discontinuous galerkin time discretization in elastoplasticity motivation numerical algorithms and applications discontinuous galerkin discretizations promise to become a very flexible tool in hp adaptive space time discretizations this is very attractive for moving interphase problems such as the free boundary between the elastic and plastic phase in elastoplastic time evolution the mathematical model of which involves variational inequalities and so the distributional time derivative is not obviously generalized to discontinuous test functions this paper motivates and introduces a discontinuous galerkin dg time discretization solution algorithms and examples are established which support feasibility and accuracy of the proposed schemes dg 0 and dg 1 the methods are compared with a backward euler and crank nicholson scheme 2002 published by elsevier science b v

discontinuous galerkin method based on peridynamic theory for linear elasticity modeling of discontinuities shock waves crack surfaces etc in solid mechanics is one of the major research areas in modeling the mechanical behavior of materials among the numerical methods the discontinuous galerkin method dgm poses some advantages in solving these problems in this study a novel formulation for dgm is derived for elastostatics based on the peridynamic theory derivation of the proposed formulation is presented numerical analyses are performed for different problems and the numerical results are compared to that of the known exact solutions of the problems the proposed weak formulation is stable and coercive peridynamic discontinuous galerkin formulation is found to be robust and successful in modeling elastostatic problems copyright 2011 john wiley sons ltd

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

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 hybridizable discontinuous galerkin method for linear elasticity this paper describes the application of the so called hybridizable discontinuous galerkin hdg method to linear elasticity problems the method has three significant features the first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements the corresponding stiffness matrix is symmetric positive definite and possesses a block wise sparse structure that allows for a very efficient implementation of the method the second feature is that when polynomials of degree k are used to approximate the displacement and the stress both variables converge with the optimal order of k 1 for any k 0 the third feature is that by using an element by element post processing a new approximate displacement can be obtained that converges at the order of k 2 whenever k 2 numerical experiments are provided to compare the performance of the hdg method with that of the continuous galerkin cg method for problems with smooth solutions and to assess its performance in situations where the cg method is not adequate that is when the material is nearly incompressible and when there is a crack copyright 2009 john wiley sons ltd

on discontinuous galerkin methods discontinuous galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present several alternatives for treating the diffusion and advective fluxes have been introduced this report summarizes some of the methods that have been proposed several numerical examples are included in the paper these present discontinuous galerkin solutions of one dimensional problems with a scalar variable results are presented for diffusion reaction problems and advection diffusion problems we discuss the performance of various formulations with respect to accuracy as well as stability of the method copyright 2003 john wiley sons ltd

homotopical variations and high dimensional zariski van kampen theorems we give a new definition of the homotopical variation operators occurring in a recent high dimensional zariski van kampen theorem a definition which opens the way to further generalizations of theorems of this kind

a discontinuous galerkin method for nonlinear shear flexible shells abstract in this paper a discontinuous galerkin method for a nonlinear shear flexible shell theory is proposed that is suitable for both thick and thin shell analysis the proposed method extends recent work on reissner mindlin plates to avoid locking without the use of projection operators such as mixed methods or reduced integration techniques instead the flexibility inherent to discontinuous galerkin methods in the choice of approximation spaces is exploited to satisfy the thin plate compatibility conditions a priori a benefit of this approach is that only generalized displacements appear as unknowns we take advantage of this to craft the method in terms of a discrete energy minimization principle thereby restoring the rayleigh ritz approach in addition to providing a straightforward and elegant derivation of the discrete equilibrium equations the variational character of the method could afford numerous advantages in terms of mesh adaptation and available solution techniques the proposed method is exercised on a set of benchmarks and example problems to assess its performance numerically and to test for shear and membrane locking

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

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

on the use of space time finite elements in the solution of elasto dynamic problems with strain discontinuities abstract the use of a discontinuous galerkin dg formulation for the solution of elasto dynamic problems with discontinuities in the displacement gradients is examined comparisons with exact solutions are presented the paper demonstrates that space time finite element methods based on a dg formulation can be very effective in the study of dynamic solid solid phase transitions as well as dynamic fracture

a fast convergent rate preserving discontinuous galerkin framework for rate independent plasticity problems abstract in this paper a finite element framework based on the incomplete interior penalty galerkin formulation a non symmetric discontinuous galerkin method is consistently formulated for modeling plasticity problems with small deformation because of its pure displacement based framework this proposed discontinuous galerkin method is possibly able to completely preserve numerical integration algorithms efficiently developed in the traditional continuous galerkin framework besides stresses on element interior quadrature points stresses on element surface quadrature points are also required to return on yielding surfaces in this discontinuous galerkin framework which is able to provide more accurate material yielding profiles than the continuous galerkin framework the performance of the proposed discontinuous galerkin framework has been evaluated in detail for j 2 and pressure dependent plasticities using perfect plasticity plasticity with hardening and associative and non associative material models quadratic convergent rates compatible to the tradition continuous galerkin method for modeling plasticity problems have been achieved within a large penalty range in a nodal based discontinuous galerkin implementation

on the spatial formulation of discontinuous galerkin methods for finite elastoplasticity abstract in this paper we present a consistent spatial formulation for discontinuous galerkin dg methods applied to solid mechanics problems with finite deformation this spatial formulation provides a general accurate and efficient dg finite element computational framework for modeling nonlinear solid mechanics problems to obtain a consistent formulation we employ the incomplete interior penalty galerkin iipg method another requirement for achieving a fast convergence rate for newton s iterations is the consistent formulation of material integrators we show that material integrators that are well developed and tested in continuous galerkin cg methods can be fully exploited for dg methods by additionally performing stress returning on element interfaces finally for problems with pressure or follower loading stiffness contributed from loaded surfaces must also be consistently incorporated in this work we propose the truesdell objective stress rate for both hypoelastoplastic and hyperelastoplastic problems two formulations based on the co rotational and multiplicative decomposition based frameworks are implemented for hypoelastoplasticity and hyperelastoplasticity respectively two new terminologies the so called standard surface geometry stiffness and the penalty surface geometry stiffness are introduced and derived through consistently linearizing the virtual work contributed from interior surface integrals the performance of our dg formulation has been demonstrated through solving a cantilever beam problem undergoing large rotations as well as a bipolar void coalescence problem where the voids grow up to several hundred times of their original volumes fast convergence rates for newton s iterations have been achieved in our iipg implementation

proof of unconditional stability for a single field discontinuous galerkin finite element formulation for linear elasto dynamics abstract a discontinuous galerkin dg finite element method fem for the solution of linear elasto dynamic problems is revisited and modified the new dg fem is based on a method originally proposed by comput methods appl mech engrg 84 1990 327 and recently adapted by comput methods appl mech engrg 191 46 2002 5315 for the solution of dynamic solid solid phase transitions as the fem formulations in both the cited works have been found not to be unconditionally stable in cases where the underlying fem grid is completely unstructured this paper offers a modification of these formulations yielding a single field dg fem that is unconditionally stable without any restrictions on the grid structure furthermore an energy conserving variant of the formulation is also suggested

explicit predictor multicorrector time discontinuous galerkin methods for non linear dynamics abstract explicit predictor multicorrector time discontinuous galerkin tdg methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non linear analysis of structural dynamics problems the formulation is intended to inherit the accuracy properties of the exact parent implicit tdg methods to this end suitable predictors and correctors are designed to achieve third order accuracy large stability limits and controllable numerical dissipation by means of an algorithmic parameter as the study of a general non linear case is rather complex the analysis of the convergence properties of the resulting algorithms are restricted to conservative duffing oscillators for which closed form solutions are available it is shown that the main properties of the underlying parent scheme can be retained finally results of representative numerical simulations relevant to duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates

a discontinuous galerkin formulation of non linear kirchhoff love shells discontinuous galerkin dg methods provide a means of weakly enforcing the continuity of the unknown field derivatives and have particular appeal in problems involving high order derivatives this feature has previously been successfully exploited comput methods appl mech eng 2008 197 2901 2929 to develop a formulation of linear kirchhoff love shells considering only the membrane and bending responses in this proposed one field method the displacements are the only unknowns while the displacement field is continuous the continuity in the displacement derivative between two elements is weakly enforced by recourse to a dg formulation it is the purpose of the present paper to extend this formulation to finite deformations and non linear elastic behaviors while the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple stress from the displacement field at the mid surface of the shell the non linear formulation considered implies the evaluation of the general stress tensor across the shell thickness leading to a reformulation of the internal forces of the shell nevertheless since the interface terms resulting from the discontinuous galerkin method involve only the resultant couple stress at the edges of the shells the extension to non linear deformations is straightforward copyright 2008 john wiley sons ltd

an element wise locally conservative galerkin lcg method for solving diffusion and convection diffusion problems an element wise locally conservative galerkin lcg method is employed to solve the conservation equations of diffusion and convection diffusion this approach allows the system of simultaneous equations to be solved over each element thus the traditional assembly of elemental contributions into a global matrix system is avoided this simplifies the calculation procedure over the standard global continuous galerkin method in addition to explicitly establishing element wise flux conservation in the lcg method elements are treated as sub domains with weakly imposed neumann boundary conditions the lcg method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging it is also shown in this paper that the proposed lcg method is identical to the standard global galerkin gg method at both steady and unsteady states for an inside node thus the method has all the advantages of the standard gg method while explicitly conserving fluxes over each element several problems of diffusion and convection diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the lcg method both linear and quadratic elements are used in the calculations for convection dominated problems petrov galerkin weighting and high order characteristic based temporal schemes have been implemented into the lcg formulation

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 one field full discontinuous galerkin method for kirchhoff love shells applied to fracture mechanics abstract in order to model fracture the cohesive zone method can be coupled in a very efficient way with the finite element method nevertheless there are some drawbacks with the classical insertion of cohesive elements it is well known that on one the hand if these elements are present before fracture there is a modification of the structure stiffness and that on the other hand their insertion during the simulation requires very complex implementation especially with parallel codes these drawbacks can be avoided by combining the cohesive method with the use of a discontinuous galerkin formulation in such a formulation all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements the recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture the purpose of this paper is to develop this formulation for kirchhoff love plates and shells it is achieved by the establishment of a full dg formulation of shell combined with a cohesive model which is adapted to the special thickness discretization of the shell formulation in fact this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations finally numerical examples demonstrate the efficiency of the method

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

a new discontinuous galerkin method for kirchhoff love shells abstract discontinuous galerkin methods dg have particular appeal in problems involving high order derivatives since they provide a means of weakly enforcing the continuity of the unknown field derivatives this paper proposes a new discontinuous galerkin method for kirchhoff love shells considering only the membrane and bending response the proposed one field method utilizes the weak enforcement in such a way that the displacements are the only unknowns while the rotation continuity is weakly enforced this work presents the formulation of the new discontinuous galerkin method for linear elastic shells demonstrates the consistency and stability of the proposed framework and establishes the method s convergence rate after a description of the formulation implementation into a finite element code these properties are demonstrated on numerical applications

a low order locking free hybrid discontinuous galerkin element formulation for large deformations abstract in this work a hybrid discontinuous galerkin dg quadrilateral element formulation is presented as usual in hybrid formulations the interior of the elements is treated separately from the skeleton which represents the element boundaries these are kinematically decoupled i e displacement jumps can occur between the skeleton and the interior of the elements the global degrees of freedom dofs are defined on the skeleton as the displacements at the corners which allows the implementation into existing finite element codes as usual in hybrid dg formulations the degrees of freedom in the interior are condensed out on the element level leading to the same number of global degrees of freedom as for continuous bilinear elements instead of using conventional shape functions in the interior the deformation gradient f is assumed constant within the element furthermore f is connected to the skeleton degrees of freedom via the weak form this leads to a very simple formulation and implementation the element is tested for several computational examples from the literature special choices of the penalty parameter are investigated which are partially derived analytically it is found that the element is free of volumetric and shear locking moreover the convergence is similar to that of other well known locking free finite element formulations

on an equivalence between a discontinuous galerkin method and reduced integration with hourglass stabilization for finite elasticity abstract discontinuous galerkin methods and methods of reduced integration with hourglass stabilization have different points of origin whereas the discontinuous galerkin idea was developed to investigate a wide range of applications among them problems of electrodynamics fluid and solid mechanics the reduced integration technology started out from the observation of the so called locking problem in linear elasticity today the two methodologies are still considered separately from each other although it has already been recognized that discontinuous galerkin methods can be also used to circumvent locking phenomena in elasticity the idea of the present contribution is to find a direct equivalence between a discontinuous galerkin method and a finite element technology based on reduced integration with hourglass stabilization the transfer between the two methods is carried out in the context of finite elasticity thus the derivation is valid for arbitrarily large deformations the main advantage of this analysis is that the choice of the free parameter in nitsche s method often also called stabilization parameter is no longer unclear but prescribed by the connection to the hourglass stabilization further important in this context is the replacement of the scalar parameter by a symmetric 2 2 matrix if two dimensional problems are considered which is here the case the result of the derivation is a discontinuous galerkin formulation which shows very good convergence behaviour not only in the limit of incompressibility but also for bending of thin structures

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

a hybrid high order method for finite elastoplastic deformations within a logarithmic strain framework we devise and evaluate numerically a hybrid high order hho method for finite plasticity within a logarithmic strain framework the hho method uses as discrete unknowns piecewise polynomials of order k 1 on the mesh skeleton together with cell based polynomi als that can be eliminated locally by static condensation the hho method leads to a primal formulation supports polyhedral meshes with non matching interfaces is free of volumetric locking the integration of the behavior law is performed only at cell based quadrature nodes and the tangent matrix in newton s method is symmetric moreover the principle of virtual work is satisfied locally with equilibrated tractions various two and three dimensional benchmarks are presented as well as comparison against known solutions with an industrial software using conforming and mixed finite elements

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

an explicit discontinuous galerkin method for non linear solid dynamics formulation parallel implementation and scalability properties an explicit dynamics spatially discontinuous galerkin dg formulation for non linear solid dynamics is proposed and implemented for parallel computation dg methods have particular appeal in problems involving complex material response e g non local behavior and failure as even in the presence of discontinuities they provide a rigorous means of ensuring both consistency and stability in the proposed method these are guaranteed the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation the semi discrete system of ordinary differential equations is integrated in time using a conventional second order central difference explicit scheme a stability criterion for the time integration algorithm accounting for the influence of the dg discretization stability is derived for the equivalent linearized system this approach naturally lends itself to efficient parallel implementation the resulting dg computational framework is implemented in three dimensions via specialized interface elements the versatility robustness and scalability of the overall computational approach are all demonstrated in problems involving stress wave propagation and large plastic deformations copyright 2007 john wiley sons ltd

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

numerical simulation of axisymmetric viscous flows by means of a particle method a vortex particle method for the simulation of axisymmetric viscous flow is presented the flow is assumed to be laminar and incompressible the navier stokes equations are expressed in an integral velocity vorticity formulation the inviscid scheme is based on nitsche s method for axisymmetric vortex sheets meanwhile two techniques are proposed for dealing with the viscous term the first uses an integral green s function method while the second is based on a diffusion velocity approach both are obtained by extension of existing methods for 2d flows the problem of satisfying boundary conditions along the axis of symmetry is specifically addressed the problem is solved by using cut off functions that are derived from the green s function of the axisymmetric diffusion equation the scheme is applied to simulate the evolution of vortex rings at intermediate reynolds number the processes of entrainment and wake formation are evident in the calculations as well as the extension of the support of vorticity due to viscous diffusion

discontinuous galerkin finite element in time for solving periodic differential equations the dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing discontinuous galerkin dg methods which are known to be higher order accurate a stable and locally conservative are used to solve such equations in this work dg methods are shown to find the steady state periodic solution of a constant coefficient structural dynamics problem with periodic forcing and b helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing convergence of the numerical solution obtained is studied for both the above equations while h and p are refined separately it is found that the dg method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes it is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p 12 two adaptive strategies based on the element level displacement jump norm i e h and hp refinement are proposed and demonstrated on problems with higher harmonic forcing primary controls and parametric variations

analysis of a family of discontinuous galerkin methods for elliptic problems the one dimensional case in this paper we analyze a family of discontinuous galerkin methods parameterized by two real parameters for elliptic problems in one dimension our main results are 1 a complete inf sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof 2 an analysis of the error in l2 where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case the analysis is consistent with numerical results and similar behavior is observed in two dimensions

on the algorithmic and implementational aspects of a discontinuous galerkin method at finite strains in this work algorithmic modifications are proposed and analyzed for a recently developed stabilized finite strain discontinuous galerkin dg method the distinguishing feature of the original method referred to as vmdg is a consistently derived expression for the numerical flux and stability tensor that account for evolving material and geometric nonlinearity in the vicinity of the interface herein the proposed modifications involve simplifications to the residual force vector and tangent stiffness matrix of the vmdg method that lead to formulations similar to other existing dg methods but retain the enhanced definition for the stability parameters the primary objective is to reduce the costs associated with implementing the method as well as executing simulations while retaining accuracy and flexibility thereby making the formulation amenable to boarder material classes such as inelasticity each simplification has associated implications on the mathematical and algorithmic properties of the method such as l 2 convergence rate solution accuracy continuity enforcement and stability of the nonlinear equation solver these implications are carefully quantified and assessed through a comprehensive numerical performance study the range of two and three dimensional problems under consideration involves both isotropic and anisotropic materials both triangular and quadrilateral element types are employed along with h and p refinement the ability of the proposed methods to produce stable and accurate results for such a broad class of problems is highlighted

a spacetime discontinuous galerkin method for scalar conservation laws abstract we present a spacetime discontinuous galerkin sdg finite element method for scalar hyperbolic conservation laws the method is consistent with the integral forms of the conservation law and its associated entropy condition written in terms of the physical godunov flux on each spacetime element the discrete basis functions are piecewise continuous in spacetime admitting discontinuities across element boundaries the resulting bubnov galerkin method is high order stable for both convex and non convex flux functions it does not require stabilization beyond the basic galerkin projection the sdg method is applicable to both layered and unstructured spacetime grids an element by element solution scheme delivers o n computational complexity where n is the number of elements when applied to spacetime meshes that conform to a special causality constraint we employ two different limiters to control the local overshoot and undershoot that the basic sdg projection generates near shocks

a scalable 3d fracture and fragmentation algorithm based on a hybrid discontinuous galerkin cohesive element method abstract a scalable algorithm for modeling dynamic fracture and fragmentation of solids in three dimensions is presented the method is based on a combination of a discontinuous galerkin dg formulation of the continuum problem and cohesive zone models czm of fracture prior to fracture the flux and stabilization terms arising from the dg formulation at interelement boundaries are enforced via interface elements much like in the conventional intrinsic cohesive element approach albeit in a way that guarantees consistency and stability upon the onset of fracture the traction separation law tsl governing the fracture process becomes operative without the need to insert a new cohesive element upon crack closure the reinstatement of the dg terms guarantee the proper description of compressive waves across closed crack surfaces the main advantage of the method is that it avoids the need to propagate topological changes in the mesh as cracks and fragments develop which enables the indistinctive treatment of crack propagation across processor boundaries and thus the scalability in parallel computations another advantage of the method is that it preserves consistency and stability in the uncracked interfaces thus avoiding issues with wave propagation typical of intrinsic cohesive element approaches a simple problem of wave propagation in a bar leading to spall at its center is used to show that the method does not affect wave characteristics and as a consequence properly captures the spall process we also demonstrate the ability of the method to capture intricate patterns of radial and conical cracks arising in the impact of ceramic plates which propagate in the mesh impassive to the presence of processor boundaries

a coupled electro thermal discontinuous galerkin method abstract this paper presents a discontinuous galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro thermal problems in this paper we discuss the fundamental equations for the transport of electricity and heat in terms of macroscopic variables such as temperature and electric potential a fully coupled nonlinear weak formulation for electro thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients the weak form can thus be formulated as a discontinuous galerkin method the existence and uniqueness of the weak form solution are proved the numerical properties of the nonlinear elliptic problems i e consistency and stability are demonstrated under specific conditions i e use of high enough stabilization parameter and at least quadratic polynomial approximations moreover the prior error estimates in the h 1 norm and in the l 2 norm are shown to be optimal in the mesh size with the polynomial approximation degree

regular article a discontinuous galerkin ale method for compressible viscous flows in moving domains we present a matrix free discontinuous galerkin method for simulating compressible viscous flows in two and three dimensional moving domains to this end we solve the navier stokes equations in an arbitrary lagrangian eulerian ale framework spatial discretization is based on standard structured and unstructured grids but using an orthogonal hierarchical spectral basis the method is third order accurate in time and converges exponentially fast in space for smooth solutions a novelty of the method is the use of a force directed algorithm from graph theory that requires no matrix inversion to efficiently update the grid while minimizing distortions we present several simulations using the new method including validation with published results from a pitching airfoil and new results for flow past a three dimensional wing subject to large flapping insect like motion

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

analysis of an hdg method for linear elasticity summary we present the first a priori error analysis for the first hybridizable discontinuous galerkin method for linear elasticity proposed in internat j numer methods engrg 80 2009 no 8 1058 1092 we consider meshes made of polyhedral shape regular elements of arbitrary shape and show that whenever piecewise polynomial approximations of degree k 0 are used and the exact solution is smooth enough the antisymmetric part of the gradient of the displacement converges with order k the stress and the symmetric part of the gradient of the displacement converge with order k 1 2 and the displacement converges with order k 1 we also provide numerical results showing that the orders of convergence are actually sharp copyright 2014 john wiley sons ltd

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

numerical analysis of growing crack problems using particle discretization scheme this paper presents the particle discretization scheme pds to analyze brittle failure of solids the scheme uses characteristic functions of voronoi and delaunay tessellations to discretize a function and its derivatives respectively a discretized function has numerous discontinuities so that these discontinuities are utilized as a candidate of crack path segment in modeling propagating cracks without making any extra computation to accommodate new displacement discontinuities when the scheme is implemented to a finite element method fem the resulting stiffness matrix coincides with the one that is obtained by using linear elements the accuracy of computing a stress intensity factor at a crack tip is examined it is shown that the accuracy is better than that of a fem with linear elements when the rotational degree of freedom is included in discretizing displacement functions three three dimensional growing crack problems are solved by means of the pds and the results are presented copyright 2009 john wiley sons ltd

an embedded mesh method for treating overlapping finite element meshes summary a new technique for treating the mechanical interactions of overlapping finite element meshes is presented such methods can be useful for numerous applications for example fluid solid interaction with a superposed meshed solid on an eulerian background fluid grid in this work we consider the interaction of two elastic domains one mesh is the foreground and defines the surface of interaction the other is a background mesh and is often a structured grid many of the previously proposed methods employ surface defined lagrange multipliers or penalties to enforce the boundary constraints it has become apparent that these methods will cause mesh locking under certain conditions appropriately applied the nitsche method can overcome this locking but in its canonical form is generally not applicable to non linear materials such as hyperelastics the relationship between interior point penalty discontinuous galerkin and nitsche s method is well known based on this relationship a nonlinear theory analogous to the nitsche method is proposed to treat nonlinear materials in an embedded mesh here a discontinuous galerkin derivative based on a lifting of the interface surface integrals provides a consistent treatment for non linear materials and demonstrates good behavior in example problems published 2012 this article is a us government work and is in the public domain in the usa

a general discontinuous galerkin method for finite hyperelasticity formulation and numerical applications a discontinuous galerkin formulation of the boundary value problem of finite deformation elasticity is presented the primary purpose is to establish a discontinuous galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the dg concept can show its superiority compared to the continuous formulation the method is based on a general hu washizu de veubeke functional allowing for displacement and stress discontinuities in the domain interior it is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation the consistency and linearized stability of the method in the non linear range as well as its convergence rate are proven an implementation in three dimensions is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner in order to demonstrate the versatility accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided copyright 2006 john wiley sons ltd

fluid structure interaction by the discontinuous galerkin method for large deformations a discontinuous galerkin method for large deformation fluid structure interaction problems is developed the fluid structure interface can be arbitrarily aligned relative to the fluid grid an eulerian description is used for the fluid with a lagrangian description of the solid results are presented for several examples that show excellent agreement with immersed fictitious element methods copyright 2008 john wiley sons ltd

on the order of functions at infinity abstract we define a new class of positive and measurable functions in terms of their asymptotic behavior at infinity this new class extends the class of regularly varying functions for broader applications we provide different characterizations of the new class and consider integrals convolutions and laplace transforms we give some applications in probability theory some natural extensions of the new class are also derived

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