a generalization of the e algorithm abstract salam a on a generalization of the algorithm journal of computational and applied mathematics 46 1993 455 464 the algorithm is a quite powerful algorithm for accelerating the convergence of some classes of sequences it was generalized in two different ways some years ago in this paper a determinantal formula for its second generalization is obtained then its kernel the set of sequences which are transformed into a constant sequence is given as the solution of a difference equation

two continuous models for the dynamics of sandpile surfaces we consider a modified bouchaud cates ravi prakash edwards model for pile surface dynamics and show that in the long scale limit this model converges to a quasistationary model of pile growth in the form of an evolutionary variational inequality

structural optimization a new dual method using mixed variables a new and powerful mathematical programming method is described which is capable of solving a broad class of structural optimization problems the method employs mixed direct reciprocal design variables in order to get conservative first order approximations to the objective function and to the constraints by this approach the primary optimization problem is replaced with a sequence of explicit subproblems each subproblem being convex and separable it can be efficiently solved by using a dual formulation an attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs examples of application to real life aerospace structures are offered to demonstrate the power and generality of the approach presented

optimality conditions for shape and topology optimization subject to a cone constraint this paper provides first order necessary optimality conditions for simultaneous shape and topology optimization subject to a cone constraint these conditions are expressed with the help of the shape and topological derivatives of the objective and constraint functionals several examples of applications are given

sandpiles and river networks extended systems with nonlocal interactions a proposed continuum model of sandpile growth is able to demonstrate instantaneous long range interactions and sudden collapses typical of real sandpiles the model is shown to be equivalent to an evolutionary quasivariational inequality in the limiting case of zero angle of repose the inequality describes water transport in a river network

a derivation of extrapolation algorithms based on error estimates in this paper we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolation processes it is shown that this approach leads to a unified derivation of many extrapolation algorithms and related devices to general results about their kernels and that it opens the way to many new algorithms their convergence and acceleration properties could also be studied within this framework

stress related topology optimization via level set approach abstract considering stress related objective or constraint functions in structural topology optimization problems is very important from both theoretical and application perspectives it has been known however that stress related topology optimization problem is challenging since several difficulties must be overcome in order to solve it effectively traditionally simp solid isotropic material with penalization method was often employed to tackle it although some remarkable achievements have been made with this computational framework there are still some issues requiring further explorations in the present work stress related topology optimization problems are investigated via a level set based approach which is a different topology optimization framework from simp numerical examples show that under appropriate problem formulations level set approach is a promising tool for stress related topology optimization problems

a reduced multiscale model for nonlinear structural topology optimization abstract this paper presents a reduced multiscale model for macroscopic structural design considering microscopic material nonlinear microstructures this work introduces reduced order model rom to alleviate the heavy computational demand of nonlinear nested multiscale procedures particularly in an optimization framework which requires multiple loops involving similar computations the surrogate model constructed using proper orthogonal decomposition pod and diffuse approximation reduces the computational effort for solving the microscopic boundary value problems multiscale analysis model fe 2 is applied to link structure and microstructures in the two scales maximum stiffness design of the macroscopic structure is realized using a discrete level set topology optimization model it is shown by means of numerical tests that the reduced multiscale model provides reasonable designs as compared to those obtained by the unreduced model while with a significantly reduced computational effort

approximation properties of the q sine bases for graphic 1 the eigenfunctions of the nonlinear eigenvalue problem associated with the one dimensional q laplacian are known to form a riesz basis of l 2 0 1 we examine in this paper the approximation properties of this family of functions and its dual and establish a non orthogonal spectral method for the p poisson boundary value problem and its corresponding parabolic time evolution initial value problem with stochastic forcing the principal objective of our analysis is the determination of optimal values of q for which the best approximation is achieved for a given p problem 1 embed inline graphic 1 gif

an enhanced aggregation method for topology optimization with local stress constraints abstract by introducing a new reduction parameter into the kreisselmeier steihauser k s function this paper presents a general k s formulation providing an approximation to the feasible region restricted by active constraints the approximation is highly accurate even when the aggregation parameter takes a relatively small value numerical difficulties such as high nonlinearity and serious violation of local constraints that may be exhibited by the original k s function are thus effectively alleviated in the considered topology optimization problem the material volume is to be minimized under local von mises stress constraints imposed on all the finite elements an enhanced aggregation algorithm based on the general k s function in conjunction with a removal and re generation strategy for selecting the active constraints is then proposed to treat the stress constraints of the optimization problem numerical examples are given to demonstrate the validity of the present algorithm it is shown that the proposed method can achieve reasonable solutions with a high computational efficiency in handling large scale stress constrained topology optimization problems

a newly developed qp relaxation method for element connectivity parameterization to achieve stress based topology optimization for geometrically nonlinear structures the aim of this work is to present a novel computational approach to employ the stress based topology optimization method stom to minimize the volume subject to the locally defined stress constraints of a geometrically nonlinear structure in the framework of the element connectivity parameterization ecp method considering the locally defined stress constraints in topology optimization to is a classic and challenging engineering problem and successful optimization procedures have recently been developed using the density based to method for linear elastic structures however no study has yet considered the static failure constraint when using to for a geometrically nonlinear structure therefore the present study develops a novel computational approach for the stom for a geometrically nonlinear structure to successfully optimize a geometrically nonlinear structure the unstable element issue must be properly addressed in addition to the stress singularity issue the existence of a large number of constraints and the highly nonlinear behavior of the local stress constraints to effectively resolve these issues this research adopts the ecp method to interpolate and optimize the connectivities among solid finite elements furthermore we find that a stress singularity issue linked to the local optima issue arises in the ecp method that is different from that of the density based to by investigating the singularity behavior in detail we develop a new qp relaxation method that is suitable for the ecp method to demonstrate the improved capability of the proposed ecp method with the modified qp relaxation several two dimensional to problems are solved

optimal design of trusses with geometric imperfections accounting for global instability abstract a topology optimization method is proposed for the design of trusses with random geometric imperfections due to fabrication errors this method is a generalization of a previously developed perturbation approach to topology optimization under geometric uncertainties the main novelty in the present paper is that the objective function includes the nonlinear effects of potential buckling due to misaligned structural members solutions are therefore dependent on the magnitude of applied loads and the direction of resulting internal member forces whether they are compression or tension direct differentiation is used in the sensitivity analysis and analytical expressions for the associated derivatives are derived in a form that is computationally efficient a series of examples illustrate how the effects of geometric imperfections and buckling may have substantial influence on truss design monte carlo simulation together with second order elastic analysis is used to verify that solutions offer improved performance in the presence of geometric uncertainties

variational inequalities in critical state problems similar evolutionary variational inequalities appear as convenient formulations for continuous quasistationary models for sandpile growth formation of a network of lakes and rivers magnetization of type ii superconductors and elastoplastic deformations we outline the main steps of such models derivation and try to clarify the origin of this similarity new dual variational formulations analogous to mixed variational inequalities in plasticity are derived for sandpiles and superconductors

on regularity of transport density in the monge kantorovich problem we show that the optimal regularity result for the transport density in the classical monge kantorovich optimal mass transport problem with the measures having summable densities is a sobolev differentiability along transport rays

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

on the variational formulation of stress constraints in isogeometric design the isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems this unification is brought about by employing the same basis functions to describe exact geometry and to approximate the physical response the isogeometric framework has recently been extended to include design optimisation a novel variational stress constraint formulation is proposed in this paper that further extends recent efforts in isogeometric design the weak form of the constraint is obtained by utilising the same basis functions not only to represent geometry response and design variables but also lagrange multipliers in this form the proposed variational formulation fits seamlessly with the underlying analysis and design environment the variational constraint formulation is compared to the widely utilised stress averaging and collocation approaches to this end volume minimisation of elastic arches subjected to stress constraints is considered results indicate that the proposed variational approach has the best convergence rates and utilises a much smaller number of discrete constraints than competing methods

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

an alternative truly mixed formulation to solve pressure load problems in topology optimization abstract the paper deals with an alternative formulation for the topology optimization of structures acted upon by pressure loads exploiting finite elements techniques that are able to handle incompressible materials the method firstly presented in o sigmund p m clausen topology optimization using a mixed formulation an alternative way to solve pressure load problems comput methods appl mech engrg 196 2007 1874 1889 33 consists in exploiting the modeling of fluid incompressibility within a topology optimization framework the implementation of a fluid phase enables to transfer pressure loads from the domain boundaries to the evolving edges of optimal design without relying on more complex techniques traditionally employed to recover the load application surfaces at each step of the minimization process in this context the main numerical trouble is therefore the application of finite elements techniques to solve incompressible materials analysis this topic in fact cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements which are well known to be affected by the locking phenomenon while sigmund and clausen 2007 uses a displacement pressure finite element discretization to solve the problem the approach herein presented consists in the adoption of a truly mixed variational formulation coupled to a discretization based on the johnson and mercier finite element that both pass the inf sup conditions of the problem even in the presence of incompressible materials the well known method of moving asymptotes mma is adopted in the numerical studies presented along with a particular density interpolation to model the presence of a fluid and solid phase within the same design the adopted scheme is especially conceived to avoid the arising of numerical instabilities that may arise within the optimization procedure when handling incompressible material moreover the accuracy and stability in stress evaluation provided by the truly mixed setting are herein exploited to introduce an alternative procedure that implements pressure constraints to avoid optimal designs that present cavities filled by fluid

variational evolution problems and nonlocal geometric motion we consider two variational evolution problems related to monge kantorovich mass transfer these problems provide models for collapsing sandpiles and for compression molding we prove the following connection between these problems and nonlocal geometric curvature motion the distance functions to surfaces moving according to certain nonlocal geometric laws are solutions of the variational evolution problems thus we do the first step of the proof of heuristics developed in earlier works the main techniques we use are differential equation methods in the monge kantorovich theory

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

nonlocal optimal design a new perspective about the approximation of solutions in optimal design abstract it is well known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations in the present work a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem the new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals the purpose of this paper is to prove existence of optimal design for the new model this work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional see 14

numerical simulations in 3d heat conduction minimizing the quadratic mean temperature gradient by an optimality criteria method we analyze an optimal design problem in three dimensional 3d heat conduction given fixed amounts of two isotropic conducting materials decide how we are to mix them in a 3d domain to minimize the quadratic mean temperature gradient by using an optimality criteria method we provide some numerical evidence that tartar s result see in calculus of variations homogenization and continuum mechanics g buttazzo g bouchitte and p suquet eds world scientific river edge nj 1994 pp 279 296 is verified in three dimensions when the target vector field is zero

the mesa problem for neumann boundary value problem abstract in this paper we study the singular limit of the porous medium equation u t u m g x u as m in a bounded domain with neumann boundary condition

stress constrained topology optimization with free form design domains abstract this paper aims at dealing with realistic and challenging design problems of stress constrained topology optimization with free form design domains first the concept of level set function lsf based modelers is introduced to transform this kind of problems into the boolean conjunction operation of a topology variation modeler tvm onto a free form design domain modeler fddm such an operation is mathematically realized by means of the so called r functions in the form of implicit lsfs within this framework topology optimization problems are classified into two general cases depending upon the existence of non designable solid feature analytical sensitivity analysis formulas are further derived compared with the existing level set based method the important sensitivity property of design domain preserving makes it possible to avoid automatically the boundary violation of the design domain caused by the zero level set movement and both the topology and boundary shape of the free form design domain can be simultaneously optimized second the implementation of the finite cell method fcm ensures the stress computing accuracy in the fixed mesh due to the use of high order shape functions and adaptive integration scheme the combination of the active set strategy and the dynamic aggregation technique also reduces the number of local stress constraints greatly finally representative examples are presented to illustrate the conveniences and effectiveness of the proposed method

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

stress based topology optimization method for steady state fluid structure interaction problems abstract this research developed a new stress based topology optimization method stom for a steady state fluid structure interaction fsi problem that minimizes the volume subject to the local stress constraints despite numerous studies on stom challenging optimization issues related to stress based topology optimization to procedures for fluid structure multiphysics systems still exist critical issues involved in creating a successful to for an fsi structure include the interpolation approach between the fluid equation and the structure equation with respect to locally defined design variables the mutual multiphysics coupling boundary conditions at dramatically evolving interfacing boundaries and a clear interpretation of the governing equations and the interaction boundary conditions for spatially varying intermediate design variables in addition to these three issues which are related to multiphysics equations there are three important considerations related to the stom the stress singularity issue the issues of multiple constraints and the highly nonlinear behavior of the stress constraints to resolve all of the aforementioned issues we applied a monolithic analysis integrating the qp relaxation method and the global p norm approach using the present method we created optimal layouts that minimize the volume constraining local stress values for a steady state fluid and structural interaction system

fast slow diffusion and collapsing sandpiles we regard the limit as p of the flow governed by the p laplacian as providing a simplistic model for the collapse of an initially unstable sandpile upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases involving nonlocal geometric curvature motion in particular we note that such models entail instantaneous mass transfer governed by monge kantorovich theory

a stochastic model for growing sandpiles and its continuum limit we study a stochastic model for the sandpile growth the height of the sandpile above every site of a d dimensional lattice is denoted by an integer the height changes randomly and after a suitable scaling of the time and space we show that the rescaled height converges to a deterministic function the limit is governed by a nonlinear evolution equation

phase field relaxation of topology optimization with local stress constraints we introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase field method in the basic formulation we have a pde constrained optimization problem where the finite element and design analysis are solved simultaneously the starting point of the relaxation is a reformulation of the material problem involving linear and 0 1 constraints only the 0 1 constraints are then relaxed and approximated by a cahn hilliard type penalty in the objective functional which yields convergence of minimizers to 0 1 designs as the penalty parameter decreases to zero a major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter the relaxation scheme yields a large scale optimization problem with a high number of linear inequality constraints we discretize the problem by finite elements and solve the arising finite dimensional programming problems by a primal dual interior point method numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach

granular flow in a three dimensional rotating container we considered a flow of a cohesionless granular material in a partially filled three dimensional rotating container a model is suggested to describe a density of the surface flow and the profile of the free surface of the granular material it is shown that when the container has an ellipsoidal shape the obtained system of partial differential equations can be reduced to a set of two ordinary differential equations

quasiconvexification in 3 d for a variational reformulation of an optimal design problem in conductivity abstract we analyze a typical 3 d conductivity problem which consists in seeking the optimal layout of two materials in a given design domain r 3 by minimizing the l 2 norm of the electric field under a constraint on the amount on each material that we can use we utilize a characterization of the 3 d divergence free vector fields which is especially appropriate for a variational reformulation by using gradient young measures as a main tool we can give an explicit form of the constrained quasiconvexification of the cost density this result is similar to the one in the 2 d situation however the characterization of the divergence free vector fields introduces a certain nonlinearity in the problem that needs to be addressed properly

brittle and ductile failure constraints of stress based topology optimization method for fluid structure interactions abstract this study considers failure theories for brittle and ductile materials in the stress based topology optimization method stom for steady state fluid structure interactions fsi in some relevant studies the subject of the stress based topology optimization to minimize volumes with local von mises stress constraints has been researched however the various failure theories for ductile and brittle materials such as the maximum shear stress theory the brittle and ductile mohr coulomb theory and the drucker prager theory have not been considered for successful stom for fsi in addition to alleviating physics interpolation issues between structure and fluid and some numerical issues related to stom the mathematical characteristics of the various failure theories should be properly formulated and constrained to resolve all the involved computational issues the present study applies the monolithic analysis method the qp relaxation method and the p norm approach to the failure constraints the present topology optimization method can create optimal layouts while minimizing volume constraining local failure constraints for ductile and brittle materials for steady state fluid and structural interaction system

optimal design in conductivity under locally constrained heat flux this paper addresses the relaxation of an optimal design problem in conductivity under a point wise constraint on the heat flux by using a variational approach developed by the authors in a previous work we obtained a complete relaxation involving the explicit computation of constrained quasiconvex envelopes this relaxed formulation turns out to be very simple in the remarkable compliance case for which numerical simulations are performed those simulations show that our approach and results are interesting from a theoretical point of view as well as from a practical perspective the emphasis of our work is placed on the local constraint on the heat flux which seems to be a new ingredient for this kind of optimal design problems

optimal design via variational principles the three dimensional case the analysis of some three dimensional optimal design problems leads us to study variational principles under curl free and divergence free constraints simultaneously we explicitly exploit the relationship between curl free and div free restrictions in order to take advantage of the accumulated experience in the classical curl case by the introduction of potentials our discussion takes place in the three dimensional situation this is a first contribution in the sense that we only deal with the most basic issues

local stress constrained and slope constrained sand topology optimisation summary we study the alternative simultaneous analysis and design sand formulation of the local stress and slope constrained topology design problem it is demonstrated that a standard trust region lagrange newton sequential quadratic programming sqp type algorithm based in this case on strictly convex and separable approximate subproblems may converge to singular optima of the local stress constrained problem without having to resort to relaxation or perturbation techniques moreover due to the negation of the sensitivity analyses in sand the density and displacement variables are independent and the immense sparsity of the sand problem solutions to large scale problem instances may be obtained in a reasonable amount of computation time this article is protected by copyright all rights reserved

eulerian shape design sensitivity analysis and optimization with a fixed grid conventional shape optimization based on the finite element method uses lagrangian representation in which the finite element mesh moves according to shape change while modern topology optimization uses eulerian representation in this paper an approach to shape optimization using eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed a continuum geometric model is defined on the fixed grid of finite elements an active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization in which each element has a unique shape density the shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements using this transformation it has been shown that the shape design problem can be treated as a parameter design problem which is a much easier method than the former a detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary the accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement two design optimization problems are presented to show the feasibility of the proposed design approach

topology optimization with local stress constraint based on level set evolution via reaction diffusion abstract this work focuses the structural topology optimization problem of mass minimization subject to local stress constraints to this aim two related issues are addressed the first one is the successful strategy used to define local stress constraints by means of an augmented lagrangian approach the second and main contribution of the present paper is the use of a reaction diffusion equation to guide via evolution of a level set the design optimization sequence the advantages of this strategy are twofold firstly it allows the creation of new holes during the optimization process a significant feature for a true topological optimization method secondly reinitialization steps usually found in classical hamilton jacobi based evolution are eliminated with a significant improvement in convergence ease a set of benchmark examples in two dimensions are presented numerical results show the efficiency of the algorithm to create new holes identify stress concentrations and to provide stable optimization sequences converging to local minima defined by stress saturated designs

evolution monge kantorovich equation abstract the paper deals with monge kantorovich equation mk for short in an open bounded domain with dirichlet boundary condition we study existence and uniqueness of a solution to the associated evolution problem emk for short and we prove the convergence to a solution of mk when time goes to a solution is a couple u where u is the potential and is the transportation flux we study the problem for a given radon measure source term and we show how to use the numerical method of dumont and igbida 2009 22 to provide numerical approximation of the solution of mk

toward a stress based topology optimization procedure with indirect calculation of internal finite element information abstract in this work a novel computational approach is developed for the gradient based stress based topology optimization method where the volume is minimized according to locally defined stress constraints of static failure criteria in the framework of tailored finite element fe software without direct access to an internal finite element information database tailored finite element codes that require substantial understanding and modification reprogramming or do not directly provide internal finite element information have rarely been used for stress based topology optimization with solid isotropic material with penalization simp methods therefore much research has been confined to two dimensional problems with rectilinear basic finite elements as complex three dimensional geometries with advanced finite element formulations are not supported to overcome this problem we developed a new computational procedure for sensitivity analysis without direct access to an internal information database a task that has previously been regarded as almost impossible since the calculation of linear strain displacement matrices is required in the sensitivity analysis the present procedure includes a node selection algorithm to efficiently calculate the matrices of irregular shaped finite elements by displacement perturbations the benefits of the present approach are that well established powerful finite element codes i e commercial or sophisticated public fe codes can be easily incorporated for linear stress based topology optimization and any type of finite element formulation can be readily employed in contrast to classical sensitivity analysis small computations are used for sensitivity analysis in this work to demonstrate the validity and efficiency of the present procedure and approach several topology optimization problems with 3d and shell elements are solved

topological derivative based topology optimization of structures subject to drucker prager stress constraints abstract an algorithm for topology optimization of elastic structures under plane stress subject to the drucker prager stress constraint is presented the algorithm is based on the use of the topological derivative of the associated objective functional in conjunction with a level set representation of the structure domain in this context a penalty functional is proposed to enforce the point wise stress constraint and a closed formula for its topological derivative is derived the resulting algorithm is of remarkably simple computational implementation it does not require post processing procedures of any kind and features only a minimal number of user defined algorithmic parameters this is in sharp contrast with current procedures for topological structural optimization with local stress constraints the effectiveness and efficiency of the algorithm presented here are demonstrated by means of numerical examples the examples show in particular that it can easily handle structural optimization problems with underlying materials featuring strong asymmetry in their tensile and compressive yield strengths

basis properties of the p q sine functions we improve the currently known thresholds for basisness of the family of periodically dilated p q sine functions our findings rely on a beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity we also determine refined bounds on the riesz constant associated with this family these results seal mathematical gaps in the existing literature on the subject

stability of finite element models for distributed parameter optimization and topology design abstract we address a problem of numerical instability that is often encountered in finite element solutions of distributed parameter optimization and variable topology shape design problems we show that the cause of this problem is numerical rather than physical in nature we consider a two field distributed parameter optimization problem involving a design field and a response field and show that the optimization problem corresponds to a mixed variational problem an improper selection of the discrete function spaces for these two fields leads to grid scale anomalies in the numerical solutions to optimization problems similar to those that are sometimes encountered in mixed formulations of the stokes problem we present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models the general theoretical framework is specialized to analyze the stability of specific optimization problems and stability results for various mixed finite element models are presented we propose patch tests that are useful in identifying unstable elements

topology optimization with mixed finite elements on regular grids abstract recently new families of mixed finite elements have been proposed to address the analysis of linear elastic bodies on regular grids adopting a limited number of degrees of freedom per element a two dimensional mixed discretization is implemented to formulate an alternative topology optimization problem where stresses play the role of main variables and both compressible and incompressible materials can be dealt with the structural compliance is computed through the evaluation of the complementary energy whereas the enforcement of stress constraints is straightforward numerical simulations investigate the features of the proposed approach comparisons with a conventional displacement based scheme are provided for compressible materials stress constrained solutions for structures made of incompressible media are introduced

stress based topology optimization of continuum structures under uncertainties abstract this work addresses the use of the topology optimization approach to the design of continuum structures with failure constraints under the hypothesis of uncertainties in the spatial distribution of young s modulus to this end the first order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to represent the random field the objective is the minimization of the amount of material used in the design subjected to local stress constraints under uncertainties the probability of failure is bounded by the one sided chebychev inequality since the exact probability distribution function of the stress constraints is not known in advance the effective probability of failure of the obtained optimal designs is validated with the use of the monte carlo simulation approach indicating that the probability of failures of the topologies obtained with the stochastic approach is within the bounds provided by the one sided chebychev inequality the optimization problem is solved by means of the augmented lagrangian method in order to address the large number of constraints associated to this kind of formulation it is shown that the correlation length and the number of standard deviations considered in the formulation play an important role in both the obtained topology and effective probability of failure

a general framework for robust topology optimization under load uncertainty including stress constraints we present a worst case approach for topology optimization under load uncertainty based on a general problem formulation involving maxima of quadratic functions of an uncertain load vector as both

explicit computation of the relaxed density coming from a three dimensional optimal design problem we analyze the relaxation of a typical three dimensional optimal design problem in conductivity consisting of minimizing the square of the l2 norm of the underlying electric potential under a volume constraint via a variational reformulation of the problem our main achievement is the explicit characterization of the constrained quasiconvexification of the cost density

discrete collapsing sandpile model abstract our main goal is to introduce and study a discrete model for the collapsing of a pile of cubes this is a typical example of self organized critical phenomena exhibited by a critical slope we prove the existence and uniqueness of the solution for the model then by using dual arguments we study the numerical computation of the solution and we present some numerical simulations

a lagrangian scheme for the solution of the optimal mass transfer problem a lagrangian method to numerically solve the l 2 optimal mass transfer problem is presented the initial and final density distributions are approximated by finite mass particles having a gaussian kernel mass conservation and the hamilton jacobi equation for the potential are identically satisfied by constant mass transport along straight lines the scheme is described in the context of existing methods to solve the problem and a set of numerical examples including applications to medical imagery are presented

singular limit of solutions of the very fast diffusion equation we prove that the distribution solutions of the very fast diffusion equation u t um m u t um m u 0u 0 in rn 0 rn 0 u x 0 u0 x u x 0 u0 x in rnrn where m n 2p n 2 and u0 x e x 2 u0 x e x 2 for any x r1 x r1 where e 0e 0 r1 0r1 0 m0 1 m0 n 2p 1 m0 n 2 we prove that the solution of the above problem will converge uniformly on every compact subset of rn 0 rn 0 to the maximal solution of the equation vt logvvt logv v x 0 u0 x v x 0 u0 x as m 0 m 0 for any smooth bounded domain rn rn m0 1 m0 max 1 n 2 p 1 m0 max 1 n 2 we prove the existence and uniqueness of solutions of the dirichlet problem u t um m u t um m u 0u 0 in 0 0 u u0u u0 in u gu g on 0 0 with either finite or infinite positive boundary value gg we also prove a similar convergence result for the solutions of the above dirichlet problem as m 0m 0

stress related topology optimization of continuum structures involving multi phase materials considering stress constraints in structural topology optimization is very important from both theoretical and application perspectives most of the corresponding studies in literature however are only focused on the problems involving single phase homogenous materials in the present paper stress constrained topology optimization of continuum structures involving multi phase heterogeneous materials is investigated a level set based variational consistent solution framework is developed numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

asymptotic properties in parabolic problems dominated by a p laplacian operator with localized large diffusion abstract this paper is concerned with upper semicontinuity of the family of attractors associated with nonlinear reaction diffusion equations with principal part governed by a degenerate p laplacian in which the diffusion d blows up in localized regions inside the domain

obstructions to existence in fast diffusion equations abstract the study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory thus in the sub field of very fast diffusion it is known that the cauchy problem can be ill posed either because of non uniqueness or because of non existence of solutions with small data the equations we consider take the general form u t d u u x u x x or its several dimension analogue fast diffusion means that d at some values of the arguments typically as u 0 or u x 0 here we describe two different types of non existence phenomena some fast diffusion equations with very singular d do not allow for solutions with sign changes while other equations admit only monotone solutions no oscillations being allowed the examples we give for both types of anomaly are closely related the most typical examples are v t v x v x and u t u xx u x for these equations we investigate what happens to the cauchy problem when we take incompatible initial data and perform a standard regularization it is shown that the limit gives rise to an initial layer where the data become admissible positive or monotone respectively followed by a standard evolution for all t 0 once the obstruction has been removed

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

topology optimization of energy absorbing structures with maximum damage constraint summary a novel density based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed this framework enables topologies to absorb large amount of energy via plastic work before failure occurs to account for the plasticity and damage during the energy absorption a coupled elastoplastic ductile damage model is incorporated with topology optimization appropriate material interpolation schemes are proposed to relax the damage in the low density regions while still ensuring the convergence of newton raphson solution process in the nonlinear finite element analyses an effective method for obtaining path dependent sensitivities of the plastic work and maximum damage via adjoint method is presented and the sensitivities are verified by the central difference method the effectiveness of the proposed methodology is demonstrated through a series of numerical examples copyright 2017 john wiley sons ltd

numerical approximations for a nonlocal evolution equation in this paper we study numerical approximations of continuous solutions to the nonlocal p laplacian type diffusion equation u t t x int omega j x y u t y u t x p 2 u t y u t x dy first we find that a semidiscretization in space of this problem gives rise to an ode system whose solutions converge uniformly to the continuous one as the mesh size goes to zero moreover the semidiscrete approximation shares some properties of the continuous problem it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity next we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties in addition we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile finally we present some numerical experiments that illustrate our results

optimal design in two dimensional conductivity for a general cost depending on the field we explore an optimal design problem in two dimensional conductivity for a rather general cost depending on the underlying field through a typical variational reformulation that has been explored recently we provide a simplified relaxed version which is amenable to numerical simulation and prove that it is a true relaxation under a main structural hypothesis several important cases are covered including a linear cost in the gradient and a convex isotropic functional in particular the pth power of the field for any p geqq1 for the isotropic non quadratic case our computations do not require an explicit form of the constrained quasiconvexification of the equivalent vector variational problem that structural assumption ties together the underlying state equation and the integral cost

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

nonlocal geometric expansion of convex planar curves abstract we consider a class of nonlocal geometric equations for expanding curves in the plane arising in the study of evolutions governed by monge kantorovich mass transfer we construct convex solutions given convex initial data in order to obtain such solutions we develop a new version of perron s method we give applications to the problem of characterizing fast slow diffusion limits

solutions to shape and topology eigenvalue optimization problems using a homogenization method a solution strategy to find the shape and topology of structures that maximize a natural frequency is presented the methodology is based on a homogenization method and the representation of the shape of the structure as a material property the problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material two dimensional plane elasticity problems are considered examples are presented for illustration

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

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

sequences of transformations and triangular recursion schemes with applications in numerical analysis abstract in many fields of numerical analysis there appear transformations of the form t k v v k i v k i v t i when v varies a sequence of transformations is obtained this approach covers for example the e and algorithms the recursion formulae for b splines bernstein polynomials and orthogonal polynomials pade approximants the divided difference scheme and projection methods in this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme the reciprocal of these results also holds furthermore we will show that t v k can be represented in terms of a complex contour integral throughout the paper we will study several examples in some detail and it will turn out that the application of our general theory leads to interesting new results in the special cases among others we will derive a new determinantal representation formula for b splines a recurrence relation for generalized bernstein polynomials a generalization of the e algorithm and we will prove that the algorithm can be represented as a quotient of determinants

stress constraints sensitivity analysis in structural topology optimization abstract sensitivity analysis is an essential issue in the structural optimization field the calculation of the derivatives of the most relevant quantities displacements stresses strains in optimum design of structures allows to estimate the structural response when changes in the design variables are introduced this essential information is used by the most frequent conventional optimization algorithms slp mma feasible directions in order to reach the optimal solution according to this idea the sensitivity analysis of the stress constraints in topology optimization problems is a crucial aspect to obtain the optimal solution when stress constraints are considered maximum stiffness approaches usually involve one linear constraint and one non linear objective function thus the computation of the required sensitivity analysis does not mean a crucial limitation however in the topology optimization problem with stress constraints efficient and accurate computation of the derivatives is needed in order to reach appropriate optimal solutions in this paper a complete analytic and efficient procedure to obtain the sensitivity analysis of the stress constraints in topology optimization of continuum structures is analyzed first order derivatives and second order directional derivatives of the stress constraints are analyzed and included in the optimization procedure in addition topology optimization problems usually involve thousands of design variables and constraints thus an efficient implementation of the algorithms used in the computation of the sensitivity analysis is developed in order to reduce the computational cost required finally the sensitivity analysis techniques presented in this paper are tested by solving some application examples

fast slow diffusion and growing sandpiles abstract we study a coupled system of ode introduced by the first author insiam j appl math 22 1972 437 458 for the heights of growing interacting sand cones we show that these ode correspond to the evolution inl2generated by the subdifferential of the convex functional which vanishes on functions whose gradient has length less than or equal to one and is infinity otherwise additionally we explain how the ode arise from evolutions governed by thep laplacian in the infinitely fast infinitely slow diffusion limit asp

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

from fast to very fast diffusion in the nonlinear heat equation we study the asymptotic behavior of the sign changing solution of the equation u t u u f when the diffusion becomes very fast i e as 1 we prove that a solution u t converges in l 1 uniformly for t in subsets with compact support in 0 t to a solution of u t u 1 u f in contrast with the case of 0 the limiting solutions are either positive or negative or identically equal to 0 in all this causes the limit to be singular in the sense that a boundary layer appears at t 0 when one lets 1

a level set approach for topology optimization with local stress constraints summary the purpose of this work is to present a level set based approach for the structural topology optimization problem of mass minimization submitted to local stress constraints the main contributions are threefold first the inclusion of local stress constraints by means of an augmented lagrangian approach within the level set context second the proposition of a constraint procedure that accounts for a continuous activation deactivation of a finite number of local stress constraints during the optimization sequence finally the proposition of a logarithmic scaling of the level set normal velocity as an additional regularization technique in order to improve the minimization sequence a set of benchmark tests in two dimensions achieving successful numerical results assesses the good behavior of the proposed method in these examples it is verified that the algorithm is able to identify stress concentrations and drive the design to a feasible local minimum copyright 2014 john wiley sons ltd

modelling topology optimization problems as linear mixed 0 1 programs this paper deals with topology optimization of discretized continuum structures it is shown that a large class of non linear 0 1 topology optimization problems including stress and displacement constrained minimum weight problems can equivalently be modelled as linear mixed 0 1 programs the modelling approach is applied to some test problems which are solved to global optimality