full analytical sensitivities in nurbs based isogeometric shape optimization abstract non uniform rational b spline nurbs has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among cad systems the advent of nurbs based isogeometric analysis has made it even more advantageous to use nurbs in shape optimization since it can potentially avoid the inaccuracy and labor tediousness in geometric model conversion from the design model to the analysis model although both positions and weights of nurbs control points affect the shape until very recently usually only control point positions are used as design variables in shape optimization thus restricting the design space and limiting the shape representation flexibility this paper presents an approach for analytically computing the full sensitivities of both the positions and weights of nurbs control points in structural shape optimization such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient based shape optimization the analytical sensitivity for both positions and weights of nurbs control points is especially beneficial for recovering optimal shapes that are conical e g ellipses and circles in 2d cylinders ellipsoids and spheres in 3d that are otherwise not possible without the weights as design variables

error analysis of trigonometric integrators for semilinear wave equations an error analysis of trigonometric integrators or exponential integrators applied to spatial semidiscretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given in particular optimal second order convergence is shown requiring only that the exact solution is of finite energy the analysis is uniform in the spatial discretization parameter it covers the impulse method which coincides with the method of deuflhard and the mollified impulse method of garcia archilla sanz serna and skeel as well as the trigonometric methods proposed by hairer and lubich and by grimm and hochbruck the analysis can also be used to explain the convergence behavior of the stormer verlet leapfrog time discretization

isogeometric structural shape optimization an old dilemma in structural shape optimization is the needed tight link between design model or geometric description and analysis model the intention of this paper is to show that isogeometric analysis offers a potential and promising way out of this dilemma to this end we show a structural shape optimization framework based on the isogeometric analysis approach with the discretization based on non uniform rational b splines nurbs the analysis model represents the structural geometry exactly furthermore nurbs enable efficient geometry control together with smooth boundaries they are the de facto standard in cad systems but are also widely used in a shape optimal design context to define the geometry representation and the design variables with the presented isogeometric approach to shape optimization the analysis model is inherently merged with the design model omitting the typically involved interplay between both we derive analytical sensitivities for nurbs discretizations which allow application of efficient gradient based optimization algorithms the present contribution is restricted to two dimensional problems of linear elasticity but the extension to three dimensions and other problem classes is straightforward some representative examples demonstrate and validate the methodology further the potential of boundary continuity control within isogeometric structural shape optimization is explored to trigger smooth or less smooth angular designs

immersing almost geodesic surfaces in a closed hyperbolic three manifold let m 3 be a closed hyperbolic three manifold we construct closed surfaces that map by immersions into m 3 so that for each one the corresponding mapping on the universal covering spaces is an embedding or in other words the corresponding induced mapping on fundamental groups is an injection

numerical simulation of a generalized zakharov system in this paper we propose and study two time splitting spectral methods for the generalized zakharov system these methods are spectrally accurate in space second order in time and unconditionally stable the unconditional stability of the methods offers greater numerical efficiency than those given in previous papers especially in the subsonic regime our numerical experiments confirm the accuracy and stability in particular we analyze their behavior in the subsonic regime the first method using the exact time integration in phase space for the wave equation for the nondispersive field converges unformly with respect to the sound speed for the dispersive wave field while the second method using the crank nicolson method in the same step with an initial layer fix by an l stable time discretization converges uniformly with respect to the sound speed for both dispersive and nondispersive fields using these new methods we also study the collision behavior of two solitons in the subsonic region as well as the transsonic region we obtain numerical results which are quantitatively different from those reported in previous papers using lower resolution numerical techniques

non lerfness of arithmetic hyperbolic manifold groups and mixed 3 manifold groups we will show that for any noncompact arithmetic hyperbolic m manifold with m 3 and any compact arithmetic hyperbolic m manifold with m 4 that is not a 7 dimensional one defined by octonions its fundamental group is not locally extended residually finite lerf the main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3 manifold groups we will also show that a compact orientable irreducible 3 manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is lerf

shape optimal design using b splines abstract shape optimal design of an elastic structure is formulated using a design element technique it is shown that bezier and b spline curves typical of the cad philosophy are well suited to the definition of design elements complex geometries can be described in a very compact way by a small set of design variables and a few design elements because of the b splines flexibility it is no longer necessary to piece design elements together in order to agree with the shape complexity nor to restrict the shape variations moreover the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh are automatically taken into account in the new formulation an analytical derivation of the sensitivity analysis will be established giving rise to numerical efficiency it will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables so that simple mathematical programming algorithms can be applied to solve it some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper

a freeform shape optimization of complex structures represented by arbitrary polygonal or polyhedral meshes in this paper we propose a new scheme for freeform shape optimization on arbitrary polygonal or polyhedral meshes the approach consists of three main steps 1 surface partitioning of polygonal meshes into different patches 2 a new freeform perturbation scheme of using the cox de boor basis function over arbitrary polygonal meshes which supports multi resolution shape optimization and does not require cad information 3 freeform shape optimization of arbitrary polygonal or polyhedral meshes numerical experiments indicate the effectiveness of the proposed approach copyright 2004 john wiley sons ltd

merits and limitations of optimality criteria method for structural optimization the merits and limitations of the optimality criteria oc method for the minimum weight design of structures subjected to multiple load conditions under stress displacement and frequency constraints were investigated by examining several numerical examples the examples were solved utilizing the oc design code that was developed for this purpose at the nasa lewis research center this oc code incorporates oc methods available in the literature with generalizations for stress constraints fully utilized design concepts and hybrid methods that combine both techniques it includes multiple choices for lagrange multiplier and design variable update methods design strategies for several constraint types variable linking displacement and integrated force method analysers and analytical and numerical sensitivities on the basis of the examples solved the optimality criteria for general application were found to be satisfactory for problems with few active constraints or with small numbers of design variables however the oc method without stress constraints converged to optimum even for large structural systems for problems with large numbers of behaviour constraints and design variables the method appears to follow a subset of active constraints that can result in a heavier design the computational efficiency of oc methods appears to be similar to some mathematical programming techniques

right angled coxeter polytopes hyperbolic six manifolds and a problem of siegel by gluing together the sides of eight copies of an all right angled hyperbolic six dimensional polytope two orientable hyperbolic six manifolds with euler characteristic 1 are constructed they are the first known examples of orientable hyperbolic six manifolds having the smallest possible volume

dual methods for optimizing finite element flexural systems abstract modern numerical methods for the optimization of large discretized systems are now well developed and highly efficient in the case of thin walled elastic structures modeled by finite elements however this is not yet true for structures whose components are subject simultaneously to bending and extension loads in this paper the idea of generalized optimality criterion goc set forth in previous papers for bar membrane and pure bending elements is extended to deal with general beam and flat shell elements the modifications brought to the goc result in explicit approximations for the behavior constraints that are correct up to the first order but that exhibit a more complex algebraic form indeed these explicit expressions are no longer merely linear in the reciprocal design variables however they continue to be additively separable and therefore dual methods remain fully applicable just as in the original statement of the goc approach numerical examples will be offered to demonstrate the efficiency of the method presented

shape optimization and its extension to topological design based on isogeometric analysis in most of structural optimization approaches finite element method fem has been employed for structural response analysis and sensitivity calculation however the approaches generally suffer certain drawbacks in shape optimization cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary in topology optimization design results are generally restricted on the initial design space and additional post processing is required for communication with cad systems these drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis spline basis functions are used in design and geometric modeling whereas lagrangian and hermitian polynomials in analysis isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional fem in structural optimization in isogeometric analysis the same spline information such as control points and spline basis functions which represent geometries in cad systems are also used in numerical analysis such unification of the mathematical languages in cad analysis and design optimization can resolve the issues mentioned above in this work structural shape optimization using isogeometric analysis is studied on 2d and shell problems the proposed framework is extended to topology optimization using trimming techniques new inner fronts are introduced by trimming spline curves in topology optimization trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization some benchmarking problems in shape and topology optimization are treated using the proposed approach

a basis change strategy for the reduced gradient method and the optimum design of large structures this paper proposes a basis change strategy within the reduced gradient method for optimization under linear constraints it ensures a non singular basis matrix at every iteration the same strategy can reliably be used within the generalized reduced gradient method for optimization under non linear constraints this method is applied to the minimum weight design of large structures under displacement and stress constraints exploiting the sparsity of the constraint jacobian matrix

finite difference method for generalized zakharov equations abstract a conservative difference scheme is presented for the initial boundary value problem for generalized zakharov equations the scheme canbe implicit or semiexplicit depending on the choice of a parameter on thebasis of a priori estimates and an inequality about norms convergence of thedifference solution is proved in order 0 h2 t2 which is better than previous results introductionthe zakharov equations 20 1 1 iet exx ne 0 1 2 ntt n e 2 xx 0describe the propagation of langmuir waves in plasmas here the complexunknown function e is the slowly varying envelope of the highly oscillatoryelectric field and the unknown real function n denotes the fluctuation of the ion density about its equilibrium value the global existence of a weak solution for the zakharov equations in one dimension is proved in 19 and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed numerical methods for the zakharov equations are studied only in 5 9 10

graphs and separability properties of groups abstract a group g is lerf locally extended residually finite if for any finitely generated subgroup s of g and for any g s there exists a finite index subgroup s 0 of g which contains s but not g using graph theoretical methods we give algorithms for constructing finite index subgroups in amalgamated free products of groups with good separability properties we prove that a free product of a free group and a lerf group amalgamated over a cyclic subgroup maximal in the free factor is lerf the maximality condition cannot be removed because adjunction of roots does not preserve property lerf we also give short proofs of some old theorems about separability properties of groups including a theorem of brunner burns and solitar that a free product of free groups amalgamated over a cyclic subgroup is lerf

minimum weight design of trusses by an optimality criteria method this work presents an optimality criteria method which accurately and efficiently solves the structural optimization problem of trusses subject to deflection and stress constraints the method is based upon the newton raphson method and a first order approximation both for deflection and stress constraints using the method of virtual work it solves the problem rigorously and no serious difficulties have yet been encountered in spite of the drawbacks inherent in the newton raphson method worked examples show its superiority to other methods and its ability to cope with trusses with many members and subject to multiple constraints

optimal shape design for thin walled beam cross sections problems of shape optimization for thin walled beam cross sections are investigated in particular shape optimization under torsional loading is treated the shape of the beam cross section is described by non uniform rational b splines nurbs biquadratic and p version finite elements are used to calculate the warping function the definition of the element geometry is carried out by isoparametric and b spline mappings both methods are discussed and compared for structural shape optimization the b spline mapping in connection with the p version finite elements is very efficient the optimization is implemented using convex linearisation and a dual method of mathematical programming sensitivity analysis is performed analytically sample calculations are included

improved sequential linear programming formulation for structural weight minimization abstract this paper presents an optimization algorithm for weight minimization of structures the algorithm denoted as lstrlp line search trust region linear programming combines sequential linear programming slp and trust region methods trm lstrlp solves a linearized sub problem in each design cycle and accepts or rejects intermediate designs based on a line search strategy which detects if the eventual improvement in cost is actually the largest possible it is to be noticed that the present work is the closure to several studies carried out by the present authors in order to improve the overall efficiency and robustness of the sequential linear programming method the lstrlp algorithm is implemented by an optimization code written in fortran 90 the optimization code is tested in eight cases of weight minimization of bar truss and frame structures the test cases include examples of large scale and configuration optimization the results obtained here are compared to those presented in literature the optimizations are run also with sequential quadratic programming sqp routines implemented in commercial software the results indicate that lstrlp is competitive with recently published algorithms and commercial software

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

a finite element method with mesh separation based approximation technique and its application in modeling crack propagation with adaptive mesh refinement summary this paper presents a fem with mesh separation based approximation technique that separates a standard element into three geometrically independent elements a dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation the novel technique makes it very easy for mesh generation of problems with complex or solution dependent varying geometry it offers a flexible way to construct displacement approximations and provides a unified framework for the fem to enjoy some of the key advantages of the hansbo and hansbo method the meshfree methods the semi analytical fems and the smoothed fem for problems with evolving discontinuities the method enables the devising of an efficient crack tip adaptive mesh refinement strategy to improve the accuracy of crack tip fields both the discontinuities due to intra element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level the effectiveness and robustness of the present method are benchmarked with several numerical examples the numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test and it is essential to apply local adaptive mesh refinement for low fracture energy problems copyright 2014 john wiley sons ltd

convergence analysis of high order time splitting pseudospectral methods for nonlinear schrodinger equations in this work the issue of favorable numerical methods for the space and time discretization of low dimensional nonlinear schrodinger equations is addressed the objective is to provide a stability and error analysis of high accuracy discretizations that rely on spectral and splitting methods as a model problem the time dependent gross pitaevskii equation arising in the description of bose einstein condensates is considered for the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed for the time integration high order exponential operator splitting methods are studied where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method the convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part essential tools in the derivation of a temporal global error estimate are furthe

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

computational procedures for topological shape design abstract this research describes a method for the design of general shapes that is able to change the shape topology during the design process and can handle geometric and behavioral constraints expressed as algebraic or partial differential equations the proposed method uses an iterative process that alternates between formulating and solving a sequence of numerical optimization problems at each iteration a design parameterization is generated and a corresponding domain discretization is then derived automatically from this design parameterization a numerical optimization problem is then formulated by expressing the objective and the problem constraints in terms of the design variables the results of this optimization update the shape and the resulting shape then initiates a new design iteration the process continues until convergence the parameterization proposed is based on a geometric abstraction called the skeleton and leads to a relatively small number of design variables the parameterization is adaptive to shape evolutions guarantees shape integrity during the design and allows the redefinition of the objective and constraints in terms of adaptively generated design variables the skeleton also forms the basis for automatic domain discretization parameterized by the design variables this parameterized discretization provides efficient means for evaluating pde constraints and computing their gradients the method overcomes some of the problems associated with current shape design procedures including boundary variation methods and homogeniz ation methods computational examples with geometric and pde constraints are presented to demonstrate the capabilities of the proposed shape design method in generating general geometries and topologies

a structural optimization method based on the level set method using a new geometry based re initialization scheme structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function and updated by solving the so called hamilton jacobi equation based on a eulerian coordinate system these new methods can allow topological alterations such as the number of holes during the optimization process whereas the boundaries of the target structure are clearly defined however the re initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures in this paper we propose a new structural optimization method based on the level set method using a new geometry based re initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the finite element method the stiffness maximization eigenfrequency maximization and eigenfrequency matching problems are considered as optimization problems several design examples are presented to confirm the usefulness of the proposed method copyright 2010 john wiley sons ltd

a fictitious energy approach for shape optimization this paper deals with shape optimization of continuous structures as in early works on shape optimization coordinates of boundary nodes of the fe domain are directly chosen as design variables convergence problems and problems with jagged shapes are eliminated by a new regularization technique an artificial inequality constraint added to the optimization problem limits a fictitious total strain energy that measures the shape change of the design with respect to a reference design the energy constraint defines a feasible design space whose size can be varied by one parameter the upper energy limit by construction the proposed regularization is applicable to a wide range of problems although in this paper the application is restricted to linear elastostatic problems

adaptive finite element methods for elliptic equations over hierarchical t meshes isogeometric analysis using nurbs non uniform rational b splines as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement in this paper we use the polynomial splines over hierarchical t meshes pht splines to construct basis functions which not only share the nice smoothness properties as the b splines but also allow us to effectively refine meshes locally we develop a residual based a posteriori error estimate for the finite element discretization of elliptic equations using pht splines basis functions and study their approximation properties in addition we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution

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

structural optimization with frequency constraints using the finite element force method a structural optimization algorithm is developed to minimize the weight of structures with truss and beam type members under single or multiple frequency constraints the cross sectional areas of the structural members are considered as the design variables the algorithm proposed combines the finite element technique based on the integrated force method with the mathematical programming technique the equilibrium matrix is generated automatically using the finite element analysis and the compatibility matrix is obtained directly using the displacement deformation relations and the single value decomposition technique when combining the equilibrium and the compatibility matrices with the force displacement relations the frequency eigenvalue equations are obtained with element forces as variables three structures composed of truss and frame type members are studied to illustrate the procedure and the results are compared with the literature it is shown that in structural problems with multiple frequency constraints the analysis procedure force or displacement method significantly affects the final optimum design the structural optimization based on the force method results in a lighter design the proposed structural optimization method is efficient to analyze and optimize both truss and beam type structures

gradient limiting shape control for efficient aerodynamic optimization local shape control methods such as b spline surfaces are well conditioned such that they allow high fidelity design optimization however this comes at the cost of degraded optimization converg

approximate solutions to the zakharov equations via finite differences abstract an energy preserving linearly implicit finite difference scheme is presented for computing solutions to the periodic initial value problem for the zakharov equations solitary waves and colliding solitary waves are computed and a comparison is made with previous calculations

weight minimisation of displacement constrained truss structures using a strain energy criterion this paper discusses a new method for the solution of the general truss weight minimisation problem under simultaneous stress and displacement constraints the method introduces a novel strain energy density criterion that refers to the ratio of the virtual strain energy per unit volume in each structural member to its average value on the whole structure the virtual strain energy comes from the unit load theorem and it is proportional to the product of the axial member forces due to both the actual loads and a virtual unit load that is applied at the node with the maximum displacement a simple recursive formula for updating the cross sectional areas based on displacement constraints is presented a general subsequent algorithm applicable to both single and multiple load cases follows this formula the results are encouraging since in all test cases the method was found to be robust and generally led to the same weight level as the literature in both small and large structures

shape design for thin walled beam cross sections using rational b splines shape optimization of thin walled beams is discussed the cross sectional shape of the beam is described by nonuniform rational b spline curves the optimization problem is defined in terms of cross sectional properties and stresses design variables are the control parameters of the rational b spline curves and the wall thickness of the beam cross sectional properties and stresses are obtained from the elasticity solution for the beam consideration of torsion and pure shear requires the solution of laplace equations which is accomplished using finite elements a design sensitivity analysis is given for the cross sectional properties and stresses a sample problem is presented to illustrate the formulations

boundary element based multiresolution shape optimisation in electrostatics we consider the shape optimisation of high voltage devices subject to electrostatic field equations by combining fast boundary elements with multiresolution subdivision surfaces the geometry of the domain is described with subdivision surfaces and different resolutions of the same geometry are used for optimisation and analysis the primal and adjoint problems are discretised with the boundary element method using a sufficiently fine control mesh for shape optimisation the geometry is updated starting from the coarsest control mesh with increasingly finer control meshes the multiresolution approach effectively prevents the appearance of non physical geometry oscillations in the optimised shapes moreover there is no need for mesh regeneration or smoothing during the optimisation due to the absence of a volume mesh we present several numerical experiments and one industrial application to demonstrate the robustness and versatility of the developed approach

a gradient based shape optimization scheme using an interface enriched generalized fem abstract a gradient based shape optimization scheme using an interface enriched generalized finite element method igfem is presented wherein the design geometry is projected onto a fixed mesh and the igfem is used for analysis this approach eliminates the mesh distortion present in conventional lagrangian shape optimization methods as well as the need for remeshing an analytical sensitivity analysis using both the adjoint or direct approaches is presented to compute derivatives of the objective and constraint functions due to the fixed nature of the mesh the so called design velocity field only needs to be computed on the structure boundary interface a comparison between igfem and conventional fem based shape optimization schemes is presented showing an improved precision for the igfem approach finally we solve various numerical examples to demonstrate the capability of the method including the computational design of particulate and microvascular composites

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

topology optimization of trusses using genetic algorithm force method and graph theory in this article size topology optimization of trusses is performed using a genetic algorithm ga the force method and some concepts of graph theory one of the main difficulties with optimization with a ga is that the parameters involved are not completely known and the number of operations needed is often quite high application of some concepts of the force method together with theory of graphs make the generation of a suitable initial population well matched with critical paths for the transformation of internal forces feasible in the process of optimization generated topologically unstable trusses are identified without any matrix manipulation and highly penalized identifying a suitable range for the cross section of each member for the ground structure in the list of profiles the length of the substrings representing the cross sectional design variables are reduced using a contraction algorithm the length of the strings is further reduced and a ga is performed in a smaller domain of design space the above process is accompanied by efficient methods for selection and by using a suitable penalty function in order to reduce the number of numerical operations and to increase the speed of the optimization toward a global optimum the efficiency of the present method is illustrated using some examples and compared to those of previous studies copyright 2003 john wiley sons ltd

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization an important feature that drives the auxetic behaviour of the star shaped auxetic structures is the hinge functional connection at the vertex connections this feature poses a great challenge for manufacturing and may lead to significant stress concentrations to overcome these problems we introduced smoothed petal shaped auxetic structures where the hinges are replaced by smoothed connections to accommodate the curved features of the petal shaped auxetics a parametrisation modelling scheme using multiple nurbs patches is proposed next an integrated shape design frame work using isogeometric analysis is adopted to improve the structural performance to ensure a minimum thickness for each member a geometry sizing constraint is imposed via piece wise bounding polynomials this geometry sizing constraint in the context of isogeometric shape optimization is particularly interesting due to the non interpolatory nature of nurbs basis the effective poisson ratio is used directly as the objective function and an adjoint sensitivity analysis is carried out the optimized designs smoothed petal auxetic structures are shown to achieve low negative poisson s ratios while the difficulties of manufacturing the hinges are avoided for the case with six petals an in plane isotropy is achieved

shape variable definition with c0 c1 and c2 continuity functions the present paper proposes a new technique for the definition of the shape design variables in 2d and 3d optimisation problems it can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the cad defining surfaces the proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori this capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process structural analysis fluid dynamic analysis crash analysis aerodynamic analysis etc

sensitivity analysis for coupled thermoelastic systems abstract this paper presents an adjoint approach derived from the reciprocal theorem for the sensitivity analysis of linear dynamic thermoelastic systems the variation of a general response functional is expressed in explicit form with respect to variations of the design fields which consist of the material properties applied loads prescribed boundary initial conditions and the structural shape the functional is dependent on these design quantities as well as the following implicity defined response fields displacement temperature stress strain heat flux temperature gradient reaction forces and reaction surface flux the formulation incorporates the reciprocal relation between variations of the real system design and response fields and an adjoint state here convolution is employed in lieu of time mappings used in other transient adjoint sensitivity derivations specializations of the formulation to uncoupled combined quasi static uncoupled and steady state thermoelasticity are also presented the finite element method is used to demonstrate the application of the formulation to a problem in automobile engine design

design sensitivity analysis for shape optimization based on the lie derivative abstract the paper presents a theoretical framework for the shape sensitivity analysis of systems governed by partial differential equations the proposed approach based on geometrical concepts borrowed from differential geometry shows that sensitivity of a performance function i e any function of the solution of the problem with respect to a given design variable can be represented mathematically as a lie derivative i e the derivative of that performance function along a flow representing the continuous shape modification of the geometrical model induced by the variation of the considered design variable theoretical formulae to express sensitivity analytically are demonstrated in detail in the paper and applied to a nonlinear magnetostatic and a linear elastic problem following both the direct and the adjoint approaches following the analytical approach one linear system of which only the right hand side needs be evaluated the system matrix being known already has to be solved for each of the design variables in the direct approach or for each performance functions in the adjoint approach a substantial gain in computation time is obtained this way compared to a finite difference evaluation of sensitivity which requires solving a second nonlinear system for each design variable this is the main motivation of the analytical approach there is some freedom in the definition of the auxiliary flow that represents the shape modification we present a method that makes benefit of this freedom to express sensitivity locally as a volume integral over a single layer of finite elements connected to both sides of the surfaces undergoing shape modification all sensitivity calculations are checked with a finite difference in order to validate the analytic approach convergence is analyzed in 2d and 3d with first and second order finite elements

comparison of b spline surface and free form deformation geometry control for aerodynamic optimization two aerodynamic shape optimization geometry control methods b spline surface control and free form deformation are applied to three optimization problems and compared on the bases of optimal shap

sensitivity analysis of linear elastic systems using domain parameterization and a mixed mutual energy principle abstract a unified approach for explicit structural design sensitivity analysis of linear elastic systems is presented first order sensitivity expression involving the complete set of design variable including shape design parameters are derived for general response functional both direct differentiation and adjoint variable methods are developed in a general variational context a domain parameterization method related to the mixed eulerian lagrangian kinematic description is introduced for the treatment of shape variation problems this approach is an alternative to the material derivative methods presented elsewhere in the literature the adjoint variable derivation is based on a mixed energy principle the mutual hu washizu energy principle which is formally stated and proved the design sensitivity derivation based on this principle unifies and extends several of the adjoint variable methods found in the literature the derivations presented here are extended to nonlinear systems in a subsequent paper the sensitivity expressions are valid for use with force displacement or mixed approximate solutions of the elasticity equations other formulations are generally only consistent with displacement solution methods numerical implementation of the design sensitivity methods is discussed and illustrative examples are included

a geometric comparison of aerofoil shape parameterisation methods a comprehensive review of aerofoil shape parameterization methods that can be used for aerodynamic shape optimization is presented seven parameterization methods are considered for a range of desi

a physics based parameterization method for shape optimization a physical approach is presented to provide shape modification of nurbs curve the movement of the control polygon of nurbs is simulated by a physical frame structure and the curve deformation is then carried out through changes of the mode coordinates which are derived from the eigenvectors of the finite element equations of the physical frame system the shape of the design boundaries is described by a linear combination of the mode shape vectors based on the presented design variable representation and p version finite element analysis an integrated approach to structural shape optimization is discussed a method is developed that enables the description of the shape to be adaptively refined during the shape optimization process numerical examples are used to demonstrate the applicability of this approach to structural shape optimization

convergence of an energy preserving scheme for the zakharov equations in one space dimension an energy preserving linearly implicit finite difference scheme is presented for approximating solutions to the periodic cauchy problem for the one dimensional zakharov system of two nonlinear partial differential equations first order convergence estimates are obtained in a standard energy norm in terms of the initial errors and the usual discretization errors

shape optimal design of elastic space frames with non linear response this paper describes an approach to shape optimal design of elastic planar frames with non linear response the foundation of the proposed approach forms an appropriate strategy of shape representation of the structure based on the design element technique a frame structure is treated as to be assembled from several frame design elements which in turn may consist of several appropriately joined beam finite elements the shape of each frame design element is defined by convenient functions involving bezier blending polynomials the original formulation of the beam finite element proposed by saje is modified in order to fit nicely into the context of the frame design element technique the formulation of the shape optimal design problem in a form of a problem of non linear mathematical programming and its solution by employing gradient based methods of mathematical programming are discussed briefly the theory is illustrated with two numerical examples

efficient and stable numerical methods for the generalized and vector zakharov system we present efficient and stable numerical methods for approximations of the generalized zakharov system gzs and vector zakharov system for multicomponent plasma vzsm with without a linear damping term the key points in the methods are based on i a time splitting discretization of a schrodinger type equation in gzs or vzsm ii discretizing a nonlinear wave type equation by a pseudospectral method for spatial derivatives and iii solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the methods are explicit unconditionally stable and of spectral order accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping term in gzs or vzsm conserve or keep the same decay rate of the wave energy as that in gzs or vzsm without a or with a linear damping term and give exact results for the plane wave solution extensive numerical tests are presented for plane waves and solitary wave collisions in one dimensional gzs and we also give the dynamics of three dimensional vzsm to demonstrate our new efficient and accurate numerical methods furthermore the methods are applied to study the convergence and quadratic convergence rates of vzsm to gzs and of gzs to the nonlinear schrodinger nls equation in the subsonic limit regime 0 varepsilon ll1 where the parameter varepsilon is inversely proportional to the acoustic speed our tests also suggest that the following meshing strategy or varepsilon resolution is admissible in this regime spatial mesh size h o varepsilon and time step k o varepsilon

error analysis of implicit euler methods for quasilinear hyperbolic evolution equations in this paper we study the convergence of the semi implicit and the implicit euler methods for the time integration of abstract quasilinear hyperbolic evolution equations the analytical framework considered here includes certain quasilinear maxwell s and wave equations as special cases our analysis shows that the euler approximations are well posed and convergent of order one the techniques will be the basis for the future investigation of higher order time integration methods and full discretizations of certain quasilinear hyperbolic problems

two level free form and axial deformation for exploratory aerodynamic shape optimization an intuitive shape parameterization and control technique suitable for high fidelity aerodynamic shape optimization is presented it relies on the principles of free form and axial deformation enabling thorough exploration of the design space while keeping the number of design variables manageable surface sensitivities to the design variables are readily available their inclusion in a highly efficient and robust adjoint based optimization methodology involving linearly elastic volume mesh deformation and a newton krylov solver for the euler equations is described the flexibility of the proposed approach is demonstrated through the exploratory shape optimization of a three pronged feathered winglet leading to a span efficiency of 1 19 under a height to span ratio constraint of 0 1 and an optimization of a regional jet wing at transonic speed where a winglet is allowed to develop starting from a planar wingtip extension leading to an 18 8 reduction in drag

a conservative difference scheme for the zakharov equations a new conservative difference scheme is presented for the periodic initial value problem of zakharov equations the scheme can be implicit or semi explicit depending on the choice of a parameter the discretization of the initial condition is of second order accuracy which is consistent with the accuracy of the scheme on the basis of a priori estimates and an inequality about norms convergence of the difference solutions is proved in the energy norm numerical experiments with the schemes are done for several test cases computational results demonstrate that the new semi explicit scheme with a new initial approximation is more accurate and computationally efficient

a new scheme for efficient and direct shape optimization of complex structures represented by polygonal meshes in this paper a new shape optimization approach is proposed to provide an efficient optimization solution of complex structures represented by polygonal meshes our approach consists of three main steps 1 surface partitioning of polygonal meshes 2 generation of shape design variables on the basis of partitioned surface patches and 3 gradient based shape optimization of the structures by reducing a weighted compliance among all load cases the main contributions of this paper include i that our approach can be directly applied on polygonal meshes with the reduction of design variables or decision variables by 10 to 1000 times compared to the conventional design variable scheme of using each mesh node ii our perturbation scheme is mathematically proven with respect to maintaining the smoothness of each surface patch except on its boundary and iii overall our approach can be used to automate time consuming shape optimization of polygonal meshes to a greater extent numerical experiments have been conducted and the results indicate the effectiveness of the approach copyright 2003 john wiley sons ltd

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

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

efficient approximation concepts using second order information the primary goal of this paper is to show how second derivative information can be used in an effective way in structural optimization problems the basic idea is to generate such an information at the expense of only one more virtual load case in the sensitivity analysis part of the finite element code to achieve this goal a primal dual approach is employed that can also be interpreted as a sequential quadratic programming method another objective is to relate the proposed method to the well known family of approximation concepts techniques where the primary optimization problem is transformed into a sequence of non linear explicit subproblems when restricted to diagonal second derivatives the new approach can be viewed as a recursive convex programming method similar to the convex linearization method conlin and to its recent generalization the method of moving asymptotes mma this new method has been successfully tested on simple problems that can be solved in closed form as well as on sizing optimization of trusses in all cases the method converges faster than conlin mma or other approximation techniques based on reciprocal variables

on structural shape optimization using an embedding domain discretization technique summary this contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique the evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization we consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements exterior elements and elements intersected by the physical domain boundary a selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component thereby we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration as it is the case for a standard lagrangian approach to structural shape optimization still we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process in order to avoid irregular and impracticable design updates we consider a geometric regularization technique to render feasible descent directions for the course of the optimization this article is protected by copyright all rights reserved

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

a characterization of virtually embedded subsurfaces in 3 manifolds this paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3 manifold which generalizes an invariant due to rubinstein and wang the subsurface is virtually embedded if and only if the almost fiber part is aspiral and in this case the subsurface is virtually a leaf of a taut foliation besides other consequences examples are exhibited that nongeometric 3 manifolds with no seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces

isogeometric sizing and shape optimisation of beam structures based on the recently proposed framework of isogeometric analysis a study of structural sizing and shape optimisation of curved beam structures is presented the idea of the isogeometric approach is to analyse the structure using its intrinsic and hence exact geometric definition the structural response is approximated using the same basis functions describing the geometry the presented isogeometric framework employs non uniform rational b splines shape changes can be represented by altering both spatial location of control points and the corresponding weights towards the optimal design adaptation of identical geometric representations for both design and analysis resolves the problem of repetitive conversion between the computer aided and facet based geometries used in conventional finite element analysis sizing variables defined at the control points are interpolated using the same basis functions due to the fact that both analysis sizing and shape design share the same underlying description direct sensitivities can be easily evaluated analytically which improves the performance of the optimisation process while sizing should reflect local stress states shape design is preferably performed on a global level thus a multilevel design approach is proposed where shape design is carried on a coarser level projecting the shape design sensitivities bridges the gap between the different levels the proposed design formulation is applied to compliance minimisation problems numerical results indicate the applicability of the method

on a splitting method for the zakharov system an error analysis of a splitting method applied to the zakharov system is given the numerical method is a lie trotter splitting in time that is combined with a fourier collocation in space to a fully discrete method first order convergence in time and high order convergence in space depending on the regularity of the exact solution are shown for this method the main challenge in the analysis is to exclude a loss of spatial regularity in the numerical solution this is done by transforming the numerical method to new variables and by imposing a natural cfl type restriction on the discretization parameters

compatibility conditions of structural mechanics for finite element analysis the equilibrium equations and the compatibility conditions are fundamental to the analyses of structures however anyone who undertakes even a cursory generic study of the compatibility conditions can discover with little effort that historically this facet of structural mechanics had not been adequately researched by the profession now the compatibility conditions cc s have been researched and are understood to a great extent for finite element discretizations the cc s are banded and can be divided into three distinct categories 1 the interface cc s 2 the cluster or field cc s and 3 the external cc s the generation of cc s requires the separating of a local region then writing the deformation displacement relation ddr for the region and finally the eliminating of the displacements from the ddr the procedure to generate all three types of cc s is presented and illustrated through examples of finite element models the uniqueness of the cc s thus generated is shown

sizing optimization of truss structures by method of centers and force formulation this paper describes a new approach to optimum weight design of truss structures the force method is incorporated in an optimization algorithm based on the method of center points design variables are the member cross sectional areas and the redundant forces evaluated for each independent loading condition acting on the structure the optimization method utilizes the largest hyperspheres inscribed within the feasible space the method of hyperspheres has been enhanced here to handle the compatibility equality constraints as well by including the analysis step in the optimization cycle there is no longer the need to perform separate structural analyses thus saving computation time the viability and efficiency of the proposed method are demonstrated for truss structures subject to multiple loading conditions and constraints on member stresses nodal displacement and minimum gage numerical results are compared with those reported in the literature

isogeometric shape optimization of vibrating membranes we consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely the main obstacle we face is the need for robust and inexpensive extension of a b spline parametrization from the boundary of a domain onto its interior a task which has to be performed in every optimization iteration we experiment with two numerical methods one is based on the idea of constructing a quasi conformal mapping whereas the other is based on a spring based mesh model for carrying out this task which turn out to work sufficiently well in the present situation we perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth optimized membrane shapes our conclusion is that isogeometric analysis fits well with shape optimization

an approximation concepts approach to shape optimal design abstract in the past much of the work done in structural optimization consisted in resizing the members of fixed configuration models in that case a powerful design procedure has now emerged which is based on the coordinate use of explicit high quality approximations of the behavior constraints and dual methods of mathematical programming there is however a large class of problems for which the main degrees of freedom for the designer correspond to the shape of the structure itself the main objectives of this paper are to recall briefly a convenient geometric representation in which the boundaries of the structure are represented by bezier or b spline curves and then to discuss the choice of optimization algorithm it is shown that cost efficient methods for structural sizing may be advantageously extended to shape optimal design problems different approximation schemes are tested and a new general optimization algorithm is presented that combines mixed approximations and dual methods many large scale applications are treated to demonstrate the generality and the efficiency of the new formulation finally considerations are given about an integrated approach including cad computer codes and finite element optimization software

a meshless compositional approach to shape optimal design meshless computational strategies for shape optimal design through the composition of behavioral fields are developed in this paper the composition of behavioral fields is carried out in a manner that is procedurally similar to composition of primitive geometrical entities through set theoretic boolean operations in constructive solid geometry the methodology is shown to lead to a tighter integration between design and analysis and is termed constructive solid analysis a meshless approximation using non uniform rational b spline basis functions is used to discretize the behavioral fields defined over the geometrical primitives remeshing which is a critical bottleneck in shape optimization using conventional finite element analysis is either eliminated or limited to those primitives that are modified further it is demonstrated that the reconstruction of stiffness matrices is required only for those geometrical primitives that are affected by the design changes design sensitivity analysis techniques are developed for the discretized solution system arising out of the compositional analysis an adaptive numerical integration scheme is developed and shown to be necessary to accurately compute the effect of small domain perturbations on the problem functions some of the common challenges in conventional shape optimization problems such as difficulty in creating and merging multiple holes are naturally resolved using the developed compositional approach several representative problems are solved to validate and to demonstrate the methodology

operator splitting for the kdv equation we provide a new analytical approach to operator splitting for equations of the type u t au b u where a is a linear operator and b is quadratic a particular example is the korteweg de vries kdv equation u t uu x u xxx 0 we show that the godunov and strang splitting methods converge with the expected rates if the initial data are sufficiently regular

isogeometric shape optimization for quasi static processes summary a framework to solve shape optimization problems for quasi static processes is developed and implemented numerically within the context of isogeometric analysis iga recent contributions in shape optimization within iga have been limited to static or steady state loading conditions in the present contribution the formulation of shape optimization is extended to include time dependent loads and responses a general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems an adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of iga the methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi axial tension and pure shear under quasi static loading conditions the method is validated using a passive control approach with an a priori known solution several applications of time dependent mechanical problems are shown to illustrate the capabilities of this approach in particular a problem is considered where an external load is allowed to move along the surface of a structure the shape of the structure is modified in order to control the time dependent displacement of the point where the load is applied according to a pre specified target copyright 2015 john wiley sons ltd

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

an optimization design method for large scale structures in this paper based on the mechanical property of structures a unified approximate formula with high precision suitable to several structural response functions and various types of finite elements is introduced first and then an algorithm for solving the nonlinear program is presented this algorithm is not only effective for solving the program constructed by this unified approximate formula but also easy to change the objective function and to solve the structural optimization problem with dual or multiple objective functions