adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of biot s consolidation model abstract this paper is concerned with the analysis of coupled mixed finite element methods applied to the biot s consolidation model we consider two mixed formulations that use the stress tensor and darcy velocity as primary variables as well as the displacement and pressure the first formulation is with a symmetric stress tensor while the other enforces the symmetry of the stress weakly through the introduction of a lagrange multiplier the well posedness of the two formulations is shown through galerkin s method and suitable a priori estimates the two formulations are then discretized with the backward euler scheme in time and with two mixed finite elements in space we present next a general and unified a posteriori error analysis which is applicable for any flux and stress conforming discretization our estimates are based on h 1 conforming reconstruction of the pressure and a suitable h 1 d conforming reconstruction of the displacement both are continuous and piecewise affine in time these reconstructions are used to infer a guaranteed and fully computable upper bound on the energy type error measuring the differences between the exact and the approximate pressure and displacement the error components resulting from the spatial and the temporal discretization are distinguished they are then used to design an adaptive space time algorithm numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm

on the numerical solution of the direct scattering problem for an open sound hard arc we derive a hypersingular integral equation which is equivalent to the scattering problem using the cosine substitution we obtain an integral equation which is closely related to the integral equation for the case of a closed boundary we solve this equation approximately by a quadrature formula method and obtain pointwise error estimates in holder norms by using a perturbation argument

block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

stable cell centered finite volume discretization for biot equations in this paper we discuss a new discretization for the biot equations the discretization treats the coupled system of deformation and flow directly as opposed to combining discretizations for the two separate subproblems the coupled discretization has the following key properties the combination of which is novel 1 the variables for the pressure and displacement are co located and are as sparse as possible e g one displacement vector and one scalar pressure per cell center 2 with locally computable restrictions on grid types the discretization is stable with respect to the limits of incompressible fluid and small time steps 3 no artificial stabilization term has been introduced furthermore due to the finite volume structure embedded in the discretization explicit local expressions for both momentum balancing forces and mass conservative fluid fluxes are available we prove stability of the proposed method with respect to all relevant limits together with consistency this proves conve

diffusion in poro elastic media existence uniqueness and regularity theory is developed for a general initial boundary value problem for a system of partial differential equations which describes the biot consolidation model in poro elasticity as well as a coupled quasi static problem in thermoelasticity additional effects of secondary consolidation and pore fluid exposure on the boundary are included this quasi static system is resolved as an application of the theory of linear degenerate evolution equations in hilbert space and this leads to a precise description of the dynamics of the system

uniformly stable discontinuous galerkin discretization and robust iterative solution methods for the brinkman problem we consider robust iterative methods for discontinuous galerkin dg h div omega conforming discretizations of the brinkman equations we describe a simple uzawa iteration for the solution of this problem which requires the solution of a nearly incompressible linear elasticity type equation with mass term on every iteration we prove the uniform stability of the dg discretization for both problems then we analyze variable v cycle and w cycle multigrid methods with nonnested bilinear forms we prove that these algorithms are robust and their convergence rates are independent of the parameters in the brinkman problem and of the mesh size the theoretical analysis is confirmed by numerical results

a comparison of numerical collocation and variational procedures to the hypersingular acoustic integral operator abstract a major difficulty in implementing many of the more innovative acoustic formulations for the solution of submerged three dimensional thin shells of arbitrary geometry in an unbounded fluid medium has been the difficulty of numerically approximating a hypersingular integral operator a variational approach given by hamdi reduces this hypersingularity cauchy singularity to a weak singularity allowing for its accurate numerical approximation recently mariem and hamdi a new boundary finite element method for fluid structure interaction internat j numer methods engrg 24 1987 1251 1267 presented an acoustic formulation for thin shells where the pressure difference across the shell is related to the shell s normal velocity solely in terms of this integral operator despite its advantage of generating a symmetric fluid matrix a double surface integration is required to form this matrix some researchers have endeavoured to develop a collocation method of approximating this highly singular operator in this paper a novel solution to the numerical approximation of this operator is given in the implementation of both the variational and collocation methodologies quadratic quadrilateral elements are used to interpolate for the acoustic and geometric variables results are then presented for thin shell scattering and radiation problems for both closed and non closed thin shells demonstrating the validity and accuracy of both methods

an accurate scheme for mixed mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models summary the interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed mode stress intensity factors sifs the present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions e g exponentially graded materials or micromechanics models e g self consistent mori tanaka or three phase model in the latter case there is no closed form expression for the material property variation and thus several quantities such as the explicit derivative of the strain energy density need to be evaluated numerically this leads to several implications in the numerical implementation the sifs are determined using conservation integrals involving known auxiliary solutions the choice of such auxil

correcting curvature density effects in the hamilton jacobi skeleton the hamilton jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two dimensional 2 d shapes the approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero while the flux is negative at the locations of skeletal points nonetheless the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and hence unbounded near endpoints this makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter this problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision however here we develop a method for 2 d skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error this is done by taking into account variations of density due to boundary curvature this yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the hamilton jacobi method

scalable algorithms for three field mixed finite element coupled poromechanics abstract we introduce a class of block preconditioners for accelerating the iterative solution of coupled poromechanics equations based on a three field formulation the use of a displacement velocity pressure mixed finite element method combined with a first order backward difference formula for the approximation of time derivatives produces a sequence of linear systems with a 3 3 unsymmetric and indefinite block matrix the preconditioners are obtained by approximating the two level schur complement with the aid of physically based arguments that can be also generalized in a purely algebraic approach a theoretical and experimental analysis is presented that provides evidence of the robustness efficiency and scalability of the proposed algorithm the performance is also assessed for a real world challenging consolidation experiment of a shallow formation

end point behaviour of solutions to hypersingular integral equations we consider one dimensional hypersingular integral equations over finite intervals the integral must be interpreted as a finite part integral such equations arise naturally in various physical situations involving thin rigid bodies or cracks examples are given a method is developed for determining the behaviour of the solution to a hypersingular integral equation near the end points of the interval of integration the method uses the mellin transform several examples are worked out in detail

a numerical method for scattering from acoustically soft and hard thin bodies in two dimensions abstract this paper presents a numerical method for predicting the acoustic scattering from two dimensional 2 d thin bodies both the dirichlet and neumann problems are considered applying the thin body formulation leads to the boundary integral equations involving weakly singular and hypersingular kernels completely regularizing these kinds of singular kernels is thus the main concern of this paper the basic subtraction addition technique is adopted the purpose of incorporating a parametric representation of the boundary surface with the integral equations is two fold the first is to facilitate the numerical implementation for arbitrarily shaped bodies the second one is to facilitate the expansion of the unknown function into a series of chebyshev polynomials some of the resultant integrals are evaluated by using the gauss chebyshev integration rules after moving the series coefficients to the outside of the integral sign others are evaluated exactly including the modified hypersingular integral the numerical implementation basically includes only two parts one for evaluating the ordinary integrals and the other for solving a system of algebraic equations thus the current method is highly efficient and accurate because these two solution procedures are easy and straightforward numerical calculations consist of the acoustic scattering by flat and curved plates comparisons with analytical solutions for flat plates are made

the postulations a la d alembert and a la cauchy for higher gradient continuum theories are equivalent a review of existing results in order to found continuum mechanics two different postulations have been used the first introduced by lagrange and piola starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients then by using the divergence theorem a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body this method assumes an a priori notion of internal work regards stress tensors as dual of virtual displacements and their gradients deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts the second method conceived by cauchy and based on the celebrated tetrahedron argument starts by postulating the type of contact interactions which can be exerted on the boundary of every suitably regular part of a body then it proceeds by proving the existence of stress tensors from a balance type postulate in this paper we review some relevant literature on the subject discussing how the two postulations can be reconciled in the case of higher gradient theories finally we underline the importance of the concept of contact surface edge and wedge s order forces

adaptive asynchronous time stepping stopping criteria and a posteriori error estimates for fixed stress iterative schemes for coupled poromechanics problems abstract in this paper we develop adaptive iterative coupling schemes for the biot system modeling coupled poromechanics problems we particularly consider the space time formulation of the fixed stress iterative scheme in which we first solve the problem of flow over the whole space time interval then exploiting the space time information for solving the mechanics two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in space and the backward euler scheme in time therefrom adaptive fixed stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions these ingredients are used to derive a posteriori error estimates for the fixed stress algorithms distinguishing the different error components namely the spatial discretization the temporal discretization and the fixed stress iteration components precisely at the iteration k 1 of the adaptive algorithm we prove that our estimate gives a guaranteed and fully computable upper bound on the energy type error measuring the difference between the exact and approximate pressure and displacement these error components are efficiently used to design adaptive asynchronous time stepping and adaptive stopping criteria for the fixed stress algorithms numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms

a numerical study of a posteriori error estimators for convection diffusion equations this paper presents a numerical study of a posteriori error estimators for convection diausion equations the study involves the gradient indicator an a posteriori error estimator which is based on gradient recovery three residual based error estimators for diaerent norms and two error estimators which are defined by solutions of local neumann problems they are compared with respect to the reliable estimation of the global error and with respect to the accuracy of the computed solutions on adaptively refined grids the numerical study shows for both criteria of comparison that none of the considered error estimators works satisfactorily in all tests 2000 elsevier science s a all rights reserved msc 65n50 65n30

an interfacially cracked orthotropic rectangular bi material subjected to antiplane shear loading abstract an interfacially cracked rectangular bi material composed of two dissimilar orthotropic elastic media is analyzed the crack is parallel to or perpendicular to the boundary surfaces under the action of antiplane shear loading the problem is reduced to a singular integral equation of the first kind by employing the fourier series method for four types of boundary conditions including the f f type c c type f c type and c f type numerical results of stress intensity factors are obtained by using the lobatto chebyshev method the results show that in addition to the geometric factor stress intensity factors are dependent on the material properties the influences of the material properties on stress intensity factors are examined and compared under different boundary conditions

stress and flux reconstruction in biots poro elasticity problem with application to a posteriori error analysis we derive equilibrated reconstructions of the darcy velocity and of the total stress tensor for biots poro elasticity problem both reconstructions are obtained from mixed finite element solutions of local neumann problems posed over patches of elements around mesh vertices the darcy velocity is reconstructed using raviartthomas finite elements and the stress tensor using arnoldwinther finite elements so that the reconstructed stress tensor is symmetric both reconstructions have continuous normal component across mesh interfaces using these reconstructions we derive a posteriori error estimators for biots poro elasticity problem and we devise an adaptive spacetime algorithm driven by these estimators the algorithm is illustrated on test cases with analytical solution on the quarter five spot problem and on an industrial test case simulating the excavation of two galleries

unfolding kernel embeddings of graphs in this paper we investigate the use of manifold learning techniques to enhance the separation properties of standard graph kernels the idea stems from the observation that when we perform multidimensional scaling on the distance matrices extracted from the kernels the resulting data tends to be clustered along a curve that wraps around the embedding space a behavior that suggests that long range distances are not estimated accurately resulting in an increased curvature of the embedding space hence we propose to use a number of manifold learning techniques to compute a low dimensional embedding of the graphs in an attempt to unfold the embedding manifold and increase the class separation we perform an extensive experimental evaluation on a number of standard graph datasets using the shortest path borgwardt and kriegel 2005 graphlet shervashidze et al 2009 random walk kashima et al 2003 and weisfeiler lehman shervashidze et al 2011 kernels we observe the most significant improvement in the case of the graphlet kernel which fits with the observation that neglecting the locational information of the substructures leads to a stronger curvature of the embedding manifold on the other hand the weisfeiler lehman kernel partially mitigates the locality problem by using the node labels information and thus does not clearly benefit from the manifold learning interestingly our experiments also show that the unfolding of the space seems to reduce the performance gap between the examined kernels highlightswe use manifold learning techniques to enhance the performance of graph kernels we try to increase the class separation by unfolding the embedding manifold we propose a procedure to find the optimal embedding of the data the kernels which neglect the locational information show the largest improvement the unfolding of the space helps to reduce the performance gap between the kernels

guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the biot problem abstract the paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi static linear biot equations the system is decoupled by employing the fixed stress split scheme which leads to an iteratively solved semi discrete system the error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations the estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method they are fully computable do not contain mesh dependent constants and provide reliable global estimates of the error measured in the energy norm moreover they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures

strain gradient elasticity for antiplane shear cracks a hypersingular integrodifferential equation approach we consider casal s strain gradient elasticity with two material lengths ell ell associated with volumetric and surface energies respectively for a mode iii finite crack we formulate a hypersingular integrodifferential equation for the crack slope supplemented with the natural crack tip conditions the full field solution is then expressed in terms of the crack profile and the green function which is obtained explicitly for ell 0 we obtain a closed form solution for the crack profile the case of small ell is shown to be a regular perturbation the question of convergence as ell ell to 0 is studied in detail both analytically and numerically

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

analysis of a discontinuous galerkin method for the biot s consolidation problem in this paper a fully discrete stabilized discontinuous galerkin method is proposed to solve the biot s consolidation problem the existence and uniqueness of the finite element solution are obtained the stability of the fully discrete solution is discussed the corresponding error estimates for the approximation of displacement and pressure in a mesh dependent norm are obtained the error estimate for the approximation of pressure in l 2 norm is also obtained

non uniqueness in two dimensional water wave problems numerical evidence and geometrical restrictions in this paper numerical evidence for the existence of eigenvalues in a particular two dimensional water wave problem is presented together with computations of the eigenvalues themselves the geometry considered is that of a pair of identical inclined surface piercing barriers arranged symmetrically about a vertical line in infinitely deep water only oscillations symmetric about this line are considered so that the geometry is equivalent to an inclined surface piercing barrier next to a vertical wall the problem is formulated as a hypersingular integral equation which is then solved numerically by expanding the unknown function corresponding to the jump in potential across the barrier as a series of chebyshev polynomials we also prove that for a more general class of bodies consisting of a pair of surface piercing bodies placed symmetrically about a vertical line midway between them there are ranges of the spectral parameter for which no eigenvalues exist the numerically computed eigenvalues for the inclined barriers are consistent with these results

temporary removal on gradient enriched elasticity theories a reply to comment on on non singular crack fields in helmholtz type enriched elasticity theories and important theoretical aspects this article has been withdrawn at the request of the editors the publisher apologizes for any inconvenience this may cause the full elsevier policy on article withdrawal can be found at https www elsevier com about our business policies article withdrawal

scale invariance in road networks we study the topological and geographic structure of the national road networks of the united states england and denmark by transforming these networks into their dual representation where roads are vertices and an edge connects two vertices if the corresponding roads ever intersect we show that they exhibit both topological and geographic scale invariance that is we show that for sufficiently large geographic areas the dual degree distribution follows a power law with exponent 2 2 or alpha or 2 4 and that journeys regardless of their length have a largely identical structure to explain these properties we introduce and analyze a simple fractal model of road placement that reproduces the observed structure and suggests a testable connection between the scaling exponent and the fractal dimensions governing the placement of roads and intersections

nonextensive information theoretic kernels on measures positive definite kernels on probability measures have been recently applied to classification problems involving text images and other types of structured data some of these kernels are related to classic information theoretic quantities such as shannon s mutual information and the jensen shannon js divergence meanwhile there have been recent advances in nonextensive generalizations of shannon s information theory this paper bridges these two trends by introducing nonextensive information theoretic kernels on probability measures based on new js type divergences these new divergences result from extending the the two building blocks of the classical js divergence convexity and shannon s entropy the notion of convexity is extended to the wider concept of q convexity for which we prove a jensen q inequality based on this inequality we introduce jensen tsallis jt q differences a nonextensive generalization of the js divergence and define a k th order jt q difference between stochastic processes we then define a new family of nonextensive mutual information kernels which allow weights to be assigned to their arguments and which includes the boolean js and linear kernels as particular cases nonextensive string kernels are also defined that generalize the p spectrum kernel we illustrate the performance of these kernels on text categorization tasks in which documents are modeled both as bags of words and as sequences of characters

asymptotic behavior of semidiscrete finite element approximations of biot s consolidation problem error estimates for spatially discrete galerkin finite element approximations of biot s model for consolidation of saturated porous media are presented the short and long time behaviors of such approximations based on both stable and unstable combinations of finite element spaces of displacement and pore pressure fields are discussed

theory of the dynamic biot allard equations and their link to the quasi static biot system we undertake establishing well posedness of the dynamic biot allard equations it is obtained using the precise properties of the dynamic permeability matrix following the homogenization derivation of the model by taking the singular limit of the contrast coefficient the quasi static biot system can be obtained from the dynamic biot equations these results can be used to formulate an efficient computational algorithm for solving dynamic biot allard equations for subsurface flows with the characteristic reservoir time scales larger than the intrinsical characteristic time this result appears to be completely new in the literature on biot s theory we conclude by showing that in the case of periodic deformable porous media the dynamic permeability has the required properties

weisfeiler lehman graph kernels in this article we propose a family of efficient kernels for large graphs with discrete node labels key to our method is a rapid feature extraction scheme based on the weisfeiler lehman test of isomorphism on graphs it maps the original graph to a sequence of graphs whose node attributes capture topological and label information a family of kernels can be defined based on this weisfeiler lehman sequence of graphs including a highly efficient kernel comparing subtree like patterns its runtime scales only linearly in the number of edges of the graphs and the length of the weisfeiler lehman graph sequence in our experimental evaluation our kernels outperform state of the art graph kernels on several graph classification benchmark data sets in terms of accuracy and runtime our kernels open the door to large scale applications of graph kernels in various disciplines such as computational biology and social network analysis

cpv and hfp integrals and their applications in the boundary element method the subject of this paper is hadamard finite part hfp and cauchy principal value cpv representations of certain singular integrals of primary concern here are hfp interpretations of strongly singular integrals for which cpv interpretations also exist a relationship between the two is stated and proved this matter is then illustrated by two examples and also discussed in the context of singular boundary integral equations bies for linear elasticity

a least squares mixed finite element method for biot s consolidation problem in porous media a least squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed in this paper for the analysis we restrict ourselves to fully saturated conditions for the flow process and to a linearly elastic material law for the deformation process this is known in the literature as biot s consolidation problem for simplicity the analysis is presented for the problem in two space dimensions our least squares approach is motivated by the fact that all process variables i e fluid pressure and flux as well as displacement field and stress tensor are approximated directly by suitable finite element spaces ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time dependent case standard h1 conforming piecewise linear and quadratic finite elements are used for the fluid pressure and for each component of the displacement respectively for the flux and stress components the h div conforming raviart thomas spaces of lowest order are used computational results are presented for some two dimensional test problems

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

the crack problem for nonhomogeneous materials under antiplane shear loading a displacement based formulation abstract this paper presents a displacement based integral equation formulation for the mode iii crack problem in a nonhomogeneous medium with a continuously differentiable shear modulus which is assumed to be an exponential function i e g x g0exp x this formulation leads naturally to a hypersingular integral equation the problem is solved for a finite crack and results are given for crack profiles and stress intensity factors the results are affected by the parameter describing the material nonhomogeneity this study is motivated by crack problems in strain gradient elasticity theories where higher order singular integral equations naturally arise even in the slope based formulation

a nonconforming high order method for the biot problem on general meshes in this work we introduce a novel algorithm for the biot problem based on a hybrid high order discretization of the mechanics and a symmetric weighted interior penalty discretization of the flow the method has several assets including in particular the support of general polyhedral meshes and arbitrary space approximation order our analysis delivers stability and error estimates that hold also when the specific storage coefficient vanishes and shows that the constants have only a mild dependence on the heterogeneity of the permeability coefficient numerical tests demonstrating the performance of the method are provided

biem analysis of dynamically loaded anti plane cracks in graded piezoelectric finite solids anti plane cracks in finite functionally graded piezoelectric solids under time harmonic loading are studied via a non hypersingular traction based boundary integral equation method biem the formulation allows for a quadratic variation of the material properties in two directions the boundary integral equation bie system is treated by using the frequency dependent fundamental solution based on radon transforms its numerical solution provides the displacements and tractions on the external boundary as well as the crack opening displacements from which the mechanical stress intensity factor sif and the electrical displacement intensity factor edif are determined several examples for single and multiple straight and curved cracks demonstrate the applicability of the method and show the influence of the different system parameters

boundary element solution for half plane problems abstract the complete fundamental solution due to unit point loads within the half plane is presented together with its application to the boundary element method this solution procedure has proved to be highly accurate and computationally more efficient than the analogue implementation of kelvin fundamental solution for problems concerning the half plane expressions for stresses at internal points are also presented and the application of the formulation to some classical problems is included

measuring graph similarity through continuous time quantum walks and the quantum jensen shannon divergence in this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs we design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous time quantum walks in order to analyze the behavior of the walks without causing wave function collapse we base our analysis on the recently introduced quantum jensen shannon divergence in particular we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic we also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the hamiltonians associated with the two original graphs have an empty intersection

characterizing graph symmetries through quantum jensen shannon divergence in this paper we investigate the connection between quantum walks and graph symmetries we begin by designing an experiment that allows us to analyze the behavior of the quantum walks on the graph without causing the wave function collapse to achieve this we base our analysis on the recently introduced quantum jensen shannon divergence in particular we show that the quantum jensen shannon divergence between the evolution of two quantum walks with suitably defined initial states is maximum when the graph presents symmetries hence we assign to each pair of nodes of the graph a value of the divergence and we average over all pairs of nodes to characterize the degree of symmetry possessed by a graph

numerical solution and its analysis for a system of hypersingular integral equations abstract a system of hypersingular integral equations occurs quite naturally in several branches of science and engineering during the formulation of many boundary value problems the analytical solution is known for the system of dominant equations however there are many real world problems such as crack problems occur in the field of fracture mechanics which may not be formulated as system of dominant equations therefore we propose a numerical method to find the approximate solution for such generalized form the convergence of the proposed method is proved in l n 2 space this convergence helps to derive theoretical error bound for the error between the exact and the approximate solution an application of the proposed method in finding numerical solution of hypersingular integral equation over the curves is shown finally the derived theoretical error bound is numerically calculated and validated with the help of numerical examples

an efficient diagonal preconditioner for finite element solution of biot s consolidation equations finite element simulations of very large scale soil structure interaction problems e g excavations tunnelling pile rafts etc typically involve the solution of a very large ill conditioned and indefinite biot system of equations the traditional preconditioned conjugate gradient solver coupled with the standard jacobi sj preconditioner can be very inefficient for this class of problems this paper presents a robust generalized jacobi gj preconditioner that is extremely effective for solving very large scale biot s finite element equations using the symmetric quasi minimal residual method the gj preconditioner can be formed inverted and implemented within an element by element framework as readily as the sj preconditioner it was derived as a diagonal approximation to a theoretical form which can be proven mathematically to possess an attractive eigenvalue clustering property the effectiveness of the gj preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three dimensional footing problem this paper casts a new perspective on the potentialities of the simple diagonal preconditioner which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix copyright 2002 john wiley sons ltd

stability and convergence of sequential methods for coupled flow and geomechanics fixed stress and fixed strain splits abstract we analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics in which the flow problem is solved first we employ the von neumann and energy methods for linear and nonlinear problems respectively we consider two sequential methods with the generalized midpoint rule for tn where is the parameter of time discretization namely the fixed strain and fixed stress splits the von neumann method indicates that the fixed strain split is only conditionally stable and that its stability limit is a coupling strength less than unity if 0 5 on the other hand the fixed stress split is unconditionally stable when 0 5 the amplification factors of the fixed stress split are different from those of the undrained split and are identical to the fully coupled method unconditional stability of the fixed stress split is also obtained from the energy method for poroelastoplasticity we show that the fixed stress split is contractive and b stable when 0 5 we also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward euler method in time from the estimates the fixed strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable the fixed stress split however is convergent for a fixed number of iterations showing better accuracy than the undrained split even when we cannot obtain the exact local bulk modulus or exact rock compressibility at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials the fixed stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus such as the dimension based estimation by which the employed local bulk modulus is less stiff than the exact local bulk modulus we provide numerical examples supporting all the estimates of stability and convergence for the fixed strain and fixed stress splits

an a posteriori based fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media in this work we develop an a posteriori based adaptive algorithm for thermal multiphase compositional flows in porous media the key ingredients are fully computable a posteriori error estimates bounding the dual norm of the residual supplemented by a nonconformity evaluation term the theory hinges on assumptions that allow the application to variety of discretization methods the estimators are then elaborated to estimate separately the space time linearization and algebraic errors this additional information is used to formulate a fully adaptive algorithm including adaptive stopping criteria for iterative solvers as well as refinement derefinement criteria for both the time step and the mesh size numerical validation is provided on an industrial case study in the context of oil recovery based on the steam assisted gravity drainage procedure implicit cell centered finite volumes with phase upwind and two point discretization of the diffusive fluxes are considered it is shown that significant gains in computational cost can be achieved in this example without hindering the quality of the results as measured by quantities of engineering interest

parameter robust discretization and preconditioning of biot s consolidation model biot s consolidation model in poroelasticity has a number of applications in science medicine and engineering the model depends on various parameters and in practical applications these parameters range over several orders of magnitude a current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations the purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete biot systems the approach taken here is to consider the stability of the problem in nonstandard or weighted hilbert spaces and employ the operator preconditioning approach we derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement the parameters of interest are small time step sizes large bulk and shear moduli and small hydraulic conductivity

analysis of block preconditioners for models of coupled magma mantle dynamics this article considers the iterative solution of a finite element discretization of the magma dynamics equations in simplified form the magma dynamics equations share some features of the stokes equations we therefore formulate analyze and numerically test an elman silvester and wathen type block preconditioner for magma dynamics we prove analytically and demonstrate numerically the optimality of the preconditioner the presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers the analysis is verified through a range of two and three dimensional numerical examples on unstructured grids from simple illustrative problems through to large problems on subduction zone like geometries the computer code to reproduce all numerical examples is freely available as supporting material

stability and convergence of sequential methods for coupled flow and geomechanics drained and undrained splits abstract we perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics in which the mechanics sub problem is solved first we consider slow deformations so that inertia is negligible and the mechanical problem is governed by an elliptic equation we use biot s self consistent theory to obtain the classical parabolic type flow problem we use a generalized midpoint rule parameter between 0 and 1 time discretization and consider two classical sequential methods the drained and undrained splits the von neumann method provides sharp stability estimates for the linear poroelasticity problem the drained split with backward euler time discretization 1 is conditionally stable and its stability depends only on the coupling strength and it is independent of time step size the drained split with the midpoint rule 0 5 is unconditionally unstable the mixed time discretization with 1 0 for mechanics and 0 5 for flow has the same stability properties as the backward euler scheme the von neumann method indicates that the undrained split is unconditionally stable when 0 5 we extend the stability analysis to the nonlinear regime poro elastoplasticity via the energy method it is well known that the drained split does not inherit the contractivity property of the continuum problem thereby precluding unconditional stability for the undrained split we show that it is b stable therefore unconditionally stable at the algorithmic level when 0 5 we also analyze convergence of the drained and undrained splits and derive the a priori error estimates from matrix algebra and spectral analysis we show that the drained split with a fixed number of iterations is not convergent even when it is stable the undrained split with a fixed number of iterations is convergent for a compressible system i e finite biot modulus for a nearly incompressible system i e very large biot modulus the undrained split loses first order accuracy and becomes non convergent in time we also study the rate of convergence of both splits when they are used in a fully iterated sequential scheme when the medium permeability is high or the time step size is large which corresponds to a high diffusion of pressure the error amplification of the drained split is lower and therefore converges faster than the undrained split the situation is reversed in the case of low permeability and small time step size we provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits in the linear and nonlinear regimes we also show that our spatial discretization finite volumes for flow and finite elements for mechanics removes the well documented spurious instability in consolidation problems at early times

three field block preconditioners for models of coupled magma mantle dynamics for a prescribed porosity the coupled magma mantle flow equations can be formulated as a two field system of equations with velocity and pressure as unknowns previous work has shown that while optimal preconditioners for the two field formulation can be obtained the construction of preconditioners that are uniform with respect to model parameters is difficult this limits the applicability of two field preconditioners in certain regimes of practical interest we address this issue by reformulating the governing equations as a three field problem which removes a term that was problematic in the two field formulation in favor of an additional equation for a pressure like field for the three field problem we develop and analyze new preconditioners and we show numerically that they are optimal in terms of problem size and less sensitive to model parameters compared to the two field preconditioner this extends the applicability of optimal preconditioners for coupled mantle magma dynamics into parameter

a von karman plate analogue for solving anti plane problems in couple stress and dipolar gradient elasticity abstract the purpose of the present work is to present a direct analogy regarding the formulation and solution of anti plane problems in the context of couple stress and dipolar gradient elasticity and the von karman plate framework we show aspects of boundary conditions in both theories of elasticity and propose a robust finite element methodology based on the von karman plate theory in order to solve complex anti plane mode iii crack problems furthermore we establish the equivalence between the anti plane gradient elasticity j integral and the plate i c integral sanders plate integral in passing we prove the path independency of the i c integral finally we examine the near tip fields for both anti plane and plate problems and establish possible strengthening effects due to the underlying microstructure the proposed analogy is achieved through the von karman plate theory where the plate is pre stressed by a constant biaxial tension the plate theory involves properties such as the plate thickness h the poisson s ratio and the bending stiffness d this information together with the pre stress n transforms into properties required by the anti plane couple stress and dipolar gradient elasticity problems the shear modulus g the internal length l and the coefficient in both problems the two dimensional space remains the same including the presence of cracks and other defects the analogy permits numerical and analytical solutions of demanding anti plane problems of gradient elasticity couple stress and dipolar utilizing the von karman plate corresponding and vice versa

evaluation of hypersingular integrals in the boundary element method by complex variable techniques strongly singular and hypersingular integrals that occur in many two dimensional applications of the boundary element method e g potential theory and liner elasticity can be treated in a unified manner by complex analysis this paper presents an elegant approach for the evaluation of such hypersingular integrals the limit to the boundary method where a source point is moved from the inside or outside of a body to its boundary is carried out using complex variables the connection of this limit with the hadamard finite part of hypersingular integrals and principal value integrals is established many examples are worked out throughout the paper to illustrate the theory

on the fixed stress split scheme as smoother in multigrid methods for coupling flow and geomechanics the fixed stress split method has been widely used as solution method in the coupling of flow and geomechanics in this work we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method in order to obtain an efficient monolithic solver for biot s problem this solver combines the advantages of being a fully coupled method with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm moreover the fixed stress split smoother is based on the physics of the problem and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori a local fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained the proposed geometric multigrid algorithm is used to solve several tests in semi structured triangular grids in order to show the good behavior of the method and its practical utility

weak solutions for nonlinear antiplane problems leading to hemivariational inequalities abstract we analyze the antiplane shear deformation of an elastic cylinder in frictional contact with a rigid foundation for static processes under the small deformation hypothesis based on the knaster kuratowski mazurkiewicz technique in the theory of the hemivariational inequalities we prove that the model has at least one weak solution moreover we present several examples of constitutive laws and friction laws for which our theoretical results are valid finally we comment on the conditions which guarantee the uniqueness of the solution

a quantum jensen shannon graph kernel for unattributed graphs in this paper we use the quantum jensen shannon divergence as a means of measuring the information theoretic dissimilarity of graphs and thus develop a novel graph kernel in quantum mechanics the quantum jensen shannon divergence can be used to measure the dissimilarity of quantum systems specified in terms of their density matrices we commence by computing the density matrix associated with a continuous time quantum walk over each graph being compared in particular we adopt the closed form solution of the density matrix introduced in rossi et al 2013 27 28 to reduce the computational complexity and to avoid the cumbersome task of simulating the quantum walk evolution explicitly next we compare the mixed states represented by the density matrices using the quantum jensen shannon divergence with the quantum states for a pair of graphs described by their density matrices to hand the quantum graph kernel between the pair of graphs is defined using the quantum jensen shannon divergence between the graph density matrices we evaluate the performance of our kernel on several standard graph datasets from both bioinformatics and computer vision the experimental results demonstrate the effectiveness of the proposed quantum graph kernel highlightswe compute a density matrix for a graph using the continuous time quantum walk we compute the quantum jensen shannon divergence between graph density matrixes we define a quantum jensen shannon graph kernel using the quantum divergence we evaluate the performance of our quantum kernel on standard graph datasets we demonstrate the effectiveness of the proposed quantum kernel

asymptotic solution of a class of strongly singular integral equations a class of strongly singular nonlinear integral equations is considered and sufficient conditions for the uniqueness of the solution and in the linear and homogeneous case the nonpositiveness of the associated eigenvalues are obtained when the singular integral is proportional to a small perturbation parameter a procedure for the construction of the uniformly valid leading term of the asymptotic solutions for both linear and nonlinear equations is presented the results are illustrated by means of an example that emerges in the analysis of microcrack growth in reinforced ceramics

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed

a hermite cubic collocation scheme for plane strain hydraulic fractures we describe a novel cubic hermite collocation scheme for the solution of the coupled integro partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain special blended cubic hermite power law basis functions with arbitrary index 0 1 are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture the implementation of blended infinite elements to model semi infinite crack problems is also described explicit formulae for the integrated kernels associated with the cubic hermite and blended basis functions are provided the cubic hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions the convergence rate of the cubic hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the o h4 error due to the hermite cubic interpolation the errors due to these two approximations need to be matched in order to achieve optimal convergence backward euler time stepping yields a robust algorithm that along with geometric increments in the time step can be used to explore the transition between propagation regimes over many orders of magnitude in time

divergence measures based on the shannon entropy a novel class of information theoretic divergence measures based on the shannon entropy is introduced unlike the well known kullback divergences the new measures do not require the condition of absolute continuity to be satisfied by the probability distributions involved more importantly their close relationship with the variational distance and the probability of misclassification error are established in terms of bounds these bounds are crucial in many applications of divergence measures the measures are also well characterized by the properties of nonnegativity finiteness semiboundedness and boundedness

convergence analysis of multirate fixed stress split iterative schemes for coupling flow with geomechanics abstract we consider multirate iterative schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium the multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence we provide a fully discrete scheme that uses backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics our analysis is based on studying the equations satisfied by the difference of iterates and using banach contraction argument to prove that the corresponding scheme is a fixed point contraction the analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms furthermore we show that the converged quantities satisfy the variational weak form for the coupled discrete system

on stability and convergence of finite element approximations of biot s consolidation problem stability and convergence analysis of finite element approximations of biot s equations governing quasistatic consolidation of saturated porous media are discussed a family of decay functions parametrized by the number of time steps is derived for the fully discrete backward euler galerkin formulation showing that the pore pressure oscillations arising from an unstable approximation of the incompressibility constraint on the initial condition decay in time error estimates holding over the unbounded time domain for both semidiscrete and fully discrete formulations are presented and a post processing technique is employed to improve the pore pressure accuracy

a sequential adaptive strategy in space time with application to consolidation of porous media issues related to space time adaptivity for a class of nonlinear and time dependent problems are discussed the dg k methods are adopted for the time integration and the a posteriori error control is based on the appropriate dual problem in space time one key ingredient is to decouple the error generation in space and time with a hierarchical decomposition of the discrete space of dual solutions the main idea put forward in the paper is to increase the computational efficiency of the adaptive scheme by avoiding recursive adaptations of the whole time mesh rather the space mesh and the time step defining each finite space time slab are defined in a truly sequential fashion the proposed adaptive strategy is applied to the coupled consolidation problem in geomechanics involving large deformations its performance is investigated with the aid of a numerical example in 2d

boundary element analysis of mode i and mixed mode i and ii crack problems of 2 d gradient elasticity abstract a boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading a simple version of mindlin s general theory of gradient elastic materials is employed and the two required boundary integral equations one for the displacement vector and the other for its normal derivative are presented use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization two representative numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the first deals with a mode i crack while the second with a mixed mode i ii crack for the second case the proposed method is used in conjunction with the method of subregions the method is employed with regular and special near the crack tip boundary elements the gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity

hypersingular and finite part integrals in the boundary element method abstract a new definition of the hadamard finite part hfp of hypersingular integrals is proposed in this paper this definition does not involve a limiting process it is completely general and is valid for one as well as higher dimensional integrals on closed as well as on open surfaces it reduces respectively to the cauchy principal value cpv and riemann integral respectively for the special cases of strongly singular and weakly singular integrands of course suitable symmetric exclusion zones must be chosen to realize cpv integrals starting with this new definition of the hfp of certain hypersingular boundary integral equations hbie that arise in potential theory and in wave scattering a regularization method is carried out in order to express the hypersingular integrals in terms of ones that are at most weakly singular the regularized versions are completely consistent with those available in the recent literature where a different definition of the hfp was employed