implicit dynamic analysis using an isogeometric reissner mindlin shell formulation summary in isogeometric analysis identical basis functions are used for geometrical representation and analysis in this work non uniform rational basis splines basis functions are applied in an isoparametric approach an isogeometric reissner mindlin shell formulation for implicit dynamic calculations using the galerkin method is presented a consistent as well as a lumped matrix formulation is implemented the suitability of the developed shell formulation for natural frequency analysis is demonstrated by a numerical example in a second set of examples transient problems of plane and curved geometries undergoing large deformations in combination with nonlinear material behavior are investigated via a zero thickness stress algorithm for arbitrary material models a j2 plasticity constitutive law is implemented in the numerical examples the effectiveness robustness and superior accuracy of a continuous interpolation method of the shell director vector is compared with experimental results and alternative numerical approaches copyright 2016 john wiley sons ltd

an adaptive method for homogenization in orthotropic nonlinear elasticity a database methodology is developed in order to characterize the constitutive behavior of macroscopically orthotropic nonlinearly elastic composites this methodology discretizes the material test space in terms of the eigenvalues of the macroscopic strain tensor and the orientation of its eigenvectors with respect to the axes of orthotropy the macroscopic properties of the composite are extracted at points of this test space by subjecting a statistically representative sample of the composite to appropriate boundary conditions and are subsequently employed in an interpolation procedure which creates the so called material map that characterizes the macroscopic behavior the macroscopic tangent required in the finite element calculations are derived either by using the tangent induced by the map for the macroscopic stress or by explicitly computing a map for the tangent the range of deformations over which a material map is applicable to a problem is enhanced by an adaptive scheme where a multi level finite element method is employed at highly deformed regions of a finite element mesh where the map limits are exceeded

on the use of nurbs based discretizations in the scaled boundary finite element method for wave propagation problems abstract we discuss the application of non uniform rational b splines nurbs in the scaled boundary finite element method sbfem for the solution of wave propagation problems at rather high frequencies we focus on the propagation of guided waves along prismatic structures of constant cross section comparisons are made between nurbs based discretizations and high order spectral elements in terms of the achievable convergence rates we find that for the same order of shape functions nurbs can lead to significantly smaller errors compared with lagrange polynomials the difference becomes particularly important at very high frequencies where spectral elements are prone to instabilities furthermore we analyze the behavior of nurbs for the discretization of curved boundaries where the benefit of exact geometry representation becomes crucial even in the low frequency range

effective properties of composite materials with periodic microstructure a computational approach abstract this study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents the theoretical background of the method is recalled first two different families of numerical methods are considered to solve the problem the first is based on the finite element method the concept of macroscopic degrees of freedom is presented the implementation of periodicity conditions is discussed a general framework permitting either a strain or stress control is proposed the second numerical method is based on fast fourier transforms it considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain the solution of this problem is based on the explicit form of the periodic green s function of the reference medium the relative merits of the two methods are compared and several examples are discussed both methods give very comparable results on test examples and their domains of applications appear to be complementary

rotation free isogeometric thin shell analysis using pht splines this paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical t meshes pht splines the method exploits the flexibility of t meshes for local refinement the main advantage of the pht splines in the context of thin shell theory is that it achieves c1 continuity so the kirchhoff love theory can be used in pristine form no rotational degrees of freedom are needed numerical results show the excellent performance of the present method

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

a reduced multiscale model for nonlinear structural topology optimization abstract this paper presents a reduced multiscale model for macroscopic structural design considering microscopic material nonlinear microstructures this work introduces reduced order model rom to alleviate the heavy computational demand of nonlinear nested multiscale procedures particularly in an optimization framework which requires multiple loops involving similar computations the surrogate model constructed using proper orthogonal decomposition pod and diffuse approximation reduces the computational effort for solving the microscopic boundary value problems multiscale analysis model fe 2 is applied to link structure and microstructures in the two scales maximum stiffness design of the macroscopic structure is realized using a discrete level set topology optimization model it is shown by means of numerical tests that the reduced multiscale model provides reasonable designs as compared to those obtained by the unreduced model while with a significantly reduced computational effort

the classification of 4 dimensional leibniz algebras abstract this paper is a contribution to the development of the non associative algebras theory more precisely this work deals with the classification of the complex 4 dimensional leibniz algebras note that the classification of 4 dimensional nilpotent complex leibniz algebras was obtained in albeverio et al 2006 1 therefore we will only consider non nilpotent case in this work

on a stress resultant geometrically exact shell model part vii shell intersections with dof finite element formulations abstract if the mid surface of a shell is smooth the classical theory describes the orientation of the director field attached to the mid surface by two independent rotational degrees of freedom at a shell intersection where this smoothness assumption no longer holds it is shown that the director field in the full nonlinear continuum shell equations must be necessarily described by three degrees of freedom this added degree of freedom however is totally unrelated to the so called drill rotation widely used as a means of tackling the shell intersection problem a computational procedure involving a trivial modification of the global singularity free update procedure described in part iii of this work is described which completely resolves the shell intersection problem without introducing drill springs or related ad hoc devices the proposed approach leaves unchanged standard finite element formulations in terms of 5 dof node affects only the global update formulae and exhibits excellent performance as illustrated by representative numerical simulations

an isogeometric continuum shell element for non linear analysis an isogeometric continuum shell formulation is proposed in which nurbs basis functions are used to construct the reference surface of the shell through the thickness behavior is interpolated using a higher order b spline which is in contrast to the standard continuum shell solid like shell formulation where a linear lagrange shape function is typically used in the thickness direction the present formulation yields a complete isogeometric representation of the continuum shell the shell element is implemented in a standard finite element code using bezier extraction which facilitates numerical integration on the reference surface of the shell through the thickness integration is done using a connectivity array which determines the support of a b spline basis function over an element the formulation has been verified using different linear and geometrically non linear examples the ability of the formulation in modelling buckling of static delaminations in composite materials is also demonstrated

the structure of the invariants of perfect lie algebras the structure of the invariants of perfect lie algebras with nontrivial centre is analysed it is shown that if the radical of a semidirect sum of a semisimple and a nilpotent lie algebra has a one dimensional centre then the defining representation r contains a copy of the trivial representation d0 of using this fact a criterion can be deduced to eliminate variables and to characterize the casimir operators of an algebra by means of certain subrepresentations r of r for rank 1 levi subalgebras all representations leading to perfect lie algebras with a radical isomorphic to a heisenberg lie algebra are determined we prove that the number of casimir operators of such an algebra is fixed for any dimension and moreover that they can be explicitly computed by means of determinantal formulae obtained from the brackets of the algebra for rank n 2 levi part a stabilization result of this nature is also given

numerical material representation using proper orthogonal decomposition and diffuse approximation from numerical point of view analysis and optimization in computational material engineering require efficient approaches for microstructure representation this paper develops an approach to establish an image based interpolation model in order to efficiently parameterize microstructures of a representative volume element rve based on proper orthogonal decomposition pod reduction of density maps snapshots when the parameters of the rve snapshot are known a priori the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters otherwise a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients we showcase the approach and its potential applications by considering a set of two phase composite snapshots the choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization

invariants of triangular lie algebras triangular lie algebras are the lie algebras which can be faithfully represented by triangular matrices of any finite size over the real complex number field in the paper invariants generalized casimir operators are found for three classes of lie algebras namely those which are either strictly or non strictly triangular and for so called special upper triangular lie algebras algebraic algorithm of boyko et al 2006 j phys a math gen 39 5749 preprint math ph 0602046 developed further in boyko et al 2007 j phys a math theor 40 113 preprint math ph 0606045 is used to determine the invariants a conjecture of tremblay and winternitz 2001 j phys a math gen 34 9085 concerning the number of independent invariants and their form is corroborated

a hierarchic family of isogeometric shell finite elements a hierarchic family of isogeometric shell finite elements based on nurbs shape functions is presented in contrast to classical shell finite element formulations inter element continuity of at least c1 enables a unique and continuous representation of the surface normal within one nurbs patch this does not only facilitate formulation of kirchhoff love type shell models for which the standard galerkin weak form has a variational index of 2 but it also offers significant advantages for shear deformable reissner mindlin type shells and higher order shell models for a 5 parameter shell formulation with reissner mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated kirchhoff love type director of the deformed configuration this split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration assumed natural strains or mixed finite element formulations the third member of the hierarchy is a 7 parameter model including thickness change and allowing for application of unmodified three dimensional constitutive laws the phenomenon of curvature thickness locking coming along with this kinematic extension again is automatically avoided by the hierarchic difference vector concept without any further treatment membrane locking and in plane shear locking are removed by two different approaches firstly elimination via the discrete strain gap dsg method and secondly removal of parasitic membrane strains using a hybrid mixed method based on the hellinger reissner variational principle the hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach

numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials the homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones this is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown at the same time the prevailing numerical methods such as concurrent methods require extensive computational efforts a simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous ma terials at small strains the proposed numerical approach comprises three steps at the rst step a representative volume element rve for a given nonlinear heterogeneous material is de ned and a loading space consisting of all the boundary conditions to be imposed on the rve is discretized into a sufficiently large number of points called nodes at the second step the boundary condition corresponding each node is prescribed on the surface of the rve and the resulting nonlinear boundary value problem is solved by the nite element method fem so as to determine the effective response of the heterogeneous material to the loading as sociated to each node of the loading space at the third step the nodal effective responses are interpolated via appropriate interpolation functions so that the effective strain energy stress strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way this leads to a non concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials the rst version of the proposed approach uses mutidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank one tensors to interpolate effective nodal responses and avoid high rank data these two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power law model are involved the numerical results given by our approach are compared with available analytical estimates exact results and full fem or concurrent multilevel fem solutions

solvable lie algebras with heisenberg ideals all finite dimensional indecomposable solvable lie algebras l n f having the heisenberg algebra h n as the nilradical are constructed the number of non nilpotent elements f that can be added to h n satisfies f or n 1 the casimir and generalized casimir operators of the algebras l n f are obtained

treatment of reissner mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework this work presents a framework for the computation of complex geometries containing intersections of multiple patches with reissner mindlin shell elements the main objective is to provide an isogeometric finite element implementation which neither requires drilling rotation stabilization nor user interaction to quantify the number of rotational degrees of freedom for every node for this purpose the following set of methods is presented control points with corresponding physical location are assigned to one common node for the finite element solution a nodal basis system in every control point is defined which ensures an exact interpolation of the director vector throughout the whole domain a distinction criterion for the automatic quantification of rotational degrees of freedom for every node is presented an isogeometric reissner mindlin shell formulation is enhanced to handle geometries with kinks and allowing for arbitrary intersections of patches the parametrization of adjacent patches along the interface has to be conforming the shell formulation is derived from the continuum theory and uses a rotational update scheme for the current director vector the nonlinear kinematic allows the computation of large deformations and large rotations two concepts for the description of rotations are presented the first one uses an interpolation which is commonly used in standard lagrange based shell element formulations the second scheme uses a more elaborate concept proposed by the authors in prior work which increases the accuracy for arbitrary curved geometries numerical examples show the high accuracy and robustness of both concepts the applicability of the proposed framework is demonstrated

a pgd based homogenization technique for the resolution of nonlinear multiscale problems abstract this paper deals with the offline resolution of nonlinear multiscale problems leading to a pgd reduced model from which one can derive microinformation which is suitable for design here we focus on an improvement to the proper generalized decomposition pgd technique which is used for the calculation of the homogenized operator and which plays a central role in our multiscale computational strategy this homogenized behavior is calculated offline using the latin multiscale strategy and pgd leads to a drastic reduction in cpu cost indeed the multiscale aspect of the latin method is based on splitting the unknowns between a macroscale and a microcomplement this macroscale is used to capture the homogenized behavior of the structure and then to accelerate the convergence of the iterative algorithm the novelty of this work lies in the reduction of the computation cost of building the homogenized operator in order to do that the problems defined within each subdomain are solved using a new pgd representation of the unknowns in which the separation of the unknowns is performed not only in time and in space but also in terms of the interface macrodisplacements the numerical example presented shows that the convergence rate of the approach is not affected by this new representation and that a significant reduction in computation cost can be achieved particularly in the case of nonlinear behavior the technique used herein is then particularly important to enhance the performances of the latin strategy but is not limited to this method the more general path which is followed in this paper is to solve the microproblems arising in the homogenization for all the configurations that can be encountered in the calculation an additional and novel benefit of this approach is that it includes the calculation of the homogenized tangent operator which gives over the whole time interval and for any cell the relation between the perturbation in terms of macroforces and the perturbation in terms of macrodisplacements in linear viscoelasticity this operator represents the homogenized behavior of the structure exactly

an efficient and robust rotational formulation for isogeometric reissner mindlin shell elements abstract this work is concerned with the development of an efficient and robust isogeometric reissner mindlin shell formulation for the mechanical simulation of thin walled structures such structures are usually defined by non uniform rational b splines nurbs surfaces in industrial design software the usage of isogeometric shell elements can avoid costly conversions from nurbs surfaces to other surface or volume geometry descriptions the shell formulation presented in this contribution uses a continuous orthogonal rotation described by rodrigues tensor in every integration point to compute the current director vector the rotational state is updated in a multiplicative manner large deformations and finite rotations can be described accurately the proposed formulation is robust in terms of stable convergence behavior in the nonlinear equilibrium iteration for large load steps and geometries with large and arbitrary curvature and in terms of insensitivity to shell intersections with kinks under small angles three different integration schemes and their influence on accuracy and computational costs are assessed the efficiency and robustness of the proposed isogeometric shell formulation is shown with the help of several examples accuracy and efficiency is compared to an isogeometric shell formulation with the more common discrete rotational concept and to lagrange based finite element shell formulations the competitiveness of the proposed isogeometric shell formulation in terms of computational costs to attain a pre defined error level is shown

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10

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

a bi value coding parameterization scheme for the discrete optimal orientation design of the composite laminate summary the discrete optimal orientation design of the composite laminate can be treated as a material selection problem dealt with by using the concept of continuous topology optimization method in this work a new bi value coding parameterization bcp scheme of closed form is proposed to this aim the basic idea of the bcp scheme is to code each material phase using integer values of 1 and 1 so that each available material phase has one unique code consisting of 1 and or 1 assigned to design variables theoretical and numerical comparisons between the proposed bcp scheme and existing schemes show that the bcp has the advantage of an evident reduction of the number of design variables in logarithmic form the benefit is particularly remarkable when the number of candidate materials becomes important in large scale problems numerical tests with up to 36 candidate material orientations are illustrated for the first time to indicate the reliability and efficiency of the bcp scheme in solving this kind of problem it proves that the bcp is an interesting and valuable scheme to achieve the optimal orientations for large scale design problems besides a four layer laminate example is tested to demonstrate that the proposed bcp scheme can easily be extended to multilayer problems copyright 2012 john wiley sons ltd

multiple scale eigendeformation based reduced order homogenization multiple scale eigendeformation based reduced order homogenization method which provides considerable computational cost saving in comparison to the direct homogenization method has been developed verified against the direct homogenization method and validated against experimental data the salient feature of the method is in the formulation of the unit cell problem in terms of residual free stresses and therefore eliminating the need for costly equilibrium calculations required by the direct homogenization method this is accomplished by introducing discretized eigendeformation fields at different scales which are collectively referred to as eigenstrains and eigenseparations and then precomputing a sequence of elasticity solutions of the unit cell problem subjected to the eigendeformation fields prior to nonlinear analysis

blended isogeometric shells abstract we propose a new isogeometric shell formulation that blends kirchhoff love theory with reissner mindlin theory this enables us to reduce the size of equation systems by eliminating rotational degrees of freedom while simultaneously providing a general and effective treatment of kinematic constraints engendered by shell intersections folds boundary conditions the merging of nurbs patches etc we illustrate the blended theory s performance on a series of test problems

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

invariants of lie algebras with fixed structure of nilradicals an algebraic algorithm is developed for computation of invariants generalized casimir operators of general lie algebras over the real or complex number field its main tools are the cartan s method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra unlike the first application of the algorithm in boyko et al 2006 j phys a math gen 39 5749 preprint math ph 0602046 which deals with low dimensional lie algebras here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable lie algebras of general dimension n restricted only by a required structure of the nilradical specifically invariants are calculated here for families of real complex solvable lie algebras these families contain with only a few exceptions all the solvable lie algebras of specific dimensions for whom the invariants are found in the literature

efficient matrix computation for tensor product isogeometric analysis the use of sum factorization abstract in this paper we discuss the use of the sum factorization for the calculation of the integrals arising in galerkin isogeometric analysis while introducing very little change in an isogeometric code based on element by element quadrature and assembling the sum factorization approach taking advantage of the tensor product structure of splines or nurbs shape functions significantly reduces the quadrature computational cost

the casimir operators of inhomogeneous groups we have found the number of invariant operators for the inhomogeneous groups igl n r isl n r iso p q iu p q isu p q isp 2n i e the inhomogeneous groups with the classical homogeneous subgroups and also for the weyl group w p q for some special cases explicit forms of the invariant operators are obtained we also discuss the methods applied together with problems in some cases possible further developments and relevance for the supersymmetric theories

all solvable extensions of a class of nilpotent lie algebras of dimension n and degree of nilpotency n 1 we construct all solvable lie algebras with a specific n dimensional nilradical of degree of nilpotency n 1 and with an n 2 dimensional maximal abelian ideal we find that for given n such a solvable algebra is unique up to isomorphisms using the method of moving frames we construct a basis for the casimir invariants of the nilradical we also construct a basis for the generalized casimir invariants of its solvable extension consisting entirely of rational functions of the chosen invariants of the nilradical

automorphisms of the lie algebra of strictly upper triangular matrices over a commutative ring let r be an arbitrary commutative ring with identity denote by n r the lie algebra over r consisting of all strictly upper triangular n 1 n 1 matrices over r with n 3 in addition for n 3 assume that the annihilator of 2 in r is zero the aim of this paper is to describe the automorphism group of n r we show that any automorphism of n r can be expressed as where and are graph extremal central and inner automorphisms respectively of n r

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

computation of invariants of lie algebras by means of moving frames a new purely algebraic algorithm is presented for computation of invariants generalized casimir operators of lie algebras it uses the cartan method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra the algorithm is applied in particular to computation of invariants of real low dimensional lie algebras a number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature bases of invariants of the real six dimensional solvable lie algebras with four dimensional nilradicals are newly calculated and listed in a table

the fundamental invariants of inhomogeneous classical groups the casimir operators of the following groups are explicitly constructed rn gl n r rn sl n r rn o p q cn u p q r2n sp n r cn gl n c cn sl n c cn o n c c2n sp n c hn u 2n hn su 2n hn o 2n hn sp p q and r r2n sp n r the method is based on a particular fiber bundle structure of the generic orbits generated by the co adjoint representation of a semidirect product

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

invariants of a semi direct sum of lie algebras we show that any semi direct sum l of lie algebras with levi factor s must be perfect if the representation associated with it does not possess a copy of the trivial representation as a consequence all invariant functions of l must be casimir operators when the number of invariants is given for all possible dimensions of l replacing the traditional method of solving the system of determining pdes by the equivalent problem of solving a system of total differential equations the invariants are found for all dimensions of the radical up to 5 an analysis of the results obtained is made and this leads to a theorem on invariants of lie algebras depending only on the elements of certain subalgebras

optimal design of material properties and material distribution for multiple loading conditions an extension of recent work 1 on the simultaneous optimization of material and structure to address the design of structures under multiple loading conditions is presented material properties are represented in the most general form possible namely as elements of the unrestricted set of positive semi definite constitutive tensors of a linearly elastic continuum existence of solutions can be shown when the objective is a weighted average of compliances and a resource constraint measured as the 2 norm or the trace of the constitutive tensors is included the optimized material properties can be derived analytically the optimization of the layout of the material leads to a sizing problem of structural optimization involving a non linear non smooth elasticity analysis the computational solution of this problem is discussed and illustrated with examples

some cohomologically rigid solvable leibniz algebras abstract in this paper we describe solvable leibniz algebras whose quotient algebra by one dimensional ideal is a lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical we prove that such leibniz algebra is unique and centerless also it is proved that the first and the second cohomology groups of the algebra with coefficients in the adjoint representation is trivial

isogeometric analysis for explicit elastodynamics using a dual basis diagonal mass formulation abstract we propose a method to obtain diagonal mass matrices for nurbs based approximation spaces by a dual lumping method the use of lumped mass matrices is of great importance in elastodynamics problems as they can be employed in explicit time integration schemes which do not require the solution of a linear system in finite elements several well established methods such as row sum diagonal scaling or nodal quadrature methods have been used to obtain lumped mass matrices for different applications however for higher order and higher continuity approximation spaces such as those derived from nurbs these approaches have only limited second order accuracy in this work we derive a dual basis which has optimal approximation and dispersion properties while maintaining local support the dual space has discontinuities at the element boundaries knots and it is used to provide the test functions in the context of a petrov galerkin method this results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high order accuracy numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity

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

multiscale structural topology optimization with an approximate constitutive model for local material microstructure abstract this paper builds on our recent work xia and breitkopf 2014 on multiscale structural topology optimization where at the microscopic scale local materials are optimized concurrently according to current loading status the former design framework requires intensive computational cost due to large number of repetitive local material optimizations to circumvent this limitation in the present work we construct a reduced database model viewing the local material optimization process as a generalized constitutive behavior using separated representations in this model the database is built from a set of numerical experiments of local material optimizations in the macroscopic strain tensor space each value in the database corresponds to the strain energy density evaluated on a material microstructure optimized according to the imposed macroscopic strain by tensor decomposition a continuous representation of the strain energy density is built as a sum of products of one dimensional interpolation functions as a result of this a priori off line step the effective strain energy and stress strain relations required for macroscopic structural evaluation and optimization are provided in a numerically explicit manner the results given by the reduced database model are compared with full scale results it is also shown that this explicit constitutive behavior representation can well serve multiscale structural design at a significantly reduced computational cost

solvable lie algebras with naturally graded nilradicals and their invariants the indecomposable solvable lie algebras with graded nilradical of maximal nilindex and a heisenberg subalgebra of codimension one are analysed and their generalized casimir invariants are calculated it is shown that rank one solvable algebras have a contact form which implies the existence of an associated dynamical system moreover due to the structure of the quadratic casimir operator of the nilradical these algebras contain a maximal non abelian quasi classical lie algebra of dimension 2n 1 indicating that gauge theories with ghosts are possible on these subalgebras

scale related topology optimization of cellular materials and structures the integrated optimization of lightweight cellular materials and structures are discussed in this paper by analysing the basic features of such a two scale problem it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell muc i e the so called representative volume element rve however with the asymptotic homogenization method used widely in actual topology optimization procedure effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases properties and spatial distribution of constituents in the microstructure regardless of scale effect from this consideration we propose the design element de concept being able to deal with conventional designs of materials and structures in a unified way by changing the scale and aspect ratio of the de scale related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns to illustrate the proposed approach numerical design problems of 2d layered structures with cellular core are investigated

nonlinear structural design using multiscale topology optimization part i static formulation we present a hierarchical multiscale design framework that couples computational homogenization with topology optimization to design a composite structure s microstructure to optimize its nonlinear elastostatic behavior to generate a well posed macroscopic topology optimization problem we use relaxation which requires homogenization to relate the macroscopic homogenized response to its microstructure and because closed form expressions for homogenized properties generally do not exist for materials with nonlinear response we rely on computational homogenization to evaluate them to optimize the homogenized properties of the unit cell we again use topology optimization and to make this unit cell optimization problem well posed we use restriction and thereby obtain a minimum microstructural length scale the coupled nonlinear analyzes and optimization problems are computationally intensive tasks that we resolve with a scalable parallel framework based on a single program multiple data programming paradigm numerical implementation is discussed and examples are provided

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

a nitsche type formulation and comparison of the most common domain decomposition methods in isogeometric analysis this paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis the methods comprise a penalty approach lagrange multiplier methods and a nitsche type method for the nitsche method a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity all these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2d in particular focus is put on non uniform rational b spline patches which join nonconformingly along their common interface thus the application of isogeometric analysis is extended to multi patches which can have an arbitrary parametrization on the adjacent edges moreover it has been shown that the unique properties provided by isogeometric analysis that is high order functions and smoothness across the element boundaries carry over for the analysis of multiple domains copyright 2013 john wiley sons ltd

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

solvable lie algebras with an mathbb n graded nilradical of maximal nilpotency degree and their invariants the class of solvable lie algebras with an graded nilradical of maximal nilpotency index is classified it is shown that such solvable extensions are unique up to isomorphism the generalized casimir invariants for the graded nilradicals and their associated solvable extensions are computed by the method of moving frames

computational analysis of nonlinear composite structures using the nonuniform transformation field analysis abstract a reduced model for the effective behavior of nonlinear composites such as metal matrix composite materials has been recently proposed by the authors 1 it extends and improves on the transformation field analysis of dvorak 2 by considering nonuniform transformation strains also called plastic modes and is referred to as the nonuniform transformation field analysis the present study is devoted to the implementation of this new homogenized model into a structural computation a brief account on the reduction procedure is given first then the time integration of the model which is required at each integration point of the structural problem is performed by means of an implicit scheme two examples are discussed the response of the homogenized structure is compared to the exact response of the actual heterogeneous structure computed with a very fine mesh it is seen that not only the structural response is accurately captured by the ntfa model but also the local stress field is correctly approximated

generalized casimir operators of solvable lie algebras with abelian nilradicals a solvable complex lie algebra l of dimension n with an abelian nilradical of dimension r is shown to have precisely 2r n generalized casimir invariants we always have r or n 2 they are constructed as invariants of the coadjoint representation of l and depend only on variables dual to elements of the nilradical their form in general involves logarithms of these variables in addition to rational and irrational functions they give rise to genuine casimir operators whenever they happen to be polynomials

novel higher order mass matrices for isogeometric structural vibration analysis abstract a set of novel higher order mass matrices are presented for isogeometric analysis of structural vibrations using nurbs the proposed method for the construction of higher order mass matrices contains two essential steps firstly based upon the standard consistent mass matrix a special reduced bandwidth mass matrix is designed this reduced bandwidth mass matrix meets the requirement of mass conservation while simultaneously preserves the same order of frequency accuracy as the corresponding consistent mass matrix subsequently a mixed mass matrix is formulated through a linear combination of the reduced bandwidth mass matrix and the consistent mass matrix the desired higher order mass matrix is then deduced from the mixed mass matrix by optimizing the linear combination parameter to achieve the most favorable order of accuracy both quadratic and cubic formulations are discussed in detail and it is shown that with regard to the vibration frequency the proposed higher order mass matrices have 6th and 8th orders of accuracy in contrast to the 4th and 6th orders of accuracy associated with the quadratic and cubic consistent mass matrices a generalization to two dimensional higher order mass matrix is further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices a series of benchmark examples congruously demonstrate that the proposed higher order mass matrices are capable of achieving the theoretically derived optimal accuracy orders for structural vibration analysis

theory and numerics of three dimensional beams with elastoplastic material behaviour a theory of space curved beams with arbitrary cross sections and an associated finite element formulation is presented within the present beam theory the reference point the centroid the centre of shear and the loading point are arbitrary points of the cross section the beam strains are based on a kinematic assumption where torsion warping deformation is included each node of the derived finite element possesses seven degrees of freedom the update of the rotational parameters at the finite element nodes is achieved in an additive way applying the isoparametric concept the kinematic quantities are approximated using lagrangian interpolation functions since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load carrying capacity of beam structures copyright 2000 john wiley sons ltd

invariants of the nilpotent and solvable triangular lie algebras invariants of the coadjoint representation of two classes of lie algebras are calculated the first class consists of the nilpotent lie algebras t m isomorphic to the algebras of upper triangular m m matrices the lie algebra t m is shown to have m 2 functionally independent invariants they can all be chosen to be polynomials and they are presented explicitly the second class consists of the solvable lie algebras l m f with t m as their nilradical and f additional linearly nilindependent elements some general results on the invariants of l m f are given and the cases m 4 for all f and f 1 or m 1 for all m are treated in detail

invariants of solvable lie algebras of dimension six all invariant functions of the coadjoint representation are determined for real solvable lie algebras of dimension six having a nilradical of dimension four many questions related to the nature of these invariants are analysed in particular we derive the general type of functions in terms of which all the invariants can be expressed and give the characterization of all solvable lie algebras with abelian nilradical that have no non trivial invariants the existence of casimir operators among these invariants is also investigated

eigendeformation based reduced order homogenization for failure analysis of heterogeneous materials the manuscript presents a novel model reduction approach for periodic heterogeneous media which combines the multiple scale asymptotic msa expansion method with the transformation field analysis tfa to reduce the computational cost of a direct homogenization approach without significantly compromising on solution accuracy the evolution of failure in micro phases and interfaces is modeled using eigendeformation adaptive model improvement strategy incorporating a hierarchical sequence of computational homogenization models is employed to control the accuracy of the model we present the model formulation and the computational details along with verification with respect to direct homogenization and validation with respect to physical experiments studies

a robust non linear mixed hybrid quadrilateral shell element in the paper a non linear quadrilateral shell element for the analysis of thin structures is presented the variational formulation is based on a hu washizu functional with independent displacement stress and strain fields the interpolation matrices for the mid surface displacements and rotations as well as for the stress resultants and strains are specified restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived the developed mixed hybrid shell element possesses the correct rank and fulfills the in plane and bending patch test using newton s method the finite element approximation of the stationary condition is iteratively solved our formulation can accommodate arbitrary non linear material models for finite deformations in the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post buckling response the essential feature of the new element is the robustness in the equilibrium iterations it allows very large load steps in comparison to other element formulations copyright 2005 john wiley sons ltd

reduced basis hybrid computational homogenization based on a mixed incremental formulation a new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed key features of the nonuniform transformation field analysis ntfa 1 2 are taken as a point of departure a short coming of the ntfa method is its limitation to simple constitutive models on the microscale a second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension both issues are related to the simple evolution law for the mode activity coefficients which are the new macroscopic internal variables of the homogenized material in the present contribution a generalization of the ntfa is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation the derivation is based on purely micro mechanical considerations the modification allows for arbitrary generalized standard materials 3 on the microscopic scale including e g crystal visco plasticity additionally the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi scale method

explicit algorithms for the nonlinear dynamics of shells abstract a finite element formulation and algorithm for the nonlinear analysis of the large deflection materially nonlinear response of impulsively loaded shells is presented a unique feature of this algorithm is the use of a bilinear four node quadrilateral element with single point quadrature and a simple hourglass control which is orthogonal to rigid body modes on an element level and does not compromise the consistency of the equations the geometric nonlinearities are treated by using a corotational description wherein a coordinate system that rotates with the material is embedded at the integration point thus the algorithm is directly applicable to anisotropic materials without any corrections for frame invariance of material property tensors this algorithm can treat about 200 element time steps per cpu second on a cyber 170 730 computer in the explicit time integration mode numerous results are presented for both elastic and elastic plastic problems with large strains that show that the method in most cases is comparable in accuracy with an earlier version of this algorithm employing a cubic triangular plate shell element but substantially faster

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

isogeometric reissner mindlin shell analysis with exactly calculated director vectors abstract an isogeometric reissner mindlin shell derived from the continuum theory is presented the geometry is described by nurbs surfaces the kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system hence the definition of nodal basis systems at the control points is necessary for the proposed formulation the control points are in general not located on the shell reference surface and thus several choices for the nodal values are possible the proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof thus the initial director vector coincides with the normal vector even for the coarsest mesh in addition to that a more accurate interpolation of the current director and its variation is proposed instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation the exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes the convergence behavior is shown to be correct for k refinement allowing the use of coarse meshes with high orders of nurbs basis functions this is potentially advantageous for applications with high numerical effort per integration point the geometrically nonlinear formulation accounts for large rotations the consistent tangent matrix is derived various standard benchmark examples show the superior accuracy of the presented shell formulation a new benchmark designed to test the convergence behavior for free form surfaces is presented despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound

critical assessment of different mass lumping schemes for higher order serendipity finite elements abstract an efficient and robust finite element based transient analysis of structures is important in many engineering applications in this context a diagonal or lumped mass matrix is an essential prerequisite in the last decades several methods to construct such a mass matrix have been proposed and therefore a comprehensive and unbiased evaluation of these approaches is needed in the present article we consequently investigate established mass lumping schemes such as the row sum method and the diagonal scaling method as well as the recently proposed manifold based method furthermore the latter approach is rigorously extended to higher order serendipity finite elements in two and three dimensions note that the manifold based method and the diagonal scaling method are very general approaches in the sense that they are applicable to arbitrary nodal based shape functions with arbitrary polynomial orders while guaranteeing the positive definiteness of the lumped mass matrix with respect to the row sum method the positivity is not ensured and negative or zero diagonal components might be computed consequently a detailed analysis of the influence and performance of these mass lumping schemes on the numerical results is conducted to this end several dynamic benchmark models including modal harmonic and transient analyses are selected showing that only suboptimal rates of convergence are attained if serendipity finite elements based on a diagonal mass matrix are employed

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

invariants of solvable lie algebras with triangular nilradicals and diagonal nilindependent elements the invariants of solvable lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on cartan s method of moving frames

nonuniform transformation field analysis the exact description of the overall behavior of composites with nonlinear dissipative phases requires an infinity of internal variables approximate models involving only a finite number of those can be obtained by considering a decomposition of the microscopic anelastic strain field on a finite set of transformation fields the transformation field analysis of dvorak proc r soc lond a 437 1992 311 corresponds to piecewise uniform transformation fields the present theory considers nonuniform transformation fields comparison with numerical simulations shows the accuracy of the proposed model

solvable lie algebras with triangular nilradicals all finite dimensional indecomposable solvable lie algebras having the triangular algebra t n as their nilradical are constructed the number of non nilpotent elements f in satisfies and the dimension of the lie algebra is

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

on the invariants of some solvable rigid lie algebras we determine fundamental systems of invariants for complex solvable rigid lie algebras having nonsplit nilradicals of characteristic sequence 3 1 1 these algebras being the natural followers of solvable algebras having heisenberg nilradicals a special case of this allows us to obtain a criterion to determine the number of functionally independent invariants of rank one subalgebras of real or complex solvable lie algebras finally we give examples of the inverse procedure obtaining fundamental systems of an algebra starting from rank one subalgebras and a criterion for the nonexistence of nontrivial invariants

dual and approximate dual basis functions for b splines and nurbs comparison and application for an efficient coupling of patches with the isogeometric mortar method abstract this contribution defines and compares different methods for the computation of dual basis functions for b splines and non uniform rational b splines nurbs they are intended to be used as test functions for the isogeometric mortar method but other fields of application are possible too three different concepts are presented and compared the first concept is the explicit formula for the computation of dual basis functions for nurbs proposed in the work of carl de boor these dual basis functions entail minimal support i e the support of the dual basis functions is equal to the support of the corresponding b spline basis functions in the second concept dual basis functions are derived from the inversion of the gram matrix these dual basis functions have global support along the interface the third concept is the use of approximate dual basis functions which were initially proposed for the use in harmonic analysis the support of these functions is local but larger than the support of the associated b spline basis functions we propose an extension of the approximate dual basis functions for nurbs basis functions after providing the general formulas we elaborate explicit expressions for several degrees of spline basis functions all three approaches are applied in the frame of the mortar method for the coupling of non conforming nurbs patches a method which allows complex discretizations with multiple intersecting interfaces is presented numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis this fact is in accordance with mathematical findings in literature which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions the dual basis functions computed from the inverse of the gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes only the approximate dual basis functions yield accurate and efficient computations where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes all basic cases of t intersections and star intersections are studied furthermore an example which combines all basic cases in a complex discretization is given the applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example

concurrent topology optimization design of material and structure within fe2 nonlinear multiscale analysis framework abstract this paper revisits concurrent design of material and structure within fe2 nonlinear multiscale analysis framework for structural stiffness maximization at macroscopic scale design variables are defined at the both scales cellular material models are defined at microscopic scale in a pointwise manner for the considered macroscopic structure they are optimized to adapt the macroscopic structural physical response though linear models are assumed at both scales the macroscopic structural equilibrium is in general nonlinear due to the adaptation of cellular material microstructures for this reason an iterative resolution based on fe2 scheme is developed to address this nonlinearity discrete topology optimization algorithm bi directional evolutionary structural optimization beso is used at the both scales it is shown by means of numerical tests that fe2 scheme can well bridge the two scales and address the nonlinearity reasonable design solutions of the macroscopic structure and its corresponding cellular materials have been obtained by the developed concurrent design framework

efficient isogeometric nurbs based solid shell elements mixed formulation and b method abstract in this work solid shell nurbs elements are developed in order to address static problems of slender structures under small perturbations a single layer of elements is considered through the thickness of the shell and the degree of approximation in that direction is chosen to be equal to two a full 3d constitutive relation is assumed the objective is to obtain highly accurate low degree elements to be used in coarse meshes in order to do that we propose a mixed method from which we derive a b projection to deal with locking the main idea is to modify the interpolation of the average stresses and strains through the thickness more precisely we develop two finite elements the first element which is based on the mixed method or equivalently on the b projection is extremely accurate but leads to a fully populated global stiffness matrix to improve the efficiency the second element uses a local least squares type procedure to define a new b projection leading to a sparse global stiffness matrix the quality and efficiency of the methods are assessed through several usual test cases and by comparison with other published techniques

the weak substitution method an application of the mortar method for patch coupling in nurbs based isogeometric analysis in this contribution a mortar type method for the coupling of non conforming nurbs surface patches is proposed the connection of non conforming patches with shared degrees of freedom requires mutual refinement which propagates throughout the whole patch due to the tensor product structure of nurbs surfaces thus methods to handle non conforming meshes are essential in nurbs based isogeometric analysis the main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation the deformations of the interface control points of adjacent patches are interrelated with a master slave relation this relation is established numerically using the weak form of the equality of mutual deformations along the interface with the help of this relation the interface degrees of freedom of the slave patch can be condensated out of the system a natural connection of the patches is attained without additional terms in the weak form the proposed method is also applicable for nonlinear computations without further measures linear and geometrical nonlinear examples show the high accuracy and robustness of the new method a comparison to reference results and to computations with the lagrange multiplier method is given

invariants of solvable rigid lie algebras up to dimension 8 the invariants of all complex solvable rigid lie algebras up to dimension 8 are computed moreover we show for rank 1 solvable algebras some criteria to deduce the non existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the casimir invariants of the associated nilradical

a class of solvable lie algebras and their casimir invariants a nilpotent lie algebra with an n 1 dimensional abelian ideal is studied all indecomposable solvable lie algebras with as their nilradical are obtained their dimension is at most n 2 the generalized casimir invariants of and of its solvable extensions are calculated for n 4 these algebras figure in the petrov classification of einstein spaces for larger values of n they can be used in a more general classification of riemannian manifolds

invariants of triangular lie algebras with one nil independent diagonal element the invariants of solvable triangular lie algebras with one nil independent diagonal element are studied exhaustively bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on cartan s method of moving frames and the special technique developed for triangular and closed algebras in boyko et al j phys a math theor 2007 40 7557 the conjecture of tremblay and winternitz j phys a math gen 2001 34 9085 on the number and form of elements in the bases is completed and proved