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

orthogonalized generalized iso geometric analysis ogiga and its applications to problems of fracture mechanics abstract ill conditioning of system matrices is one of the drawbacks of generalized extended isogeometric analysis giga xiga which results in increase in computational cost in the case of iterative solvers or erroneous results in the case of direct solvers linear dependency between the basis functions which can be measured in terms of smallest angle among all the bases is one of the attributes of ill conditioning in xiga in the present study a simplified method to find the smallest angle among all the bases is introduced also to reduce the problem of ill conditioning orthogonalized generalized isogeometric analysis ogiga is proposed in which the standard and the corresponding enriched basis functions are orthogonalized in order to study the performance of present method various fracture mechanics problems like 2 d homogenous cracks bi material interface cracks inclined cracks and 3 d crack problems are chosen the effect of crack position and the radius of enrichment zone on accuracy scaled condition number and smallest angle among all the bases is studied the results obtained from the proposed ogiga method are compared with results obtained from standard xiga and xiga with enrichment function orthogonalization oe xiga agathos et al 2019 in all the cases considered the conditioning of system matrices is significantly improved by employing present methodology and it can be easily extended to wide variety of problems

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

computing finite galois groups arising from automorphic forms abstract we study modulo p reduction of compatible systems of p adic representations of the absolute galois group of q arising from an algebraic automorphic representation in particular we prove that there is a field extension of q with the galois group g 2 p ramified at 5 and p only for a set of primes p of density one

geometry independent field approximation cad analysis integration geometrical exactness and adaptivity in isogeometric analysis iga the same spline representation is employed for both the geometry of the domain and approximation of the unknown fields over this domain this identity of the geometry and field approximation spaces was put forward in the now classic 2005 paper 20 as a key advantage on the way to the integration of computer aided design cad and subsequent analysis in computer aided engineering cae 20 claims indeed that any change to the geometry of the domain is automatically inherited by the approximation of the field variables without requiring the regeneration of the mesh at each change of the domain geometry yet in finite element versions of iga a parameterization of the interior of the domain must still be constructed since cad only provides information about the boundary the identity of the boundary and field representation decreases the flexibility in which this parameterization can be generated and somewhat constrains the modeling and simulation process because an approximation able to represent the domain geometry accurately need not be adequate to also approximate the field variables accurately in particular when the solution is not smooth we propose here a new paradigm called geometry independent field approximation gift where the spline spaces used for the geometry and the field variables can be chosen and adapted independently while preserving geometric exactness and tight cad integration gift has the following features 1 it is possible to flexibly

a hierarchical approach to adaptive local refinement in isogeometric analysis abstract adaptive local refinement is one of the key issues in isogeometric analysis in this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical b splines we investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity furthermore we use concepts well established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting this also allows us to access a posteriori error estimation techniques numerical results for several different examples are given and they turn out to be very promising

local refinement of analysis suitable t splines we develop a local refinement algorithm for analysis suitable t splines which does not produce excessive propagation of control points we then demonstrate its use as an adaptive framework for isogeometric analysis analysis suitable t splines are a class of t splines which are linearly independent and form a partition of unity these properties coupled with local refinement make this class of t splines appealing as a basis for isogeometric analysis

isogeometric analysis using polynomial splines over hierarchical t meshes for two dimensional elastic solids isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries one of the major drawbacks of nurbs based isogeometric finite elements is the inefficiency of local refinement in this study we present an alternative to nurbs based isogeometric analysis that allows for local refinement the idea is based on polynomial splines and exploits the flexibility of t meshes for local refinement the shape functions satisfy important properties such as non negativity local support and partition of unity several numerical examples are used to demonstrate the reliability of the present method

two dimensional domain decomposition based on skeleton computation for parameterization and isogeometric analysis abstract this paper proposes a method for decomposing two dimensional domains into subdomains for parameterization and isogeometric analysis given a complex domain in a plane the skeleton of the domain is computed to guide the domain decomposition a continuous parameterization of the domain is then obtained by parameterizing each respective subdomain this parameterization method is applied with isogeometric analysis to solve numerical pdes over two dimensional domains examples are provided to demonstrate that the new parameterization method is superior to other state of the art parameterization techniques and that it performs better in isogeometric analysis

smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points geometric design and isogeometric analysis considerations abstract we present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes acknowledging the differing requirements posed by design e g the convenience of an intuitive control net and analysis e g good approximation behavior we propose the construction of a separate smooth spline space for each while ensuring isogeometric compatibility requiring the geometric models to be members of the analysis suitable spaces the methodology is simple and is presented for bi cubic splines extensions to higher degrees are possible and are briefly discussed the presentation has been structured to show compatibility with t splines a state of the art cad technology but the approach should extend to other locally refinable spline technologies based on local tensor product structures an instantiation of the framework is presented and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework and show significantly higher convergence rates than attained previously in the considered setting

on maximal eigenfrequency separation in two material structures the 1d and 2d scalar cases abstract we present a method to maximize the separation of two adjacent eigenfrequencies in structures with two material components the method is based on finite element analysis and topology optimization in which an iterative algorithm is used to find the optimal distribution of the materials results are presented for eigenvalue problems based on the 1d and 2d scalar wave equations two different objectives are used in the optimization the difference between two adjacent eigenfrequencies and the ratio between the squared eigenfrequencies in the 1d case we use simple interpolation of material parameters but in the 2d case the use of a more involved interpolation is needed and results obtained with a new interpolation function are shown in the 2d case the objective is reformulated into a double bound formulation due to the complication from multiple eigenfrequencies it is shown that some general conclusions can be drawn that relate the material parameters to the obtainable objective values and the optimized designs

isogeometric generalized n th order perturbation based stochastic method for exact geometric modeling of composite structures static and dynamic analysis with random material parameters abstract the contribution herein proposes an isogeometric generalized n th order perturbation based stochastic method for exactly modeling representing composite structures comprising of different materials with particular attention to both static and dynamic analysis of structures with random material characteristics herein we exactly represent the geometric model via isogeometric analysis iga such as exactly modeling the interfaces in composite structures with dissimilar materials and also continuous variable thicknesses that cannot be achieved by existing traditional methods such as fem besides only a limited or scant work has been conducted thus far with iga and stochastic methods we consider the uncertainties of elastic modulus and mass density into account as stochastic inputs in static and dynamic isogeometric stochastic analyses moreover we derive and expand the iga based random input parameter and all state functions included in static and dynamic equilibrium equations around their expectations via a generalized n th order taylor series using a small perturbation parameter e in addition we determine the probabilistic moments of the stochastic solution that satisfy the given accuracy requirement by expanding to n th order the results obtained by the proposed method the finite element method wherever feasible and monte carlo simulations for both benchmark and engineering applications verify the following a the proposed methodology can achieve more accurate deterministic solutions with improved efficiency thereby strengthening the effectiveness and efficiency brought by the stochastic method this is in contrast to the fem based method which weakens them b on the other hand it can more efficiently acquire reliable and more accurate stochastic results both for expected values and standard deviations in the static and dynamic free vibration analyses note that the larger the problem size is the more efficient the proposed method will be

an iga discontinuous galerkin method on the union of overlapped patches abstract in isogeometric analysis iga the discontinuous galerkin method iga dg is often applied when the domain is composed of multiple non uniform rational b splines nurbs patches these patches are often required to be c 0 continuous to make the iga dg available and this requirement brings some problems to the cad process this paper extends the iga dg method to solve problems on the union of the overlapped patches with this method igaodg a problem on the overlapped domain is seen as multiple sub problems on the non trimmed patches and then the continuities of traces of the numerical solutions are penalized on the boundaries of non trimmed elements with the coefficients deduced from the classical dg method comparing with the iga methods which apply the cad boolean operations to deal with the overlapped domains the presented method carries out the integrations in the non trimmed patches thus no additional integration scheme needs to be considered to achieve the gaussian integration precision discussions on the presented discretization scheme and its convergence analysis show that this method is available on the domain created by the union of the matched mismatched and overlapped patches as predicted in the convergence analysis the presented method gives the optimal convergence rates in our numerical examples finally through one of its engineering applications this paper shows how to extend igaodg to solve problems that are defined on the trimmed patches

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

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

hierarchically refined and coarsened splines for moving interface problems with particular application to phase field models of prostate tumor growth abstract moving interface problems are ubiquitous in science and engineering to develop an accurate and efficient methodology for this class of problems we present algorithms for local h adaptivity of hierarchical b splines to be utilized in isogeometric analysis we extend bezier projection an efficient quadrature free local projection technique to the hierarchical setting in this case extraction operators may not be invertible to address this issue we develop a multi level reconstruction operator which maintains the locality properties of the projection we also introduce a balance parameter to control the overlap of hierarchical functions leading to improved numerical conditioning we apply our algorithms to the simulation of localized prostate cancer growth we model this disease using the phase field method and a set of diffusion reaction equations to account for the dynamics of nutrients and a key biomarker termed prostate specific antigen our results include examples on simple 2d and 3d domains and a more compelling tissue scale patient specific simulation which is run over a prostate anatomy extracted from medical images our methods for local h adaptivity efficiently capture the evolving interface between the tumor and the neighboring healthy tissue with remarkable accuracy in all cases

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

simultaneous isogeometrical shape and material design of functionally graded structures for optimal eigenfrequencies abstract a new optimization strategy for eigenfrequency optimization of functionally graded fg structures within the framework of isogeometric analysis iga approach is introduced the proposed methodology which utilizes a concurrent procedure by combining the shape and material composition optimization of these structures employs an extended form of the standard iga method by allowing for gradation of material properties through patches the distributions of the graded material properties are considered as imaginary surfaces over the computational domain and captured in a fully isogeometric formulation using the same nurbs based parameterization which is employed for the geometry modeling as well as the solution approximation considering the in plane coordinates of the control points defining the design boundary surfaces as well as the applicates of all the control points describing the variations of material properties as design variables we subsequently adopt a mathematical programming algorithm to simultaneously find the optimum shape and material composition of fg structures a couple of illustrative numerical examples in 2d elasticity with eigenfrequencies as their either constraints or objective functions are presented to demonstrate the high performance of the proposed methodology it will be seen that the obtained results by this concurrent optimization procedure have much better dynamic performance compared to the optimal results of the simple shape or material composition design

on the variational formulation of stress constraints in isogeometric design the isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems this unification is brought about by employing the same basis functions to describe exact geometry and to approximate the physical response the isogeometric framework has recently been extended to include design optimisation a novel variational stress constraint formulation is proposed in this paper that further extends recent efforts in isogeometric design the weak form of the constraint is obtained by utilising the same basis functions not only to represent geometry response and design variables but also lagrange multipliers in this form the proposed variational formulation fits seamlessly with the underlying analysis and design environment the variational constraint formulation is compared to the widely utilised stress averaging and collocation approaches to this end volume minimisation of elastic arches subjected to stress constraints is considered results indicate that the proposed variational approach has the best convergence rates and utilises a much smaller number of discrete constraints than competing methods

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

isogeometric analysis of free surface flow abstract this paper presents the first application of isogeometric analysis a new computational technology built on higher order and higher continuity basis functions employed in computer aided design and computer graphics to the computation of free surface phenomena described using the level set approach the method is based on the variational framework that is suitable for discretization by standard finite elements as well as the basis functions employed in isogeometric analysis the underlying numerical formulation globally conserves mass and preserves a sharp air water interface for the entire length of the simulation the numerical tests indicate that the proposed methodology gives an accurate description of the free surface behavior in both quasi steady and dynamic regimes furthermore very good per degree of freedom accuracy is obtained when higher order and higher continuity isogeometric discretizations are employed in free surface computations

a nurbs based immersed methodology for fluid structure interaction abstract we introduce an isogeometric immersed and fully implicit formulation for fluid structure interaction fsi the method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids which are a common case in various fields such as for example biomechanics in our fsi method we utilize an eulerian mesh on the whole domain and a lagrangian mesh on the solid domain the lagrangian mesh is arbitrarily located on top of the eulerian mesh in a non conforming fashion due to the formulation of our problem based on the immersed finite element method ifem we do not need mesh update or remeshing algorithms the fluid structure interface is the boundary of the lagrangian mesh but cuts arbitrarily the eulerian mesh the generalized method is used for time discretization and nurbs based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains the information transfer between the two meshes is carried out using the nurbs functions which avoids the use of the so called discretized delta functions the higher order and especially the higher continuity of nurbs functions allow us to deal with severe mesh distortion in the lagrangian mesh in comparison with classical c 0 linear piecewise functions as we prove numerically our numerical solutions attain good agreement with theoretical data for free falling objects in two and three dimensions which confirms the feasibility of our methodology

a quadratically convergent unstructured remeshing strategy for shape optimization a novel unstructured remeshing environment for gradient based shape optimization using triangular finite elements is presented the remeshing algorithm is based on a truss structure analogy in solving for the equilibrium position of the truss system the quadratically convergent newton s method is used exact analytical sensitivity information is therefore made available to the shape optimization algorithm the overall computational efficiency in gradient based shape optimization is very high in solving the truss structure analogy we compare our quadratically convergent newton solver with a previously proposed forward euler solver this includes notes regarding mesh uniformity element quality convergence rates and efficiency we present three numerical examples it is then shown that remeshing may introduce discontinuities and local minima we demonstrate that the effects of these on gradient based algorithms are alleviated to some extent through mesh refinement and may largely be overcome with a simple multi start strategy copyright 2005 john wiley sons ltd

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

isogeometric shape optimization of photonic crystals via coons patches abstract in this paper we present an approach that extends isogeometric shape optimization from optimization of rectangular like nurbs patches to the optimization of topologically complex geometries we have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps salient features of this approach include the following 1 multi patch coons representation of design geometry the design geometry is represented as a collection of coons patches where the four boundaries of each patch are represented as nurbs curves the use of multiple patches is motivated by the need for representing topologically complex geometries the coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization 2 exact boundary conversion to the analysis geometry with guaranteed mesh injectivity the analysis geometry is a collection of nurbs patches that are converted from the multi patch coons representation with geometric exactness in patch boundaries the internal nurbs control points are embedded in the parametric domain of the coons patches with a built in mesh rectifier to ensure the injectivity of the resulting b spline geometry i e every point in the physical domain is mapped to one point in the parametric domain 3 analytical sensitivities sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities the nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding coons patches

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

isogeometric spline forests in this paper we present isogeometric spline forests an isogeometric spline forest is a hierarchical spline representation capable of representing surfaces or volumes of arbitrarily complex geometry and topological genus spline forests can accomodate arbitrary degree and smoothness in the underlying hierarchical basis as well as non uniform knot interval configurations we describe adaptive h refinement and coarsening algorithms for isogeometric spline forests and develop a bezier extraction framework which provides a simple and efficient single level finite element description of the complex multi level unstructured hierarchical spline basis we then demonstrate the potential of spline forests as a basis for analysis in the context of transient advection diffusion problems where fully integrated adaptivity is demonstrated for the first time in an isogeometric simulation in all cases the adaptive process remains local even in the case of moving fronts and preserves exact geometry at the coarsest level of the discretization the accuracy and robustness of the approach is demonstrated in all cases

parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology abstract isogeometric analysis iga over a mesh of arbitrary topology is a challenging task since the concept of iga was proposed in the literature one can find some results in iga over an irregular mesh based on approximating schemes while little work is reported so far using interpolatory schemes this paper presents a parametric mesh regularization pmr method for iga using mapped interpolatory basis functions over a mesh of arbitrary topology given a geometric model defined over a mesh of arbitrary topology a global parametric mesh is first defined from the given control mesh using a gravity center method gcm yuan and ma 2014 the resulting parametric mesh is further regularized such that it ensures absolute interpolatory modeling with mapped interpolatory basis functions necessary geometric rules are developed for the regularization process while maintaining the same mesh topological structure the interpolatory basis function used in this paper is a kind of truncated interpolatory basic function tibf studied in our early work yuan and ma 2013 it can also be applied to other mapped interpolatory basis functions in addition to the capability of interpolatory modeling the proposed pmr method can also be applied to approximating schemes which further enriches the capability of such schemes in both modeling and isogeometric analysis of some models in a more appropriate and convenient manner the proposed pmr method has also the potential in achieving higher order continuity at extraordinary vertices using higher order mapped interpolatory basis functions

planar domain parameterization for isogeometric analysis based on teichmuller mapping abstract given the boundary curve of a planar domain finding a parametric spline representation for the domain is called domain parameterization a good parameterization of the computational domain plays a key role in isogeometric analysis since it influences the accuracy of the subsequent analysis in this paper we propose a new approach for planar domain parameterization based on teichmuller map a special map in the class of quasi conformal map under given correspondence of four boundary curves a unique teichmuller map between a unit square and a computational domain can be obtained which guarantees a bijection map and minimizes the maximal conformality distortion we propose an efficient iterative algorithm to compute the teichmuller map based on alternating direction method of multipliers admm experimental results show that our method can produce more uniform parameterization and increase the accuracy as well as decrease the condition numbers of the stiffness matrices in isogeometric analysis than other state of the art approaches

shape optimization directly from cad an isogeometric boundary element approach using t splines abstract we develop a t spline isogeometric boundary element method igabem simpson et al 2012 scott et al 2013 simpson et al 2014 to shape sensitivity analysis and gradient based shape optimization in three dimensional linear elasticity contrary to finite element based isogeometric analysis iga approaches no parametrization of the volume is required hence the iterative optimization algorithm can be implemented directly from cad without any mesh generation or postprocessing step for returning the resulting structure to cad designers t splines also guarantee a water tight geometry without the manual geometrical repair work as with non uniform rational b splines nurbs we demonstrate the worth of the method by analysing problems with and without analytical solutions including engineering examples involving complex shapes additionally we provide all the derivations of the required sensitivities and the details pertaining to the geometries examined in the benchmarking to provide helpful reference problems for 3d shape optimization

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

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

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

collaborative design of fiber path and shape for complex composite shells based on isogeometric analysis abstract composite shells with complex geometry are widely used in aerospace structures due to the complexity of geometry and curvilinear fiber path the analysis and optimization based on finite element analysis fea for complex variable stiffness vs shells is extremely time consuming by comparing with fea isogeometric analysis iga exhibits higher prediction efficiency of buckling load in this work the formula of geometric stiffness matrix for complex vs shells is derived for the first time based on degenerated shell method using iga which is the basis of performing linear buckling analysis then a new variable curvature quasi linear function vcqlf to describe curvilinear fiber path is proposed which can further expand the design space of vs shells after that two frameworks for shape optimization of complex shells are put forward and then compared and it is found that the one based on loft function can provide representative control variables of shape and effectively reduces the number of design variables for complex shells finally a novel collaborative optimization framework of fiber path and shell shape using iga is established by comparison of traditional methods it is demonstrated that the proposed framework can greatly improve the efficiency of optimization and fully explore the buckling load of complex vs shells

overlapping schwarz methods for isogeometric analysis we construct and analyze an overlapping schwarz preconditioner for elliptic problems discretized with isogeometric analysis the preconditioner is based on partitioning the domain of the problem into overlapping subdomains solving local isogeometric problems on these subdomains and solving an additional coarse isogeometric problem associated with the subdomain mesh we develop an h analysis of the preconditioner showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and overlap sizes for fixed polynomial degree p and regularity k of the basis functions numerical results in two and three dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h p k number of subdomains n overlap size and also jumps in the coefficients of the elliptic operator

spline based meshfree method summary in the present work the spline based meshfree method sbmfm is presented the sbmfm uses spline basis functions for field variables and boundary representation because of the inherent properties of splines that is nurbs nonuniform rational b splines the analysis domain used in the sbmfm has its own parametric domain unlike the classical concepts in the computer aided design cad community in which the nurbs patches are used to model the body the nurbs patch is treated as a background mesh in the present work the real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single nurbs patch building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible from a deformation point of view the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation in the sbmfm boundary spline curves freely move with complete independence from the quadrilateral background nurbs patch the so called element connectivity is unnecessary for geometry description in this paper geometrically nonlinear problems are considered implementation strategy is presented and the effectiveness of the present sbmfm is shown through various numerical examples copyright 2012 john wiley sons ltd

isogeometric shape optimization of shells using semi analytical sensitivity analysis and sensitivity weighting abstract we present isogeometric shape optimization for shell structures applying sensitivity weighting and semi analytical analysis we use a rotation free shell formulation and all involved geometry models i e initial design analysis model optimization model and final design use the same geometric basis in particular nurbs a sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update a multilevel design approach is applied such that the design space can be chosen independently from the analysis space the use of semi analytical sensitivities allows having different polynomial degrees for design and analysis model different numerical examples are performed which confirm the applicability of the proposed method furthermore a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods

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

computations of siegel modular forms of genus two we explain the basic notions and theorems for doing computations in the theory of siegel modular forms of degree two on the full modular group and of even weight this synopsis concludes with a handy and computationally realistic algorithm for tabulating the fourier coefficients of such forms and the euler factors of their spinor zeta functions in the second part of this paper we present and discuss some of the results of actual computations which we performed following this algorithm we point out two theoretically striking phenomena that are implied by the results of these computations

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

hp finite element discretisation of the electrical impedance tomography problem abstract this paper is concerned with the accurate recovery of the permittivity and conductivity from the solution of the electrical impedance tomography eit inverse problem given the knowledge of either the neumann to dirichlet or the dirichlet to neumann map the linearised eit inverse problem for the fourier regime is discretised by hp finite elements which allows the accurate calculation of the sensitivity matrix entries the variational algorithm of knowles 1998 31 is extended to allow the recovery of both material parameters by solution of the eit problem in the laplace regime the associated forward problem is again discretised by hp finite elements a series of numerical results are included to demonstrate the effectiveness of the proposed techniques

cad inspired hierarchical partition of unity constructions for nurbs based meshless design analysis and optimization in this paper we develop a unified representational paradigm for design and analysis that is inspired by the set theoretic boolean operations of the constructive solid geometry csg procedure of computer aided design cad we develop the notion of a primitive design state corresponding to a csg primitive region which is characterized by a description of the geometrical shape of the primitive and by a description of the material distribution within the primitive we analogously define a primitive behaviour state that is associated with a primitive design state and that is determined through a global analysis problem we define a global multi level design problem to determine the primitive design states we define boolean operations on the fields belonging to primitive design and behaviour spaces and show that the compositions of these fields amount to a hierarchical partition of unity construction we propose to use non uniform rational b splines nurbs to discretize the geometry material and behavioural fields local approximations defined on the primitive regions we show that this leads to recursive partition of unity constructions on the knot spaces sub local approximations due to the partition of unity property of nurbs basis functions since the primitive shapes are parametrically defined using nurbs the numerical quadrature is carried out on the parametric sub regions of the nurbs entities this eliminates the need for either a background mesh or a pseudo mesh for quadrature the developed methodology is implemented in a symbolic meshless computational framework written using the java language jnurbs and was demonstrated on two classes of problems the first class is the simultaneous optimal design of shape and multi material distributions the procedure is argued as being a generalization of classical topology optimization to designing objects made of functionally graded materials with pre defined geometrical constraints such as holes or fixed boundary shape the second class of problems demonstrated in the paper is optimal design to mitigate the effects of cracks here we identify the optimal locations of holes introduced to reduce the stress intensity factor of an edge crack copyright 2007 john wiley sons ltd

a mesh regularization scheme to update internal control points for isogeometric shape design optimization abstract this paper presents a variational method to update internal control points in isogeometric shape optimization the important properties of domain parameterization such as bijective mapping between parametric and physical domains uniform mesh and orthogonal mesh are enforced simultaneously the bijective mapping is achieved by minimizing a dirichlet energy functional to prevent the divergent behavior of the minimizing process due to the severely distorted initial mesh a constraint is introduced to enforce the positive jacobian of mapping from parametric to physical domains in spite of adding the constraint that might increase computational costs the proposed method is more efficient due to the convexity of dirichlet energy functional compared with the other unconstrained methods also it turns out that the proposed method is more effective to achieve the bijective mapping especially near a concave boundary the uniform parameterization of the domain is achieved by minimizing the dirichlet energy functional and the orthogonality of mesh is performed by minimizing a dimensionless functional the required design sensitivity of the employed functional and constraint is derived with respect to the position of internal control points the developed scheme of mesh regularization is effective to maintain the high quality of domain parameterization during the shape optimization process

isogeometric analysis enhanced by the scaled boundary finite element method abstract by leveraging the information of a typical cad model in the analysis the intensive process of discretization can be circumvented this unification has led to the isogeometric analysis iga hughes et al 2005 however as the cad model provides information only of the boundary a 2d 3d stress analysis is still one major step away in this work the concepts of isogeometric analysis and the scaled boundary finite element method sbfem are combined the sbfem requires only the boundary information and hence provides a seamless integration with the cad modeling within the proposed framework the nurbs basis functions are used to discretize the unknown fields in the circumferential direction whilst analytical solution is sought in the radial direction we further extend the framework to problems with singularities and to dynamic analysis the accuracy and the convergence properties of the proposed method are demonstrated with benchmark problems in the context of linear elasticity and linear elastic fracture mechanics

isogeometric topology optimization using trimmed spline surfaces in the present work a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed in the proposed approach the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by cad systems is employed for structural response analysis and sensitivity calculation in the topology optimization the outer and inner boundaries of design models are represented by a spline surface and trimming curves design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves new sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified the creation of new inner fronts during optimization is allowed for the topological flexibility an inner front merging algorithm is also presented the proposed spline based topology optimization is used to solve some benchmarking problems design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one it is also shown that post processing effort for converting to cad model is eliminated by using the same spline information in numerical analysis and design optimization

a hybrid igafem igabem formulation for two dimensional stationary magnetic and magneto mechanical field problems abstract isogeometric analysis iga can bridge the gap between geometrical and numerical modelling to this end the same functions used in computer aided design are applied to represent geometry and approximate field variables in the numerical model the concept has already been implemented to solve field problems using finite element fem and boundary element methods bem but coupling of both methods has not been applied in the current work an isogeometric fem bem coupling is proposed and applied to two dimensional stationary magnetic field problems while fem is used to model magnetisable bodies allowing for heterogeneous structures and non linear constitutive behaviour the bem domain accounts for the surrounding free space both methods are coupled on the surface of the magnetisable body due to this hybrid igafem bem approach no meshing of free space is necessary and truncation errors are avoided for problems to be solved on open infinite or semi infinite domains once the solution for the magnetic problem is obtained using the hybrid method igafem is used to solve a magneto mechanical field problem with one sided coupling in a subdomain of the magnetic problem this one sided coupling is realised by a magnetic stress tensor computed from the solution of the stationary magnetic field problem from the comparison of error norms and convergence rates for nurbs and discretisations based on lagrangian polynomials smaller errors and similar convergence rates are found for the proposed method for the same polynomial order of the basis functions and a comparable mesh size

initial solution estimation for one step inverse isogeometric analysis in sheet metal stamping abstract recently isogeometric analysis iga based on incremental methods for simulating the stamping process has been researched to the best of our knowledge however few studies have combined iga and one step inverse approach which is based on total deformation theory of plasticity a key step for one step inverse iga is to estimate a good initial solution traditional mesh based initial solution algorithms for one step inverse approach are not suitable for one step inverse nurbs based iga in this paper we presented a method which can rapidly unfold the undevelopable nurbs surface onto a planar domain and obtain a good initial solution estimation for one step inverse iga the key idea of the presented method is unfolding the control net of a nurbs surface for isogeometric analysis by energy based initial solution estimation algorithm in addition we developed a cutting stitching algorithm which can separate a complex control net into several parts with simple shapes numerical examples illustrate the initial solutions using the presented method are approaching the final results by one step inverse finite element method this implies that the iterative steps and computational time of one step inverse iga will be reduced significantly compared with that of one step inverse finite element method

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

geometric families of 4 dimensional galois representations with generically large images we study compatible families of four dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the images will be generically large if certain conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field we apply our result to an example constructed by jasper scholten a non selfdual 4 dimensional galois representation www math uiuc edu algebraic number theory 0183 1999 obtaining a family of linear groups and one of unitary groups as galois groups over mathbb q

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

adjoint based airfoil optimization with adaptive isogeometric discontinuous galerkin method abstract in this work an adjoint based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous galerkin method for compressible euler equations to investigate the significance of each design variable of airfoil b spline parameterization we first parameterize the airfoil by b spline curve approximation with some control points viewed as design variables and build the b spline representation of the flow field with the curve to apply the goal oriented h adaptive isogeometric dg method for flow solution then we compute and employ the discrete adjoint solutions for both multi target error estimation in adaptive mesh refinement with the isogeometric nature not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables indicating that the numerical solutions and objective could be differentiable with respect to those variables consequently the gradient is totally computed in an accurate approach and the sensitivity analysis is thus improved by reducing the spatial discretization error and introducing the analytical expression of derivative to reveal the key parameters for optimization in an intuitive and efficient manner although the sqp optimization algorithm is adopted in the paper the given accurate gradient can be applied to any gradient based optimization algorithm the proposed algorithm is demonstrated on rae2822 airfoil with inviscid transonic flow where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area the numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

on the images of the galois representations attached to genus 2 siegel modular forms we address the problem of the determination of the images of the galois representations attached to genus 2 siegel cusp forms of level 1 having multiplicity one these representations are symplectic we prove that the images are as large as possible for almost every prime if the siegel cusp form is not a maass spezialform and verifies two easy to check conditions mathematics subject classification 11f80 11f46

fundamental aspects of shape optimization in the context of isogeometric analysis we develop a mathematical foundation for shape optimization problems under state equation constraints where both state and control are discretized by b splines or nurbs in other words we use isogeometric analysis iga for solving the partial differential equation and a nodal approach to change domains where control points take the place of nodes and where thus a quite general class of functions for representing optimal shapes and their boundaries becomes available the minimization problem is solved by a gradient descent method where the shape gradient will be defined in isogeometric terms this gradient is obtained following two schemes optimize first discretize then and reversely discretize first optimize then we show that for isogeometric analysis the two schemes yield the same discrete system moreover we also formulate shape optimization with respect to nurbs in the optimize first ansatz which amounts to finding optimal control points and weights simultaneously numerical tests illustrate the theory

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

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

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

isogeometric analysis with bezier tetrahedra abstract this paper presents an approach for isogeometric analysis of 3d objects using rational bezier tetrahedral elements in this approach both the geometry and the physical field are represented by trivariate splines in bernstein bezier form over the tetrahedrangulation of a 3d geometry given a nurbs represented geometry either untrimmed or trimmed we first convert it to a watertight geometry represented by rational triangular bezier splines rtbs for trimmed geometries a compatible subdivision scheme is developed to guarantee the watertightness the rtbs geometry preserves exactly the original nurbs surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs from the watertight rtbs geometry a bezier tetrahedral partition is generated automatically by imposing continuity constraints on bezier ordinates of the elements we obtain a set of global c r smooth basis functions and use it as the basis for analysis numerical examples demonstrate that our method achieves optimal convergence in c r spaces and can handle complicated geometries

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

continuity and convergence in rational triangular bezier spline based isogeometric analysis abstract this paper presents a method for isogeometric analysis using rational triangular bezier splines rtbs where optimal convergence rates are achieved in this method both the geometry and the physical field are represented by bivariate splines in bernstein bezier form over the triangulation of a domain from a given physical domain bounded by nurbs curves a parametric domain and its triangulation are constructed by imposing continuity constraints on bezier ordinates we obtain a set of global c r smooth basis functions convergence analysis shows that isogeometric analysis with such c r rtbs basis can deliver the optimal rate of convergence provided that the c r geometric map remains unchanged during the refinement process this condition can be satisfied by constructing a pre refinement geometric map that is sufficiently smooth numerical experiments verify that optimal rates of convergence are achieved for poisson and linear elasticity problems

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

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

isogeometric shape design sensitivity analysis of stress intensity factors for curved crack problems abstract an isogeometric shape design sensitivity analysis method is developed for the stress intensity factors sifs in curved crack problems in order to obtain an enhanced shape sensitivity of sifs exact higher order geometric information of curvature and normal vector is embedded in the isogeometric shape sensitivity expressions a direct differentiation method is employed for the design sensitivity analysis and the size and orientation of the crack are selected as design variables in the isogeometric approach the nurbs basis functions in cad system are directly utilized in the response analysis which enables a seamless incorporation of exact geometry and higher continuity into the computational framework the precise evaluation of the crack face integral as well as the interaction integral is essential for the computation of the stress intensity factors for the curved crack problems in a mixed mode the cad based exact representation of tangential and normal vectors enables us to exactly define a local coordinate system at the crack tip whose shape dependency naturally leads to configuration design variations that include the change of crack orientation compared with the conventional finite element approach a higher continuity of stress and strain fields are expected in the interaction integral domain the design dependencies of auxiliary field and q function are considered in the design sensitivity formulation reflecting the convection effects due to the non constant design velocity in the interaction integral domain various numerical examples of curved crack problems are presented to verify the developed isogeometric sensitivity analysis method through the comparison with exact and finite element solutions