an explicit dynamics extended finite element method part 2 element by element stable explicit explicit dynamic scheme abstract this paper presents a general explicit time integration scheme for dynamics simulations using the extended finite element method with standard critical time step we use the generalized mass lumping technique proposed in part i of this paper this technique allows us to consider arbitrary enrichment functions in the x fem for explicit dynamics simulations in this second part the proposed approach allows the use of standard finite elements critical time step estimates for that purpose we develop a classical element by element strategy that couples the standard central difference scheme with the unconditionally stable explicit scheme proposed by chang s y chang an explicit method with improved stability property int j numer methods engrg 77 8 2008 1100 1120 this scheme coupled with x fem allows us to recover fe critical time steps independently of the enrichment functions considered furthermore a study of the stability property of this new explicit scheme is proposed in the x fem framework both with fixed enrichments and evolving enrichments with time some examples illustrate the good properties of the new explicit numerical time scheme and some applications to dynamics crack growth are given

realizing stable categories as derived categories abstract in this paper we discuss a relationship between representation theory of graded self injective algebras and that of algebras of finite global dimension for a positively graded self injective algebra a such that a 0 has finite global dimension we construct two types of triangle equivalences first we show that there exists a triangle equivalence between the stable category of z graded a modules and the derived category of a certain algebra of finite global dimension secondly we show that if a has gorenstein parameter l then there exists a triangle equivalence between the stable category of z l z graded a modules and a derived orbit category of which is a triangulated hull of the orbit category of the derived category

a characterization of finite auslander reiten quivers a we give necessary and sufficient conditions for a finite k modulated translation quiver to be an auslander reiten quiver in terms of certain homology groups associated to the quiver section 3 b we show that whether a finite k modulated translation quiver is an auslander reiten quiver depends only on the underlying valued translation quiver and the characteristic of k and give necessary and sufficient conditions only in terms of the valued quiver and char k we also show that any valued quiver with valuations 1 n or n l n q 3 admits a modulation over any prime field sections 5 6 7 c to a valued translation quiver r we associate an unvalued translation quiver r i e all valuations are 1 and show that a finite k modulated translation quiver is an auslander reiten quiver if and only if the associated unvalued translation quiver with the trivial k modulation k prime field of k is an auslander reiten quiver section 8

enriched space time finite element method a new paradigm for multiscaling from elastodynamics to molecular dynamics summary the main objective of this paper is to present an enriched version of the space time fem method to incorporate multiple temporal scale features with a focus on dynamics problems the method is established by integrating the basic framework of the space time discontinuous galerkin method with the extended finite element method two versions of the method have been developed one at the continuum scale elastodynamics and the other focuses on the dynamics at the molecular level after an initial outline of the formulation we explore the incorporation of different types of enrichments based on the length scale of interest the effects of both continuous and discontinuous enrichments are demonstrated through numerical examples involving wave propagations and dynamic fracture in harmonic lattice the robustness of the method is evaluated in terms of convergence and the ability to capture the fine scale features it is shown that the enriched space time fem leads to an improvement in the convergence properties over the traditional space time fem for problems with multiple temporal features it is also highly effective in integrating atomistic with continuum representations with a coupled framework copyright 2012 john wiley sons ltd

covering theory for linear categories with application to derived categories abstract we extend the galois covering theory introduced by bongartz gabriel for skeletal linear categories to general linear categories we show that a galois covering between krull schmidt categories preserves irreducible morphisms and almost splits sequences specializing to derived categories we study when a galois covering between locally bounded linear categories induces a galois covering between the bounded derived categories of finite dimensional modules as an application we show that each locally bounded linear category with radical squared zero admits a gradable galois covering which induces a galois covering between the bounded derived categories of finite dimensional modules and a galois covering between the auslander reiten quivers of these bounded derived categories in a future paper this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero

a discontinuous galerkin finite element method for dynamic and wave propagation problems in non linear solids and saturated porous media a time discontinuous galerkin finite element method dgfem for dynamics and wave propagation in non linear solids and saturated porous media is presented the main distinct characteristic of the proposed dgfem is that the specific p3 p1 interpolation approximation which uses piecewise cubic hermite s polynomial and linear interpolations for both displacements and velocities in the time domain is particularly proposed consequently continuity of the displacement vector at each discrete time instant is exactly ensured whereas discontinuity of the velocity vector at the discrete time levels still remains the computational cost is then obviously saved particularly in the materially non linear problems as compared with that required for the existing dgfem both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non linear problems numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional galerkin finite element method using the newmark algorithm in the time domain copyright 2003 john wiley sons ltd

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

generalized continuum modeling of 2 d periodic cellular solids abstract a generalized continuum representation of two dimensional periodic cellular solids is obtained by treating these materials as micropolar continua linear elastic micropolar constants are obtained using an energy approach for square equilateral triangular mixed triangle and diamond cell topologies the constants are obtained by equating two different continuous approximations of the strain energy function furthermore the effects of shear deformation of the cell walls on the micropolar elastic constants are also discussed a continuum micropolar finite element approach is developed for numerical simulations of the cell structures the solutions from the continuum representation are compared with the exact discrete simulations of these cell structures for a model problem of elastic indentation of a rectangular domain by a point force the utility of the micropolar continuum representation is illustrated by comparing various cell structures with respect to the stress concentration factor at the root of a circular notch

wave propagation in two dimensional periodic lattices plane wave propagation in infinite two dimensional periodic lattices is investigated using floquet bloch principles frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies hexagonal honeycomb kagome lattice triangular honeycomb and the square honeycomb these topologies exhibit dramatic differences in their long wavelength deformation properties long wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices the slenderness ratio of the constituent beams of the lattice or relative density has a significant influence on the band structure the techniques developed in this work can be used to design lattices with a desired band structure the observed spatial filtering effects due to anisotropy at high frequencies short wavelengths of wave propagation are consistent with the lattice symmetries

dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment summary a methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial dierential equation loses hyperbolicity the approach is limited to rate independent materials so that the transition occurs on a set of measure zero the discrete discontinuity is treated by the extendednite element method xfem whereby arbitrary discontinuities can be incorporated in the model without remeshing loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model a new method was developed for the case when the discontinuity ends within an element it facilitates the modelling of crack tips that occur within an element in a dynamic setting the method is applied to several dynamic crack growth problems including the branching of cracks copyright 2003 john wiley sons ltd

higher nakayama algebras i construction we introduce higher dimensional analogues of the nakayama algebras from the viewpoint of iyama s higher auslander reiten theory more precisely for each nakayama algebra a and each positive intege

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

a further look at newmark houbolt etc time stepping formulae this paper looks at some well known methods for the numerical integration of the vibration equation from the point of view of various groupings of the independent equations involved new forms are given of the general three parameter method introduced by zienkiewicz 1 some new one step methods are introduced these are of great practical importance because of the ease with which the size of time step can be changed

x fem explicit dynamics for constant strain elements to alleviate mesh constraints on internal or external boundaries this paper deals with the use of the extended finite element method x fem for rapid dynamic problems to solve the equations of motion a common technique is the explicit direct integration with a newmark scheme since this temporal scheme is only conditionally stable the critical time step must be determined it is generally induced by mesh constraints the idea of the paper is to weaken constraints on mesh generation algorithms so that the critical time step is as large as possible using the x fem one allows a non conformity between mesh and discontinuities such as cracks holes or interfaces in a first part we present a summary about direct integration schemes and about the extended finite element method then we focus on the theoretical description of a 1d x fem finite element and its generalization to 2d and 3d finite elements then dynamic numerical simulations are shown they concern structures under impact with holes or external boundaries not exactly matched by the mesh comparisons are made with numerical results coming from the abaqus software it shows that developments are satisfactory we conclude with some outlooks concerning this work

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

an explicit dynamics extended finite element method part 1 mass lumping for arbitrary enrichment functions this paper presents a general mass lumping technique for explicit dynamics simulations using the extended finite element method with arbitrary enrichment functions the proposed mass lumping technique is a generalization of previously published results for cracks and holes time step estimates are studied for crack singular enrichment functions and for hole enrichment in both cases we show that the critical time step does not tend to zero and is of the same order as that of the same unenriched element the performance of the method is demonstrated on several numerical examples that compare well with classical finite elements and previously published x fem results

the relationship between homological properties and representation theoretic realization of artin algebras we will study the relationship of quite different object in the theory of artin algebras namely auslander regular rings of global dimension two torsion theories categories and almost abelian categories we will apply our results to characterization problems of auslander reiten quivers 0 1 there exists a bijection between equivalence classes of krull schmidt categories c with additive generators m and morita equivalence classes of semiperfect rings which is given by c 7 c m m and the converse is given by 7 pr for the category pr of finitely generated projective modules although th is bijection itself is rather formal it will be very fruitful to study the relationship of a d below the object of this paper is to study it under the assumption that is an arti n algebra

wave propagation in non centrosymmetric beam lattices with lumped masses discrete and micropolar modeling the in plane acoustic behavior of non centrosymmetric lattices having nodes endowed with mass and rotational inertia and connected by massless ligaments with asymmetric elastic properties has been analyzed through a discrete model and a continuum micropolar model in the first case the propagation of harmonic waves and the dispersion functions have been obtained by the discrete floquet bloch approach it is shown that the optical branch departs from a critical point with vanishing group velocity and is decreasing for increasing the norm of the wave vector a micropolar continuum model has been derived through a continualization method based on a down scaling law from a second order taylor expansion of the generalized macro displacement field it is worth noting that the second order elasticity tensor coupling curvatures and micro couples turns out to be negative definite also in the general case of non centrosymmetric lattice the eigenvalue problem governing the harmonic propagation in the micropolar non centrosymmetric continuum results in general characterized by a hermitian full matrix that is exact up to the second order in the wave vector examples concerning square and equilateral triangular lattices have been analyzed and their acoustic properties have been derived with the discrete and continuum models the dependence of the floquet bloch spectra on the lattice non centrosymmetry is shown together with validity limits of the micropolar model finally in consideration of the negative definiteness of the second order elastic tensor of the micropolar model the loss of strong hyperbolicity of the equation of motion has been investigated

the relevance of conservation for stability and accuracy of numerical methods for fluid structure interaction numerical simulation of fluid structure interactions has typically been done using partitioned solution methods however partitioned methods are inherently non conservative and generally numerically unstable the deficiencies of partitioned methods have motivated the investigation of monolithic solution methods conservation is possible for monolithic methods the conditions have recently been presented e h van brummelen s j hulshoff r de borst energy conservation under incompatibility for fluid structure interaction problems comput methods appl mech engrg 192 2003 2727 2748 in the present paper we investigate the relevance of maintaining conservation for a model fluid structure interaction problem viz the piston problem to distinguish the effect of the error induced by the interface coupling from the fluid and structure discretization errors we use fluid subcycling and an exact time integration method for the structure a comparison between conservative and non conservative monolithic methods as well as partitioned methods is made we show that maintaining conservation has considerable impact on the stability and accuracy of the numerical method these results also indicate that only for a conservative monolithic scheme the improvement in accuracy over partitioned methods warrants the computational cost associated with a monolithic solution moreover we illustrate the implications that particular combinations of fluid and structure discretizations can have on the conservation properties of the fluid structure interaction problem

on time integration in the xfem the extended finite element method xfem is often used in applications that involve moving interfaces examples are the propagation of cracks or the movement of interfaces in two phase problems this work focuses on time integration in the xfem the performance of the discontinuous galerkin method in time space time finite elements fes and time stepping schemes are analyzed by convergence studies for different model problems it is shown that space time fe achieve optimal convergence rates special care is required for time stepping in the xfem due to the time dependence of the enrichment functions in each time step the enrichment functions have to be evaluated at different time levels this has important consequences in the quadrature used for the integration of the weak form a time stepping scheme that leads to optimal or only slightly sub optimal convergence rates is systematically constructed in this work copyright 2009 john wiley sons ltd

on the construction of blending elements for local partition of unity enriched finite elements for computational efficiency partition of unity enrichments are preferably localized to the sub domains where they are needed it is shown that an appropriate construction of the elements in the blending area the regionwhere the enriched elements blend to unenriched elements is often crucial for good performance of local partition of unity enrichments an enhanced strain formulation is developed which leads to good performance the optimal rate of convergence is achieved for polynomial enrichments it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence the methods are illustrated by several examples the examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element but other enrichments are also considered results show that both methods provide optimal rates of convergence

complete symmetry classification and compact matrix representations for 3d strain gradient elasticity abstract strain gradient elasticity sge is now often used in mechanics and physics owing to its capability to model some non classical phenomena such as size effects in materials and structures however certain fundamental questions about it have not yet received complete responses in its linear setting the constitutive law of sge is characterized by a fourth order elasticity tensor a fifth order one and a sixth order one even if the matrix representations for the 3d fourth and sixth order elasticity tensors are available for all possible symmetry classes the counterparts for the 3d fifth order tensor whose presence is unavoidable for materials with non centrosymmetric microstructure are still lacking in addition although the symmetry classes for each of the fourth fifth and sixth order elasticity tensor spaces are known the symmetry classes for these tensor spaces as a whole have never been reported and clarified in the literature the present work solves these two fundamental problems preventing the full understanding and exploitation of the linear constitutive law of sge precisely the matrix representations of the fifth order tensor for all of its 29 symmetry classes are provided in a compact and well structured way further linear sge is shown to possess 48 symmetry classes and for each of these symmetry classes the matrix representations of its fourth fifth and sixth order tensors are now available

homogenization and equivalent in plane properties of two dimensional periodic lattices abstract the equivalent in plane properties for hexagonal and re entrant auxetic lattices are investigated through the analysis of partial differential equations associated with their homogenized continuum models the adopted homogenization technique interprets the discrete lattice equations according to a finite differences formalism and it is applied in conjunction with the finite element description of the lattice unit cell it therefore allows handling of structures with different levels of complexity and internal geometry within a general and compact framework which can be easily implemented the estimation of the mechanical properties is carried out through a comparison between the derived differential equations and appropriate elasticity models equivalent young s moduli poisson s ratios and relative density are estimated and compared with analytical formulae available in the literature in plane wave propagation characteristics of honeycombs are also investigated to evaluate phase velocity variation in terms of frequency and direction of propagation comparisons are performed with the values obtained through the application of bloch theorem for two dimensional periodic structures to show the accuracy of the technique and highlight limitations introduced by the long wavelength approximation associated with the homogenization technique

a new fast finite element method for dislocations based on interior discontinuities a new technique for the modelling of multiple dislocations based on introducing interior discontinuities is presented in contrast to existing methods the superposition of infinite domain solutions is avoided interior discontinuities are specified on the dislocation slip surfaces and the resulting boundary value problem is solved by a finite element method the accuracy of the proposed method is verified and its efficiency for multi dislocation problems is illustrated bounded core energies are incorporated into the method through regularization of the discontinuities at their edges though the method is applied to edge dislocations here its extension to other types of dislocations is straightforward copyright 2006 john wiley sons ltd

comparison of wave propagation characteristics of the cosserat continuum model and corresponding discrete lattice models a comparison is made between two dimensional elastic discrete lattices and a corresponding cosserat continuum firstly it is demonstrated how the equations of motion of the cosserat model can be retrieved from those of a lattice model for this purpose two lattice geometries are elaborated a 7 cell hexagonal lattice and a 9 cell square lattice for both lattices the individual cells have two translational spring interactions and a rotational spring interaction with their neighbouring cells secondly the dispersion relations for the lattice models and the cosserat continuum model are examined in order to determine up to which wavelength of the deformation field the cosserat model accurately represents the underlying discrete micro structure the effect of the lattice anisotropy and inhomogeneity on this accuracy is also discussed

new development in extended finite element modeling of large elasto plastic deformations this paper presents new achievements in the extended finite element modeling of large elasto plastic deformation in solid problems the computational technique is presented based on the extended finite element method x fem coupled with the lagrangian formulation in order to model arbitrary interfaces in large deformations in x fem the material interfaces are represented independently of element boundaries and the process is accomplished by partitioning the domain with some triangular sub elements whose gauss points are used for integration of the domain of elements the large elasto plastic deformation formulation is employed within the x fem framework to simulate the non linear behavior of materials the interface between two bodies is modeled by using the x fem technique and applying the heaviside and level set based enrichment functions finally several numerical examples are analyzed including arbitrary material interfaces to demonstrate the efficiency of the x fem technique in large plasticity deformations copyright 2008 john wiley sons ltd

time integration for extended discontinuous galerkin methods with moving domains summary we study time integration schemes for discontinuous galerkin discretizations of problems with moving immersed interfaces two approaches have been discussed in literature so far splitting schemes and space time methods splitting schemes are cheap and easy to implement but are non conservative inherently limited to low orders of accuracy and require extremely small time steps space time methods on the other hand are conservative allow for large time steps and generalize to arbitrary orders of accuracy however these advantages come at the expense of a severe growth the systems to be solved within this work we present a generic strategy that combines the advantages of both concepts by evaluating numerical fluxes in a moving reference frame and by making use of a conservative cell agglomeration strategy we study the performance of this strategy in combination with backward difference formulas bdf and explicit runge kutta schemes in the context of the scalar transport equation the burgers equation and the heat equation our results indicate that higher order spatial and temporal convergence rates can be achieved without increasing the size of the systems to be solved this article is protected by copyright all rights reserved

τ2 stable tilting complexes over weighted projective lines abstract let x be a weighted projective line and coh x the associated category of coherent sheaves we classify the tilting complexes t in d b coh x such that 2 t t where is the auslander reiten translation in d b coh x as an application of this result we classify the 2 representation finite algebras which are derived equivalent to a canonical algebra this complements iyama oppermann s classification of the iterated tilted 2 representation finite algebras by passing to 3 preprojective algebras we obtain a classification of the selfinjective cluster tilted algebras of canonical type this complements ringel s classification of the selfinjective cluster tilted algebras

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

d representation finite self injective algebras abstract in this paper we initiate the study of higher dimensional auslander reiten theory of self injective algebras we give a systematic construction of weakly d representation finite self injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the serre functor the condition holds in particular for all fractionally calabi yau algebras of global dimension at most d this generalizes riedtmann s classical construction of representation finite self injective algebras our method is based on an adaptation of gabriel s covering theory for k linear categories to the setting of higher dimensional auslander reiten theory applications include n fold trivial extensions and classical and higher preprojective algebras which are shown to be d representation finite in many cases we also get a complete classification of all d representation finite self injective nakayama algebras for arbitrary d

arbitrary discontinuities in space time finite elements by level sets and x fem an enriched finite element method with arbitrary discontinuities in space time is presented the discontinuities are treated by the extended finite element method x fem which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper surface which is described by level sets a space time weak form for conservation laws is developed where the rankine hugoniot jump conditions are natural conditions of the weak form the method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non linear burgers equations by capturing the discontinuity in time as well as space results are improved over capturing the discontinuity in space alone and the method is remarkably accurate implications to standard semi discretization x fem formulations are also discussed copyright 2004 john wiley sons ltd

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian

a combined space time extended finite element method the newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities zienkiewicz proposed an approach using finite element concept in time which allows a new look at the newmark method the idea of this paper is to propose thanks to this approach the use of a time partition of the unity method denoted time extended finite element method tx fem for improved numerical simulations of time discontinuities an enriched basis of shape functions in time is used to capture with a good accuracy the non polynomial part of the solution this formulation allows a suitable form of the time stepping formulae to study stability and energy conservation the case of an enrichment with the heaviside function is developed and can be seen as an alternative approach to time discontinuous galerkin method t dgm stability and accuracy properties of which can be derived from those of the tx fem then space and time x fem stx fem are combined to obtain a unified space time discretization this combined stx fem appears to be a suitable technique for space time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities copyright 2005 john wiley sons ltd

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

extensions over hereditary artinian rings with self dualities i in this paper we study the finitely generated indecomposable modules over an arbitrary extension over an artinian ring with self morita duality let a be an artinian ring with self duality and t an extension over a with kernel q see 11 chap xiv sect 21 such that qa and a q are isomorphic to injective hulls of top a and topla respectively such an a module q will be called quasi frobenius then it will be proved that