anisotropic mesh adaptation for crack detection in brittle materials the quasi static brittle fracture model proposed by g francfort and j j marigo can be gamma approximated at each time evolution step by the ambrosio tortorelli functional in this paper we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field second we build on this variational model an adapted discretization to numerically compute the time evolving minimizing solution we present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure the main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack but only far away from the crack tip while they show a highly isotropic behavior in a neighborhood of the crack tip instead as a consequence of these properties the resulting discretizations follow very closely the propagation of the fracture

cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

isogeometric analysis and hierarchical refinement for higher order phase field models abstract while the interest in higher order models in physics and mechanics grows their numerical simulation still poses a challenge especially for arbitrary shaped three dimensional domains this contribution presents the mathematical framework as well as the application to different problems in the field of material science fracture mechanics and diffusion problems all models under consideration require at least c 1 continuity which prevents the application of standard finite element analysis and local mesh refinements introducing isogeometric analysis iga for the discretization in a finite element framework enables us to deal with these requirements moreover a general hierarchical refinement scheme based on a subdivision projection is presented here for one two and three dimensional simulations this technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization using this mathematical framework the improved convergence of a kuramoto sivashinsky model a mesh adapted thermal diffusion simulation and computations of a priori unknown crack propagation in different fracture modes underline the versatility of the presented hierarchical refinement scheme

a primal dual active set method and predictor corrector mesh adaptivity for computing fracture propagation using a phase field approach abstract in this paper we consider phase field based fracture propagation in elastic media the main purpose is the development of a robust and efficient numerical scheme to enforce crack irreversibility as a constraint we use a primal dual active set strategy which can be identified as a semi smooth newton method the active set iteration is merged with the newton iteration for solving the fully coupled nonlinear partial differential equation discretized using finite elements resulting in a single rapidly converging nonlinear scheme it is well known that phase field models require fine meshes to accurately capture the propagation dynamics of the crack because traditional estimators based on adaptive mesh refinement schemes are not appropriate we develop a predictor corrector scheme for local mesh adaptivity to reduce the computational cost this method is both robust and efficient and allows us to treat temporal and spatial refinements and to study the influence of model regularization parameters finally our proposed approach is substantiated with different numerical tests for crack propagation in elastic media and pressurized fracture propagation in homogeneous and heterogeneous media

stable ranks of split extensions of banach algebras in this paper we estimate the stable ranks of a banach algebra in terms of the stable ranks of its quotient algebra and ideal under the assumption that the quotient map splits as an application several results about the bass and the connected stable ranks of nest algebras are obtained

a phase field multirate scheme with stabilized iterative coupling for pressure driven fracture propagation in porous media abstract phase field methods have the potential to simulate large scale evolution of networks of fractures in porous media without the need to explicitly track interfaces practical field simulations require however that robust and efficient decoupling techniques can be applied for solving these complex systems in this work we focus on the mechanics step that involves the coupling of elasticity and the phase field variable we develop a multirate scheme in which a coarser time grid is employed for the mechanics equation i e the displacements and a finer time grid is taken for the phase field problem the performance of this algorithm is demonstrated for two test cases

the nuclear dimension of graph c algebras abstract consider a graph c algebra c e with a purely infinite ideal i possibly all of c e such that i has only finitely many ideals and c e i is approximately finite dimensional we prove that the nuclear dimension of c e is 1 if i has infinitely many ideals then the nuclear dimension of c e is either 1 or 2

relatively spectral morphisms and applications to k theory abstract spectral morphisms between banach algebras are useful for comparing their k theory and their noncommutative dimensions as expressed by various notions of stable ranks in practice one often encounters situations where the spectral information is only known over a dense subalgebra we investigate such relatively spectral morphisms we prove a relative version of the density theorem regarding isomorphism in k theory we also solve swan s problem for the connected stable rank in fact for an entire hierarchy of higher connected stable ranks that we introduce

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

k homology and k theory for the lamplighter groups of finite groups let f be a finite group we consider the lamplighter group l f z over f we prove that l has a classifying space for proper actions el which is a complex of dimension 2 we use this to give an explicit proof of the baum connes conjecture without coefficients that states that the assembly map il kil el ki c l i 0 1 is an isomorphism actually k0 c l is free abelian of countable rank with an explicit basis consisting of projections in c l while k1 c l is infinite cyclic generated by the unitary of c l implementing the shift finally we show that for f abelian the c algebra c l is completely characterized by f up to isomorphism

banach algebras and rational homotopy theory let a be a unital commutative banach algebra with maximal ideal space max a we determine the rational h type of gln a the group of invertible n n matrices with coefficients in a in terms of the rational cohomology of max a we also address an old problem of j l taylor let lcn a denote the space of last columns of gln a we construct a natural isomorphism max a q 2n 1 s lcn a q for n 1 2 s 1 which shows that the rational cohomology groups of max a are determined by a topological invariant associated to a as part of our analysis we determine the rational h type of certain gauge groups f x g for g a lie group or more generally a rational h space

topological stable rank of nest algebras we establish a general result about extending a right invertible row over a banach algebra to an invertible matrix this is applied to the computation of right topological stable rank of a split exact sequence we also introduce a quantitative measure of stable rank these results are applied to compute the right left topological stable rank for all nest algebras this value is either 2 or infinity and rtsr t n 2 occurs only when n is of ordinal type less than 2 and the dimensions of the atoms grows sufficiently quickly we introduce general results on partial matrix algebras over a banach algebra this is used to obtain an inequality akin to rieffel s formula for matrix algebras over a banach algebra this is used to give further insight into the nest case

an adaptive finite element approximation of a variational model of brittle fracture the energy of the francfort marigo model of brittle fracture can be approximated in the sense of gamma convergence by the ambrosio tortorelli functional in this work we formulate and analyze two adaptive finite element algorithms for the computation of its local minimizers for each algorithm we combine a newton type method with residual driven adaptive mesh refinement we present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the ambrosio tortorelli functional

a higher order phase field model for brittle fracture formulation and analysis within the isogeometric analysis framework phase field models based on the variational formulation for brittle fracture have recently been gaining popularity these models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions in this work we propose a fourth order model for the phase field approximation of the variational formulation for brittle fracture we derive the thermodynamically consistent governing equations for the fourth order phase field model by way of a variational principle based on energy balance assumptions the resulting model leads to higher regularity in the exact phase field solution which can be exploited by the smooth spline function spaces utilized in isogeometric analysis this increased regularity improves the convergence rate of the numerical solution and opens the door to higher order convergence rates for fracture problems we present an analysis of our proposed theory and numerical examples that support this claim we also demonstrate the robustness of the model in capturing complex three dimensional crack behavior

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

iterative coupling of flow geomechanics and adaptive phase field fracture including level set crack width approaches in this work we present numerical studies of fixed stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium specifically fracture propagations are described by employing a phase field approach the extension to fixed stress splitting to propagating phase field fractures and systematic investigation of its properties are important enhancements to existing studies moreover we provide an accurate computation of the fracture opening using level set approaches and a subsequent finite element interpolation of the width the latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time fixed stress formulations for fluid filled phase field fractures accurate fracture width finite element computations using level sets numerical examples in 2d and 3d with spatial and temporal convergence studies

a global local approach for hydraulic phase field fracture in poroelastic media abstract in this work phase field modeling of hydraulic fractures in porous media is extended towards a global local approach therein the failure behavior is solely analyzed in a small local domain in the surrounding medium a simplified and linearized system of equations is solved both domains are coupled with robin type interface conditions the fractures inside the local domain are allowed to propagate and consequently both subdomains change within time here a predictor corrector strategy is adopted in which the local domain is dynamically adjusted to the current fracture pattern the resulting framework is algorithmically described in detail and substantiated with some numerical tests

on the c algebra of matrix finite bounded operators abstract let h be a separable hilbert space with a fixed orthonormal basis let b k h denote the set of operators whose matrices have no more than k non zero entries in each line and in each column the closure of the union over k n of b k h is a c algebra we study some properties of this c algebra we show that this c algebra is not an a w algebra its group of invertibles is contractible and it gives rise to an example of a c algebra with a dense subalgebra and a maximal closed ideal with zero intersection

thin spectra and stable range conditions abstract the topological stable rank of a banach algebra a is the least integer n tsr a such that un a is dense in an where un a a1 an an i 1naai a we give a spectral characterization of the condition tsr a n valid for arbitrary banach algebras for a commutative banach algebra a the condition tsr a n is related to the presence of some kind of discs in the spectrum x a

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

an optimal control problem governed by a regularized phase field fracture propagation model this paper is concerned with an optimal control problem governed by a regularized fracture model using a phase field technique to avoid the nondifferentiability due to the irreversibility constraint on the fracture growth the phase field fracture model is relaxed using a penalization approach the existence of a solution to the penalized fracture model is shown and the existence of at least one solution for the regularized optimal control problem is established moreover the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss qp approximations to the nonlinear optimization problem a numerical example suggests that these can be used to obtain a fast convergent algorithm

quad equations and auto backlund transformations of nls type systems treating an integrable quad equation along with its two generalized symmetries as a compatible system allows one to construct an auto backlund transformation for solutions of the related nls type system a fixed periodic reduction of the quad equation yields a quasi periodic reduction of its generalized symmetries that turns them into differential constraints compatible with the nls type system

on penalization in variational phase field models of brittle fracture abstract irreversible evolution is one of the central concepts as well as implementation challenges of both the variational approach to fracture by francfort and marigo 1998 and its regularized counterpart by bourdin francfort and marigo 2000 2007 and 2008 which is commonly referred to as a phase field model of brittle fracture irreversibility of the crack phase field imposed to prevent fracture healing leads to a constrained minimization problem whose optimality condition is given by a variational inequality in our study the irreversibility is handled via penalization provided the penalty constant is well tuned the penalized formulation is a good approximation to the original one with the advantage that the induced equality based weak problem enables a much simpler algorithmic treatment we propose an analytical procedure for deriving the optimal penalty constant more precisely its lower bound which guarantees a sufficiently accurate enforcement of the crack phase field irreversibility our main tool is the notion of the optimal phase field profile as well as the convergence result it is shown that the explicit lower bound is a function of two formulation parameters the fracture toughness and the regularization length scale but is independent of the problem setup geometry boundary conditions etc the formulation ingredients degradation function tension compression split etc as well as the discretization mesh size the optimally penalized formulation is tested for three benchmark problems including one with available analytical solution we also compare our results with those obtained by using the alternative irreversibility technique based on the notion of the history field by miehe et al 2010

covering dimension for nuclear c algebras ii the completely positive rank is an analogue of the topological covering dimension defined for nuclear c algebras via completely positive approximations these may be thought of as simplicial approximations of the algebra which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes we introduce a technical variation of completely positive rank and show that the two theories coincide in many important cases furthermore we analyze some of their properties in particular we show that both theories behave nicely with respect to ideals and that they coincide with the covering dimension of the spectrum for certain continuous trace c algebras

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

an augmented lagrangian method for the phase field approach for pressurized fractures abstract in the modeling of pressurized fractures using phase field approaches the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase field function in comparison to the classical case in elasticity the presence of the pressure requires the additional constraint and makes the problem much harder to analyze after temporal discretization this induces a minimization problem in each time step over a solution dependent admissible set to avoid solving the resulting variational inequality corresponding to the first order necessary conditions a penalization approach is used commonly to remove the inequality constraint it is well known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process consequently to avoid such a drawback we propose an augmented lagrangian algorithm for the discrete in time and continuous in space phase field problems the final set of equations is solved in a decoupled fashion the proposed method is substantiated with several benchmark and prototype tests in two and three dimensions

an infinite analogue of rings with stable rank one abstract replacing invertibility with quasi invertibility in bass first stable range condition we discover a new class of rings the qb rings these constitute a considerable enlargement of the class of rings with stable rank one b rings and include examples like end f v the ring of endomorphisms of a vector space v over some field f and b f the ring of all row and column finite matrices over f we show that the category of qb rings is stable under the formation of corners ideals and quotients as well as matrices and direct limits we also give necessary and sufficient conditions for an extension of qb rings to be a qb ring and show that extensions of b rings often lead to qb rings specializing to the category of exchange rings we characterize the subset of exchange qb rings as those in which every von neumann regular element extends to a maximal regular element i e a quasi invertible element finally we show that the c algebras that are qb rings are exactly the extremally rich c algebras studied by l g brown and the second author

stable rank and approximation theorems in it is conjectured that for h the bass stable rank bsr is 1 and the topological stable rank tsr is 2 bsr h 1 if and only if for every f f2 e hx x h00 which is a corona pair i e there exist g g2 e h00 such that f gl f2g2 1 there exists a g e hco such that f f2g e h 1 the invertibles in h however it suffices to consider corona pairs f 1 f2 where f is a blaschke product it is also shown that there exists a g e h00 such that f f2g e exp h00 if and only if logf1 can be boundedly analytically defined on z e d if2 z i 0 tsr h 2 if and only if the corona pairs are uniformly dense in h00 x h however it suffices to show that the corona pairs are uniformly dense in pairs of blaschke products this condition would be satisfied if the interpolating blaschke products were uniformly dense in the blaschke products for b an inner function k h2 e bh2 is an h module via the compressed toeplitz operators cf pktf ik for f e h00 where tf is the 2 toeplitz operator tfg fg for g e h some stable rank questions can be recast as lifting questions for fi n c h 0 there exist gi hi n c h such that en1 f bgi hi 1 if and only if the compressed toeplitz operators cs 1 may be lifted to toeplitz operators tf 1 which generate b h2 as an ideal

coupling fluid structure interaction with phase field fracture abstract in this work a concept for coupling fluid structure interaction with brittle fracture in elasticity is proposed the fluid structure interaction problem is modeled in terms of the arbitrary lagrangian eulerian technique and couples the isothermal incompressible navier stokes equations with nonlinear elastodynamics using the saint venant kirchhoff solid model the brittle fracture model is based on a phase field approach for cracks in elasticity and pressurized elastic solids in order to derive a common framework the phase field approach is re formulated in lagrangian coordinates to combine it with fluid structure interaction a crack irreversibility condition that is mathematically characterized as an inequality constraint in time is enforced with the help of an augmented lagrangian iteration the resulting problem is highly nonlinear and solved with a modified newton method e g error oriented that specifically allows for a temporary increase of the residuals the proposed framework is substantiated with several numerical tests in these examples computational stability in space and time is shown for several goal functionals which demonstrates reliability of numerical modeling and algorithmic techniques but also current limitations such as the necessity of using solid damping are addressed

tensor products of c algebras with the ideal property abstract combining a construction of dadarlat of a unital simple non exact c algebra c of real rank zero and stable rank one which is shape equivalent to a uhf algebra with results of kirchberg and a result obtained by dadarlat and the first named author we show that b h c contains an ideal that is not generated by its projections we also find a unital separable sub c algebra a of b h such that a is of real rank zero and a c has an ideal that is not generated by its projections

modified newton methods for solving fully monolithic phase field quasi static brittle fracture propagation abstract our goal in this work is to develop and compare modified newton methods for fully monolithic quasi static brittle phase field fracture propagation in variational phase field fracture a smoothed phase field indicator variable denoting the crack path is coupled to elasticity moreover a crack irreversibility condition is incorporated to develop a fully monolithic scheme is an extremely challenging task since the underlying problem is non convex and the jacobian of newton s method is indefinite to split the problem using alternating minimization thus a partitioned approach is a possible resort however there are good reasons to consider monolithic approaches such as for example robustness and efficiency although an error oriented newton method can cope with a larger variety of configurations it appears that this method is not always robust and also not always efficient inspired by nonlinear flow problems as alternative we develop a modified newton scheme in which globalization is based on a dynamic modification of the jacobian matrix rather than utilizing line search or trust region strategies this variation switches smoothly between full newton and newton like steps in several 2d and 3d numerical examples all of them with different characteristic features our modified newton solver is compared to a backtracking line search newton method another line search method monitoring the global energy and allowing for negative curvatures and to already published results of an error oriented version these computations also include further modifications of newton s method and detailed discussions why certain schemes either work or fail revisiting all findings the main outcome of this paper is that the modified newton scheme with jacobian modification is currently the only method that works in a robust and efficient way for all provided examples whereas line search schemes or the error oriented scheme show deficiencies for certain configurations

trivial gleason parts and the topological stable rank of h we show that the subset of the maximal ideal space of h formed by the trivial gleason parts is totally disconnected a pair g h h is called a corona pair if inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i 0 where inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i is the open unit disc we prove that the set u 2 h of corona pairs is dense in h h

the automorphism group of a simple tracially ai algebra the structure of the automorphism group of a simple tai algebra is studied in particular we show that frac overline rm inn a overline rm inn 0 a is isomorphic as a topological group to an inverse limit of discrete abelian groups for a unital simple ah algebra with bounded dimension growth consequently frac overline rm inn a overline rm inn 0 a is totally disconnected

a phase field model for cohesive fracture in this paper a phase field model for cohesive fracture is developed after casting the cohesive zone approach in an energetic framework which is suitable for incorporation in phase field approaches the phase field approach to brittle fracture is recapitulated the approximation to the dirac function is discussed with particular emphasis on the dirichlet boundary conditions that arise in the phase field approximation the accuracy of the discretisation of the phase field including the sensitivity to the parameter that balances the field and the boundary contributions is assessed at the hand of a simple example the relation to gradient enhanced damage models is highlighted and some comments on the similarities and the differences between phase field approaches to fracture and gradient damage models are made a phase field representation for cohesive fracture is elaborated starting from the aforementioned energetic framework the strong as well as the weak formats are presented the latter being the starting point for the ensuing finite element discretisation which involves three fields the displacement field an auxiliary field that represents the jump in the displacement across the crack and the phase field compared to phase field approaches for brittle fracture the modelling of the jump of the displacement across the crack is a complication and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack the sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated as well as how the results depend on the orders of the discretisation of the three fields finally examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure the latter example being somewhat limited because only straight crack propagation is considered

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

crack initiation in brittle materials in this paper we study the crack initiation in a hyper elastic body governed by a griffith type energy we prove that during a load process through a time dependent boundary datum of the type t tg x and in the absence of strong singularities e g this is the case of homogeneous isotropic materials the crack initiation is brutal that is a big crack appears after a positive time ti 0 conversely in the presence of a point x of strong singularity a crack will depart from x at the initial time of loading and with zero velocity we prove these facts for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components the main tool we employ to address the problem is a local minimality result for the functional epsilon nu gamma int omega f x nabla v rm d x k mathcal h 1 gamma where omega subseteq mathbb r 2 k 0 and f is a suitable caratheodory function we prove that if the uncracked configuration u of relative to a boundary displacement has at most uniformly weak singularities then configurations u with mathcal h 1 gamma small enough are such that epsilon u emptyset epsilon u gamma gamma

metric completions of ordered groups and k0 of exchange rings we give a description of the closure of the natural affine continuous function representation of k0 r for any exchange ring r this goal is achieved by extending the results of goodearl and handelman about metric completions of dimension groups to a more general class of pre ordered groups which includes k0 of exchange rings as a consequence the results about k 0 of regular rings which the author gave in an earlier paper can be extended to a wider class of rings which includes c algebras of real rank zero among others also the framework of pre ordered groups developed here allows other potential applications

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

an iterative staggered scheme for phase field brittle fracture propagation with stabilizing parameters abstract this paper concerns the analysis and implementation of a novel iterative staggered scheme for quasi static brittle fracture propagation models where the fracture evolution is tracked by a phase field variable the model we consider is a two field variational inequality system with the phase field function and the elastic displacements of the solid material as independent variables using a penalization strategy this variational inequality system is transformed into a variational equality system which is the formulation we take as the starting point for our algorithmic developments the proposed scheme involves a partitioning of this model into two subproblems phase field and mechanics with added stabilization terms to both subproblems for improved efficiency and robustness we analyze the convergence of the proposed scheme using a fixed point argument and find that under a natural condition the elastic mechanical energy remains bounded and if the diffusive zone around crack surfaces is sufficiently thick monotonic convergence is achieved finally the proposed scheme is validated numerically with several bench mark problems

the cancellation theorem for projective modules over irrational rotation c algebras on montre que deux modules projectifs sur a qui sont isomorphes de facon stable sont en fait isomorphes

when is multiplication in a banach algebra open abstract we develop the theory of banach algebras whose multiplication regarded as a bilinear map is open we demonstrate that such algebras must have topological stable rank 1 however the latter condition is strictly weaker and implies only that products of non empty open sets have non empty interior we then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in l 1 n 0 is open by appealing to ultraproduct techniques we demonstrate that neither in l 1 z nor in l 1 q convolution is uniformly open the problem of openness of multiplication in banach algebras of bounded operators on banach spaces and their calkin algebras is also discussed

when almost multiplicative morphisms are close to homomorphisms absrract it is shown that approximately multiplicative contractive positive morphisms from c x with dim x 2 into a simple c algebra a of real rank zero and of stable rank one are close to homomorphisms provided that certain k theoretical obstacles vanish as a corollary we show that a homomorphism h c x a is approximated by homomorphisms with finite dimensional range if h gives no k theoretical obstacle

an infinite family of non isomorphic c algebras with identical k theory we exhibit a countably infinite family of simple separable nuclear and mutually non isomorphic c algebras which agree on k theory and traces the algebras do not absorb the jiang su algebra z tensorially answering a question of n c phillips they are also pairwise shape and morita equivalent confirming a conjecture from our earlier work the distinguishing invariant is the radius of comparison a non stable invariant of the cuntz semigroup

on the stable rank of algebras of operator fields over metric spaces let be a finitely generated torsion free two step nilpotent group let c denote the universal c algebra of we show that sr c sr c 1 where for a unital c algebra a sr a is the stable rank of a and where 1 is the space of one dimensional representations of in process we give a stable rank estimate for maximal full algebras of operator fields over metric spaces