on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

sequence dependent scheduling with order deliveries this paper studies a sequence dependent scheduling problem incorporating order delivery motivated by satellite imaging scheduling a set of jobs is to be processed on a single machine and each job belongs to a specific group the completion time of a group is the moment when all jobs belonging to this group are completed the problem is to determine a processing sequence of the jobs such that the sum of weighted completion times over all groups is minimized we present a binary integer program to formulate the studied problem and then develop an o n 22 n dynamic programming algorithm for determining optimal solutions to produce approximate solutions within an acceptable time we design a tabu search algorithm an iterated local search algorithm and a genetic algorithm computational experiments are conducted to study the performance of the integer program and the solution algorithms numerical statistics suggest that the binary integer program can reach optimal solutions faster than the integer program existing in the literature and the iterated local search algorithm outperforms other approaches when the number of jobs increases

the delivery man problem and cumulative matroids given a complete directed graph g v a the delivery man problem dmp consists of determining a hamiltonian circuit minimizing the sum of distances along the circuit from a given vertex v1 to every vertex of v including v1 itself there exists a number of applications of the dmp in the fields of distribution and machine scheduling the dmp is np hard the objective of this paper is to develop new theoretical results and an exact algorithm for the problem a new integer linear programming formulation is provided and results on the matroidal structure of a class of combinatorial problems are developed these are used to derive lower bounds for the dmp these bounds are embedded into an enumerative algorithm the largest problems solved to optimality with the proposed algorithm involve 60 vertices this compares favorably with previously published methods

dynamic crack propagation in a 2d elastic body the out of plane case abstract already in 1920 griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials nowadays a generalized energy balance law is used in mechanics f erdogan crack propagation theories in h liebowitz ed fracture vol 2 academic press new york 1968 pp 498 586 l b freund dynamic fracture mechanics cambridge univ press cambridge 1990 d gross bruchmechanik springer verlag berlin 1996 in order to predict how a running crack will grow we discuss this situation in a rigorous mathematical way for the out of plane state this model is described by two coupled equations in the reference configuration a two dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law we handle both equations separately assuming at first that the crack position is known then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip

thermodynamics and analysis of rate independent adhesive contact at small strains we address a model for adhesive unilateral frictionless signorini type contact between bodies of heatconductive viscoelastic material in the linear kelvin voigt rheology undergoing thermal expansion the flow rule for debonding the adhesion is considered rate independent and unidirectional and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

low dimensional description of the dynamics in separated flow past thick airfoils results are presented for the numerical simulation of unsteady viscous incompressible flow past thick airfoils specifically flow past a naca 4424 at an angle of attack of 2 5 deg and reynolds numbers in the range of 1700 4000 has been simulated using the spectral element method at these conditions the flow is separated and an unsteady wake is formed application of the method of empirical eigenfunctions reveals the structure of the most energetic components of the flow these are found to occur in pairs that through phase exchange are responsible for the vortex shedding a set of ordinary differential equations is obtained for the amplitudes of these eigenfunc tions by a galerkin projection of the navier stokes equations the solutions of the model system are compared with the full simulation the work is of relevance to the transition process and observed routes to chaos in airfoil wakes he unsteady flow past an airfoil is an example of a complex flow of great engineering relevance that involves many spa tial and temporal scales at reynolds numbers of practical signifi cance the flow is so complicated that numerical simulations of the flow represent major research efforts apart from coupling invis cid viscous interactions through the boundary layer turbulence modeling is employed to account for the small scale behavior direct numerical simulations on the other hand are limited to mod est reynolds numbers which are usually too low for engineering purposes nevertheless these simulations are useful in investigat ing the nonlinear interaction in transitional and weakly turbulent flows since other than the usual numerical approximations no additional assumptions are necessary the spectral element method 1 which combines the advantages of finite element meth ods in allowing complex geometries and the fast convergence properties of spectral methods has become a practical tool for this investigation 2 recently the method of empirical eigenfunctions3 4 has been utilized as a method of analyzing complicated turbulent flow data when applied using an ensemble of flow realizations the method yields a set of orthogonal functions that are best able to account for the energy of the system as represented by the ensem ble much can be learned about the flow by consideration of the structure and interaction of these functions or modes in this paper and in other work 5 we combine the methodologies of direct spec tral element simulations and empirical eigenfunctions in applying the method of empirical eigenfunctions we wish to understand the wake system of flow past an airfoil both in terms of the nature of the large scale structures and that of the disturbances that induce transition as a start we restrict our attention to a rela tively thick airfoil at modest reynolds numbers with the chosen geometry and parameter values the flow is separated over the rear of the airfoil and appears aperiodic we apply the method to data obtained from numerical simulations the derived eigenfunctions reveal the nature of the large scale structures observable in the flow by using the set of the derived orthogonal functions as a

anisotropic mesh adaptation for the generalized ambrosio tortorelli functional with application to brittle fracture abstract quasi static crack propagation in brittle materials is modeled via the ambrosio tortorelli approximation the crack is modeled by a smooth phase field defined on the whole computational domain since the crack is confined to a thin layer the employment of anisotropic adapted grids is shown to be a really effective tool in containing computational costs we extend the anisotropic error analysis applied to the classical ambrosio tortorelli approximation by artina et al to the generalized ambrosio tortorelli functional where a unified framework for several elasticity laws is dealt with as well as a non convex fracture energy can be accommodated after deriving an anisotropic a posteriori error estimator we devise an algorithm which alternates optimization and mesh adaptation both anti plane and plane strain configurations are numerically checked

for the parabolic lame system on polygonal domains the transport equation abstract we study the parabolic lame system with initial and boundary conditions on non convex plane polygonal domains we express the solution by the inverse of the sum of two operators taken from g da prato p grisvard sommes d operateurs lineaires et equations differentielles operationnelles j math pures appl 54 1975 305 387 and split the solution into a singular part and a regular part by applying to the inverse the corner singularity result of the lame system with parameter we show that the remainder has the h 2 q regularity and that the coefficients of the corner singularities so called the stress intensity factors have the fractional order regularities on the time interval also we investigate the transport equation directed by the vector field having the corner singularity decomposition

on the cauchy problem for the wave equation on time dependent domains abstract we introduce a notion of solution to the wave equation on a suitable class of time dependent domains and compare it with a previous definition we prove an existence result for the solution of the cauchy problem and present some additional conditions which imply uniqueness

a variational formulation of rate independent phase transformations using an extremum principle we propose a rate independent mesoscopic model for the hysteretic evolution of phase transformations in shape memory alloys the model uses the deformation and phase indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws using the associated functionals admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality

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

from discrete visco elasticity to continuum rate independent plasticity rigorous results we show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco elastic crystal lattice with quenched disorder the constitutive structure changes as a result of two concurrent limiting procedures the vanishing viscosity limit and the discrete to continuum limit in the course of these limits a non convex elastic problem transforms into a convex elastic problem while the quadratic rate dependent dissipation of visco elastic lattice transforms into a singular rate independent dissipation of an ideally plastic solid in order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape memory alloys the approach however is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

stochastic homogenisation of free discontinuity problems in this paper we study the stochastic homogenisation of free discontinuity functionals assuming stationarity for the random volume and surface integrands we prove the existence of a homogenised random free discontinuity functional which is deterministic in the ergodic case moreover by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise subadditive ergodic theorem by akcoglou and krengel we characterise the limit volume and surface integrands in terms of asymptotic cell formulas

a quasistatic evolution model for the interaction between fracture and damage this paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture in both cases the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process the minimality at each time and the energy balance from a mechanical point of view it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium the brutal character of the damage process is then replaced by a progressive one where the original damage internal variable that is the characteristic function of the damaged part is replaced by the local volume fraction the analysis rests on a locality property for mixtures which enables us to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the g closure set

existence and convergence for quasi static evolution in brittle fracture this paper investigates the mathematical well posedness of the variational model of quasi static growth for a brittle crack proposed by francfort and marigo in 15 the starting point is a time discretized version of that evolution which results in a sequence of minimization problems of mumford and shah type functionals the natural weak setting is that of special functions of bounded variation and the main difficulty in showing existence of the time continuous quasi static growth is to pass to the limit as the time discretization step tends to 0 this is performed with the help of a jump transfer theorem which permits under weak convergence assumptions for a sequence un of sbv functions to its bv limit u to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence un in particular it is shown that the notion of minimizer of a mumford and shah type functional for its own jump set is stable under weak convergence assumptions furthermore our analysis justifies numerical methods used for computing the time continuous quasi static evolution 2003 wiley periodicals inc

existence and uniqueness of dynamic evolutions for a peeling test in dimension one abstract in this paper we present a one dimensional model of a dynamic peeling test for a thin film where the wave equation is coupled with a griffith criterion for the propagation of the debonding front our main results provide existence and uniqueness for the solution to this coupled problem under different assumptions on the data

a weak legendre collocation spectral method for the solution of the incompressible navier stokes equations in unstructured quadrilateral subdomains a weak legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible navier stokes equations in general two dimensional and axisymmetric geometries a semi implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field a series of elliptic boundary value problems arises from the above procedure each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains in particular a modified variational formulation of the partial differential equations is presented which leads after discretization to a weak multidomain approximation of the corresponding problems a weak formalism for the influence matrix technique is also developed which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns a method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization exponential convergence is demonstrated for a plethora of stokes and navier stokes simulations

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

ground states of simple bodies that may undergo brittle fractures abstract equilibrium states of elastic brittle solids that may suffer nucleation of cracks in finite deformation setting are analyzed crack patterns are described in terms of appropriate radon measures namely curvature varifolds with boundary a new form of the energy is presented it includes terms associated with the curvature of margins and tips of possible cracks existence of minima of the energy is established in classes of pairs of deformation and families of varifolds configurational balances in weak form are determined with reference to generic curvature varifolds with boundary they include non standard terms associated with the curvatures involved in the energy pointwise balances of configurational actions are also evaluated in a special case new pointwise balances at the tips and along the margins of the crack pattern emerge

fracture surfaces and the regularity of inverses for bv deformations motivated by nonlinear elasticity theory we study deformations that are approximately differentiable orientation preserving and one to one almost everywhere and in addition have finite surface energy this surface energy mathcal e was used by the authors in a previous paper and has connections with the theory of currents in the present paper we prove that mathcal e measures exactly the area of the surface created by the deformation this is done through a proper definition of created surface which is related to the set of discontinuity points of the inverse of the deformation in doing so we also obtain an sbv regularity result for the inverse

a density result in two dimensional linearized elasticity and applications we show that in a two dimensional bounded open set whose complement has a finite number of connected components the vector fields u h 1 2 are dense in the space of fields whose symmetrized gradient e u is in l 2 4 this allows us to show the continuity of some linearized elasticity problems with respect to variations of the set with applications to shape optimization or the study of crack evolution

the brezis ekeland principle for doubly nonlinear equations the celebrated brezis ekeland principle c r acad sci paris ser a b 282 1976 pp ai a1197 a1198 aii and a971 a974 characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems this note extends this characterization to doubly nonlinear evolution equations driven by convex potentials the characterization is exploited in order to establish approximation results for gradient flows doubly nonlinear equations and rate independent evolutions

maximally dissipative local solutions to rate independent systems and application to damage and delamination problems abstract the system of two inclusions u e t u t z t 0 and r z z e t u t z t 0 with the dissipation potential r degree 1 homogeneous and with the stored energy e t separately convex is considered the relation between conventional weak solutions and local solutions is shown and a suitably integrated maximal dissipation principle is devised to select force driven local solutions and eliminate solutions with too early jumps as it may occur in energy driven ones this is illustrated on scalar examples an approximation by a simple and efficient semi implicit time discretization of the fractional step type is shown to converge to local solutions on the scalar examples the approximate solutions are shown to satisfy the integrated maximal dissipation principle asymptotically while in general it is devised only to serve as an a posteriori tool to justify or possibly adaptively adjust thus obtained approximate solutions as in fact force driven applications of such solutions are illustrated on specific examples from continuum mechanics at small strains involving inelastic processes in a bulk or on a surface namely damage and delamination

epsilon stable quasi static brittle fracture evolution we introduce a new definition of stability e stability that implies local minimality and is robust enough for passing from discrete time to continuous time quasi static evolutions even with very irregular energies we use this to give the first existence result for quasi static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time but not necessarily global minimizers the key ingredient in our model is the physically reasonable property absent in global minimization models that whenever there is a jump in time from one state to another there must be a continuous path from the earlier state to the later along which the energy is almost decreasing it follows that these evolutions are much closer to satisfying griffith s criterion for crack growth than are solutions based on global minimization and initiation is more physical than in global minimization models 2009 wiley periodicals inc

two level discretizations of nonlinear closure models for proper orthogonal decomposition proper orthogonal decomposition has been successfully used in the reduced order modeling of complex systems its original promise of computationally efficient yet accurate approximation of coherent structures in high reynolds number turbulent flows however still remains to be fulfilled to balance the low computational cost required by reduced order modeling and the complexity of the targeted flows appropriate closure modeling strategies need to be employed since modern closure models for turbulent flows are generally nonlinear their efficient numerical discretization within a proper orthogonal decomposition framework is challenging this paper proposes a two level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced order models the two level method computes the nonlinear terms of the reduced order model on a coarse mesh compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step the two level method attains the same level of accuracy while dramatically reducing the computational cost we numerically illustrate these improvements in the two level method by using it in three settings the one dimensional burgers equation with a small diffusion parameter n 10 3 the two dimensional flow past a cylinder at reynolds number re 200 and the three dimensional flow past a cylinder at reynolds number re 1000

crack growth in polyconvex materials we discuss a model for crack propagation in an elastic body where the crack path is described a priori in particular we develop in the framework of finite strain elasticity a rate independent model for crack evolution which is based on the griffith fracture criterion due to the nonuniqueness of minimizing deformations the energy release rate is no longer continuous with respect to time and the position of the crack tip thus the model is formulated in terms of the clarke differential of the energy generalizing the classical crack evolution models for elasticity with strictly convex energies we prove the existence of solutions for our model and also the existence of special solutions where only certain extremal points of the clarke differential are allowed

an incremental minimization principle suitable for the analysis of low cycle fatigue in metals a coupled ductile brittle damage model abstract the present paper is concerned with a novel variational constitutive update suitable for the analysis of low cycle fatigue in metals the underlying constitutive model originally advocated in 1 accounts for plastic deformation as well as for damage accumulation the latter is captured by a combination of two constitutive models while the first of those is associated with ductile damage the second material law is related to a quasi brittle response the complex overall model falls into the range of so called generalized standard materials and thus it is thermodynamically consistent however since the evolution equations are non associative it does not show an obvious variational structure by enforcing the flow rule as well as the evolution equations through a suitable parameterization a minimization principle can be derived nevertheless discretized in time this principle is employed for developing an effective numerical implementation since the mechanical subproblems corresponding to ductile damage and that of quasi brittle damage are uncoupled an efficient staggered scheme can be elaborated within both steps newton s method is applied while the evolution of the quasi brittle damage requires only the computation of a one dimensional optimization problem the ductile damage model is defined by a numerically more expensive tensor valued variable for further increasing the numerical performance of the respective minimization principle a closed form solution for the inverse of the hessian matrix is derived by numerically analyzing the prediction of mesocrack initiation in low cycle fatigue simulations the performance of the resulting algorithm is demonstrated

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

a variational view of partial brittle damage evolution under time dependent loading an elastic material undergoes the simplest form of damage in which it passes from its original state to a weaker elastic state elaborating on prior work 14 we establish the existence of a relaxed variational evolution where at each time the two states of the material combine to form a fine mixture optimal from the standpoint of the applied load at that time yet preserving the irreversibility of the damaging process

minimization of the k th eigenvalue of the robin laplacian abstract the paper is concerned with the minimization of the k th eigenvalue of the laplace operator with robin boundary conditions among all open sets of r n satisfying a volume constraint we prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets the main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation one of the key difficulties for k 3 comes from the fact that the eigenvalues are critical points

optimal regularity and structure of the free boundary for minimizers in cohesive zone models we study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics under smoothness assumptions on the boundary conditions and on the fracture energy density we show that minimizers are c 1 1 2 and that near non degenerate points the fracture set is c 1 alpha for some alpha in 0 1

threshold based quasi static brittle damage evolution we introduce models for static and quasi static damage in elastic materials based on a strain threshold and then investigate the relationship between these threshold models and the energy based models introduced in francfort and marigo eur j mech a solids 12 149 189 1993 and francfort and garroni ration mech anal 182 1 125 152 2006 a somewhat surprising result is that while classical solutions for the energy models are also threshold solutions this is shown not to be the case for nonclassical solutions that is solutions with microstructure a new and arguably more physical definition of solutions with microstructure for the energy based model is then given in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors rather than by the tensors themselves we prove existence for this energy based problem and show that these solutions are also threshold solutions a by product of this analysis is that all local minimizers in both the classical setting and for the new microstructure definition are also global minimizers

stability results for doubly nonlinear differential inclusions by variational convergence we present a stability result for a wide class of doubly nonlinear equations featuring general maximal monotone operators and possibly nonconvex and nonsmooth energy functionals the limit analysis consists in the reformulation of the differential evolution as a scalar energy conservation equation with the aid of the so called fitzpatrick theory for the representation of monotone operators in particular our result applies to the vanishing viscosity approximation of rate independent systems

quasistatic evolution problems for linearly elastic perfectly plastic materials the problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate independent processes existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation this approach provides a new approximation result for the solutions of the quasistatic evolution problem which are shown to be absolutely continuous in time four equivalent formulations of the problem in rate form are derived a strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain

approximation of rate independent optimal control problems this work introduces an abstract approximation scheme for optimal control problems in the energetic theory of rate independent systems this scheme builds upon the usual temporal semidiscretization and a gamma convergence approximation of the functionals in a full discretization this approximation consists of a sequence of discrete functionals defined on finer and finer discrete spaces the main new result of this work is a rigorous convergence analysis for this scheme interestingly it turns out that the usual time discretization is incomplete as a basis for a fully discrete approximation of the optimal control problem and a new nonlocal condition has to be introduced since this new condition is necessary and sufficient for a one to one correspondence of discrete time and continuous time solutions to rate independent systems we also gain a new theoretical insight which might be of independent interest at the heart of the derivation are so called reverse approximation results for which we provide extensions and refinements to give a concrete example and to demonstrate how the abstract theory can be applied in a concrete problem we present a full discretization with finite elements of the optimal control problem for a rate independent partial differential inclusion

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

a γ convergence approach to stability of unilateral minimality properties in fracture mechanics and applications we prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by francfort marigo in 14 then we give an application to the quasistatic evolution of cracks in composite materials the main tool in the analysis is a convergence result for energies of the form

an eigenerosion approach to brittle fracture the present work is concerned with the verification and validation of a variant of the eigenfracture scheme of schmidt et al 2009 based on element erosion which we refer to as eigenerosion eigenerosion is derived from the general eigenfracture scheme by restricting the eigendeformations in a binary sense they can be either zero in which case the local behavior is elastic or they can be equal to the local displacement gradient in which case the corresponding material neighborhood is failed or eroded when combined with a finite element approximation this scheme gives rise to element erosion i e the elements can be either intact in which case their behavior is elastic or be completly failed or eroded and have no load bearing capacity we verify the eigenerosion scheme through comparisons with analytical solutions and through convergence studies for mode i fracture propagation both in two and three dimensions and for structured and random meshes finally by way of validation we apply the eigenerosion scheme to the simulation of mixed modes i iii experiments in poly methyl methacrylate plates

non interpenetration of matter for sbv deformations of hyperelastic brittle materials we prove that the ciarlet necas non interpenetration of matter condition 9 can be extended to the case of sbv deformations of hyperelastic brittle materials and can be taken into account for some variational models in fracture mechanics in order to formulate such a condition we define the deformed configuration under an sbv map by means of the approximately differentiable representative and we prove some connected stability results under weak convergence we provide an application to the case of brittle ogden s materials

a model for the quasistatic growth of cracks with fractional dimension we study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials we give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution both the antiplane and the planar cases are treated

young measure quasi static damage evolution an existence result for the quasi static evolution of incomplete damage in elastic materials is presented the absence of gradient terms in the damage variable causes a critical lack of compactness therefore the analysis is developed in the framework of young measures where a notion of solution is defined presenting some improvements with respect to previous contributions the main new feature in the proof of the existence result regards a delicate construction of the joint recovery sequence

global existence results for viscoplasticity at finite strain we study a model for rate dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor and investigate the existence of global in time solutions to the related pde system we reveal its underlying structure as a generalized gradient system where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite strain elasticity as well as the multiplicative decomposition of finite strain plasticity moreover the dissipation potential depends on the left invariant plastic rate and thus depends on the plastic state variable the existence theory is developed for a class of abstract nonsmooth and nonconvex gradient systems for which we introduce suitable notions of solutions namely energy dissipation balance and energy dissipation inequality solutions hence we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory

a model for the quasi static growth of brittle fractures existence and approximation results we give a precise mathematical formulation of a variational model for the irreversible quasi static evolution of brittle fractures proposed by g a francfort and j j marigo and based on griffith s theory of crack growth in the two dimensional case we prove an existence result for the quasi static evolution and show that the total energy is an absolutely continuous function of time although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities this existence result is proved by a time discretization process where at each step a global energy minimization is performed with the constraint that the new crack contains all cracks formed at the previous time steps this procedure provides an effective way to approximate the continuous time evolution

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

fracture paths from front kinetics relaxation and rate independence crack fronts play a fundamental role in engineering models for fracture they are the location of both crack growth and the energy dissipation due to growth however there has not been a rigorous mathematical definition of crack front nor rigorous mathematical analysis predicting fracture paths using these fronts as the location of growth and dissipation here we give a natural weak definition of crack front and front speed and consider models of crack growth in which the energy dissipation is a function of the front speed that is the dissipation rate at time t is of the form int f t psi v x t rm d mathcal h n 2 x where f t is the front at time t and v is the front speed we show how this dissipation can be used within existing models of quasi static fracture as well as in the new dissipation functionals of mielke ortiz an example of a constrained problem for which there is existence is shown but in general if there are no constraints or other energy penalties this dissipation must be relaxed we prove a general relaxation formula that gives the surprising result that the effective dissipation is always rate independent

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

an improved approximation ratio for the minimum latency problem networks and matroids given a tour visiting n points in a metric space the latency of one of these points p is the distance traveled in the tour before reaching p the minimum latency problem mlp asks for a tour passing through n given points for which the total latency of the n points is minimum in effect we are seeking the tour with minimum average arrival time this problem has been studied in the operations research literature where it has also been termed the delivery man problem and the traveling repairman problem the approximability of the mlp was first considered by sahni and gonzalez in 1976 however unlike the classical traveling salesman problem tsp it is not easy to give any constant factor approximation algorithm for the mlp recently blum et al a blum p chalasani d coppersimith w pulleyblank p raghavan m sudan proceedings of the 26th acm symposium on the theory of computing 1994 pp 163 171 gave the first such algorithm obtaining an approximation ratio of 144 in this work we develop an algorithm which improves this ratio to 21 55 moreover combining our algorithm with a recent result of garg n garg proceedings of the 37th ieee symposium on foundations of computer science 1996 pp 302 309 provides an approximation ratio of 10 78 the development of our algorithm involves a number of techniques that seem to be of interest from the perspective of the tsp and its variants more generally