an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

a computational framework for polyconvex large strain elasticity abstract this paper presents a novel computational formulation for large strain polyconvex elasticity the formulation based on the original ideas introduced by schroder et al 2011 introduces the deformation gradient the fibre map its adjoint the area map and its determinant the volume map as independent kinematic variables of a convex strain energy function compatibility relationships between these variables and the deformed geometry are enforced by means of a multi field variational principle with additional constraints this process allows the use of different approximation spaces for each variable the paper extends the ideas presented in schroder et al 2011 by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the hellinger reissner type where the new conjugate stresses are primary variables together with the deformed geometry both compressible and incompressible or nearly incompressible elastic models are considered a key element to the developments presented in the paper is the new use of a tensor cross product presented for the first time by de boer 1982 page 76 which facilitates the algebra associated with the adjoint of the deformation gradient for the numerical examples quadratic interpolation of the displacements piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the jacobian and its stress conjugate are considered for compressible cases in the case of incompressible materials two formulations are presented first continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second linear continuous interpolation for both displacement and pressure stabilised via a petrov galerkin technique

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity analytical estimates this is the second of two papers in which we motivate introduce and analyze a new type of strategy for the stabilization of discontinuous galerkin dg methods in nonlinear elasticity problems the foremost goal behind it is to enhance the robustness of the method without deteriorating the accuracy of the resulting solutions its distinctive property is that for nonlinear elastic problems the stabilization term is solution dependent and hence it is termed an adaptive stabilization mechanism the key contribution of this paper is the construction of a stabilization strategy for which the method is perfectly stable since the stabilization parameters can be explicitly computed as part of the numerical solution this is accomplished through the main result of this paper which consists of a theorem that provides lower bounds for the size of the stabilization parameters numerical examples confirm the guaranteed stability of the resulting method however they also show that the computed lower bounds overestimate the minimum amount of stabilization needed negatively affecting the approximation properties of the method these results underscore the fact that a better understanding of the stabilization mechanisms is needed in order to construct a method that is both robust and efficient

a stabilized mixed finite element method for finite elasticity formulation for linear displacement and pressure interpolation abstract a stabilized mixed finite element method for finite elasticity is presented the method circumvents the fulfillment of the ladyzenskaya babuska brezzi condition by adding mesh dependent terms which are functions of the residuals of the euler lagrange equations to the usual galerkin method the weak form and the linearized weak form are presented in terms of the reference and current configuration numerical experiments using a tetrahedral element with linear shape functions for the displacements and for the pressure show that the method successfully yields a stabilized element

stabilized finite element method for viscoplastic flow formulation and a simple progressive solution strategy abstract this paper presents a stabilized finite element formulation for steady state viscoplastic flow and a simple strategy for solving the resulting non linear equations with a newton raphson algorithm an eulerian stabilized finite element formulation is presented where mesh dependent terms are added element wise to enhance the stability of the mixed finite element formulation a local reconstruction method is used for computing derivatives of the stress field needed when higher order elements are used linearization of the weak form is derived to enable a newton raphson solution procedure of the resulting non linear equations in order to get convergence in the newton raphson algorithm a trial solution is needed which is within the radius of convergence an effective strategy for progressively moving inside the radius of convergence for highly non linear viscoplastic constitutive equations typical of metals is presented numerical experiments using the stabilization method with both linear and quadratic shape functions for the velocity and pressure fields in viscoplastic flow problems show that the stabilized method and the progressive convergence strategy are effective in non linear steady forming problems finally conclusions are inferred and extensions of this work are discussed

penalty function formulations for incompressible nonlinear elastostatics abstract the analysis of incompressible nonlinear elastic solids is considered by a penalty function approach in particular the necessary compatibility conditions for developing penalty forms for isotropic nonlinear elasticity are addressed and the commonly used form of strain energy functional leading to a penalty formulation of the incompressibility constraint extended the mooney rivlin model is used to show how previous developments can lead to physically meaningless situations for a certain class of incompressible materials which includes the important mooney rivlin model a new and particularly simple formulation is proposed

an adaptive stabilization strategy for enhanced strain methods in non linear elasticity this paper proposes and analyzes an adaptive stabilization strategy for enhanced strain es methods applied to quasistatic non linear elasticity problems the approach is formulated for any type of enhancements or material models and it is distinguished by the fact that the stabilization term is solution dependent the stabilization strategy is first constructed for general linearized elasticity problems and then extended to the non linear elastic regime via an incremental variational principle a heuristic choice of the stabilization parameters is proposed which in the numerical examples proved to provide stable approximations for a large range of deformations different problems and material models we also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable these are not advocated since they are generally larger than the ones based on heuristics and hence prone to deteriorate the locking free behavior of es methods numerical examples with two different non linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations finally the method is strongly based on earlier developments for discontinuous galerkin methods and hence throughout the paper we offer a perspective about the similarities between the two

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

implicit finite incompressible elastodynamics with linear finite elements a stabilized method in rate form abstract we propose a stabilization method for linear tetrahedral finite elements suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics in particular we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models in this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work which was instead focused on explicit time integration and simple neo hookean models for nearly incompressible elasticity with the goal of computational efficiency we also present a two step block gauss seidel strategy for the time update of displacements velocities and pressures specifically a mixed system of equations for the velocity and pressure is updated implicitly in a first stage and the displacements are updated explicitly in a second stage the proposed mixed formulation is then embedded in newton type strategies for the nonlinear solution of the equations of motion various implicit time integration strategies are considered and particularly we focus on high frequency dissipation time integrators which are preferable in transient mechanics applications an extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach

a new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems abstract in order to accurately model the behaviour of geostructures it is usually not possible to neglect the interaction between the soil skeleton and the pore fluid classical finite element models taking into account this interaction are formulated in terms of the displacement and pore pressure fields and are based on the assumption that the fluid acceleration relative to the soil skeleton is negligible this type of mixed problems is similar to others found in solid and fluid mechanics and might give rise to numerical instabilities unless certain requirements are met there are two classical approaches to this problem the first is usually known as the zienkiewicz taylor patch test for mixed formulations as a consequence of this test the interpolation degree of the displacement field is required to be higher than the corresponding one of the pressure field mathematically speaking this is a necessary condition for stability the second approach mathematically more involved is usually known as the bab ska brezzi inf sup condition and constitutes a sufficient condition for stability however it is possible to obtain stable formulations circumventing the interpolation degree requirement through the so called stabilization techniques these techniques were initially applied in the context of fluid mechanics and later extended to solid mechanics this article presents a new formulation in which stabilization is achieved through an approach based on the simo rifai enhanced strain element

mixed linear linear simplicial elements for incompressible elasticity and plasticity this paper exploits the concept of orthogonal sub grid scales to stabilize the behaviour of mixed linear linear simplicial elements triangles and tetrahedra in incompressible or nearly incompressible situations both incompressible elastic and j2 plastic constitutive behaviours have been considered the different assumptions and approximations used to derive the method are exposed implementation and computational aspects are also discussed showing that a robust application of the proposed formulation is feasible numerical examples show that the elements derived are free of volumetric locking and spurious oscillations of the pressure and that the results obtained compare favourably with those obtained with the q1p0 quadrilateral

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

an iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics abstract the discretization of the u p model for saturated porous media results in the semi discrete system of mixed type in displacements and pressures the babuska brezzi condition or the simpler patch test proposed by zienkiewicz and taylor precludes the use of elements with the equal low order of interpolation for u and p unless special stabilization techniques such as the fractional step algorithms are used the purpose of this paper is to present a modified version of the fractional step algorithm which allows much larger time step sizes than those for the existing ones the method is based on introducing an iteration algorithm the numerical experiments demonstrate the effectiveness and improved performance of the proposed modified version of the fractional step algorithm

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

efficient and principled method for detecting communities in networks a fundamental problem in the analysis of network data is the detection of network communities groups of densely interconnected nodes which may be overlapping or disjoint here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models we show how the method can be implemented using a fast closed form expectation maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times we test the method both on real world networks and on synthetic benchmarks and find that it gives results competitive with previous methods we also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem

a vertex centred finite volume jameson schmidt turkel jst algorithm for a mixed conservation formulation in solid dynamics a vertex centred finite volume algorithm is presented for the numerical simulation of fast transient dynamics problems involving large deformations a mixed formulation based upon the use of the linear momentum the deformation gradient tensor and the total energy as conservation variables is discretised in space using linear triangles and tetrahedra in two dimensional and three dimensional computations respectively the scheme is implemented using central differences for the evaluation of the interface fluxes in conjunction with the jameson schmidt turkel jst artificial dissipation term the discretisation in time is performed by using a total variational diminishing tvd two stage runge kutta time integrator the jst algorithm is adapted in order to ensure the preservation of linear and angular momenta the framework results in a low order computationally efficient solver for solid dynamics which proves to be very competitive in nearly incompressible scenarios and bending dominated applications

a stabilised petrov galerkin formulation for linear tetrahedral elements in compressible nearly incompressible and truly incompressible fast dynamics a mixed second order stabilised petrov galerkin finite element framework was recently introduced by the authors lee et al 2014 46 the new mixed formulation written as a system of conservation laws for the linear momentum and the deformation gradient performs extremely well in bending dominated scenarios even when linear tetrahedral elements are used yielding equal order of convergence for displacements and stresses in this paper this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties first a new conservation law for the jacobian of the deformation is added into the system providing extra flexibility to the scheme second a variationally consistent petrov galerkin stabilisation methodology is derived third an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics for completeness and ease of understanding these three improvements are presented both in small and large strain regimes studying the eigenstructure of the resulting systems a series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

multiple positive solutions of the one dimensional p laplacian in this work we investigate the existence of positive solutions of the p laplacian using the quadrature method we prove the existence of multiple solutions of the one dimensional p laplacian for 0 and determine their exact number for 0 2005 elsevier inc all rights reserved

triangles and tetrahedra in explicit dynamic codes for solids explicit dynamic codes which are used currently for the study of plastic deformations in impact or with some modification for metal forming suffer two serious limitations first only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing second even with the use of such elements special devices such as reduced integration must be introduced to avoid locking and reduce costs these necessitate complex hour glass control mending type procedures the main difficulties are those due to the need of treatment of almost incompressible deformation modes recently similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem thus allowing an almost unrestricted choice of element interpolations 1998 john wiley sons ltd

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

a hybridizable discontinuous galerkin formulation for non linear elasticity abstract we revisit the hybridizable discontinuous galerkin method for non linear elasticity introduced by s c soon 2008 we show that it can be recast as a minimization problem of a non linear functional over a space of discontinuous approximations to the displacement the functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter element jumps of the displacement we then show that if this new energy is not properly weighted the minimizers might not converge to the exact solution we construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter element jumps finally we explore the performance of the method for the case of piecewise linear approximations in rather demanding situations in both two dimensional and for the first time three dimensional situations they include almost incompressible materials large deformations with large shear layers and cavitation we also compare the method with the continuous galerkin method and a previously explored discontinuous galerkin method and show that when using piecewise linear approximations and a moderate number of degrees of freedom the current method turns out to be more efficient for the computation of the gradient