax operators we study integrable hierarchies associated with spectral problems of the form p q where p and q are difference operators the corresponding nonlinear differential difference equations can be viewed as inhomogeneous generalizations of the bogoyavlensky type lattices while the latter turn into the korteweg de vries equation under the continuous limit the lattices under consideration provide discrete analogs of the sawada kotera and kaup kupershmidt equations the r matrix formulation and several of the simplest explicit solutions are presented

design parameterization for topology optimization by intersection of an implicit function abstract this paper introduces a new approach to topology optimization where the structural boundary is defined by the intersection of an implicit signed distance function and a cutting surface the cutting surface is discretized using finite element shape function polynomials and the nodal values become the design variables during optimization thus the design parameterization is separated from the analysis discretization which enables a reduction in the number of design variables compared with methods that use element wise variables the proposed method can obtain solutions with smooth boundaries and the parameterization allows solution by standard optimization methods several 2d and 3d examples are used to demonstrate the effectiveness of the method including minimization of compliance and complaint mechanism problems the results show that the method can obtain good solutions to well known problems with smooth clearly defined boundaries and that this can be achieved using significantly fewer design variables compared with element based methods

extensions of convexity models the notion of convexity model is introduced to provide a general frame for statistical theories of physical interest this frame encompasses in particular the classical and the quantum cases in a convexity model the states of the physical system and the convex structure they form play a basic role observables and related quantities are then naturally defined the notion of extensions of a convexity model is studied it appears physically relevant to cope with several needs paradigmatically with the one of viewing the physical system as a part of a compound system we focus attention on quantum like extensions of both the usual classical and quantum convexity models as well as on classical like extensions of the quantum model the behavior of state overlapping and state superposition under model extension is briefly examined

mastersymmetries angle variables and recursion operator of the relativistic toda lattice conserved quantities bi hamiltonian formulation and recursive structure of the relativistic toda lattice rt are obtained in an algorithmic way without making use of the lax representation furthermore for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries a new hierarchy of completely integrable systems is discovered which turns out to correspond to the negative of the hierarchy of rt thus it is shown that the full algebra of time dependent symmetry group generators for each member of the rt hierarchy is isomorphic to the algebra of first order differential operators with laurent polynomials as coefficients the surprising phenomenon is revealed that the members of the rt hierarchy are connected to their negative counterparts by explicit backlund transformation

topology optimization with multiple phase projection topology optimization is a powerful tool capable of generating new solutions to engineering design problems while these designs may offer optimal performance in a computational setting it is not uncommon for them to be impractical or unrealizable from engineering or fabrication points of view this challenge motivates the present work a methodology is proposed for restricting the minimum length scale of each material phase used in the design the technique allows a designer to for example prescribe a minimum allowable length scale of structural members solid phase as well as the minimum allowable length scale on holes void phase the proposed approach utilizes the heaviside projection method hpm to continuum topology optimization each material phase is associated with a design variable field that is projected onto element space using regularized heaviside functions the fields are independently projected and the resulting distributions are assembled to yield topology the technique is demonstrated on classic topology optimization problems including minimum compliance heat conduction and compliant mechanisms solutions are shown to be near discrete topologies satisfying minimum length scale criterion placed on each phase control over length scale is achieved implicitly and therefore the technique does not require additional constraints

algebraic entropy for semi discrete equations we extend the definition of algebraic entropy to semi discrete difference differential equations calculating the entropy for a number of integrable and non integrable systems we show that its vanishing is a characteristic feature of integrability for this type of equation

multiple scale analysis of discrete nonlinear partial difference equations the reduction of the lattice potential kdv we consider multiple lattices and functions defined on them we introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices we use these results to perform multiple scale reduction of the lattice potential korteweg de vries equation

structural topology optimization based on non local shepard interpolation of density field abstract this paper presents a non local density interpolation strategy for topology optimization based on nodal design variables in this method design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes by using the shepard family of interpolants the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point the employed interpolation scheme has an explicit form and satisfies non negative and range restricted properties required by a physically significant density interpolation since the discretizations of the density field and the displacement field are implemented on two independent sets of points the method is well suited for a topology optimization problem with a design domain containing higher order elements or non quadrilateral elements moreover it has the ability to yield mesh independent solutions if the radius of the influence domain is reasonably specified numerical examples demonstrate the validity of the proposed formulation and numerical techniques it is also confirmed that the method can successfully avoid checkerboard patterns as well as islanding phenomenon

an enhanced genetic algorithm for structural topology optimization genetic algorithms gas have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures in this paper a versatile robust and enhanced ga is proposed for structural topology optimization by using problem specific knowledge the original discrete black and white 0 1 problem is directly solved by using a bit array representation method to address the related pronounced connectivity issue effectively the four neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns a simpler version of the perimeter control approach is developed to obtain a well posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge free design to handle the problem of representation degeneracy effectively a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner an efficient fem based function evaluation method is developed to reduce the computational cost a dynamic penalty method is presented for the ga to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy with all these enhancements and appropriate choice of the ga operators the present ga can achieve significant improvements in evolving into near optimum solutions and viable topologies with checkerboard free mesh independent and hinge free characteristics numerical results show that the present ga can be more efficient and robust than the conventional gas in solving the structural topology optimization problems of minimum compliance design minimum weight design and optimal compliant mechanisms design it is suggested that the present enhanced ga using problem specific knowledge can be a powerful global search tool for structural topology optimization copyright 2005 john wiley sons ltd

a mixed fem approach to stress constrained topology optimization we present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion the main idea is to adopt a mixed finite element discretization scheme wherein not only displacements as usual but also stresses are the variables entering the formulation by doing so any stress constraint may be handled within the optimization procedure without resorting to post processing operation typical of displacement based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed two dual variational principles of hellinger reissner type are presented in continuous and discrete form that which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes int j numer meth engng 1984 24 3 359 373 extensive numerical simulations are performed and ongoing extensions outlined including the optimization of elastoplastic and incompressible media copyright 2007 john wiley sons ltd

design space optimization using a numerical design continuation method summary a generalized optimization problem in which design space is also a design to be found is de9ned and a numerical implementation method is proposed in conventional optimization only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted a design space is speci9ed by the number of design variables and the layout or con9guration to solve this type of design space problems a simple initial design space is selected and gradually improved while the usual design variables are being optimized to make the design space evolve into a better one one may increase the number of design variables but in this transition there are discontinuities in the objective and constraint functions accordingly the sensitivity analysis methods based on continuity will not apply to this discontinuous stage to overcome the di culties a numerical continuation scheme is proposed based on a new concept of a pivot phase design space two new categories of structural optimization problems are formulated and concrete examples shown copyright 2001 john wiley sons ltd

the lattice schwarzian kdv equation and its symmetries in this paper we present a set of results on the symmetries of the lattice schwarzian korteweg de vries 1skdv equation we construct the lie point symmetries and using its associated spectral problem an infinite sequence of generalized symmetries and master symmetries we finally show that we can use master symmetries of the 1skdv equation to construct non autonomous non integrable generalized symmetries

topology optimization using a continuous density field and adaptive mesh refinement summary a new method of topology optimization is introduced in which a continuous material field is combined with adaptive mesh refinement using a continuous material field with separate analysis and design meshes allows the method to produce optimal designs that are free of numerical artifacts like checkerboard patterns and material islands adaptive mesh refinement is then applied to both meshes to precisely locate the optimal boundary of the final structure a helmholtz type density filter is used to prevent the appearance of small topological features as the mesh refinement proceeds results are presented for several test problems including problems with geometrically complex domain boundaries this article is protected by copyright all rights reserved

quad equations and auto backlund transformations of nls type systems treating an integrable quad equation along with its two generalized symmetries as a compatible system allows one to construct an auto backlund transformation for solutions of the related nls type system a fixed periodic reduction of the quad equation yields a quasi periodic reduction of its generalized symmetries that turns them into differential constraints compatible with the nls type system

spectral properties of observables and convex mappings in quantum mechanics observables of a physical system can be identified with convex mappings sending states into probability measures the properties of these mappings determine the spectral properties of the observables the mutually exclusive cases of injective and surjective mappings are characterized and the convex structure of their ranges are investigated in particular the validity of the krein milman property for these convex sets is studied

a discrete time relativistic toda lattice for each of the two simplest hamiltonian flows from the relativistic toda hierarchy we introduce two integrable symplectic discretizations all four discrete time systems are demonstrated to belong to the same hierarchy and to exemplify the general scheme for symplectic maps on groups equipped with quadratic poisson brackets the initial value problem for the difference equations is solved in terms of a factorization problem in a group interpolating hamiltonian flows are found for all maps

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

a polytree based adaptive polygonal finite element method for multi material topology optimization abstract this study presents a polytree based adaptive methodology for multi material topology optimization mmtop polytree data structure is introduced as a general recursive multi level mesh that is automatically refined in processing based on error analysis in order to resolve hanging nodes in element edges the wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons a new definition of filter radius is also proposed to improve the efficiency of filters and optimized results the combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases including void phase but also decreases the computing time of the overall process in comparison to using the regular fine meshes several benchmark and practical problems are considered to show distinct features of the proposed method

on the integrability of a new discrete nonlinear schrodinger equation we consider the nonlinear schrodinger equation on the lattice introduced by leon and manna two years ago to describe the slowly varying envelope approximation of some nonlinear differential difference equations onlinear klein gordon chain etc we show that this equation does not admit local generalized symmetries of order greater than three in such a way we prove that the leon and manna discrete nonlinear schrodinger equation does not have the same integrability properties as the toda lattice equation from which it has been derived at the end we provide some reasoning to justify the result obtained

multiple scale analysis of dynamical systems on the lattice we propose a new approach to the multiple scale analysis of difference equations in the context of the finite operator calculus we derive the transformation formulae that map any given dynamical system continuous or discrete into a rescaled discrete system by generalizing a classical result due to jordan under suitable analytical hypotheses on the function space we consider the rescaled equations are of finite order our results are applied to the study of multiple scale reductions of dynamical systems and in particular to the case of a discrete nonlinear harmonic oscillator

improving the efficiency of large scale topology optimization through on the fly reduced order model construction summarytopology optimization of large scale structures is computationally expensive notably because of the cost of solving the equilibrium equations at each iteration reduced order models by projection also known as reduced basis models have been proposed in the past for alleviating this cost we propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures the novel approach is based on constructing the reduced basis on the fly using previously calculated solutions of the equilibrium equations the reduced basis is thus adaptively constructed and enriched based on the convergence behavior of the topology optimization a direct approach and an approach with adjusted sensitivities are described and their algorithms provided the approaches are tested and compared on various 2d and 3d minimum compliance topology optimization benchmark problems computational cost savings by up to a factor of 12 are demonstrated using the proposed methods copyright textcopyright 2014 john wiley sons ltd

the structure of classical extensions of quantum probability theory on the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models we prove that every such classical extension is essentially given by the so called misra bugajski reduction map we consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective hilbert space as phase space and discuss features of the induced hidden variable model moreover some relevant technical results on the topology and borel structure of the projective hilbert space are reviewed

peculiar symmetry structure of some known discrete nonautonomous equations we study the generalized symmetry structure of three known discrete nonautonomous equations one of them is the semidiscrete dressing chain of shabat two others are completely discrete equations defined on the square lattice the first one is a discrete analogue of the dressing chain introduced by levi and yamilov the second one is a nonautonomous generalization of the potential discrete kdv equation or in other words the h1 equation of the well known adler bobenko suris list we demonstrate that these equations have generalized symmetries in both directions if and only if their coefficients depending on the discrete variables are periodic the order of the simplest generalized symmetry in at least one direction depends on the period and may be arbitrarily high we substantiate this picture by some theorems in the case of small periods in case of an arbitrarily large period we show that it is possible to construct two hierarchies of generalized symmetries and conservation laws the same picture should take place in case of any nonautonomous equation of the adler bobenko suris list

a multi resolution method for 3d multi material topology optimization abstract this paper presents a multi resolution implementation in 3d for multi material topology optimization problem an alternating active phase algorithm where the problem at hand is divided into a series of the traditional material void phase topology optimization is employed for the multi material problem different levels of discretization are used for the displacement mesh design variable mesh and density mesh which provides higher resolution designs for the solutions a projection scheme is employed to compute the element densities from design variables and control the length scale of the material density simple block coordinate descent method similar to the gauss seidel technique is used to solve the subproblems several 3d numerical examples are presented to demonstrate the ease and the effectiveness of the proposed implementation incorporating the multi resolution method into the multi material approach robust designs with improved resolution can be achieved for real life problems with complex geometries

towards classification of dimensional integrable equations integrability conditions i in this paper we attempt to extend the symmetry approach well developed in the case of 1 1 dimensional equations to the 2 1 dimensional case presence of nonlocal terms in symmetries and conservation laws is the main feature of integrable 2 1 dimensional equations we have introduced a concept of quasi local functions to characterize nonlocalities we have found a few first integrability conditions for a class of scalar equations in terms of quasi local functions and have demonstrated that they are suitable for testing integrability

are there contact transformations for discrete equations we define infinitesimal contact transformations for ordinary difference schemes as transformations that depend on k 1 lattice points k 1 and can be integrated to form a local or global lie group we then prove that such contact transformations do not exist

extension of the module of invertible transformations classification of integrable systems we demonstrate that for the systems of equations which are invariant under a point group or possess conservation laws of the zeroth or first order a nontrivial extension of the module of invertible transformations is possible that simplifies greatly a classification of the integrable systems of equations here we present an exhaustive list and a classification of the second order systems of the formu t u xx f u v u x v x v t v xx g u v u x v x which possess the conservation laws of higher order the reduction group approach allows us to define the lax type representations for some new equations of our list

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

on the integrability of a new lattice equation found by multiple scale analysis in this paper we discuss the integrability properties of a nonlinear partial difference equation on the square obtained by the multiple scale integrability test from a class of multilinear dispersive equations defined on a four point lattice

multiscale expansion of the lattice potential kdv equation on functions of an infinite slow varyness order we present a discrete multiscale expansion of the lattice potential korteweg de vries lpkdv equation on functions of an infinite order of slow varyness to do so we introduce a formal expansion of the shift operator on many lattices holding at all orders the lowest secularity condition from the expansion of the lpkdv equation gives a nonlinear lattice equation depending on shifts of all orders of the form of the nonlinear schrodinger equation

miura type transformations for lattice equations and lie group actions associated with darboux lax representations miura type transformations mts are an essential tool in the theory of integrable nonlinear partial differential and difference equations we present a geometric method to construct mts for differential difference lattice equations from darboux lax representations dlrs of such equations the method is applicable to parameter dependent dlrs satisfying certain conditions we construct mts and modified lattice equations from invariants of some lie group actions on manifolds associated with such dlrs using this construction from a given suitable dlr one can obtain many mts of different orders the main idea behind this method is closely related to the results of drinfeld and sokolov on mts for the partial differential kdv equation considered examples include the volterra narita itoh bogoyavlensky toda and adler postnikov lattices some of the constructed mts and modified lattice equations seem to be new

on the hierarchy of the landau lifshitz equation abstract the landau lifshitz equation ll equation is shown to have infinitely many commuting one parameter symmetry groups and constants of motion in involution the infinitesimal generators of these one parameter symmetry groups define the hierarchy of the ll equation we give explicit formulas for these generators and for the conserved densities furthermore infinitely many time dependent symmetry groups and conservation laws are constructed all results are obtained via a simple direct lie algebra approach and without making any use of the quantum inverse scattering method based on the ingenious results of r j baxter

a polytree based adaptive polygonal finite element method for topology optimization summary we propose a highly effective approach using a novel adaptive methodology to perform topology optimization with polygonal meshes called polytree meshes polytree is a hierarchical data structure based on the principle of recursive spatial decomposition of each polygonal element with n nodes into n 1 arbitrary new polygonal elements enabling more efficient utilization of unstructured meshes and arbitrary design domains in topology optimization in order to treat hanging nodes after each optimization loop we define the wachspress coordinate on a reference element and then utilize an affine map to obtain shape functions and their gradients on arbitrary polygons with n vertices and m hanging nodes called side nodes as the polygonal element with n m vertices the polytree meshes do not only improve the boundary description quality of the optimal result but also reduce the computational cost of optimization process in comparison with the use of uniformly fine meshes several numerical examples are investigated to show the high effectiveness of the proposed method this article is protected by copyright all rights reserved

reducing dimensionality in topology optimization using adaptive design variable fields topology optimization methodologies typically use the same discretization for the design variable and analysis meshes analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense this paper proposes leveraging properties of the heaviside projection method hpm to separate the design variable field from the analysis mesh in continuum topology optimization hpm projects independent design variables onto element space over a prescribed length scale a single design variable therefore influences several elements creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space the algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses the technique is demonstrated on minimum compliance maximum stiffness problems solved using continuous optimization and genetic algorithms for the former the proposed algorithm typically identifies solutions having objective functions within 1 of those found using full design variable fields computational savings are minor to moderate for the minimum compliance formulation with a single constraint and are substantial for formulations having many local constraints when using genetic algorithms solutions are consistently obtained on mesh resolutions that were previously considered intractable copyright 2009 john wiley sons ltd

effect algebras and statistical physical theories the dichotomic physical quantities of a physical system can be naturally hosted in a mathematical structure called effect algebra of which orthomodular posets and boolean algebras are particular examples we examine how effect algebras arise inside statistical physical theories and conversely we study to what extent an effect algebra can be taken as a primitive structure on which a satisfactory statistical physical model equipped with a convex set of states can be constructed

linearization through symmetries for discrete equations we show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete cole hopf transformation if the equation is linearizable then the symmetry provides the linearizing transformation at the end we present a few examples of applications for equations defined on four lattice points

multi scale expansions in the theory of systems integrable by the inverse scattering transform abstract it is shown that using multi scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the ist method into other systems of this type

discrete reductive perturbation technique we expand a partial difference equation p e on multiple lattices and obtain the p e which governs its far field behavior the perturbative reductive approach is here performed on well known nonlinear p es both integrable and nonintegrable we study the cases of the lattice modified korteweg de vries mkdv equation the hietarinta equation the lattice volterra kac van moerbeke equation and a nonintegrable lattice kdv equation such reductions allow us to obtain many new p es of the nonlinear schrodinger type

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations