inverse avalanches in the abelian sandpile model we define and study the inverse of particle addition process in the abelian sandpile model we show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches we study the probability distribution of s 1 the number of untopplings in the first inverse avalanche for a square lattice we determine prob s 1 exactly for s 1 0 1 2 and 3 for large s 1 we show that prob s 1 varies as s 1 11 8 in the direct avalanches this is related to the probability distribution of the number of sites which topple as often as the site where the particle was added these results are verified by numerical simulations

solving second order initial value problems by a hybrid multistep method without predictors abstract a three step seventh order hybrid linear multistep method hlmm with three non step points is proposed for the direct solution of the special second order initial value problems ivps of the form y f x y with an extension to y f x y y the main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures the stability properties of the methods are discussed by expressing them as a one step method in higher dimension the methods are then applied in block form as simultaneous numerical integrators over non overlapping intervals numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature

loop erased self avoiding random walk and the laplacian random walk comments that the laplacian random walk with eta 1 recently introduced by lyklema and evertsz 1986 is the same process as the loop erased self avoiding random walk analysed previously by the author 1980 some rigorous results about this process are reviewed

numerical solution for high order differential equations using a hybrid neural network optimization method this paper reports a novel hybrid method based on optimization techniques and neural networks methods for the solution of high order ordinary differential equations here neural networks is considered as a part of large field called neural computing or soft computing this means that we propose a new solution method for the approximated solution of high order ordinary differential equations using innovative mathematical tools and neural like systems of computation this hybrid method can result in improved numerical methods for solving initial boundary value problems without using preassigned discretisation points the mixture of feed forward neural networks and optimization techniques based on nelder mead method is used to introduce the close analytic form of the solution for the differential equation excellent test results are obtained for the solution of lower and higher order differential equations the model finds approximation solution for the differential equation inside and outside the domain of consideration for the close enough neighborhood of initial boundary points numerical examples are described to demonstrate the method

a semi implicit level set method for structural shape and topology optimization this paper proposes a new level set method for structural shape and topology optimization using a semi implicit scheme structural boundary is represented implicitly as the zero level set of a higher dimensional scalar function and an appropriate time marching scheme is included to enable the discrete level set processing in the present study the hamilton jacobi partial differential equation pde is solved numerically using a semi implicit additive operator splitting aos scheme rather than explicit schemes in conventional level set methods the main feature of the present method is it does not suffer from any time step size restriction as all terms relevant to stability are discretized in an implicit manner the semi implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance hence the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency resultantly it allows enhanced relaxation on the time step size originally limited by the courant friedrichs lewy cfl condition of the explicit schemes the stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements furthermore the present method avoids additional cost to globally reinitialize the level set function for regularization purpose it is noted that the periodically applied reinitializations are time consuming procedures in particular the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating splitting and merging processes which makes the final design independent of initial guess and helps reduce the probability of converging to a local minimum the availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs

discussion of second order topological sensitivity analysis by j rocha de faria et al abstract the aim of this discussion is to expose incorrect results in a previous ijss article

infinite volume limit of the abelian sandpile model in dimensions d 3 we study the abelian sandpile model on mathbb z d in d 3 we prove existence of the infinite volume addition operator almost surely with respect to the infinite volume limit of the uniform measures on recurrent configurations we prove the existence of a markov process with stationary measure and study ergodic properties of this process the main techniques we use are a connection between the statistics of waves and uniform two component spanning trees and results on the uniform spanning forest measure on mathbb z d

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

topology optimization with closed b splines and boolean operations abstract in this work free form curves of closed b splines cbs are introduced as basic design primitives to realize structural topology optimization in the way of shape optimization unlike the existing density and level set methods the cbs takes full advantage of its parametric form for the design deformability and flexibility with a few number of design variables and its implicit form for boolean operations of the structural topology accounting for any overlapping merging and separation of different cbs no matter how geometric complexities of the cbs and the design domain are the cbs is implemented within the computing framework of finite cell method fcm so that structural analysis and topology optimization can efficiently be carried out using the fixed mesh typical examples are tested to demonstrate the effectiveness and generality of the proposed method in dealing with engineering design problems

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions this paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions the macroscopic response is assumed to be predicted by a well established multi scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element rve of material the proposed formula a symmetric fourth order tensor field over the rve domain is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the rve in the limits when the inclusion matrix phase contrast ratio tends to zero and infinity the sensitivities to the insertion of a hole and a rigid inclusion respectively are rigorously obtained the derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail the derived fundamental formula is of interest to many areas of applied and computational mechanics to illustrate its potential applicability a simple finite element based example is presented where the topological derivative information is used to automatically generate a bi material microstructure to meet pre specified macroscopic properties

non criticality criteria for abelian sandpile models with sources and sinks we prove that the abelian sandpile model on a random binary and binomial tree as introduced in redig ruszel and saada j stat phys 147 653 677 2012 is not critical for all branching probabilities p 1 by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps we obtain some more information about the exponential tail of the avalanche radius next we study the sandpile model on zd with some additional dissipative sites we provide examples and sufficient conditions for non criticality we also make a connection with the parabolic anderson model finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non criticality in the presence of a finite number of sources using a connection with the homogeneous pinning model

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

waves of topplings in an abelian sandpile abstract we study the structure of the avalanche process in the 2d abelian sandpile model an avalanche may be represented as a sequence of waves each consisting of sites that toppled only once in that wave it is shown that the waves are in one to one correspondence with the set of two rooted spanning trees and can be described in terms of green functions we argue that the probability distribution of waves of size s varies as 1 s for large s we prove also that the avalanches started at the open boundary consist of only one wave and find the asymptotic distribution of their sizes which varies as s 3 2 we establish the equivalence between waves and inverse avalanches introduced by dhar and manna and reproduce their result 11 8 for the critical exponent of the size distribution of the first inverse avalanche or in our terms the last wave

development of pareto topology optimization considering thermal loads abstract in this research we developed a level set based topology optimization with a topological derivative formulation considering thermal load thermo elasticity equations were utilized to obtain the sensitivity of the objective function after inserting a small hole in the domain total strain energy and the maximum stress in the design domain were taken as the objective functions after taking the thermal loading effect into account the total strain energy density function became a nonhomogeneous function of the strain in addition temperature variation changed hooke s law from a linear homogeneous to a linear nonhomogeneous expression including a zero order term we derived the sensitivity value of the selected objective functions with respect to a perturbation in the structural domain under mechanical and thermal loads while considering these changes in the governing equations we performed several numerical optimization problems to demonstrate the validity of the present level set based pareto topology optimization two types of examples compliance and stress minimization were solved based on the chosen objective functions furthermore in the stress minimization examples the derived formula was extended to consider thermal effects in the failure theories for pressure independent and dependent materials

intersections of random walks a direct renormalization approach various intersection probabilities of independent random walks ind dimensions are calculated analytically by a direct renormalization method adapted from polymer physics this heuristic approach based on edwards continuum model leads to a straightforward derivation and also to refinements of lawler s results for the simultaneous intersections of two walks in 4 or three walks in 3 these results are generalized top walks in d d frac 2p p 1 p 2 ford 4 an infinite set of universal critical exponents l l 1 are derived they govern the asymptotic probability z l sim s sigma l thatl star walks in d with a common origin do not intersect before times the l s are calculated up to ordero e2 whered 4 e this information is used to calculate the probability z g that a set of independent random walks in d or d d 4 respectivelyd 3 form a given topological networks g of multiple intersection points in the absence of any other double point respectively triple point this is generalized to a network in d leqq frac 2p p 1 dimension with exclusion ofp tuple points the method is quite general and can be used to calculate any critical intersection probability and provides the probabilist with a large variety of exact results yet to be proven rigorously

sensitivity analysis in kinematically incompatible models abstract a natural question that arises when working with the so called kinematically incompatible models is about the positioning of the internal artificial boundary that splits the domain of analysis into the sub domains for which the different incompatible models are established although the experience in modeling might provide an ad hoc solution to such matter it may also draw misleading conclusions here the sensitivity analysis furnishes a systematic and reliable framework to study the impact of the positioning of the artificial boundary over the solution of the problem in this sense the present work is concerned with the sensitivity analysis of the variational formulation corresponding to the kinematically incompatible models when the boundary over which the incompatibility between the models takes place is changed assuming the presence of a discontinuity in the fields over a given interface between two incompatible models this analysis allows to measure the sensitivity of a given cost functional when the location of such interface is changed the application for which this analysis is envisaged is to assess the correctness or incorrectness in the definition of the positioning of such coupling interface between dimensionally heterogeneous domains also some numerical results are provided to give numerical evidence of the usefulness of the present tool

higher order topological derivatives in elasticity the topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain in this work we compute higher order topological derivatives for elasticity problems so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain a critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space we present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion we also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown

optimal topological design through insertion and configuration of finite sized heterogeneities in this paper we develop a procedure for optimal topological design by sequentially inserting finite sized non spherical inclusions or holes within a homogeneous domain we propose a new criterion for topology change that results in a trade off problem to achieve the greatest least change in the objective for the least greatest change in the size of the inclusion hole respectively we derive the material derivative of the proposed objective termed as the configurational derivative that describes sensitivity of arbitrary functionals to arbitrary motions of the inclusion hole as well as the domain boundaries we specifically utilize the sensitivity to position orientation and scaling of finite sized heterogeneities to effect topological design we simplify the configurational derivative to the special case of infinitesimally small spherical inclusions or holes and show that the developed derivative is a generalization of the classical topological derivative the computational implementation relies on b spline isogeometric approximations we demonstrate through a series of examples optimal topology achieved through sequential insertion of a heterogeneity of fixed shape and optimization of its configuration location orientation and scale

a staggered approach to shape and topology optimization using the traction method and an evolutionary type advancing front algorithm abstract we investigate a novel approach for structural shape optimization on the basis of complementary shape and topological sensitivity analysis as in early approaches to shape optimization the domain variation is specified by modification of boundary nodal points hence leading to an updated lagrangian description for the course of optimization to overcome the formation of oscillating boundaries in the optimal design trials we employ the traction method to establish smooth descent directions for shape variation therein an auxiliary elastic problem is solved in which the shape sensitivity constitutes the external loading and the deformation of the auxiliary elastic body is used as a descent direction for shape variation rather than the shape sensitivity itself we complement this method through an evolutionary type element removal procedure that is based on the topological sensitivity such that an advancing front algorithm is gradually removing elements from the design boundary of the domain once the minimum topological sensitivity is no longer encountered at the design boundary we create a hole in the domain again using the topological sensitivity to specify its exact location and resume the element removal procedure for the newly established design boundary since this approach yields only a vague estimate of the true optimal shape of newly established holes the traction method is again used for shape variation of the extended design boundary

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

topology optimization of steady navier stokes flow with body force abstract this paper presents the topology optimization of steady navier stokes flows with body forces that influence the optimal shape and topology of fluid flows based on the implicit expression of the fluid flow with the level set method an optimization problem is formulated and analyzed using the continuous adjoint method the shape and topological sensitivities are computed based on the adjoint and asymptotic analysis of the optimization problem in the optimization procedure the level set surface is evolved based on the shape sensitivity and nucleated based on the topological sensitivity simultaneously three kinds of body forces that are commonly used in the design of fluid devices i e constant nonuniform and solution dependent body forces are considered in the two dimensional and three dimensional numerical examples numerical results demonstrate that this method can effectively achieve the topology optimization of the navier stokes flows with body forces

incorporating topological derivatives into shape derivatives based level set methods shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems the shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain the combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated

topological and shape gradient strategy for solving geometrical inverse problems abstract in this paper we present a technique for shape reconstruction based on the topological and shape gradients the shape in consideration is a solution of an inverse conductivity problem to solve such a problem numerically we compute the topological gradient of a kohn vogelius type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole the reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation then we use a gradient type algorithm to perform a good reconstruction various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm

combination of topology optimization and optimal control method this paper presents the combination of topology optimization and optimal control method to find the optimal match between the material topology and control in the presented method the material topology is determined using the simp solid isotropic material with penalization method which has been popularly used in topology optimization in the simp method the design variable is relaxed and bounded in the interval 0 1 and the evolution of the design variable is usually implemented by the method of moving asymptotes mma which can be used to deal with optimization problem with multiple integral constraints and bound constraint of the design variable in the combination of topology optimization and optimal control method the control variable appears along with the design variable in order to evolve the control variable and design variable using mma simultaneously the control variable is regularized using a bound constraint and the corresponding bound constraint is projected onto the interval 0 1 which is the same as the bound constraint of the design variable the optimization problem is analyzed using the adjoint method to obtain the adjoint sensitivity during the optimization procedure the design and control variables are filtered by the helmholtz filters to ensure the smoothness of the distribution to ensure the minimum scale length and remove the gray area in the material topology the filtered design variable is projected by the threshold method the feasibility and robustness of the combination of these two methods are demonstrated by several test problems including heat transfer fluid flow and compliance minimization

approaching criticality via the zero dissipation limit in the abelian avalanche model the discrete height abelian sandpile model was introduced by bak tang wiesenfeld and dhar as an example for the concept of self organized criticality when the model is modified to allow grains to disappear on each toppling it is called bulk dissipative we provide a detailed study of a continuous height version of the abelian sandpile model called the abelian avalanche model which allows an arbitrarily small amount of dissipation to take place on every toppling we prove that for non zero dissipation the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure we show that in the whole non zero dissipation regime the model is not critical i e spatial covariances of local observables decay exponentially we then study the zero dissipation limit and prove that the self organized critical model is recovered both for the stationary measure and for the dynamics we obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality we rigorously establish the mean field value of this exponent for d 4

topology optimization of unsteady incompressible navier stokes flows this paper discusses the topology optimization of unsteady incompressible navier stokes flows an optimization problem is formulated by adding the artificial darcy frictional force into the incompressible navier stokes equations the optimization procedure is implemented using the continuous adjoint method and the finite element method the effects of dynamic inflow reynolds number and target flux on specified boundaries for the optimal topology of unsteady navier stokes flows are presented numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady navier stokes flows

second order topological sensitivity analysis abstract the topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain classically this derivative comes from the second term of the topological asymptotic expansion dealing only with infinitesimal holes however for practical applications we need to insert holes of finite size therefore we consider one more term in the expansion which is defined as the second order topological derivative in order to present these ideas in this work we apply the topological shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the poisson s equations taking the total potential energy as cost function and the state equation as constraint furthermore we also study the effects of different boundary conditions on the hole neumann and dirichlet both homogeneous finally we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion which has two main features it allows us to deal with hole of finite size and provides a better descent direction in optimization process

a binary level set model and some applications to mumford shah image segmentation in this paper we propose a pde based level set method traditionally interfaces are represented by the zero level set of continuous level set functions instead we let the interfaces be represented by discontinuities of piecewise constant level set functions each level set function can at convergence only take two values i e it can only be 1 or 1 thus our method is related to phase field methods some of the properties of standard level set methods are preserved in the proposed method while others are not using this new method for interface problems we need to minimize a smooth convex functional under a quadratic constraint the level set functions are discontinuous at convergence but the minimization functional is smooth we show numerical results using the method for segmentation of digital images

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

the upper critical dimension of the abelian sandpile model the existing estimation of the upper critical dimension of the abelian sandpile model is based on a qualitative consideration of avalanches as self avoiding branching processes we find an exact representation of an avalanche as a sequence of spanning subtrees of two component spanning trees using equivalence between chemical paths on the spanning tree and loop erased random walks we reduce the problem to determination of the fractal dimension of spanning subtrees then the upper critical dimension du 4 follows from lawler s theorems for intersection probabilities of random walks and loop erased random walks

topological sensitivity analysis for three dimensional linear elasticity problem in this work we use the topological shape sensitivity method to obtain the topological derivative for three dimensional linear elasticity problems adopting the total potential energy as cost function and the equilibrium equation as constraint this method based on classical shape sensitivity analysis leads to a systematic procedure to calculate the topological derivative in particular firstly we present the mechanical model later we perform the shape derivative of the corresponding cost function and finally we calculate the final expression for the topological derivative using the topological shape sensitivity method and results from classical asymptotic analysis around spherical cavities in order to point out the applicability of the topological derivative in the context of topology optimization problems we use this information as a descent direction to solve a three dimensional topology design problem furthermore through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm

modelling of topological derivatives for contact problems the problem of topology optimization is considered for free boundary problems of thin obstacle types the formulae for the first term of asymptotics for energy functionals are derived the precision of obtained terms is verified numerically the topological differentiability of solutions to variational inequalities is established in particular the so called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis such results lead to the topological derivatives of shape functionals for contact problems the topological derivatives are used in numerical methods of simultaneous shape and topology optimization

waves in the sandpile model on fractal lattices abstract the scaling properties of waves of topplings of the sandpile model on a fractal lattice are investigated the exponent describing the asymptotics of the distribution of last waves in an avalanche is found predictions for scaling exponents in the forward and backward conditional probabilities for two consecutive waves are given all predictions were examined by simulations on the sierpinski gasket and were found to be in a reasonable agreement with the numerical data

shape optimisation with multiresolution subdivision surfaces and immersed finite elements abstract we develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness during shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information by virtue of the multiresolution editing semantics updating the coarse control mesh vertex coordinates leads to large scale geometry changes and conversely updating the fine control mesh coordinates leads to small scale geometry changes in our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum this approach effectively prevents the appearance of non physical boundary geometry oscillations and control mesh pathologies like inverted elements independent of the fineness of the control mesh used for optimisation on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution with the immersed finite element method there is no need to maintain an analysis suitable domain mesh in some of the presented two and three dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain the merging or removing of holes is not considered

a level set method for shape and topology optimization of both structure and support of continuum structures abstract we present a level set method for the shape and topology optimization of both structure and support two level set functions are used to implicitly represent a structure the traction free boundary and the dirichlet boundary are represented separately and are allowed to be continuously changed during the optimization the optimization problem of minimum compliance is considered the shape derivatives of both boundaries are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and we do not use the artificial weak material numerical examples in two dimensions are investigated

shape and topology optimization of compliant mechanisms using a parameterization level set method in this paper a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms the structural shape boundary is implicitly embedded into a higher dimensional scalar function as its zero level set resultantly establishing the level set model by applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function the temporal and spatial hamilton jacobi equation from the conventional level set method is then discretized into a series of algebraic equations accordingly the original shape and topology optimization is now fully transformed into a parameterization problem namely size optimization with the expansion coefficients of interpolants as a limited number of design variables design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints the structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method it is highlighted that the present method can not only inherit the merits of the implicit boundary representation but also avoid some unfavorable features of the conventional discrete level set method such as the cfl condition restriction the re initialization procedure and the velocity extension algorithm finally an extensively investigated example is presented to demonstrate the benefits and advantages of the present method especially its capability of creating new holes inside the design domain

topological sensitivity analysis the so called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems this derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain however the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study because they have not the same topology therefore some specific mathematical framework should be developed in order to obtain the derivatives this work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts the main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative this idea leads to a more simple and constructive formulation than the ones found in the literature further to point out the straightforward use of the proposed methodology it is applied for solving some design problems in steady state heat conduction

strain energy change to the insertion of inclusions associated to a thermo mechanical semi coupled system abstract the topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation according to the literature the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations in addition the topological asymptotic analysis associated to multi physics problems has been reported in the literature only on the level of mathematical analysis of singularly perturbed geometrical domains in this work we present the topological derivative in its closed form for the total potential mechanical energy associated to a thermo mechanical semi coupled system when a small circular inclusion is introduced at an arbitrary point of the domain in particular we consider the linear elasticity system modeled by the navier equation coupled with the steady state heat conduction problem modeled by the laplace equation the mechanical coupling term comes out from the thermal stress induced by the temperature field since this term is non local we introduce a non standard adjoint state which allows to obtain a closed form for the topological derivative finally we provide a full mathematical justification for the derived formulas and develop precise estimates for the remainders of the topological asymptotic expansion

numerical methods for ordinary differential equations in the 20th century abstract numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons the first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of bashforth and adams for linear multistep methods and runge for runge kutta methods other reasons which of course apply to numerical analysis in general are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics this survey paper follows many of the main strands in the developments of these methods both for general problems stiff systems and for many of the special problem types that have been gaining in significance as the century draws to an end

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

isogeometric topology optimization using trimmed spline surfaces in the present work a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed in the proposed approach the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by cad systems is employed for structural response analysis and sensitivity calculation in the topology optimization the outer and inner boundaries of design models are represented by a spline surface and trimming curves design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves new sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified the creation of new inner fronts during optimization is allowed for the topological flexibility an inner front merging algorithm is also presented the proposed spline based topology optimization is used to solve some benchmarking problems design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one it is also shown that post processing effort for converting to cad model is eliminated by using the same spline information in numerical analysis and design optimization

asymptotic analysis polarization matrices and topological derivatives for piezoelectric materials with small voids asymptotic formulae for the mechanical and electric fields in a piezoelectric body with a small void are derived and justified such results are new and useful for applications in the field of design of smart materials in this way the topological derivatives of shape functionals are obtained for piezoelectricity the asymptotic formulae are given in terms of the so called polarization tensors matrices which are determined by the integral characteristics of voids the distinguishing feature of the piezoelectricity boundary value problems under consideration is the absence of positive definiteness of a differential operator which is non self adjoint two specific gibbs functionals of the problem are defined by the energy and the electric enthalpy the topological derivatives are defined in different manners for each of the governing functionals actually the topological derivative of the enthalpy functional is local i e defined by the pointwise values of the mechanical and electric fields which is contrary to the energy functional and some other suitable shape functionals which admit nonlocal topological derivatives i e depending on the whole problem data an example with weak interaction between mechanical and electric fields provides the analytic asymptotic expansions and can be used in numerical procedures of optimal design for smart materials

sounding of finite solid bodies by way of topological derivative this paper is concerned with an application of the concept of topological derivative to elastic wave imaging of finite solid bodies containing cavities building on the approach originally proposed in the elastostatic theory of shape optimization the topological derivative which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity free reference body is used as a void indicator through an assembly of sampling points where it attains negative values the computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary value problems for the reference body which are here formulated as boundary integral equations for a comprehensive treatment of the subject formulas for topological sensitivity are obtained using three alternative methodologies namely i direct differentiation approach ii adjoint field method and iii limiting form of the shape sensitivity analysis the competing techniques are further shown to lead to distinct computational procedures methodologies i and ii are implemented within a bem based platform and validated against an analytical solution a set of numerical results is included to illustrate the utility of topological derivative for 3d elastic wave sounding of solid bodies an approach that may perform best when used as a pre conditioning tool for more accurate gradient based imaging algorithms despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion abstract the aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept this technique will be applied as a systematic procedure to obtain the eshelby s energy momentum tensor associated to the problem under consideration in order to highlight special features of this procedure and without loss of generality we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross section shape change are achieved these integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order eshelby s energy momentum tensors also the null divergence property of these tensors is easily proved by comparing in each case the domain and boundary integral shape derivative arrived at finally an example with a known exact solution corresponding to an elastic bar with elliptical transverse cross section submitted to twist is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives

qualitative identification of cracks using 3d transient elastodynamic topological derivative formulation and fe implementation a time domain topological derivative td approach is developed for transient elastic wave imaging of buried cracks the td which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal trial crack is expressed in terms of the time convolution of the free field and an adjoint field as a function of that specified location and of the trial crack shape following previous studies on cavity identification in similar conditions the td field is here considered as a natural and computationally efficient approach for defining a crack location indicator function this study emphasizes the implementation and exploitation of td fields using the standard displacement based fem a straightforward exploitation of the relevant sensitivity formulation established here results on several numerical experiments on 3d elastodynamic and acoustic configurations are reported and discussed allowing to assess and highlight many features of the proposed td based fast qualitative crack identification including its ability to identify multiple cracks and its robustness against data noise

a level set based topology optimization method for simultaneous design of elastic structure and coupled acoustic cavity using a two phase material model abstract this papers proposes a level set based topology optimization method for the simultaneous design of acoustic and structural material distributions in this study we develop a two phase material model that is a mixture of an elastic material and acoustic medium to represent an elastic structure and an acoustic cavity by controlling a volume fraction parameter in the proposed model boundary conditions at the two phase material boundaries are satisfied naturally avoiding the need to express these boundaries explicitly we formulate a topology optimization problem to minimize the sound pressure level using this two phase material model and a level set based method that obtains topologies free from grayscales the topological derivative of the objective functional is approximately derived using a variational approach and the adjoint variable method and is utilized to update the level set function via a time evolutionary reaction diffusion equation several numerical examples present optimal acoustic and structural topologies that minimize the sound pressure generated from a vibrating elastic structure

a level set method for topology optimization of heat conduction problem under multiple load cases in this paper we present a numerical approach of topology optimization under multiple load cases for heat conduction problem this framework is based on the theories of topological derivative and shape derivative for elliptic system we employ level set model to implicitly represent geometric boundary of thermal conductive material introducing topological derivative will generate new topology in the design domain which suppresses the dependence of initial topology guess to some extent the shape optimization is obtained by combining shape derivative with level set method the functional of quadratic temperature gradient is taken as the objective function in our analysis which is subjected to the state equation of steady heat conduction and volume constrain the shape of material domain is treated as the design variable and the final result is achieved by updating level set function gradually we develop an effective numerical technique to implement the optimal design with multiple load cases for heat conduction problem numerical examples demonstrate that our proposed approach is effective and robust for topology optimization of heat conduction problem

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

solving directly special fourth order ordinary differential equations using runge kutta type method in this paper an explicit runge kutta method for solving directly fourth order ordinary differential equations odes is constructed and denoted as rkfd we present a relevant colored tree theory and the associated b series theory for the order conditions based on the order conditions a three stage fourth order rkfd method and a three stage fifth order rkfd method are constructed numerical illustrations are presented to show the efficiency of the new rkfd methods by comparing them with other existing runge kutta nystrom rkn and runge kutta rk methods in the scientific literature after converting the problem into a system of second order odes and a system of first order odes respectively

a new method for inverse electromagnetic casting problems based on the topological derivative the inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape in this paper we propose a new method for the topology design of such inductors the inverse electromagnetic casting problem is formulated as an optimization problem and topological derivatives are considered in order to locate new wires in the right position several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

higher order topological sensitivity for 2 d potential problems application to fast identification of inclusions abstract this article concerns an extension of the topological derivative concept for 2 d potential problems involving penetrable inclusions whereby a cost function j is expanded in powers of the characteristic size e of a small inclusion the o e4 approximation of j is established for a small inclusion of given location shape and conductivity embedded in a 2 d region of arbitrary shape and conductivity and then generalized to several such inclusions simpler and more explicit versions of this result are obtained for a centrally symmetric inclusion and a circular inclusion numerical tests are performed on a sample configuration for i the o e4 expansion of potential energy and ii the identification of a hidden inclusion for the latter problem a simple approximate global search procedure based on minimizing the o e4 approximation of j over a dense search grid is proposed and demonstrated

abelian sandpile model on the bethe lattice the authors study bak tang and wiesenfeld s abelian sandpile model 1987 of self organised criticality on the bethe lattice exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained they also determine the probability distribution of the number of distinct sites that topple at least once the number of toplings at the origin and the total number of toplings in an avalanche the probability that an avalanche consists of more than n toplings varies as n 1 2 for large n the probability that its duration exceeds t decreases as 1 t for large t these exponents are the same as for the critical percolation clusters in mean field theory

regularized perimeter for topology optimization the perimeter functional is known to oppose serious difficulties when it has to be handled within a topology optimization procedure in this paper a regularized perimeter functional mbox per varepsilon defined for two and three dimensional domains is introduced on one hand the convergence of mbox per varepsilon to the exact perimeter when varepsilon tends to zero is proved on the other hand the topological differentiability of mbox per varepsilon for varepsilon 0 is analyzed these features lead to the design of a topology optimization algorithm suitable for perimeter dependent objective functionals several numerical results illustrate the method

defeaturing a posteriori error analysis via feature sensitivity it is well known that small geometric features within a cad model can significantly impact the computational cost and often undermine the reliability of finite element analysis engineers therefore resort to defeaturing or detail removal wherein the offending features are suppressed prior to computational analysis however this results in a defeaturing induced analysis error in this paper we estimate this error in an a posteriori sense through the novel concept of feature sensitivity the latter determines the first order change in quantities of interest when an arbitrary cluster of small geometric features is deleted from a model a formal theory and a set of associated algorithms are provided to compute the feature sensitivity associated with a scalar elliptic partial differential equation the theory is supported through numerical experiments in 2 d involving both internal and boundary features copyright 2008 john wiley sons ltd

on the discrete variant of the traction method in parameter free shape optimization abstract one of the major challenges in the parameter free approach to computational shape optimization is the avoidance of oscillating i e non smooth boundaries in the optimal design trials in order to achieve this we investigate a method for regularization that corresponds to the discrete variant of the so called traction method in this approach the design updates are generated in terms of a displacement field which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain the main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function we propose an algorithm in which constraint functions will be taken into account by using an augmented lagrangian formulation and a step length control ensures a sufficient decrease condition in terms of the objective function within each iteration we examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest descent algorithm

a topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method

a level set based topology optimisation for acoustic elastic coupled problems with a fast bem fem solver abstract this paper presents a structural optimisation method in three dimensional acoustic elastic coupled problems the proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points in the process of the optimisation configuration of the elastic materials is expressed with a level set function and the distribution of the level set function is iteratively updated with the help of the topological derivative the topological derivative is associated with state and adjoint variables which are the solutions of the acoustic elastic coupled problems in this paper the acoustic elastic coupled problems are solved by a bem fem coupled solver in which the fast multipole method fmm and a multi frontal solver for sparse matrices are efficiently combined along with the detailed formulations for the topological derivative and the bem fem coupled solver we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves from which we confirm the effectiveness of the present method

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

a semismooth newton method for topology optimization abstract this paper deals with elliptic optimal control problems for which the control function is constrained to assume values in 0 1 based on an appropriate formulation of the optimality system a semismooth newton method is proposed for the solution convergence results are proved and some numerical tests illustrate the efficiency of the method

logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension we study numerically the dynamical properties of the btw model on a square lattice for various dimensions the aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean field exponents our results are consistent with the assumption that the scaling behavior of the four dimensional btw model is characterized by the mean field exponents with additional logarithmic corrections we benefit in our analysis from the exact solution of the directed btw model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension similar logarithmic corrections forms fit the numerical data for the four dimensional btw model strongly suggesting that the value of the upper critical dimension is four