on the stability of the reduced basis method for stokes equations in parametrized domains we present an application of reduced basis method for stokes equations in domains with affine parametric dependence the essential components of the method are i the rapid convergence of global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n selected points in parameter space ii the off line on line computational procedures decoupling the generation and projection stages of the approximation process the operation count for the on line stage in which given a new parameter value we calculate an output of interest depends only on n typically very small and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation design optimization and real time control particular attention is given i to the pressure treatment of incompressible stokes problem ii to find an equivalent inf sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem iii to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions iv to reduce computational costs in order to allow real time solution of parametrized problem

vector level sets for description of propagating cracks in finite elements a new level set method is developed for describing surfaces that are frozen behind a moving front such as cracks in this formulation the level set is described in two dimensions by a three tuple the sign of the level set function and the components of the closest point projection to the surface the update of the level set is constructed by geometric formulas which are easily implemented results are given for growth of lines in two dimensions that show the method is very accurate the method combines very naturally with the extended finite element method xfem where the discontinuous enrichment for cracks is best described in terms of level set functions examples of crack growth simulations obtained by combining this level set method with the extended finite element method are given copyright 2003 john wiley sons ltd

a new certification framework for the port reduced static condensation reduced basis element method abstract in this paper we introduce a new certification framework for the port reduced static condensation reduced basis element pr scrbe method which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides in an offline computational stage we construct a library of interoperable parametrized reference components in the subsequent online stage we instantiate and connect the components at the interfaces ports to form a system of components to compute a truth finite element approximation of the say coercive elliptic partial differential equation on the component based system we use a domain decomposition approach for an efficient simulation we employ two different types of model reduction a reduced basis rb approximation within the interior of the component huynh et al 2013 and empirical port reduction eftang and patera 2013 on the ports where the components connect we demonstrate the well posedness of the pr scrbe approximation and introduce a new certification framework to assess the quality of the port reduction we use conservative fluxes we adapt the standard estimators from rb methods to the scrbe setting to derive an a posteriori error estimator for the rb error contribution in order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for pr scrbe we adapt techniques from multi scale methods and component mode synthesis finally we prove that the effectivity of the derived estimator can be bounded we provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational online costs

extra dof free and linearly independent enrichments in gfem abstract a gfem without extra dof is developed without the extra dof the resulting linear system becomes independent in size of the order of local approximation the resulting pu approximation becomes free of linear dependence and the resulting stiffness matrix becomes good conditioned numerical studies show the new gfem s excellent convergence properties and excellent stability

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

combined extended and superimposed finite element method for cracks a combination of the extended finite element method xfem and the mesh superposition method s version fem for modelling of stationary and growing cracks is presented the near tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity the two displacement fields are matched through a transition region the method can robustly deal with stationary crack and crack growth it simplifies the numerical integration of the weak form in the galerkin method as compared to the s version fem numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method copyright 2004 john wiley sons ltd

topology optimization with implicit functions and regularization summary topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape the implicit function is constrained by upper and lower bounds so that only a band of nodal variables needs to be considered in each step of the optimization the weak form of the equilibrium equation is expressed as a heaviside function of the implicit function the heaviside function is regularized to permit the evaluation of sensitivities we show that the method is a dual of the bendshe kikuchi method the method is applied both to problems of optimizing single material and

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

extended isogeometric analysis for simulation of stationary and propagating cracks summary a novel approach based on a combination of isogeometric analysis iga and extended fem is presented for fracture analysis of structures the extended isogeometric analysis is capable of an efficient analysis of general crack problems using nonuniform rational b splines as basis functions for both the solution field approximation and the geometric description and it can reproduce crack tip singular fields and discontinuity across a crack iga has attracted a lot of interest for solving different types of engineering problems and is now further extended for the analysis of crack stability and propagation in two dimensional isotropic media concepts of the extended fem are used in iga to avoid the necessity of remeshing in crack propagation problems and to increase the solution accuracy around the crack tip crack discontinuity is represented by the heaviside function and isotropic analytical displacement fields near a crack tip are reproduced by means of the crack tip enrichment functions also the lagrange multiplier method is used to impose essential boundary conditions moreover the subtriangles technique is utilized for improving the accuracy of integration by the gauss quadrature rule several two dimensional static and quasi static crack propagation problems are solved to demonstrate the efficiency of the proposed method and the results of mixed mode stress intensity factors are compared with analytical and extended fem results copyright 2011 john wiley sons ltd

the intrinsic xfem a method for arbitrary discontinuities without additional unknowns a new method for treating arbitrary discontinuities in a finite element fe context is presented unlike the standard extended fe method xfem no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities the method constructs an approximation space consisting of mesh based enriched moving least squares mls functions near discontinuities and standard fe shape functions elsewhere there is only one shape function per node and these functions are able to represent known characteristics of the solution such as discontinuities singularities etc the mls method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional thereby weight functions are involved and special mesh based weight functions are proposed in this work the enrichment is achieved through the intrinsic basis the method is illustrated for linear elastic examples involving strong and weak discontinuities and matches optimal rates of convergence even for crack tip applications copyright 2006 john wiley sons ltd

spectral based discontinuous galerkin reduced basis element method for parametrized stokes problems in this work we extend to the stokes problem the discontinuous galerkin reduced basis element dgrbe method introduced in antonietti etal 2015 by this method we aim at reducing the computational cost for the approximation of a parametrized stokes problem on a domain partitioned into subdomains during an offline stage expensive but performed only once a low dimensional approximation space is built on each subdomain for any new value of the parameter the rapid evaluation of the solution takes place during the online stage and consists in a galerkin projection onto the low dimensional subspaces computed offline the high fidelity discretization on each subdomain used to build the local low dimensional subspaces is based on spectral element methods the continuity of both the velocity and the normal component of the cauchy stress tensor at subdomain interfaces is weakly enforced by a discontinuous galerkin approach

numerically determined enrichment functions for the extended finite element method and applications to bi material anisotropic fracture and polycrystals strain singularities appear in many linear elasticity problems a very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method fem special enrichment functions describing this singular behavior can be used in the extended finite element method x fem to circumvent this problem these functions have to be known in advance but their analytical form is unknown in many cases li et al described a method to calculate singular strain fields at the tip of a notch numerically a slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain we will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the x fem the convergence rates have been compared with the ones obtained by the fem this was done for a series of problems including a polycrystalline structure copyright 2010 john wiley sons ltd

the extended finite element method for rigid particles in stokes flow a new method for the simulation of particulate flows based on the extended finite element method x fem is described in this method the particle surfaces need not conform to the finite element boundaries so that moving particles can be simulated without remeshing the near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh we present a weak form of the equations of motion useful for the simulation of freely moving particles and solve example problems for particles with prescribed and unknown velocities copyright 2001 john wiley sons ltd

stable generalized iso geometric analysis sgiga for problems with discontinuities and singularities abstract numerical analysis of physical mathematical problems based on generalized extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks the presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed in order to alleviate these drawbacks three different stable generalized isogeometric analysis sgiga methods are proposed in this paper in sgiga the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function problems with weak and strong discontinuities singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed sgiga methods from the results it is observed that sgiga yields optimal convergence rates as well as better conditioning of system matrices the results obtained from the proposed sgiga methods are also compared with the results from the established methods stable generalized finite element method sgfem and extended isogeometric analysis xiga to study the relative performances with respect to accuracy and conditioning

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

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

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

extended finite element method with edge based strain smoothing esm xfem for linear elastic crack growth abstract this paper presents a strain smoothing procedure for the extended finite element method xfem the resulting edge based smoothed extended finite element method esm xfem is tailored to linear elastic fracture mechanics and in this context to outperform the standard xfem in the xfem the displacement based approximation is enriched by the heaviside and asymptotic crack tip functions using the framework of partition of unity this eliminates the need for the mesh alignment with the crack and re meshing as the crack evolves edge based smoothing es relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes and produces a softening effect leading to a close to exact stiffness super convergence and ultra accurate solutions the present method takes advantage of both the es fem and the xfem thanks to the use of strain smoothing the subdivision of elements intersected by discontinuities and of integrating the singular derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

a discontinuous galerkin method with nonoverlapping domain decomposition for the stokes and navier stokes problems a family of discontinuous galerkin finite element methods is formulated and analyzed for stokes and navier stokes problems an inf sup condition is established as well as optimal energy estimates for the velocity and l 2 estimates for the pressure in addition it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces

monotonicity methods for nonlinear diffusion equations and their approximations with error estimates abstract this paper deals with the initial boundary value problem p for the nonlinear diffusion equation u t 1 u g in 0 t in a general domain r n with smooth bounded boundary where n n t 0 and is a single valued maximal monotone function on r e g r r q 1 r q 0 the above equation represents a number of well known models e g porous media equation colli and fukao 6 studied the above problem p and the approximate problem p e which consists of the cahn hilliard system with error estimates when is a bounded domain n 2 or 3 and 1 is replaced with they considered one more approximation p e to solve p e they used compactness methods for abstract doubly nonlinear evolution equations to solve the second approximate problem p e they established the error estimate saying that the solution of p e converges to the solution of p as e 0 the present work asserts that one can solve the original problem p and the approximate problem p e individually and directly even if is a general domain moreover this paper gives an error estimate between the solution of p e and the solution of p

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

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

a reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks abstract in this paper we propose a reduced basis hybrid method rbhm for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent the rbhm is applied to stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled the rbhm is built upon the reduced basis element method rbem and it takes advantage from both the reduced basis methods rb and the domain decomposition method we move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes on the latter representative solutions corresponding to the same governing partial differential equations are computed for different values of some parameters of interest representing for example the deformation of the blocks a generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration the desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions the geometrical parametrization of the domain by transfinite mapping induces non affine parameter dependence an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline online decomposition of the reduced basis procedure this computational decomposition yields a considerable reduction of the problem complexity results computed on some combinations of 2d and 3d geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses pressure and velocity at subdomain interfaces

stability of finite elements under divergence constraints a finite dimensional stability test for checking velocity pressure finite element trial spaces is presented applications are made to a new class of element pairs proposed in this paper as well as to existing spaces

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

a static condensation reduced basis element method complex problems abstract we extend the static condensation reduced basis element scrbe method to treat the class of parametrized complex helmholtz partial differential equations the main ingredients are i static condensation at the interdomain level ii a conforming eigenfunction port representation at the interface level iii the reduced basis rb approximation of finite element fe bubble functions at the intradomain level and iv rigorous system level error bounds which reflect rb perturbation of the fe schur complement we then incorporate these ingredients in an offline online computational strategy to achieve rapid and accurate prediction of parametric systems formed from instantiations of interoperable parametrized archetype components from a library we introduce a number of extensions with respect to the original scrbe framework first primal dual rb methods for general non symmetric complex problems second stability constant procedures for weakly coercive problems at both the interdomain level and intradomain level third treatment of non port linear functional outputs as well as functions of outputs fourth consideration of particular components and outputs relevant to acoustic applications we consider several numerical examples in acoustics in particular focused on mufflers and horns to demonstrate that the approach can handle models with many parameters and or topology variations with particular reference to waveguide like applications

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries the aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques the main idea behind this approach is to compute once locally and for few reference shapes some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries these functions will represent the basis of a reduced space where the global solution is sought for the continuity of the latter is assured by a classical domain decomposition approach test results on poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed or the set of boundary conditions used for defining locally the basis functions the proposed approach simplifies the pre computation of the reduced basis space by splitting the global problem into smaller local subproblems thanks to this feature it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed c 2015 elsevier ltd all rights reserved

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

a combined extended finite element and level set method for biofilm growth this paper presents a computational technique based on the extended finite element method xfem and the level set method for the growth of biofilms the discontinuous derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm fluid interface the xfem is coupled with a comprehensive level set update scheme with velocity extensions which makes updating the biofilm interface fast and accurate without need for remeshing the kinetics of biofilms are briefly given and the non linear strong and weak forms are presented the non linear system of equations is solved using a newton raphson scheme example problems including 1d and 2d biofilm growth are presented to illustrate the accuracy and utility of the method the 1d results we obtain are in excellent agreement with previous 1d results obtained using finite difference methods our 2d results that simulate finger formation and finger tip splitting in biofilms illustrate the robustness of the present computational technique copyright 2007 john wiley sons ltd

reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries roles of the inf sup stability constants in this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady stokes flows in affinely parametrized geometries focusing on the role played by the brezzi s and babuska s stability constants the crucial ingredients of the methodology are a galerkin projection onto a low dimensional space of basis functions properly selected an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables the combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction ideally suited for real time simulation and many query contexts e g optimization control or parameter identification in particular in this work we focus on i the stability of the reduced basis approximation based on the brezzi s saddle point theory and the introduction of a supremizer operator on the pressure terms ii a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the babuska s inf sup constant including residuals calculations iii the computation of a lower bound of the stability constant and iv different options for the reduced basis spaces construction we present some illustrative results for both interior and external steady stokes flows in parametrized geometries representing two parametrized classical poiseuille and couette flows a channel contraction and a simple flow control problem around a curved obstacle

a unified enrichment approach addressing blending and conditioning issues in enriched finite elements abstract we present a combination of techniques to improve the convergence and conditioning properties of partition of unity pu enriched finite element methods by applying these techniques to different types of enrichment functions namely polynomial discontinuous and singular higher order convergence rates can be obtained while keeping condition number growth rates similar to the ones corresponding to standard finite elements

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

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

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

enrichment of the method of finite spheres using geometry independent localized scalable bubbles summary in this paper we report the development of two new enrichment techniques for the method of finite spheres a truly meshfree method developed for the solution of boundary value problems on geometrically complex domains in the first method the enrichment functions are multiplied by a weight function with compact support while in the second one a floating enrichment node is introduced the scalability of the enrichment bubbles offers flexibility in localizing the spatial extent to which the enrichment field is applied the bubbles are independent of the underlying geometric discretization and therefore provide a means of achieving convergence without excessive refinement several numerical examples involving problems with singular stress fields are provided demonstrating the effectiveness of the enrichment schemes and contrasting them to traditional geometry dependent enrichment strategies in which one or more nodes associated with the geometric discretization of the domain are enriched an additional contribution of this paper is the use of a meshfree numerical integration technique for computing the j integral using the domain integral method copyright 2006 john wiley sons ltd

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

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

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

three dimensional improved xfem ixfem for static crack problems abstract an improved xfem ixfem for three dimensional linear elastic fracture mechanics lefm problems is developed it utilizes an extra dof free pu approximation to fundamentally overcome the linear dependence and the ill conditioning issues of the standard xfem and the corrected xfem cxfem the introduced pu approximation is based on local least squares fitting with a one point interpolation constraint the resulting pu approximation interpolates and has no difficulties in boundary treatments detailed comparisons are made with the current xfems in 3d in terms of accuracy convergence and conditioning properties or solver efficiency better accuracy and convergence are demonstrated the remarkably better efficiency with a subspace iterative solver the de facto in large scale simulation is observed the number of iterations for the system of linear equations is reduced by orders of magnitude compared with the current xfems

convergence of domain integrals for stress intensity factor extraction in 2 d curved cracks problems with the extended finite element method summary the aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors sif when solving 2 d curved crack problems with the extended fem xfem domain integrals specially the j integral and the interaction integral are widely used for sif extraction and provide high accurate estimations with fems the crack description in xfem is usually realized using level sets this allows to define a local basis associated with the crack geometry in this work the effect of the level set local basis definition on the domain integral has been studied the usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems and we introduce some modifications to improve the behavior in curved crack analyses despite the good accuracy of domain integrals convergence rates are not always optimal and convergence to the exact solution cannot be assured for curved cracks the lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields with our modified integral proposal the optimal convergence rate is achieved by controlling the q function and the size of the extraction domain copyright 2013 john wiley sons ltd