sensitivity analysis with the modified heaviside function for the optimal layout design of multi component systems two kinds of design variables i e pseudo density variables associated with the framework structure and location design variables associated with connected components are involved in the layout design of multi component systems although sensitivities with respect to the first ones can easily be carried out as in topology optimization the semi analytical method sam is often used for sensitivity analysis with respect to the location design variables due to the geometric perturbation of the finite element mesh the latter can then be regarded as a geometric perturbation model gpm in this paper we propose a material perturbation model mpm using fixed finite element fe mesh for sensitivity analysis with respect to location design variables the material discontinuity across the boundary between each component and the framework structure is smoothed approximately by means of a modified heaviside function when a location design variable of a certain component is perturbed attached finite elements to the component boundary are assumed to undertake only a shift of material properties while the finite element mesh itself remains geometrically unchanged as a result analytical sensitivities with respect to location design variables are achieved as easily as for pseudo density variables the computing efficiency is thus improved because the velocity field for the mesh perturbation in the semi analytical scheme is no longer needed the mpm is illustrated by means of numerical tests especially the design optimization of 3d multi component systems

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

isogeometric collocation methods with generalized b splines we introduce isogeometric collocation methods based on generalized b splines and we analyze their performance through numerical examples for univariate and multivariate scalar and vector valued problems in particular advection diffusion and linear elasticity model problems are addressed the resulting method combines the favorable properties of isogeometric collocation and the geometrical and analytical problem oriented advantages of generalized b splines

stress constrained shape and topology optimization with fixed mesh a b spline finite cell method combined with level set function abstract in this paper we develop an efficient and flexible design method that integrates the b spline finite cell method b spline fcm and the level set function lsf for stress constrained shape and topology optimization any structure of complex geometry is embedded within an extended regular and fixed eulerian mesh no matter how the structure is optimized high order b spline shape functions are further implemented to ensure precisions of stress analysis and sensitivity analysis meanwhile level set functions i e implicit functions are used to enable topological changes of the considered structure through smooth boundary variations involved parameters rather than the conventional discrete form of lsf are directly taken as design variables to facilitate the numerical computing process to be specific the lsf is constructed by means of r functions that incorporate cubic splines as implicit functions to offer flexibilities for shape optimization within the framework of fixed mesh while the compactly supported radial basis functions cs rbfs are employed as implicit functions for stress constrained topology optimization it is shown the proposed fcm lsf method is a convenient approach that makes it possible to calculate stress and stress sensitivities with high precision representative examples of shape and topology optimization with and without stress constraints are solved with success demonstrating the advantages of the fcm lsf method

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

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

topology optimization in b spline space abstract in this paper we present a new form of density based topology optimization where the design space is restricted to the b spline space an arbitrarily shaped design domain is embedded into a rectangular domain in which tensor product b splines are used to represent the density field we show that with proper choice of b spline degrees and knot spans the b spline design space is free from checkerboards without extraneous filtering or penalty we further reveal that the b spline representation provides an intrinsic filter for topology optimization where the filter size is controlled by b spline degrees and knot spans this b spline filter is effective in removing numerical artifacts and controlling minimal feature length in optimized structures when the b spline basis functions span multiple analysis elements we demonstrate that the b spline filter is linear in storage cost and does not require neighboring element information further this b spline based density representation decouples the design representation of density distribution from the finite element mesh thus multi resolution designs can be obtained without re meshing the design domain in particular successive optimization with respect to design resolutions leads to topologically simple features obtainable in either coarse or fine design resolutions thus achieving a form of mesh independency with respect to design representation this approach is versatile in the sense a variety of finite element and isogeometric analysis techniques can be used for solution of equilibrium equations and a variety of projection methods can be used to approximate b spline density in analysis numerical studies have been conducted over several representative topology optimization problems including minimal compliance of mbb beams compliant mechanism inverters and heat conductions

a rescaled localized radial basis function interpolation on non cartesian and nonconforming grids in this paper we propose a rescaled localized radial basis function rl rbf interpolation method based on the use of compactly supported radial basis functions starting from the classical rbf interpolation technique we introduce a rescaling that allows for exact interpolation of constant fields between nonconforming meshes without the use of an extra polynomial term we also present two dimensional and three dimensional numerical examples on arbitrary finite element meshes to show that the rl rbf interpolation leads to accurate results fast evaluation and easy parallelization of the algorithm all the computations are carried out using the open source finite element library lifev

optimal order isogeometric collocation at galerkin superconvergent points abstract in this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least squares collocation method presented in anitescu et al 2015 and the variational collocation method presented in gomez and de lorenzis 2016 the focus is on smoothest b splines nurbs approximations i e having global c p 1 continuity for polynomial degree p within the framework of gomez and de lorenzis 2016 we select as collocation points a subset of those considered in anitescu et al 2015 which are related to the galerkin superconvergence theory with our choice that features local symmetry of the collocation stencil we improve the convergence behavior with respect to gomez and de lorenzis 2016 achieving optimal l 2 convergence for odd degree b splines nurbs approximations the same optimal order of convergence is seen in anitescu et al 2015 where however a least squares formulation is adopted further careful study is needed since the robustness of the method and its mathematical foundation are still unclear

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

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

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

non conforming spectral element finite element approximations for partial differential equations abstract the aim of spectral element methods is to combine the relative advantages of finite elements complex geometries and spectral techniques high accuracy we present here a new non conforming method which allows for greater flexibility in spectral element domain decomposition as well as direct spectral element finite element coupling theoretical justification and numerical examples are given

a level set based parameterization method for structural shape and topology optimization this paper presents an effective parametric approach by extending the conventional level set method to structural shape and topology optimization using the compactly supported radial basis functions rbfs and the optimality criteria oc method the structural design boundary is first represented implicitly by embedding into a higher dimensional level set function as its zero level set and the rbfs of a favorable smoothness are then applied to interpolate the level set function the original initial value problem is thus converted to a parametric optimization with the expansion coefficients of the interplant posed as the design variables the oc method is then applied to advance the structure boundary in terms of the velocity field derived from the parametric optimization hence the structural shape and topology optimization is now transformed into a process of iteratively finding coefficients to update the level set function to achieve an optimal configuration the numerical considerations of the conventional level set method including upwind schemes velocity extension and reinitialization are eliminated the proposed scheme is capable of addressing structural shape fidelity and topology change simultaneously and of keeping the boundary smooth during the optimization process furthermore numerical convergence is expected to be improved a widely investigated example in the framework of structural stiffness designs is applied to demonstrate the efficiency and accuracy of the proposed approach copyright 2007 john wiley sons ltd

weakly enforced essential boundary conditions for nurbs embedded and trimmed nurbs geometries on the basis of the finite cell method summary enforcing essential boundary conditions plays a central role in immersed boundary methods nitsche s idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints we formulate an extension of nitsche s method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the b spline and non uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries furthermore we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3d geometries with several benchmark problems we demonstrate the overall good convergence behavior of the proposed method and its good accuracy we provide extensive studies on the stability of the method its influence parameters and numerical properties and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two a newly composed boundary integration concept further enhances the modeling process and allows a flexible discretization independent introduction of boundary conditions finally we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non uniform rational basis spline geometries copyright 2013 john wiley sons ltd

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

the finite cell method for three dimensional problems of solid mechanics this article presents a generalization of the recently proposed finite cell method to three dimensional problems of linear elasticity the finite cell method combines ideas from embedding or fictitious domain methods with the p version of the finite element method besides supporting a fast simple generation of meshes it also provides high convergence rates mesh generation for a boundary representation of solids and for voxel based data obtained from ct scans is addressed in detail in addition the implementation of non homogeneous neumann boundary conditions and the computation of cell matrices based on a composed integration is presented the performance of the proposed method is demonstrated by three numerical examples including the elastostatic computation of a human bone biopsy

shape optimization with topological changes and parametric control recent advances in shape optimization rely on free form implicit representations such as level sets to support boundary deformations and topological changes by contrast parametric shape optimization is formulated directly in terms of meaningful geometric design variables but usually does not support free form boundary and topological changes we propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques the shapes in the design space are represented implicitly as level sets of a higher dimensional function that is constructed using b splines to allow free form deformations and parameterized primitives combined with r functions to support desired parametric changes our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free form shapes the resulting method is also general in that it subsumes most other types of shape optimization as special cases we describe an implementation of the proposed technique with attractive numerical properties the explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient based optimization methods furthermore our implementation does not require any error prone polygonization or approximation of level sets isocurves and isosurfaces the effectiveness of the method is demonstrated by several numerical examples

ill posedness of the 3d navier stokes equations in a generalized besov space near bmo 1 abstract the ill posedness of the 3d navier stokes equations in a generalized besov space which is smaller than b q 1 q 2 is considered in 2008 bourgain pavlovic proved that the 3d navier stokes equation is ill posed in b 1 by showing norm inflation phenomena of the solution for some initial data on the other hand in 2008 germain proved that the flow map is not c 2 in the space b q 1 for q 2 however he did not treat ill posed problem in such spaces thus our result is an extension of these previous results

stress based shape and topology optimization with the level set method abstract this paper proposes a level set method to solve minimum stress and stress constrained shape and topology optimization problems the method solves a sub optimization problem every iteration to obtain optimal boundary velocities a p norm stress functional is used to aggregate stresses in a single constraint the shape sensitivity function is derived and a computational procedure based on a least squares interpolation approach is devised in order to compute sensitivities at the boundaries adaptive constraint scaling is used to enforce exact control of stress limits numerical results show that the method is able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries

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

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

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

isogeometric mortar methods the application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically for the lagrange multiplier two choices of uniformly stable spaces are presented both of them are spline spaces but of a different degree in one case we consider an equal order pairing for which a cross point modification based on a local degree reduction is required in the other case the degree of the dual space is reduced by two compared to the primal this pairing is proven to be inf sup stable without any necessary cross point modification several numerical examples confirm the theoretical results and illustrate additional aspects

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

mesh grading in isogeometric analysis this paper is concerned with the construction of graded meshes for approximating so called singular solutions of elliptic boundary value problems by means of multipatch discontinuous galerkin isogeometric analysis schemes such solutions appear for instance in domains with re entrant corners on the boundary of the computational domain in problems with changing boundary conditions in interface problems or in problems with singular source terms making use of the analytic behavior of the solution we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach we prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in isogeometric analysis

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

isogeometric analysis and domain decomposition methods in the present work we use the mortar finite element method for the coupling of nonconforming discretized sub domains in the framework of nonlinear elasticity the mortar method has been shown to preserve optimal convergence rates see laursen 2002 25 for details and is variationally consistent we show that the method can be applied to isogeometric analysis with little effort once the framework of nurbs based shape functions has been implemented furthermore a specific coordinate augmentation technique allows the design of an energy momentum scheme for the constrained mechanical system under consideration the excellent performance of the redesigned mortar method as well as the energy momentum scheme is illustrated in representative numerical examples

a finite element based level set method for structural optimization a finite element based level set method is implemented for structural optimization the streamline diffusion finite element method is used for solving both the level set equation and the reinitialization equation the lumped scheme is addressed and the accuracy is compared with the conventional finite difference based level set method a dirichlet boundary condition is enforced during the reinitialization to prevent the boundary from drifting numerical examples of minimum mean compliance design illustrate the reliability of the proposed optimization method copyright 2009 john wiley sons ltd

radial basis functions and level set method for structural topology optimization level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures in this paper the popular radial basis functions rbfs in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization rbf implicit modelling with multiquadric mq splines is developed to define the implicit level set function with a high level of accuracy and smoothness a rbf level set optimization method is proposed to transform the hamilton jacobi partial differential equation pde into a system of ordinary differential equations odes over the entire design domain by using a collocation formulation of the method of lines with the mathematical convenience the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients a physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods the proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted numerical examples show the success of the present rbf level set method in the accuracy convergence speed and insensitivity to initial designs in topology optimization of two dimensional 2d structures it is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization

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

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

the internodes method for the treatment of non conforming multipatch geometries in isogeometric analysis abstract in this paper we apply the internodes method to solve second order elliptic problems discretized by isogeometric analysis methods on non conforming multiple patches in 2d and 3d geometries internodes is an interpolation based method that on each interface of the configuration exploits two independent interpolation operators to enforce the continuity of the traces and of the normal derivatives internodes easily handles both parametric and geometric nurbs non conformity we specify how to set up the interpolation matrices on non conforming interfaces how to enforce the continuity of the normal derivatives and we give special attention to implementation aspects the numerical results show that internodes exhibits optimal convergence rate with respect to the mesh size of the nurbs spaces and that it is robust with respect to jumping coefficients

a level set method for topology optimization of continuum structures with bounded design domains in practice a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary the design domains of practical structures are often limited and significantly affect the final optimal design of the structures structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods firstly to overcome the limitations of current level set methods and the stopping issue of structural boundary movements this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains a set of new normal velocities required by level set movements is given a level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques which include the level set regularization algorithm gradient projection algorithm nonlinear velocity mapping algorithm and return mapping algorithm secondly in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers a mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints the validity and effectiveness of the proposed method are demonstrated with two examples

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

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

a large deformation frictional contact formulation using nurbs based isogeometric analysis this paper focuses on the application of nurbs based isogeometric analysis to coulomb frictional contact problems between deformable bodies in the context of large deformations a mortar based approach is presented to treat the contact constraints whereby the discretization of the continuum is performed with arbitrary order nurbs as well as c0 continuous lagrange polynomial elements for comparison purposes the numerical examples show that the proposed contact formulation in conjunction with the nurbs discretization delivers accurate and robust predictions results of lower quality are obtained from the lagrange discretization as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface copyright 2011 john wiley sons ltd

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

stress based topology optimization with discrete geometric components abstract in this paper we introduce a framework for the stress based topology optimization of structures made by the assembly of discrete geometric components such as bars and plates that are described by explicit geometry representations to circumvent re meshing upon design changes we employ the geometry projection method to smoothly map the geometric components onto a continuous density field defined over a uniform finite element grid for analysis the geometry projection is defined in a manner that prevents the singular optima phenomenon widely reported in the literature and that effectively considers stresses only on the geometric components and not on the void region as in previous work a size variable is ascribed to each geometry component and penalized in the spirit of solid isotropic material with penalization simp allowing the optimizer to entirely remove geometric components from the design we demonstrate our method on the l bracket benchmark for stress based optimization problems in 2 d and 3 d

weak coupling for isogeometric analysis of non matching and trimmed multi patch geometries nitsche s method can be used as a coupling tool for non matching discretizations by weakly enforcing interface constraints we explore the use of weak coupling based on nitsche s method in the context of higher order and higher continuity b splines and nurbs we demonstrate that weakly coupled spline discretizations do not compromise the accuracy of isogeometric analysis we show that the combination of weak coupling with the finite cell method opens the door for a truly isogeometric treatment of trimmed b spline and nurbs geometries that eliminates the need for costly reparameterization procedures we test our methodology for several relevant technical problems in two and three dimensions such as gluing together trimmed multi patches and connecting non matching meshes that contain b spline basis functions and standard triangular finite elements the results demonstrate that the concept of nitsche based weak coupling in conjunction with the finite cell method has the potential to considerably increase the flexibility of the design through analysis process in isogeometric analysis

dual primal isogeometric tearing and interconnecting solvers for multipatch dg iga equations abstract in this paper we consider a new version of the dual primal isogeometric tearing and interconnecting ieti dp method for solving large scale linear systems of algebraic equations arising from discontinuous galerkin dg isogeometric analysis of diffusion problems on multipatch domains with non matching meshes the dg formulation is used to couple the local problems across patch interfaces the purpose of this paper is to present this new method and provide numerical examples indicating a polylogarithmic condition number bound for the preconditioned system and showing an incredible robustness with respect to large jumps in the diffusion coefficient across the interfaces

analysis of the internodes method for non conforming discretizations of elliptic equations abstract internodes is a general method to deal with non conforming discretizations of second order partial differential equations on regions partitioned into two or several subdomains it exploits two intergrid interpolation operators one for transferring the dirichlet trace across the interface the others for the neumann trace in every subdomain the original problem is discretized by the finite element method using a priori non matching grids and piece wise polynomials of different degree in this paper we provide several interpretations of the method and we carry out its stability and convergence analysis showing that internodes exhibits optimal convergence rate with respect to the finite element sizes finally we propose an efficient algorithm for the solution of the corresponding algebraic system

ill posedness of the navier stokes equations in a critical space in 3d abstract we prove that the cauchy problem for the three dimensional navier stokes equations is ill posed in b 1 in the sense that a norm inflation happens in finite time more precisely we show that initial data in the schwartz class s that are arbitrarily small in b 1 can produce solutions arbitrarily large in b 1 after an arbitrarily short time such a result implies that the solution map itself is discontinuous in b 1 at the origin

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

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

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

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

a unified parametric design approach to structural shape optimization this paper presents a general parametric design approach for 2 d shape optimization problems this approach has been achieved by integrating practical design methodologies into numerical procedures it is characterized by three features i automatic selection of a minimum number of shape design variables based on the cad geometric model ii integration of sequential convex programming algorithms to solve equality constrained optimization problems iii efficient sensitivity analysis by means of the improved semi analytical method it is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities numerical solutions are performed to demonstrate the applicability of the proposed approach a discussion of the results is also given

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

weak solutions for the navier stokes equations with b 1 ln bxr 1 r 21 r l2 initial data one of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained in the book recent developments in the navier stokes problems lemarie rieusset proved that the navier stokes equations have global weak solutions for initial data in the space bxr 1 r 21 r rn l2 rn 0 r 1 where xr is the space of functions whose pointwise products with hr functions belong to l2 xr denotes the closure of c0 rn in xr and bxr 1 r 21 r rn is the besov space over xr in this paper we partially extend this result of lemarie rieusset to the larger initial value space b 1 ln rn bxr 1 r 21 r rn l2 rn 0 r 1 where b 1 ln rn is a logarithmically modified version of the usual besov space b 1 rn

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

optimization of structures with thin layer functional device on its surface through a level set based multiple type boundary method abstract functional structure i e native base structure equipped with thin layer functional devices on its surface is important in many engineering applications the performance of such a functional structure is seriously affected by the configuration of the base structure and the surface functional devices and the optimization of configuration is a simultaneous topology optimization of the boundaries that belong to different types in the present study load cell is taken as an example of the functional structure and strain gauge is taken as an example of the functional device the task of designing a load cell is accomplished by solving a multi objective topology optimization problem whose objective makes a compromise between stiffness and flexibility of the base structure the design variables include two types of boundaries functional boundary where a strain gauge is installed the free boundary where there does not exist any connection to the environment the optimization problem is solved by using a level set based multiple type boundary method two level set functions are used to represent a functional structure and the two types of boundaries evolution of the two boundaries is modeled by two independent hamilton jacobi equations re meshing is conducted in each iteration of optimization quadrilateral body fitted mesh is used for the finite element analysis and the artificial weak material is not used numerical examples in two dimensions are investigated

integrated layout design of multi component systems using xfem and analytical sensitivity analysis this study presents the integrated layout optimization of multi component systems using a fixed mesh the optimization formulation is established under the framework of the extended finite element method xfem the level set method is used to represent components and is combined with the xfem to describe material discontinuities across elements sensitivity analysis is proposed with respect to geometric variables of components and pseudo densities of the basic structure an analytical shape sensitivity analysis method with respect to positions and shapes of components is developed both solid and void components are considered to show the efficiency and accuracy of the proposed shape sensitivity analysis method furthermore a revised finite circle method that adapts shape changes of elliptical components is proposed for the definition of non overlapping constraints finally numerical examples of maximizing the structural stiffness are tested to demonstrate the proposed method

isogeometric analysis using t splines on two patch geometries we develop optimal approximation estimates for t splines in the case of geometries obtained by gluing two standard tensor product patches we derive results both for the t spline space in the parametric domain and the mapped t nurbs in the physical one a set of numerical tests in complete accordance with the theoretical developments is also presented

shape optimization of dirichlet boundaries based on weighted b spline finite cell method and level set function abstract this paper addresses extended shape optimization problems where structural supports i e dirichlet boundaries and free boundaries are simultaneously optimized unlike traditional fem the weighted b spline finite cell method fcm is applied as structural analysis tool and combined with the level set function lsf to take into account dirichlet boundary condition dbc automatically through penalization of the displacement field the proposed shape optimization method achieves a comprehensive integration of fixed grid b spline shape function and level set function as both the structure and dirichlet boundaries are described in the form of lsf any modification of dirichlet boundaries can be made in a straightforward way as easily as free boundaries by changing continuous design variables meanwhile the computing accuracy is ensured within the framework of fixed grid owing to the quadtree refinements of boundary cells stress related shape optimization problems are finally solved to demonstrate the merit and validity of the proposed optimization method in dealing with shape optimization of dirichlet boundaries

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