the sl3 selberg integral abstract using an extension of the well known evaluation symmetry a new cauchy type identity for macdonald polynomials is proved after taking the classical limit this yields a new sl 3 generalisation of the famous selberg integral closely related results obtained in this paper are an sl 3 analogue of the askey habsieger kadell q selberg integral and an extension of the q selberg integral to a transformation between q integrals of different dimensions

a geometry projection method for shape optimization we present a new method for shape optimization that uses an analytical description of the varying design geometry as the control in the optimization problem a straightforward filtering technique projects the design geometry onto a fictitious analysis domain to support simplified response and sensitivity analysis however the analytical geometry model is referenced directly for all purely geometric calculations the method thus combines the advantages of direct geometry representations with the simplified analysis procedures that are possible with fictitious domain analysis methods such as the material distribution methods commonly used in topology optimization the projected geometry measure converges to the indicator function of the analytical geometry model in the limit of numerical mesh refinement consequently optimal designs obtained with the new method converge to solutions of well defined continuum optimization problems in the limit of mesh refinement this property is confirmed in example computations for minimum compliance design of an elastic structure subject to a volume constraint and for minimum volume design subject to a maximum stress constraint copyright 2004 john wiley sons ltd

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

a selberg integral for the lie algebra an a new q binomial theorem for macdonald polynomials is employed to prove an a n analogue of the celebrated selberg integral this confirms the mathfrak g rm a n case of a conjecture by mukhin and varchenko concerning the existence of a selberg integral for every simple lie algebra mathfrak g

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

shape optimization using the finite element method on multiple meshes with nitsche coupling an important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration common strategies are to employ mesh deformation

a nurbs based generalized finite element scheme for 3d simulation of heterogeneous materials a 3d nurbs based interface enriched generalized finite element method nigfem is introduced to solve problems with complex discontinuous gradient fields observed in the analysis of heterogeneous materials the method utilizes simple structured meshes of hexahedral elements that do not necessarily conform to the material interfaces in heterogeneous materials by avoiding the creation of conforming meshes used in conventional fem the nigfem leads to significant simplification of the mesh generation process to achieve an accurate solution in elements that are crossed by material interfaces the nigfem utilizes non uniform rational b splines nurbs to enrich the solution field locally the accuracy and convergence of the nigfem are tested by solving a benchmark problem we observe that the nigfem preserves an optimal rate of convergence and provides additional advantages including the accurate capture of the solution fields in the vicinity of material interfaces and the built in capability for hierarchical mesh refinement finally the use of the nigfem in the computational analysis of heterogeneous materials is discussed development of generalized finite element method for 3d mesoscale modeling of materials with complex microstructures novel nurbs based enrichment functions built in capability to perform hierarchical local mesh refinement method is applicable to engineering problems characterized by discontinuous gradient and complex internal geometry

singular vectors for the wn algebras in this paper we use free field realisations of the a type principal or casimir wn algebras to derive explicit formulae for singular vectors in fock modules these singular vectors are constructed by applying screening operators to fock module highest weight vectors the action of the screening operators is then explicitly evaluated in terms of jack symmetric functions and their skew analogues the resulting formulae depend on sequences of pairs of integers that completely determine the fock module as well as the jack symmetric functions

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

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

3 d shape optimal design and automatic finite element regridding a unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed a domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed for numerical implementation of shape optimal design parameterization of the boundary shape of mechanical components is defined and illustrated using a bezier surface in shape design problems nodal points of the finite element model move as the shape changes a method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid for numerical implementation of the continuum shape design sensitivity analysis and automatic regridding an established finite element analysis code is used to demonstrate the feasibility of the method developed shape design optimization of a main engine bearing cap is carried out as an example

on a cluster category of infinite dynkin type and the relation to triangulations of the infinity gon let k be a field and let d be a k linear algebraic triangulated category with split idempotents let be the suspension functor of d and let s be a 2 spherical object of d that is the morphism space d s i s is k for i 0 and i 2 and vanishes otherwise assume that s classically generates d that is each object of d can be built from s using de suspensions direct sums direct summands and distinguished triangles it was proved in 15 thm 2 1 that d is uniquely determined by these properties as we will explain d is a good candidate for a cluster category of dynkin type a for instance we show that there is a bijection between the cluster tilting subcategories of d and certain triangulations of the gon we use this to give an example of a subcategory a which is weak cluster tilting that is satisfies a 1a a but fails to be functorially finite perpendicular

stress adapted numerical form finding of pre stressed surfaces by the updated reference strategy in this paper we present the updated reference strategy for numerical formfinding of pre stressed membranes which is based on standard finite elementdiscretization the singularities of the inverse problem are regularizedby a homotopy mapping a projection scheme is proposed where anisotropicpre stress is defined with respect to an additional reference plane whichreflects the initially developable surface of membrane strips in the productionprocess physically problematic combinations of edge geometry and surfacestress are solved by a self adaptive stress correction scheme the algorithmis based on a local criterion derived from differential geometry severalexamples illustrate the success of each idea and implementation

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

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

optimal shapes of mechanically motivated surfaces abstract subject of this contribution is form finding of optimal structural shapes with regard to the load carrying behaviour of surface structures under certain load cases in general those optimal shapes prefer a membrane state of stress to transfer loading bending is omitted as much as possible it will be focused on two different disciplines and related numerical approaches which deal with solutions of the mentioned task form finding of prestressed membranes and general shape optimization as design is an inverse problem both approaches share similar problematic properties as e g indeterminate in plane location of surface discretization or necessary regularization and filtering of sensitivity and other data as it will turn out those remedies found for the very special methods of membrane design can be abstracted and transferred to general optimization procedures that merges into elegant numerical shape optimal design techniques which combine advantages of both approaches and allow for effective and efficient shape optimization of free formed surfaces directly on the finite element mesh and for a large number of variables typical applications are for example membrane design free form architecture and structural engineering and metal sheet design

maximal rigid subcategories and cluster tilting subcategories in 2 cy categories abstract let c be a connected hom finite 2 cy triangulated category we prove that if t is a functorially finite maximal rigid subcategory of c without loops in its quiver then t is cluster tilting in particular this gives a positive answer to a conjecture proposed in 5

eulerian shape design sensitivity analysis and optimization with a fixed grid conventional shape optimization based on the finite element method uses lagrangian representation in which the finite element mesh moves according to shape change while modern topology optimization uses eulerian representation in this paper an approach to shape optimization using eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed a continuum geometric model is defined on the fixed grid of finite elements an active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization in which each element has a unique shape density the shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements using this transformation it has been shown that the shape design problem can be treated as a parameter design problem which is a much easier method than the former a detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary the accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement two design optimization problems are presented to show the feasibility of the proposed design approach

a 3d interface enriched generalized finite element method for weakly discontinuous problems with complex internal geometries an interface enriched generalized finite element method gfem is introduced for 3d problems with discontinuous gradient fields the proposed method differs from conventional gfem by assigning the generalized degrees of freedom to the interface nodes i e nodes generated along the interface when creating integration subdomains instead of the nodes of the original mesh a linear combination of the lagrangian shape functions in these integration subelements are then used as the enrichment functions to capture the discontinuity in the gradient field this approach provides a great flexibility for evaluating the enrichment functions including for cases where elements are intersected with multiple interfaces we show that the method achieves optimal rate of convergence with meshes which do not conform to the phase interfaces at a computational cost similar to or lower than that of conventional gfem the potential of the method is demonstrated by solving several heat transfer problems with discontinuous gradient field encountered in particulate and fiber reinforced composites and in actively cooled microvascular materials

bc_n symmetric abelian functions we construct a family of bc n symmetric biorthogonal abelian functions generalizing koornwinder s orthogonal polynomials and prove a number of their properties most notably analogues of macdonald s conjectures the construction is based on a direct construction for a special case generalizing okounkov s interpolation polynomials we show that these interpolation functions satisfy a collection of generalized hypergeometric identities including new multivariate elliptic analogues of jackson s summation and bailey s transformation

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

shape design optimization of hyperelastic structures using a meshless method ashapedesignoptimizationprocedureforhyperelasticstructuresisdevelopedusing ameshlessmethodforanalysis and a continuum based design sensitivity analysis dsa method the meshless method greatly reduces the mesh distortion or entanglement encountered in using the e nite element method for large deformation nonlinear analysis and structural shape design optimization the dsa method of grindeanu et al grindeanu i chang k h choi k k and chen j s design sensitivity analysis of hyperelastic structures using a meshless method aiaa journal vol 36 no 4 1998 pp 618 627 is extended by using a pressure projection method to avoid volumetric locking for nearly incompressible materials without the need for large support sizes for the meshless shape functions and to reduce the cpu time the lagrange multiplier method is used to impose the essential boundary conditions an engine mount is employed as an example to demonstrate the feasibility of the proposed optimization method the mass is minimized subject to constraints on hydrostatic pressure and stiffness characteristics of the component the design velocity e elds corresponding to theshape design parameters are obtained using the design sensitivity analysis and optimization tool shape design optimization is carried out using the modie ed feasible direction of the design optimization tool

a numerical methodology to assess the quality of the design velocity field computation methods in shape sensitivity analysis the velocity field in shape sensitivity analysis is not uniquely defined although it must meet numerous theoretical and practical criteria these practical criteria can be used to compare the existing velocity field computation methods which meet the theoretical criteria but only in qualitative terms when the fem is used in design sensitivity analysis dsa due to the fe discretization error the dsa errors will depend on the design velocity field considered this paper presents a numerical methodology for quality evaluation of design velocity field computation methods in quantitative terms based on the analysis of the dsa discretization error the sensitivity of the squared energy norm m u 2 am am being a design variable has been taken as the magnitude to measure the error of the dsa for h refinements the squared error in energy norm e u 2 and the error in m e m are theoretically related by a constant which is independent of the refinement degree of the fe model the quality of the design velocity field computation methods can therefore be assessed in terms of the stability of e m e u 2 in sequences of meshes an example of the use of this methodology where six design velocity field computation methods are compared is presented copyright 2004 john wiley sons ltd

a systematic approach to shape sensitivity analysis abstract a fully detailed development of the domain parameterization method is presented for shape sensitivity analysis it is shown that equivalent results are obtained from the material derivative method in fact the material derivative method may be viewed as a special case of the domain parameterization method which occurs when the reference configuration coincides with the body configuration the method is illustrated for the laplace problem in which explicit shape sensitivities are derived by the adjoint and direct differentiation methods both finite element and boundary element applications are discussed the similarities between this approach and the isoparametric finite boundary element method are transparent in the finite element approach it is shown that the sensitivity integrals may be transformed to the boundary as is commonly done in the material derivative method for the adjoint method however this does not seem possible for the direct differentiation method finally in the boundary element approach the sensitivities do not require the differentiation of the fundamental solutions

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

an interface enriched generalized fem for problems with discontinuous gradient fields summary a new generalized fem is introduced for solving problems with discontinuous gradient fields the method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges the proposed approach has several advantages over conventional generalized fem formulations such as a lower computational cost easier implementation and straightforward handling of dirichlet boundary conditions a detailed convergence study of the proposed method and a comparison with the standard fem are presented for heat transfer problems the method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard fem with conforming meshes various application problems are presented including the conjugate heat transfer problem encountered in microvascular materials copyright 2011 john wiley sons ltd

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

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

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

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

optimal design of nonlinear parabolic systems part ii variable spatial domain with applications to casting optimization abstract in part i design sensitivity expressions for nonlinear parabolic systems were derived and then specialized for the analysis for thermal systems which govern many processing operations however that presentation did not account for shape variations here we present the shape sensitivity analysis for the thermal system the sensitivities are derived via the domain parameterization method and computed with the finite element method various means for parameterizing the shape are also discussed finally the sensitivity analysis is coupled with nonlinear programming to optimize the design of a sand casting

an a r threesome matrix models 2d conformal field theories and 4d n 2 gauge theories we explore the connections between three classes of theories a quiver matrix models d 2 conformal a r toda field theories and d 4 n 2 supersymmetric conformal a r quiver gauge theories in particular we analyze the quiver matrix models recently introduced by dijkgraaf and vafa unpublished and make detailed comparisons with the corresponding quantities in the toda field theories and the n 2 quiver gauge theories we also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions c 2010 american institute of physics doi 10 1063 1 3449328

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway