search games with multiple hidden objects we consider a class of zero sum search games in which a searcher seeks to minimize the expected time to find several objects hidden by a hider we begin by analyzing a game in which the searcher wishes to find k balls hidden among n k boxes there is a known cost of searching each box and the searcher seeks to minimize the total expected cost of finding all the objects in the worst case we show that it is optimal for the searcher to begin by searching a k subset h of boxes with probability nu h which is proportional to the product of the search costs of the boxes in h the searcher should then search the n k remaining boxes in a random order a worst case hider distribution is the distribution nu we distinguish between the case of a smart searcher who can change his search plan as he goes along and a normal searcher who has to set out his plan from the beginning we show that a smart searcher has no advantage we then show how the game can be formulated in terms of a more general ne

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

stable gfem sgfem improved conditioning and accuracy of gfem xfem for three dimensional fracture mechanics abstract in this paper we present an extension of the stable generalized fem sgfem for 3 d fracture mechanics problems numerical experiments show that the use of available singular enrichment bases derived from the 2 d asymptotic crack tip solution leads to a severe loss of accuracy in the sgfem an enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the sgfem the proposed sgfem for 3 d fractures delivers the same rate of growth in condition number as the standard fem for proper choice of singular enrichment functions the accuracy and conditioning obtained with the sgfem is compared with the generalized fem gfem for different types of singular enrichment bases a fully 3 d fracture mechanics problem is considered to highlight issues that arise only in 3 d problems the convergence of stress intensity factors sifs extracted from gfem and sgfem solutions and the effect of the size of enrichment sub domains on the accuracy of extracted sifs are also studied it is shown that the accuracy of sifs may deteriorate with mesh refinement when the topological enrichment strategy is adopted the proposed sgfem delivers much better accuracy than the gfem for both geometrical and topological enrichment strategies

searching a variable speed network a point lies on a network according to some unknown probability distribution starting at a specified root of the network a searcher moves to find this point at speeds that depend on his location and direction he seeks the randomized search algorithm that minimizes the expected search time this is equivalent to modeling the problem as a zero sum hide and seek game whose value is called the search value of the network we make a new and direct derivation of an explicit formula for the search value of a tree proving that it is equal to half the sum of the minimum tour time of the tree and a quantity called its incline the incline of a tree is an average over the leaf nodes of the difference between the time taken to travel from the root to a leaf node and the time taken to travel from a leaf node to the root this difference can be interpreted as height of a leaf node assuming uphill is slower than downhill we then apply this formula to obtain numerous results for general networks we also introduce a new general method of comparing the search value of networks that differ in a single arc some simple networks have very complicated optimal strategies that require mixing of a continuum of pure strategies many of our results generalize analogous ones obtained for constant velocity in both directions by s gal but not all of those results can be extended

rendezvous search on the labeled line the rendezvous search problem is the problem of finding optimal search strategies for two people who are placed randomly on a known search region and want to meet each other in minimal expected time we focus on initial location distributions that are centrally symmetric and nonincreasing as one moves away from the center including the discretized and or truncated gaussian densities when the search region is a discrete or a continuous interval and the interval is labeled so that the searchers know their own location at all times we prove that the optimal strategy for both searchers is to go directly to the center and wait there the same result also holds for rendezvous search on the infinite line

stable enrichment and local preconditioning in the particle partition of unity method this paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods in particular we focus on the particle partition of unity method ppum yet the presented results hold for any partition of unity based enrichment scheme the goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h version independent of the regularity of the solution hence we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme to this end we enrich our ppum function space in an enrichment zone hierarchically near the singularities of the solution this initial enrichment however can lead to a severe ill conditioning and can compromise the stability of the discretization to overcome the ill conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well conditioned basis independent of the employed initial enrichment the construction of this preconditioner is of linear complexity with respect to the number of discretization points we obtain optimal error bounds for an enriched ppum discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super convergence the results of our numerical experiments clearly show that our enriched ppum with local preconditioning recovers the optimal convergence rate of o h p of the uniform h version globally for the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of o h p with delta geq frac 1 2 for p 1 the convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation

large time behavior of the solution of the heat equation with nonlinear strong absorption abstract we give an estimate for the asymptotic behavior of the solution of a semilinear heat equation and of its free boundary

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

a stable and optimally convergent generalized fem sgfem for linear elastic fracture mechanics abstract in this paper we investigate the accuracy and conditioning of the stable generalized fem sgfem and compare it with standard generalized fem gfem for a 2 d fracture mechanics problem the sgfem involves localized modifications of enrichments used in the gfem and the conditioning of the stiffness matrix in this method is of the same order as in the fem numerical experiments show that using the sgfem with only the modified heaviside functions which are used as enrichments in the gfem to approximate the solution of fracture problems in 2 d gives inaccurate results however the sgfem using an additional set of enrichment function yields accurate results while not deteriorating the conditioning of the stiffness matrix rules for the selection of the optimal set of enrichment nodes based on the definition of enrichment functions used in the sgfem are also presented this set leads to optimal convergence rates while keeping the number of degrees of freedom equal to or close to the gfem we show that it is necessary to enrich additional nodes when the crack line is located along element edges in 2 d the selection of these nodes depends on the definition of the enrichment functions at the crack discontinuity a simple and yet generic implementation strategy for the sgfem in an existing gfem xfem software is described the implementation can be used with 2 d and 3 d elements it leads to an efficient evaluation of sgfem enrichment functions

an improved stable xfem is xfem with a novel enrichment function for the computational modeling of cohesive cracks abstract owing to the solution space enriched with a priori known knowledge through the partition of unity the extended or generalized finite element method xfem gfem has been widely applied to the modeling of localized failure in solids as far as cohesive cracks are concerned in this work on the one hand the xfem enriched with heaviside function or the shifted one with respect to nodal values is of optimal accuracy but is prone to the issue of ill conditioned or even singular system matrix for arbitrary crack propagation on the other hand the stable xfem s xfem circumvents the ill conditioning issues by introducing the residual heaviside function with respect to its linear interpolant at the cost of losing accuracy e g spurious stress locking for discontinuities with non uniform displacement jumps aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix this work addresses a simple yet effective improved stable xfem is xfem with a novel enrichment function both the xfem and s xfem are recovered as its particular cases with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale in the proposed is xfem as the enrichment basis functions are not contained in the standard polynomial space conditioning of the resulting system matrix is nearly insensitive to the mesh discontinuity configuration even if the crack path gets close to element nodes furthermore the is xfem with a small stabilization parameter is almost of the same accuracy as the xfem particularly pathological numerical results polluted by spurious stress locking exhibited in the s xfem are not observed across a fully softened discontinuity with non uniform displacement jumps as only the enrichment function is modified the proposed is xfem together with appropriate crack propagation criterion and tracking algorithm can be implemented easily in the existing xfem codes several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed is xfem regarding both accuracy of the solution and conditioning of the system matrix

search for an immobile hider in a known subset of a network abstract a unit speed searcher constrained to start in a given closed set s wishes to quickly find a point x known to be located in a given closed subset h of a metric network q this defines a game g g q h s where the payoff to the maximizing hider is the time for the searcher path to reach x lengths on q are defined by a measure which then defines distance as least length of connecting path for trees q we find that the minimax search time value v of g is given by v h d h s 2 where d h s is what we call the h diameter of s and equals the usual diameter d s of s in the case h q for the classical case of gal where the s is a singleton and h q our formula reduces to his result v q if s h q our formula gives dagan and gal s result v q d q 2 in all other cases our result is new optimal searches consist of minimum length paths covering h which start and end at points of s traversed equiprobably in either direction

conditioning of finite element equations with arbitrary anisotropic meshes this is the published version also available here http dx doi org 10 1090 s0025 5718 2014 02822 6 first published in math comput in 2014 published by the american mathematical society

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

symmetric weak form integral equation method for three dimensional fracture analysis a symmetric galerkin boundary element method is developed for the analysis of linearly elastic isotropic three dimensional solids containing fractures the formulation is based upon a weak form displacement integral equation and a weak form traction integral equation recently developed by li and mear 1997 these integral equations are only weakly singular and their validity requires only that the boundary displacement data be continuous hence allowing standard co elements to be employed as part of the numerical implementation a special crack tip element is developed which has a novel feature in that there exist degrees of freedom associated with the nodes at the crack front as a result a higher degree of approximation is achieved for the relevant displacement data on the crack and further the stress intensity factors are obtained directly in terms of the crack front nodal data various examples are treated for cracks in unbounded domains and for cracks in finite domains including both embedded and surface breaking cracks and it is demonstrated that highly accurate results can be achieved using relatively coarse meshes

an implicit geometrical approach to level sets update for 3d non planar x fem crack propagation abstract in the x fem framework the crack geometry is described independently of the structural mesh usually by means of two level set functions thanks to the versatility and wide applicability of the level set methodology the main difficulty in the adoption of the level set approach is the formulation of an efficient and robust update scheme allowing to calculate the new level set values associated to the new crack configuration after a propagation step in this paper an implicit geometrical methodology is formulated in order to overcome the stability and performance issues affecting the resolution of the hamilton jacobi partial differential equations governing the level set update the calculation of the new level sets values is based on geometrical considerations only which enormously reduces the computational time and increases the robustness of the update calculations in contrast to other geometrical approaches this methodology can be applied both to 2d and 3d problems and the crack front and surface are always implicitly described by means of the two level sets this avoids both the need to describe somehow the crack surface and front and the creation of algorithms to track their evolution which improves further the performance of the methodology the numerical tests reported in the paper show that the proposed methodology is 30 times faster in 2d and 3 6 times faster in 3d than classic approaches based on the resolution of the hamilton jacobi equations and 2 7 times faster than explicit geometrical approaches the proposed methodology can also manage the separation of a crack front in several fronts during the propagation and the merging of several fronts into one single front

convergence analysis of linear or quadratic x fem for curved free boundaries abstract the aim of this paper is to provide a priori error estimates for problems involving curved interfaces and solved with the linear or quadratic extended finite element method x fem with particular emphasis on the influence of the geometry representation and the quadrature we focus on strong discontinuity problems which covers the case of holes in a material or cracks not subjected to contact as the main applications the well known approximation of the curved geometry based on the interpolated level set function and straight linear or curved quadratic subcells is used whose accuracy is quantified by means of an appropriate error measure a priori error estimates are then derived which depend upon the interpolation order of the displacement and foremost upon the above error measure and the quadrature scheme in the subcells the theoretical predictions are successfully compared with numerical experiments

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

strong and weak arbitrary discontinuities in spectral finite elements methods for constructing arbitrary discontinuities within spectral finite elements are described and studied we use the concept of the extended finite element method xfem which introduces the discontinuity through a local partition of unity so there is no requirement for the mesh to be aligned with the discontinuities a key aspect of the implementation of this method is the treatment of the blending elements adjacent to the local partition of unity we found that a partition constructed from spectral functions one order lower than the continuous approximation is optimal and no special treatment is needed for higher order elements for the quadrature of the galerkin weak form since the integrand is discontinuous we use a strategy of subdividing the discontinuous elements into 6 and 10 node triangles the order of the element depends on the order of the spectral method for curved discontinuities several numerical examples are solved to examine the accuracy of the methods for straight discontinuities we achieved the optimal convergence rate of the spectral element for the curved discontinuity the convergence rate in the energy norm error is suboptimal we attribute the suboptimality to the approximations in the quadrature scheme we also found that modification of the adjacent elements is only needed for lower order spectral elements copyright 2005 john wiley sons ltd

a robust preconditioning technique for the extended finite element method the extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions but the resulting stiffness matrices can become ill conditioned in that case iterative solvers need a large number of iterations to obtain an acceptable solution in this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments a domain decomposition is employed and the algorithm is very well suited for parallel computations the method was tested in numerical experiments to show its effectiveness the experiments have been conducted for structures containing cracks and material interfaces we show that the corresponding enrichments can result in arbitrarily ill conditioned matrices the method proposed here however provides well conditioned matrices and can be applied to any sort of enrichment the complexity of this approach and its relation to the domain decomposition is discussed computation times have been measured for a structure containing multiple cracks for this structure the computation times could be decreased by a factor of 2

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

a study of the representation of cracks with level sets the level set method has been used for a few years to represent cracks in fracture mechanics simulations instead of an explicit description of the cracks faces geometry this paper studies in detail the shape of the level set functions around a crack in two dimensions that is propagating with a sharp kink obtained both with level set update methods found in the literature and with several innovative update methods developed by the author a criterion based on the computation of a j integral of a virtual displacement field obtained with the values of the level set functions is proposed in order to assess the quality of these update methods with the help of this criterion two optimal approaches are identified which predict an accurate evolution of the crack with smooth and consistent level set functions these methods are then applied in three dimensions to the case of an initially penny shaped crack that propagates out of its plane copyright 2006 john wiley sons ltd

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

the rendezvous search problem the author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time this time is called the rendezvous value of the region it is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise a general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players these concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network the discrete rendezvous problem on a cycle graph for players restricted to symmetric markovian strategies is then solved finally the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other

find and fetch search on a tree we introduce a new type of search game called the find and fetch game f q o the hider simply picks any point h in the network q the searcher starts at time zero at a given point o of q moving at unit speed until he reaches h finds the hider then he returns at a given speed along the shortest path back to o arriving at time r the payoff this models the problem faced in many types of search including search and rescue problems and foraging problems of animals where food must be found and returned to the lair when q is a binary tree we derive optimal probabilities for the searcher to branch at the nodes these probabilities give a positive bias towards searching longer branches first we show that the minimax value of the return time r the game value of f q o is d where is the total length of q and d is the mean distance from the root o to the leaves terminal nodes of q where the mean is taken with respect to what is known as the equal branch density distribution as goes to infinity our problem reduces to the search game model where the payoff is simply the time to reach the hider and our results tend to those obtained by gal gal s 1979 search games with mobile and immobile hider siam j control optim 17 1 99 122 and anderson and gal anderson e j s gal 1990 search in a maze probab engrg inform sci 4 3 311 318 for that model we also apply our return time formula d to determine the ideal location for the root lair or rescue center o assuming it can be moved in the traditional find only model the location of o does not matter

asymptotic behavior of the solutions of the dead core problem u u pup 0 1 1 this equation originates from the study of a class of models for the reaction diffusion processes of a gas inside a chemical reactor 2 81 from the mathematical point of view the major interest of equation 1 1 lies in the competition of the linear diffusion and the nonlinear fast p 0 t 0 with initial and boundary conditions

rendezvous search on the interval and the circle two people are placed randomly and independently on a street of unit length they attempt to find each other in the shortest possible expected time we solve this problem assuming each searcher knows where he or she is on the street for monotonic density functions for the initial placement this includes the uniform pdf as a special case this gives an example of a rendezvous search problem where there is no advantage in being allowed to use asymmetric strategies we also solve some corresponding problems for the circle when asymmetric strategies are permitted one of these shows that it can sometimes be optimal for one player to wait for the other to find him

a well conditioned and optimally convergent xfem for 3d linear elastic fracture summary a variation of the extended finite element method for three dimensional fracture mechanics is proposed it utilizes a novel form of enrichment and point wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy optimal convergence rates and improved conditioning for two dimensional and three dimensional crack problems a bespoke benchmark problem is introduced to determine the method s accuracy in the general three dimensional case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40 for moderately refined meshes and topological enrichment moreover when a fixed enrichment volume is used the number of iterations required grows at a rate which is reduced by a factor of 2 compared with standard extended finite element method diminishing the number of iterations by almost one order of magnitude copyright 2015 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

a stable generalized extended p hierarchical fem for three dimensional linear elastic fracture mechanics abstract in this paper the quadratic stable generalized finite element method sgfem proposed in sanchez rivadeneira and duarte 2019 is extended to 3 d fracture problems with non planar crack surfaces and curved crack fronts this sgfem is based on p hierarchical fem enrichments and its approximation space is the same as in its gfem counterpart singular enrichments are modified using a discontinuous finite element interpolant for conditioning control this leads with the proper choice of enrichments to stiffness matrices with a scaled condition number of o h 2 the same order as in the standard fem the robustness of the method with respect to the position of a crack relative to the mesh is demonstrated a scalar implementation of singular enrichments that can exactly reproduce the first term of the mode i ii and iii of the asymptotic expansion of the elasticity solution is presented improved approximations of signed distance functions based on integration sub elements are also presented convergence studies of a fully 3 d fracture problem with a non planar crack surface and a curved crack front are presented they show that the convergence rate of the proposed method on a sequence of uniform meshes is three times higher than quadratic quarter point finite elements

high order extended finite element method for cracked domains the aim of the paper is to study the capabilities of the extended finite element method xfem to achieve accurate computations in non smooth situations such as crack problems although the xfem method ensures a weaker error than classical nite element methods the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity the di culty can be overcome by modifying the enrichment of the nite element basis with the asymptotic crack tip displacement solutions as well as with the heaviside function numerical simulations show that the modi ed xfem method achieves an optimal rate of convergence i e like in a standard nite element method for a smooth problem

hide and seek games on a network using combinatorial search paths this paper introduces a new search paradigm to hide and seek games on networks the hider locates at any point on any arc the searcher adopts a combinatorial path when searching the network a sequence of arcs each adjacent to the last and traced out at unit speed in previous literature the searcher was allowed simple motion any unit speed path including ones that turn around inside an arc the new approach more closely models real problems such as search for improvised explosive devices using vehicles that can only turn around at particular locations on a road the search game is zero sum with the time taken to find the hider as the payoff using a lemma giving an upper bound for the expected search time on a semi eulerian network we solve the search game on a network q3 consisting of two nodes connected by three arcs of arbitrary lengths when two q3 networks with unit length arcs are linked by two small central arcs incident at the start node one of these arcs must be traversed at least th

extended finite element method with global enrichment a variation of the extended finite element method for 3d fracture mechanics is proposed it utilizes global enrichment and point wise as well as integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy optimal convergence rates and improved conditioning for two and three preprint submitted to elsevier science 29 january 2015 page 1 of 81 http mc manuscriptcentral com nme international journal for numerical methods in engineering

mining coal or finding terrorists the expanding search paradigm we show how to optimize the search for a hidden object terrorist or simply hider located at a point h according to a known or unknown distribution v on a rooted network q we modify the traditional pathwise search approach to a more general notion of expanding search when the hider is restricted to the nodes of q an expanding search s consists of an ordering a1 a2 of the arcs of a spanning subtree such that the root node is in a1 and every arc ai is adjacent to a previous arc aj j i if ak contains h the search time t is a1 ak5 where is length measure on q for more general distributions v an expanding search s is described by the nested family of connected sets s4t5 that specify the area of q that has been covered by time t s 0 is the root s t t and t min t h 2 s t for a known hider distribution v on a tree q the expected time minimizing strategy sn begins with the rooted subtree q maximizing the density v q q for arbitrary networks we use this criterion on all spanning subtrees the search sn can be interpreted as the optimal method of mining known coal seams when the time to move miners or machines is negligible compared to digging time when the hider distribution is unknown we consider the zero sum search game where the hider picks h the searcher s and the payoff is t for trees q the value is v q d 2 where d is a mean distance from root to leaf nodes if q is 2 arc connected v q 2 applications and interpretations of the expanding search paradigm are given particularly to multiple agent search

rendezvous search on the line we present two new results for the asymmetric rendezvous problem on the line we first show that it is never optimal for one player to be stationary during the entire search period in the two player rendezvous then we consider the meeting time of n players in the worst case and show that it has an asymptotic behavior of n 2 o log n

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

search games on trees with asymmetric travel times a point h is hidden in a rooted tree q which is endowed with asymmetric distances travel times between nodes we determine the randomized search strategy starting from the root which minimizes the expected time to reach h in the worst case this is equivalent to a zero sum search game gamma left q right with minimizing searcher maximizing hider and payoff equal to the capture time the worst hiding distribution over the leaves from the searcher s viewpoint is one where at every node i the probability of each branch is proportional to the minimum time required to tour it from i the optimal randomized search is a mixture over depth first searches we also consider briefly some other networks and the possibility of a mobile hider our formulation with asymmetric travel times generalizes that of gal siam j control optim 17 1979 pp 99 122 for symmetric travel times and also the search games of kikuta j oper res 38 1995 pp 70 88 and kikuta and ruckle naval res logist 41 1994 pp 821 831 who posited search costs c i at each node i which were added to the travel time to obtain the payoff we also briefly consider what happens if we allow the searcher hider to start hide at any leaf node we determine when properties found by dagan and gal networks 52 2008 pp 156 161 for the symmetric version of such games hold in our asymmetric context

optimal trade off between speed and acuity when searching for a small object a searcher seeks to find a stationary hider located at some point h not necessarily a node on a given network q the searcher can move along the network from a given starting point at unit speed but to actually find the hider she must pass it while moving at a fixed slower speed which may depend on the arc in this bimodal search game the payoff is the first time the searcher passes the hider while moving at her slow speed this game models the search for a small or well hidden object e g a contact lens improvised explosive device predator search for camouflaged prey we define a bimodal chinese postman tour as a tour of minimum time i which traverses every point of every arc at least once in the slow mode for trees and weakly eulerian networks networks containing a number of disjoint eulerian cycles connected in a tree like fashion the value of the bimodal search game is i 2 for trees the optimal hider strategy has full support on the network this differs from traditional search games where it is optimal for him to hide only at leaf nodes we then consider the notion of a lucky searcher who can also detect the hider with a positive probability q even when passing him at her fast speed this paper has particular importance for demining problems

on error estimator and p adaptivity in the generalized finite element method this paper addresses the issue of a p adaptive version of the generalized finite element method gfem the technique adopted here is the equilibrated element residual method but presented under the gfem approach i e by taking into account the typical nodal enrichment scheme of the method such scheme consists of multiplying the partition of unity functions by a set of enrichment functions these functions in the case of the element residual method are monomials and can be used to build the polynomial space one degree higher than the one of the solution in which the error functions is approximated global and local measures are defined and used as error estimator and indicators respectively the error indicators calculated on the element patches that surrounds each node are used to control a refinement procedure numerical examples in plane elasticity are presented outlining in particular the effectivity index of the error estimator proposed finally the p adaptive procedure is described and its good performance is illustrated by the last numerical example

on the conditioning of finite element equations with highly refined meshes it is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally provided the mesh remains nondegenerate and a natural scaling of the basis functions is used bounds for the euclidean condition number as a function of the number of degrees f freedom are derived in n geq 2 dimensions when n geq 3 the bound is the same as for the regular mesh case but when n 2 a factor appears in the bound for the condition number that is logarithmic in the ratio of the maximum and minimum mesh sizes applications of the results to the conjugate gradient iterative method for solving such linear systems are given

higher order stable generalized finite element method the generalized finite element method gfem is a galerkin method where the trial space is obtained by augmenting the trial space of the standard finite element method fem by non polynomial functions called enrichments that mimic the local behavior of the unknown solution of the underlying variational problem the gfem has excellent approximation properties but its conditioning could be much worse than that of the fem however if the enrichments satisfy certain properties then the conditioning of the gfem is not worse than that of the standard fem and the gfem is referred to as the stable gfem sgfem in this paper we address the higher order sgfem that yields higher order convergence and suggest a specific modification of the enrichment function that guarantees the required conditioning yielding a robust implementation of the higher order sgfem

the orthonormalized generalized finite element method ogfem efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions abstract an extension of the generalized finite element method gfem is proposed with which we efficiently reduce approximation errors the new method constructs a stiffness matrix with a conditioning that is significantly better than the stable generalized finite element method sgfem and the finite element method fem accordingly the risk of a severe loss of accuracy in the computed solution which burdens the gfem is prevented furthermore the computational cost of the inversion of the associated stiffness matrix is significantly reduced the gfem employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution the stiffness matrix constructed from these basis functions is often ill conditioned and the accuracy of the solution cannot be guaranteed we prevent this by orthonormalizing the basis functions and refer to the method as the orthonormalized generalized finite element method ogfem because the ogfem has the flexibility to orthonormalize either a part or all of the basis functions the method can be considered as a generalization of the gfem the method is applicable with single or multiple global and or local enrichment functions problems in blending elements are avoided by a modification of the enrichment functions the method is demonstrated for the one dimensional modified helmholtz and poisson equations and compared with the fem gfem and sgfem