topology optimization for effective energy propagation in rate independent elastoplastic material systems abstract a topology optimization framework for effective energy management in dynamically loaded structures with rate independent elastoplastic material behavior is presented the elastoplastic constitutive equations are integrated using an implicit numerical scheme based on a multi level newton procedure this primal analysis procedure requires the computation of the incremental algorithmic consistent tangent operator derived from the mappings that enforce the evolving constitutive relations for our gradient based optimization we develop discrete adjoint sensitivity expressions finite element implementation is discussed and the optimization framework is exemplified via the design of three dimensional structures subjected to dynamic loads

global well posedness of the dynamic φ4 model in the plane we show global well posedness of the dynamic 4 4 model in the plane the model is a nonlinear stochastic pde that can only be interpreted in a renormalised sense solutions take values in suitable weighted besov spaces of negative regularity

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

voigt reuss topology optimization for structures with nonlinear material behaviors this work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization in the proposed framework alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain since two or more materials are permitted to simultaneously occupy local regions of the design domain small strain integration algorithms for general two material mixtures of solids are developed for the voigt isostrain and reuss isostress assumptions and hybrid combinations thereof structural topology optimization problems involving non linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis dsa of energy type functionals are presented the consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials elastoplastic materials and viscoelastic materials the good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures it is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance 1997 john wiley sons ltd

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

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 and shape optimization for elastoplastic structural response abstract it is a common practice to base both material topology optimization as well as a subsequent shape optimization on linear elastic response however in order to obtain a realistic design it might be essential to base the optimization on a more realistic physical behavior i e to consider geometrically or and materially nonlinear effects in the present paper an elastoplastic von mises material model with linear isotropic hardening softening for small strains is used the objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed with respect to the specific features of either topology or shape optimization for example the number of optimization variables or their local global influence on the structural response different methods are applied for topology optimization problems the gradient of the ductility is determined by the variational adjoint approach in shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach the topology optimization problem is solved by an optimality criteria oc method and the shape optimization problem by a mathematical programming mp method in topology optimization a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities the procedures are verified by 2d design problems under plane stress conditions

cyclicity of bicyclic operators and completeness of translates we study cyclicity of operators on a separable banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions a simple consequence of our main result is that a bicyclic unitary operator on a banach space with separable dual is cyclic our results also imply that if s a n n in mathbb z longmapsto a n 1 n in mathbb z is the shift operator acting on the weighted space of sequences ell omega 2 mathbb z if the weight satisfies some regularity conditions and n 1 for nonnegative n then s is cyclic if rm lim n rightarrow infty rm log omega n sqrt n 0 on the other hand one can see that s is not cyclic if the series sum n geq 1 rm log omega n n 2 diverges we show that the question of herrero whether either s or s is cyclic on ell omega 2 mathbb z admits a positive answer when the series sum n in mathbb z rm log s n n 2 1 is convergent we also prove completeness results for translates in certain banach spaces of functions on mathbb r

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

the dynamic phi 4_3 φ 3 4 model comes down from infinity we prove an a priori bound for the dynamic phi 4 3 model on the torus which is independent of the initial condition in particular this bound rules out the possibility of finite time blow up of the solution it also gives a uniform control over solutions at large times and thus allows one to construct invariant measures via the krylov bogoliubov method it thereby provides a new dynamic construction of the euclidean phi 4 3 field theory on finite volume our method is based on the local in time solution theory developed recently by gubinelli imkeller perkowski and catellier chouk the argument relies entirely on deterministic pde arguments such as embeddings of besov spaces and interpolation which are combined to derive energy inequalities

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

ergodicity and exponential mixing of the real ginzburg landau equation with a degenerate noise abstract in this paper we establish the existence uniqueness and attraction properties of an invariant measure for the real ginzburg landau equation in the presence of a degenerate stochastic forcing acting only in four directions the main challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system to achieve this we propose a condition which only requires four noises

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

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

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

tangent operators and design sensitivity formulations for transient non linear coupled problems with applications to elastoplasticity tangent operators and design sensitivities are derived for transient non linear coupled problems the solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods the derived formulations are suitable for finite element implementations analyses of systems with materials that exhibit history dependent response may be obtained directly by applying the analyses of transient non linear coupled systems rate independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

nonlinear fluctuations of weakly asymmetric interacting particle systems we introduce what we call the second order boltzmann gibbs principle which allows one to replace local functionals of a conservative one dimensional stochastic process by a possibly nonlinear function of the conserved quantity this replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states as an application of this second order boltzmann gibbs principle we introduce the notion of energy solutions of the kpz and stochastic burgers equations under minimal assumptions we prove that the density fluctuations of one dimensional stationary weakly asymmetric conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic burgers equation we also show that the fluctuations of the height function associated to these models are given by energy solutions of the kpz equation in this sense unfortunately we lack a uniqueness result for these energy solutions we conjecture these solutions to be unique and we show some regularity results for energy solutions of the kpz burgers equation supporting this conjecture

a littlewood paley description of modelled distributions abstract we exhibit a fundamental link between hairer s theory of regularity structures 11 and the paracontrolled calculus of 9 by using paraproducts we provide a littlewood paley description of the spaces of modelled distributions in regularity structures that is similar to the besov description of classical holder spaces

topology optimization of structures under variable loading using a damage superposition approach we present an original algorithm and accompanying mathematical formulation for topology optimization of structures that can sustain material damage and are subject to multiple load cases with varying configurations damage accumulation is simulated using a coupled non linear brittle damage model the structures are optimized for minimum mass subject to stiffness constraints defined as the compliance evaluated at the end of each loading sequence to achieve robustness of the optimized structures the respective damage fields caused by each individual load case are computed and combined using superposition to simulate a worst case damage field all load cases are then run a second time using the worst case damage distribution as a starting point in this way one effectively accounts for the spectrum of possible load sequences to which the structure may be subjected results from this method are compared with an exhaustive brute force approach in which all non repeating load sequences are analyzed individually for each method the corresponding sensitivities are derived and implemented analytically using a path dependent adjoint method the two approaches are implemented on a series of numerical examples which demonstrate that the superposition method produces structures that are as robust as those obtained using the exhaustive method but require significantly less computational effort copyright 2014 john wiley sons ltd

cyclicity in lp spaces and zero sets of the fourier transforms abstract we study the cyclicity of vectors u in l p z it is known that a vector u is cyclic in l 2 z if and only if the zero set z u of its fourier transform u has lebesgue measure zero and log u l 1 t where t is the unit circle here we show that unlike l 2 z there is no characterization of the cyclicity of u in l p z 1 p 2 in terms of z u and the divergence of the integral t log u moreover we give both necessary conditions and sufficient conditions for u to be cyclic in l p z 1 p 2

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

failure mitigation in optimal topology design using a coupled nonlinear continuum damage model abstract this paper presents a novel framework for performing topology optimization of structures while accounting for material damage a quasi static non local brittle damage model is coupled to a linear finite element analysis code for modeling the structural response and the initiation and propagation of damage the optimization problem is parameterized using simp variables and failure mitigation is achieved through the enforcement of constraints on the maximum local damage intensity we derive a computationally efficient adjoint sensitivity formulation that accounts for the path dependence of the coupled analysis problem the algorithm is validated through a series of 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

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

wiener s closure of translates problem and piatetski shapiro s uniqueness phenomenon n wiener characterized the cyclic vectors with respect to translations in p z and l p r p 1 2 in terms of the zero set of the fourier transform he conjectured that a similar characterization should be true for 1 p 2 our main result contradicts this conjecture