a level set method for shape and topology optimization of large displacement compliant mechanisms a parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements a level set model is established mathematically as the hamilton jacobi equation to capture the motion of the free boundary of a continuum structure the structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension the radial basis function with compact support is then applied to interpolate the level set function leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation in doing so the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients as a result the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method with the concept of the shape derivative the geometrical non linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms several numerical benchmark examples illustrate the effectiveness of the present level set method in particular its capability of generating new holes inside the material domain the proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme the reinitialization procedure and the velocity extension algorithm in the conventional level set method

conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

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

topology optimization with implicit functions and regularization summary topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape the implicit function is constrained by upper and lower bounds so that only a band of nodal variables needs to be considered in each step of the optimization the weak form of the equilibrium equation is expressed as a heaviside function of the implicit function the heaviside function is regularized to permit the evaluation of sensitivities we show that the method is a dual of the bendshe kikuchi method the method is applied both to problems of optimizing single material and

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

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

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

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

elimination of fast chaotic degrees of freedom on the accuracy of the born approximation we apply standard projection operator techniques known from nonequilibrium statistical mechanics to eliminate fast chaotic degrees of freedom in a low dimensional dynamical system through the usual perturbative approach we end up in second order with a stochastic system where the fast chaotic degrees of freedom are modelled by gaussian white noise the accuracy of the perturbation expansion is analysed in detail by the discussion of an exactly solvable model

stochastic modelling replacing fast degrees of freedom by noise in systems with timescale separation where the fast degrees of freedom exhibit chaotic motion the latter are replaced by suitable stochastic processes a projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones the resulting equations can be approximated in a controlled way by fokker planck equations or equivalently by stochastic differential equations for slow degrees of freedom we discuss some model situations and explore the accuracy of the approximations by numerical simulations

topology optimization of plane structures using smoothed particle hydrodynamics method summary this paper presents an alternative topology optimization method based on an efficient meshless smoothed particle hydrodynamics sph algorithm to currently calculate the objective compliance the deficiencies in standard sph method are eliminated by introducing corrective smoothed particle method and total lagrangian formulation the compliance is established relative to a designed density variable at each sph particle which is updated by optimality criteria method topology optimization is realized by minimizing the compliance using a modified solid isotropic material with penalization approach some numerical examples of plane elastic structure are carried out and the results demonstrate the suitability and effectiveness of the proposed sph method in the topology optimization problem copyright 2016 john wiley sons ltd

averaging in dynamical systems and large deviations the paper treats ordinary differential equations of the form frac dz varepsilon left t right dt varepsilon b z varepsilon t f t y wheref t is a hyperbolic flow large deviations bounds for the averaging principle are obtained here in the form appeared previously in f1 f2 for the case when the flowf t is replaced by a markov process

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

from the perron frobenius equation to the fokker planck equation we show that for certain classes of deterministic dynamical systems the perron frobenius equation reduces to the fokker planck equation in an appropriate scaling limit by perturbative expansion in a small time scale parameter we also derive the equations that are obeyed by the first and second order correction terms to the fokker planck limit case in general these equations describe non gaussian corrections to a langevin dynamics due to an underlying deterministic chaotic dynamics for double symmetric maps the first order correction term turns out to satisfy a kind of inhomogeneous fokker planck equation with a source term for a special example we are able solve the first and second order equations explicitly

stochastic homogenization for an energy conserving multi scale toy model of the atmosphere we study a hamiltonian toy model for a lagrangian fluid parcel in the semi geostrophic limit which exhibits slow and fast dynamics we first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system in a second step we use stochastic singular perturbation theory to derive an effective reduced stochastic differential equation for the slow dynamics we verify the results in numerical simulations

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

driven chaotic motion in single and double well potentials abstract the stochastic motions in stable and bistable potentials are compared on the basis of a langevin like equation with a deterministic chaotic process replacing gaussian white noise ergodic and mixing properties are obtained under rather general circumstances in the case of the symmetric double well potential the system exhibits a behavior quite different from what one observes for stable potentials as the control parameters are varied the system undergoes a phase transition from an ergodic to a nonergodic phase lyapunov exponents and fractal dimensions are studied for more detailed insight

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

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

concentration and limit behaviors of stationary measures abstract in this paper we study limit behaviors of stationary measures of the fokker planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative noise including additive white noise case as the noises are vanishing various results on the invariance and concentration of the limit measures are obtained in particular we show that if the noise perturbed systems admit a uniform lyapunov function then the stationary measures form a relatively sequentially compact set whose weak limits are invariant measures of the unperturbed system concentrated on its global attractor in the case that the global attractor contains a strong local attractor we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor and on the contrary in the presence of a strong local repeller in the global attractor there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller moreover we show that if there is a strongly repelling equilibrium in the global attractor then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium as applications of these results an example of stochastic hopf bifurcation and an example with non decomposable limit sets are provided our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system

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

best constant in sobolev trace inequalities on the half space using a mass transportation method we study optimal sobolev trace inequalities on the half space and prove a conjecture made by escobar in 1988 about the minimizers

a bridge between sobolev and escobar inequalities and beyond abstract the classical sobolev and escobar inequalities are embedded into the same one parameter family of sharp trace sobolev inequalities on half spaces equality cases are characterized for each inequality in this family by tweaking a well known mass transportation argument and lead to a new comparison theorem for trace sobolev inequalities the case p 2 corresponds to a family of variational problems on conformally flat metrics which was previously settled by carlen and loss with their method of competing symmetries in this case minimizers interpolate between conformally flat spherical and hyperbolic geometries passing through the euclidean geometry defined by the fundamental solution of the laplacian

design sensitivity analysis and topology optimization of displacement loaded non linear structures abstract a continuum based design sensitivity analysis dsa method for geometrically nonlinear systems with nonhomogeneous boundary conditions is developed to topologically optimize the displacement loaded nonlinear structures in the adjoint variable method the solution space requires just homogeneous boundary conditions even if the original system has nonhomogeneous ones a design sensitivity expression for the instantaneous rigidity functional is derived for the displacement loaded nonlinear topology optimization the tangent stiffness is obtained at the end of the equilibrium iterations in the nonlinear analysis of the original system this stiffness is used in the dsa so that no iteration would be necessary to evaluate the design sensitivity expressions in force loaded systems the solution dose not converge easily because the material distributes sparsely sometimes during optimization however when the displacement loaded system is used there is no convergence difficulty

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

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

indistinguishable states ii the imperfect model scenario given a perfect model of a chaotic system and a set of noisy observations of arbitrary duration it is not possible to determine the state of this system precisely rather one must consider a set of states which are indistinguishable from one another given the observations k judd l a smith indistinguishable states i physica d 151 2001 yet the perfect model scenario is a fiction in practice all models are imperfect how do the results from the perfect model scenario change under imperfect models it is shown to be essential to take even small model imperfections into account failure to do so can systematically degrade state estimation or prediction of nonlinear systems with an imperfect model the system state space and model state space are rarely if ever equivalent and so one must consider a projection of the system state into the model state space furthermore it is almost certain that no trajectory of the model is consistent with an infinite series of observations thus there is no consistent way to estimate the projection of system state using trajectories there are pseudo orbits however that are consistent with observations and these can be used to estimate the projection of the system state using pseudo orbits one finds that as in the perfect model scenario there is a set of states that are indistinguishable from the projection of the system state estimation of this set of indistinguishable states and the probability density on these states is discussed the main conclusions are i that there is no state of the model that can be identified with the state of the system and ii that great care must be taken when using an imperfect model to forecast the system because the initialization of the model state from observations can provide a poor analogue for the system the forecast may not shadow the future behaviour of the system for very long even if one were able to obtain a noise free projection of the system state the ultimate aims of probability forecasts should be re examined in light of these results

topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique abstract this paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density based topology optimization of elastic structures undergoing large displacements by adding a special hyperelastic material to the design domain excessive distortion and numerical instability occurred in the low density or intermediate density elements are thus effectively alleviated during the optimization process the properties of the additional hyperelastic material are established based on a new interpolation scheme which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure in conjunction with the adjoint variable scheme for sensitivity analysis the topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given to demonstrate the effectiveness of the proposed method

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

fast chaos versus white noise entropy analysis and a fokker planck model for the slow dynamics abstract we show in detail how methods of time series analysis such as dimension and entropy estimates support the idea that fast low dimensional chaos can be modelled properly by noise motivated by this observation we derive a formalism by which fast chaotic degrees of freedom in systems with time scale separation can be replaced by a suitable stochastic process a fokker planck equation for the phase space density of the slow degrees of freedom is derived from the full deterministic system by a projection operator technique together with a perturbation expansion we compare two different projection strategies and illustrate the resulting equations by specifying them for a simple model system

multi resolution multi scale topology optimization a new paradigm abstract the purpose of this work is to present a new concept multi resolution multi scale topology optimization the key idea of the present strategy is that design optimization should be performed progressively from low to high resolution not at a single resolution level to achieve the multi resolution strategy design optimization is formulated in a wavelet based variable space not in a direct density variable space the major advantages of the multi resolution design optimization include 1 topologically simple and close to the global optimum structures may be obtained without any explicit constraint and 2 the convergence is not sensitive to mathematical programming methods for the efficient numerical implementation of the multi resolution approach the side constraints imposed on the direct density variables are removed by mapping the density variables into intermediate variables these intermediate variables are then wavelet transformed to new design variables it is addressed that the present multi resolution topology optimization can resolve major numerical instability problems such as mesh dependencies and local minima the usefulness of the multi scale nature of the wavelets in the present multi resolution multi scale optimization formulation is also discussed

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem