level set based structural topology optimization for minimizing frequency response abstract for the purpose of structure vibration reduction a structural topology optimization for minimizing frequency response is proposed based on the level set method the objective of the present study is to minimize the frequency response at the specified points or surfaces on the structure with an excitation frequency or a frequency range subject to the given amount of the material over the admissible design domain the sensitivity analysis with respect to the structural boundaries is carried out while the extended finite element method x fem is employed for solving the state equation and the adjoint equation the optimal structure with smooth boundaries is obtained by the level set evolution with advection velocity derived from the sensitivity analysis and the optimization algorithm a number of numerical examples in the frameworks of two dimension 2d and three dimension 3d are presented to demonstrate the feasibility and effectiveness of the proposed approach

lattice sums for off axis electromagnetic scattering by gratings we consider both the spatial domain and spectral domain forms of the green s function appropriate in the electromagnetic diffraction of a plane wave incident at a general angle in the xy plane on a singly periodic structure or grating oriented along the x axis we equate the spatial and spectral forms of the green s function and so establish expressions from which grating lattice sums can be evaluated for oblique incidence we also obtain a set of identities among the lattice sums we use these lattice sums in an expression for the green s function which we show to be computationally fast if knowledge of this function at several points is required for small values of y

numerical methods for the topology optimization of structures that exhibit snap through the inclusion of non linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response e g the geometric non linear response of compliant mechanisms in previous work the non linear response is computed by standard non linear elastic finite element analysis here we incorporate a load displacement constraint method to traverse non linear equilibrium paths with limit points to design structures that exhibit snap through behaviour to accomplish this we modify the basic arc length algorithm and embed this analysis into the topology optimization problem copyright 2002 john wiley sons ltd

a comparison of acceleration procedures for the two dimensional periodic green s function alternate acceleration procedures are evaluated for the two dimensional free space periodic green s function two of these techniques yield exponential convergence rates and are preferable if high accuracies are required numerical aspects of these procedures are described

efficient prediction of marsaglia zaman random number generators we show that the random number generator of marsaglia and zaman 1991 produces the successive digits of a rational b adic number the b adic number system generalizes p adic numbers to an arbitrary integer base b using continued fractions we derive an efficient prediction algorithm for this generator

a meshfree level set method for topological shape optimization of compliant multiphysics actuators abstract this paper proposes a topology optimization method for compliant multiphysics actuators of geometrically nonlinear structures using meshfree galerkin weak forms and level set methods the design boundary is implicitly represented as the zero level set of a higher dimensional level set function leading to a level set model capable of handling complex shape and topological changes with flexibilities a family of compactly supported radial basis functions csrbfs is firstly used to interpolate the level set function of lipschitz continuity and then augmented to construct the shape function for meshless approximation by satisfying basic requirements in particular the predetermined consistency and the kronecker delta function property a meshless galerkin method mgm with global weak forms is established to implement the discretization of the state equations the design of actuators is transformed into an easier size optimization from a more difficult shape and topology optimization the design boundary evolution is just a question of advancing the discrete level set function in time by updating the design variables of the size optimization compared to most conventional level set methods the proposed meshless level set method is able to implement the free moving boundary discontinuities without remeshing and unify two different numerical procedures in propagating the discrete level set function e g eulerian grid and approximating the state equation e g lagrangian mesh respectively this method can also avoid numerical difficulties in solving a series of complicate hamilton jacobi partial differential equations pdes with explicit time schemes two typical numerical examples are used to demonstrate the effectiveness of the proposed method

fast iterative division of p adic numbers a fast iterative scheme based on the newton method is described for finding the reciprocal of a finite segment p adic numbers hensel code the rate of generation of the reciprocal digits per step can be made quadratic or higher order by a proper choice of the starting value and the iterating function the extension of this method to find the inverse transform of the hensel code of a rational polynomial over a finite field is also indicated

second order reliability method based inverse reliability analysis using hessian update for accurate and efficient reliability based design optimization summary first order reliability method form has been mostly utilized for solving reliability based design optimization rbdo problems efficiently however second order reliability method sorm is required in order to estimate a probability of failure accurately in highly nonlinear performance functions despite accuracy of sorm its application to rbdo is quite challenging due to unaffordable numerical burden incurred by a hessian calculation for reducing the numerical efforts a quasi newton approach to approximate the hessian is introduced in this study instead of calculating the true hessian the proposed sorm with the approximated hessian requires computations only used in form leading to very efficient and accurate reliability analysis the proposed sorm also utilizes a generalized chi squared distribution in order to achieve better accuracy furthermore sorm based inverse reliability method is proposed in this study an accurate reliability index corresponding to a target probability of failure is updated using the proposed sorm two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed the proposed sorm based inverse analysis is then extended to rbdo in order to obtain a reliability based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system the numerical study results show that the proposed reliability analysis and rbdo achieve efficiency of form and accuracy of sorm at the same time copyright 2014 john wiley sons ltd

topology design optimization of geometrically non linear structures using meshfree method a topology optimization method using meshfree methods is developed for geometrically non linearly structures a density approach is used to optimally distribute the bulk material densities in a continuum design domain the reproducing kernel rk shape function is employed to discretize both displacement and bulk density fields thus the points can be located at any part of the design domain and be easily included in or excluded from the response analysis the well known convergence difficulty which normally occurs in the sub domain of low bulk density can be avoided by excluding the sub domain the numerical instability of bulk density solution due to the rank deficiency of system matrix is effectively prevented by properly selecting the number of independent design variables through numerous numerical examples we confirm that the developed method works very well even in the large deformation problems the sub domain exclusion scheme is found to be greatly beneficial in terms of computational costs compared with the results without the scheme

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

fast algorithms for helmholtz green s functions the formal representation of the quasi periodic helmholtz green s function obtained by the method of images is only conditionally convergent and thus requires an appropriate summation convention for its evaluation instead of using this formal sum we derive a candidate green s function as a sum of two rapidly convergent series one to be applied in the spatial domain and the other in the fourier domain as in ewald s method we prove that this representation of green s function satisfies the helmholtz equation with the quasi periodic condition and furthermore leads to a fast algorithm for its application as an operator we approximate the spatial series by a short sum of separable functions given by gaussians in each variable for the series in the fourier domain we exploit the exponential decay of its terms to truncate it we use fast and accurate algorithms for convolving functions with this approximation of the quasi periodic green s function the resulting method yields a fast solver for the helmholtz equation with the quasi periodic boundary condition the algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when green s function is applied to discontinuous functions or potentials with singularities we also construct helmholtz green s functions with dirichlet neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi periodic green s function to apply them the complexity in dimension d 2 of these algorithms is graphic 1 d log c log 1 d where is the desired accuracy is proportional to the number of wavelengths contained in the computational domain and c is a constant we illustrate our approach with examples 1 embed inline graphic 1 gif

two dimensional phase modulated lattice sums with application to the helmholtz green s function a class of two dimensional phase modulated lattice sums in which the denominator is an indefinite quadratic polynomial q is expressed in terms of a single exponentially convergent series of elementary functions this expression provides an extremely efficient method for the computation of the quasi periodic green s function for the helmholtz equation that arises in a number of physical contexts when studying wave propagation through a doubly periodic medium for a class of sums in which q is positive definite our new result can be used to generate representations in terms of functions which are significant generalisations of known results

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

structural shape and topology optimization using a meshless galerkin level set method summary this paper aims to propose a meshless galerkin level set method for shape and topology optimization of continuum structures to take advantage of the implicit free boundary representation scheme the design boundary is represented as the zero level set of a scalar level set function to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface compactly supported radial basis functions csrbfs are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes the meshless galerkin method with global weak form is used to implement the discretization of the state equations this provides a pathway to unify the two different numerical stages in most conventional level set methods 1 the propagation of discrete level set function on a set of eulerian grid and 2 the approximation of discrete equations on a set of lagrangian mesh the original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization to which many efficient optimization algorithms can be applied the proposed level set method can describe the moving boundaries without remeshing for discontinuities the motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization several benchmark examples are used to demonstrate the effectiveness of the proposed method the numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods it is straightforward to apply the proposed method to more advanced shape and topology optimization problems copyright 2011 john wiley sons ltd

optimal layout design of three dimensional geometrically non linear structures using the element connectivity parameterization method the topology design optimization of three dimensional geometrically non linear continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization to overcome these difficulties the element connectivity parameterization ecp method with two implementation formulations is proposed in ecp structural layouts are represented by inter element connectivity which is controlled by the stiffness of element connecting zero length links depending on the link location ecp may be classified as an external ecp e ecp or an internal ecp i ecp in this paper i ecp is newly developed to substantially enhance computational efficiency the main idea in i ecp is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level as for ecp implementation with commercial software e ecp developed earlier for two dimensional problems is easier to use even for three dimensional problems because it requires only numerical analysis results for design sensitivity calculation the characteristics of the i ecp and e ecp methods are compared and these methods are validated with numerical examples copyright 2006 john wiley sons ltd

spectral stochastic isogeometric analysis of free vibration abstract a novel spectral stochastic isogeometric analysis ssiga is proposed for the free vibration analysis of engineering structures involving uncertainties the proposed ssiga framework treats the stochastic free vibration problem as a stochastic generalized eigenvalue problem the stochastic young s modulus and material density of the structure are modelled as random fields with gaussian and non gaussian distributions the basis functions the non uniform rational b spline nurbs and t spline within computer aided design cad system are adopted within the ssiga which can eliminate geometric errors between design model and uncertainty analysis model the arbitrary polynomial chaos apc expansion is implemented to investigate the stochastic responses i e eigenvalues and eigenvectors of the structure a galerkin based method is freshly proposed to solve the stochastic generalized eigenvalue problems the statistical moments probability density function pdf and cumulative distribution function cdf of the eigenvalues can be effectively obtained two numerical examples with irregular geometries are investigated to illustrate the applicability accuracy and efficiency of the proposed ssiga for free vibration analysis of engineering structures

lattice sums for the helmholtz equation a survey of different representations for lattice sums for the helmholtz equation is made these sums arise naturally when dealing with wave scattering by periodic structures one of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d lambda lattice sums are related to and can be calculated from the quasi periodic green s function and this object serves as the starting point of the analysis

optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval an optimization method for uncertain structures is suggested based on convex model and a satisfaction degree of interval in the investigated problem the uncertainty only exists in constraints convex model is used to describe the uncertainty in which the intervals of the uncertain parameters are only needed not necessarily to know the precise probability distributions a satisfaction degree of interval which represents the possibility that one interval is smaller than another is employed to deal with the uncertain constraints based on a predetermined satisfaction degree level the uncertain constraints are transformed to deterministic ones and the transformed optimization problem can be solved by traditional optimization methods for complex structural problems that the optimization model cannot be expressed in an explicit form the interval analysis method is adopted to calculate the intervals of the constraints efficiently and whereby eliminate the optimization nesting two numerical examples have been presented to demonstrate the efficiency of the suggested method

rapidly convergent two dimensional quasi periodic green function throughout the spectrum including wood anomalies we introduce a new methodology based on new quasi periodic green functions which converge rapidly even at and around wood anomaly configurations for the numerical solution of problems of scattering by periodic rough surfaces in two dimensional space as is well known the classical quasi periodic green function ceases to exist at wood anomalies the approach introduced in this text produces fast green function convergence throughout the spectrum on the basis of a certain finite differencing approach and smooth windowing of the classical green function lattice sum the resulting green function convergence is super algebraically fast away from wood anomalies and it reduces to an arbitrarily high user prescribed algebraic order of convergence at wood anomalies

a constructive method for plane wave representations of special functions abstract a general constructive scheme for the derivation of plane wave representations of special functions is proposed illustrative examples of the construction are given as one case study new integral representations of the elliptic weierstrass function are derived these complement and generalize similar new plane wave integral representations of the same function recently found by dienstfrey and huang 2006 8 using other techniques our approach is inspired by recent developments in the so called fokas transform for the solution of boundary value problems for partial differential equations

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

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

sensitivity analyses of form based and drm based performance measure approach pma for reliability based design optimization rbdo in a gradient based design optimization it is necessary to know sensitivities of the constraint with respect to the design variables in a reliability based design optimization rbdo the constraint is evaluated at the most probable point mpp and called the probabilistic constraint thus it requires the sensitivities of the probabilistic constraints at mpp this paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at mpp for both first order reliability method form based performance measure approach pma and dimension reduction method drm based pma numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method fdm however since the sensitivity calculation at the true drm based mpp requires the second order derivatives and additional mpp search the sensitivity derivation at the approximated drmbased mpp which does not require the second order derivatives and additional mpp search to find the drm based mpp is proposed in this paper a convergence study illustrates that the sensitivity at the approximated drm based mpp converges to the sensitivity at the true drm based mpp as the design approaches the optimum design hence the sensitivity at the approximated drm based mpp is proposed to be used for the drm based rbdo to enhance the efficiency of the optimization

source neutral green functions for periodic problems in electrostatics and their equivalents in electromagnetism we consider source neutral problems for periodic problems in electrostatics these may be used to derive an identity due to lord rayleigh entirely within the context of the unit cell and without use of reasoning based on contributions from charges at an outer boundary of the periodic system or employing conditionally convergent lattice sums we also consider the equivalent of source free green9functions for the helmholtz equation which are green9functions with their specular parts removed we show that as the wavenumber tends to zero the non specular dynamic green9functions tend with no divergences to the source neutral electrostatic green9function

a new integral representation for quasi periodic fields and its application to two dimensional band structure calculations in this paper we consider band structure calculations governed by the helmholtz or maxwell equations in piecewise homogeneous periodic materials methods based on boundary integral equations are natural in this context since they discretize the interface alone and can achieve high order accuracy in complicated geometries in order to handle the quasi periodic conditions which are imposed on the unit cell the free space green s function is typically replaced by its quasi periodic cousin unfortunately the quasi periodic green s function diverges for families of parameter values that correspond to resonances of the empty unit cell here we bypass this problem by means of a new integral representation that relies on the free space green s function alone adding auxiliary layer potentials on the boundary of the unit cell itself an important aspect of our method is that by carefully including a few neighboring images the densities may be kept smooth and convergence rapid this framework results in an integral equation of the second kind avoids spurious resonances and achieves spectral accuracy because of our image structure inclusions which intersect the unit cell walls may be handled easily and automatically our approach is compatible with fast multipole acceleration generalizes easily to three dimensions and avoids the complication of divergent lattice sums

optimization based on reliability and confidence interval design for the structural acoustic system with interval probabilistic variables abstract in practice the uncertain parameters whose unique distributions cannot be identified on the basis of limited data existed in the structural acoustic system extensively the uncertain parameters with limited data can be modeled as interval probabilistic variables whose distribution parameters are expressed as interval variables instead of unique values for the optimization of the structural acoustic system with interval probabilistic variables a new optimization technique named as the optimization based on reliability and confidence interval design o rcid is developed o rcid is a nested loop optimization whose computational cost during the evaluation of the confidence interval of sound pressure response and the system reliability is considerable to improve the computational efficiency of o rcid a change of variable interval probabilistic perturbation method cv ippm which is based on the perturbation approach and change of variable technique is proposed to calculate the confidence interval of sound pressure response and the system reliability by introducing cv ippm the nested loop o rcid is converted into an approximate single loop one which can be efficiently solved numerical examples demonstrate the effectiveness and efficiency of o rcid based on cv ippm for the optimization of the structural acoustic system with interval probabilistic variables

a polynomial expansion approach for response analysis of periodical composite structural acoustic problems with multi scale mixed aleatory and epistemic uncertainties abstract the response analysis of periodical composite structural acoustic problem with multi scale mixed aleatory and epistemic uncertainties is investigated based on homogenization method in this paper the aleatory uncertainties are presented by bounded random variables whereas the epistemic uncertainties are presented by interval variables and evidence variables when dealing with the combination of bounded random variables interval variables and evidence variables enormous computation is needed to estimate the output probability bounds of the sound pressure response of the periodical composite structural acoustic system to reduce the involved computational cost but without losing accuracy by transforming all of the bounded random variables and interval variables into evidence variables appropriately an evidence theory based polynomial expansion method epem is developed in which the gegenbauer series expansion is employed to approximate the variation range of the response with respect to evidence variables by using epem the probability bounds of the response can be obtained efficiently a numerical example is used to validate the proposed method and two engineering examples are given to demonstrate its efficiency

robust topology optimization under multiple independent unknown but bounded loads abstract the optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable or even lead to catastrophic failures a two level optimization formulation is often used in the robust topology optimization rto under uncertain loads various approaches have been reported to identify the critical loads associated with the worst structure responses because convex model approaches apply convex approximations to the original non convex model at the lower level the optimal designs obtained by these methods are greatly dependent on the quality of the approximation in this paper a new formulation based on the wolfe duality for the rto problems with multiple independent unknown but bounded loads is proposed following the two level formulation the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the wolfe duality both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non convexity associated with the lower level formulation numerical examples are also presented to demonstrate the proposed approach

construction and application of an ellipsoidal convex model using a semi definite programming formulation from measured data abstract as a set theory based convex model the ellipsoidal model provides an attractive framework for treating uncertain but bounded variations in the structural reliability analysis and design optimization however improper modeling of the uncertainties may give rise to misleading non probabilistic reliability analysis thus result in either unsafe or over conservative designs this paper presents a systematic study on the mathematical formulation for constructing the minimum volume ellipsoidal convex model using a given set of sample data and shows its application in existing methods of non probabilistic reliability analysis and design optimization of structures with bounded uncertainties in this method the uncertain parameters are first divided into groups according to their sources for each individual group of uncertainties the minimum volume ellipsoid problem is reformulated into a semi definite programming sdp problem and thus can be efficiently solved to its global optimum further a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space this uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations moreover it has another useful property the affine invariance which is shown to be necessary for meaningful definition of a non probabilistic reliability index the effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources

reliability based design optimization with dependent interval variables summary reliability based design optimization rbdo ensures design requirements are satisfied at required reliability levels in the presence of uncertainty the recent rbdo methodologies can accommodate two types of uncertainty aleatory type represented by random variables and epistemic type represented by interval variables however only independent interval variables can be handled the objective of this research is to introduce dependent interval variables into rbdo the dependency of interval variables comes from equality and inequality constraints with the intervals the reliability is also an interval and its lower bound is considered to preserve conservativeness an efficient optimization method is developed to solve rbdo problems the method is an extension of the existing framework of sequential optimization and reliability analysis with the inclusion of inequality constraints of interval variables the new method is efficient because it decouples the optimization process from the process of reliability and interval analyses the new method is applied to the rbdo of a slider mechanism with joint clearances which are known as dependent interval variables copyright 2012 john wiley sons ltd

non probabilistic reliability based topology optimization of geometrically nonlinear structures using convex models this paper describes a non probabilistic reliability based topology optimization method for the design of continuum structures undergoing large deformations the variation of the structural system is treated with the multi ellipsoid convex model which is a realistic description of the parameters being inherently uncertain but bounded or lacking sufficient probabilistic data the formulation of the optimal design is established as a volume minimization problem with non probabilistic reliability constraints on the geometrically nonlinear structural behaviour in order to circumvent numerical difficulties in solving the nested double loop optimization problem a performance measure based approach is employed to transform the constraint on the reliability index into one on the concerned performance in conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints the optimization problem is solved by gradient based mathematical programming methods three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method the obtained optimal solutions show the importance of incorporating various uncertainties in the design problem moreover it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation

probabilistic interval reliability of structural systems the probabilistic reliability approach is the most widely used method for reliability analysis the recent research shows that the reliabilities of structural systems strongly depend on the parameters of the probability model it is possible that the little error in the estimation of the parameters may lead to the remarkable error of the resulting probability in this study we introduce the interval approach into the conventional reliability theory we present a novel approach which allows us to obtain the system failure probability interval from the statistical parameter intervals of the basic variables this approach is a combination of the two techniques namely the classical reliability theory and the interval analysis in the end of this paper we show the feasibility of the proposed approach through two examples of the truss systems

a homogenization method for shape and topology optimization abstract shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by bendsoe and kikuchi together with various examples which may justify validity and strength of the present approach for plane structures

on the conversion of hensel codes to farey rationals three different algorithms are described for the conversion of hensel codes to farey rationals the first algorithm is based on the trial and error factorization of the weight of a hensel code inversion and range test the second algorithm is deterministic and uses a pair of different p adic systems for simultaneous computation from the resulting weights of the two different hensel codes of the same rational two equivalence classes of rationals are generated using the respective primitive roots the intersection of these two equivalence classes uniquely identifies the rational both the above algorithms are exponential in time and or space

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

lattice sums and the two dimensional periodic green s function for the helmholtz equation many algorithms that are currently used for the solution of the helmholtz equation in periodic domains require the evaluation of the green s function g x x0 the fact that the natural representat

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

integral representations of harmonic lattice sums we present new integral formulas for coulombic lattice sums that arise in solid state physics material science and complex analysis unlike ewald summation which can yield approximate integral representations the formulas described here are exact simple quadrature rules with modest numbers of nodes yield highly accurate results

reliability analysis for multidisciplinary systems with random and interval variables tremendous efforts have been devoted to developing efficient approaches to reliability analysis for multidisciplinary systems most of the approaches are only capable of dealing with random variables modeled by probability distributions both random and interval variables however may exist in multidisciplinary systems their propagation through coupled subsystems make s reliability analysis computationally expensive in this work a unified reliability analysis framework is proposed to deal with both random and interval variables in multidisciplinary systems the framework is an extension of an existing unified uncertainty analysis framework for single disciplinary problems the new framework involves probabilistic analysis and interval analysis both probabilistic analysis and interval analysis are decoupled from each other and are performed sequentially the first order reliability method is used for probabilistic analysis three supporting algorithms are developed the effectiveness of the algorithms is demonstrated with a mathematical example and an engineering application

antioptimization of structures with large uncertain but non random parameters via interval analysis non probabilistic interval modeling of uncertain but non random parameters for structures is developed in this paper for antioptimization analysis consisting in determining the least favorable responses the uncertain but non random parameter is considered to be a deterministic variable belonging to a set modeled as an interval the least favorable static displacement bound estimation for structures with uncertain but non random parameters is transformed into solving interval linear equations for small interval parameters the width of interval being small the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters by means of the interval matrix central notation and the natural interval extension interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by qiu et al 1 for large interval parameters a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study the numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method

efficient calculation of lattice sums for free space periodic green s function an efficient method to calculate the lattice sums is presented for a one dimensional 1 d periodic array of line sources the method is based on the recurrence relations for hankel functions and the fourier integral representation of the zeroth order hankel function the lattice sums of arbitrary high order are then expressed by an integral of elementary functions which is easily computed using a simple scheme of numerical integration the calculated lattice sums are used to evaluate the free space periodic green s function the numerical results show that the proposed method provides a highly accurate evaluation of the green s function with far less computation time even when the observation point is located near the plane of the array

an adaptive fast solver for the modified helmholtz equation in two dimensions in this paper we present a fast multipole accelerated integral equation method for solving the modified helmholtz equation du x b 2u x f x in two dimensions the method is direct and unlike classical fft based fast solvers it allows for adaptive mesh refinement but with comparable amount of work per grid point when the computational domain is rectangular dirichlet neumann periodic and free space boundary conditions can be imposed analytically without the need to solve a system of linear equations several important features of the algorithm are discussed including the use of precomputed tables diagonal translation operators and lattice sums to impose periodic boundary conditions numerical experiments show that for a wide range of the parameter b the algorithm is stable and high order accurate

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

a novel method of non probabilistic reliability based topology optimization corresponding to continuum structures with unknown but bounded uncertainties abstract a new non probabilistic reliability based topology optimization nrbto method for detailed design of continuum structures is proposed in this paper in which the unknown but bounded ubb uncertainties existing in material and external loads are considered simultaneously based on the interval set theoretic model traditional displacement constraint is transformed into the non probabilistic reliability format in which a novel measuring index namely the optimization feature distance is defined by a combination of the area ratio ideas thus the optimal layout issue is converted by the lightweight design problem with displacement reliability constraints moreover the interval parametric vertex approach and the adjoint vector method are also employed to obtain the sensitivity information between the reliability measurement and design variables by utilizing the method of moving asymptotes mma the optimization problem that is investigated can be solved iteratively two engineering examples are eventually presented to demonstrate the validity and effectiveness of the methodology that has been developed and numerical results show that considering the ubb uncertainty effects during the whole procedure of topology optimization may have a significant influence on the final structural configurations

spectral stochastic isogeometric analysis for linear stability analysis of plate abstract a spectral stochastic isogeometric analysis ssiga scheme is proposed for the stochastic linear stability analysis of plate with uncertain material properties within the proposed ssiga scheme the first order shear deformation theory of plate is adopted for modelling the kinematic relationship both homogeneous and functionally graded material fgm models can be incorporated the considered spatially dependent uncertainties i e young s modulus and poisson s ratio are modelled as random fields with gaussian and lognormal distributions and the spatially independent uncertainty i e gradient index of fgm is modelled as random variable the generalized isogeometric basis function is adopted for both the random field geometry representation and random field discretization through the karhunen loeve k l expansion an extended support vector regression x svr with a new generalized gegenbauer polynomial kernel is developed to model the nonlinear relationship between the structural uncertainties and the buckling load by further implementing various nonparametric statistical inference methods the mean standard deviation probability density function pdf and cumulative distribution function cdf of the buckling load can be effectively established to determine the strength limit of the plate the accuracy efficiency and applicability of the proposed approach are illustrated through two numerical examples

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

a fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions we present a new integral equation method for the calculation of two dimensional scattering from periodic structures involving triple points multiple materials meeting at a single point the combination of a robust and high order accurate integral representation and a fast direct solver permits the efficient simulation of scattering from fixed structures at multiple angles of incidence we demonstrate the performance of the scheme with several numerical examples

hybrid uncertain analysis of acoustic field with interval random parameters abstract for the acoustic field prediction without sufficient information an interval random model is introduced to treat with the uncertain parameters in which the probability distribution parameters of random variables are modeled as interval variables instead of given precise values based on the interval random model the hybrid uncertain acoustic dynamic equilibrium equation is constructed and a hybrid uncertain analysis method is proposed in the proposed method the interval random matrix and vector are expanded by the first order taylor series and the sound pressure response vector is obtained by the matrix perturbation technique a monotonic analysis is conducted for the sound pressure response vector and based on it the lower and upper bounds of the sound pressure response vector are obtained by the vertex method according to the response bounds the intervals of expectation and variance of the sound pressure response vector are calculated by the random interval moment method two numerical examples which include a simple problem with explicit expression and a complex engineering application are presented to demonstrate the effectiveness of the present method