a discontinuous galerkin finite element method for time dependent partial differential equations with higher order derivatives in this paper we develop a new discontinuous galerkin dg finite element method for solving time dependent partial differential equations pdes with higher order spatial derivatives unlike the traditional local discontinuous galerkin ldg method the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system stability is ensured by a careful choice of interface numerical fluxes the method can be designed for quite general nonlinear pdes and we prove stability and give error estimates for a few representative classes of pdes up to fifth order numerical examples show that our scheme attains the optimal k 1 th order of accuracy when using piecewise k th degree polynomials under the condition that k 1 is greater than or equal to the order of the equation

a conservative free energy dissipating and positivity preserving finite difference scheme for multi dimensional nonlocal fokker planck equation abstract in this work we design and analyze a conservative positivity preserving and free energy dissipating finite difference method for the multi dimensional nonlocal fokker planck fp equation based on a non logarithmic landau transformation a central differencing spatial discretization using harmonic mean approximations is developed both forward and backward euler discretizations in time are employed to derive an explicit scheme and a linearized semi implicit scheme respectively three desired properties that are possessed by analytical solutions i mass conservation ii free energy dissipation and iii positivity are proved to be maintained at discrete level remarkably numerical analysis demonstrates that the semi implicit time discretization ensures the property of positivity preserving unconditionally due to the advantages brought by the harmonic mean approximations an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi implicit scheme extensive numerical tests are performed to validate aforementioned properties numerically

analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials large strain theory for standard dissipative solids we propose a fundamentally new approach to the treatment of shearband localizations in strain softening elastic plastic solids at finite strains based on energy minimization principles associated with microstructure developments the point of departure is a general internal variable formulation that determines the finite inelastic response as a standard dissipative medium consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasi hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables the existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments of finite elasticity furthermore localization phenomena are interpreted as microstructure developments on multiple scales associated with non convex incremental stress potentials in analogy to elastic phase decomposition problems these microstructures can be resolved by the relaxation of non convex energy functionals based on a convexification of the stress potential the relaxed problem provides a well posed formulation for a mesh objective analysis of localizations as close as possible to the non convex original problem based on an approximated rank one convexification of the incremental stress potential we develop a computational two scale procedure for a mesh objective treatment of localization problems at finite strains it constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the microshearing of a rank one laminate aligned to the shear band this problem is sufficiently robust with regard to applications to large scale inhomogeneous deformation processes of elastic plastic solids the performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which report on the mesh independence of the results copyright 2003 john wiley sons ltd

a finite volume scheme for nonlinear degenerate parabolic equations we propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional for some of these models porous media equation drift diffusion system for semiconductors it has been proved that the transient solution converges to a steady state when time goes to infinity the present scheme preserves steady states and provides a satisfying long time behavior moreover it remains valid and second order accurate in space even in the degenerate case after describing the numerical scheme we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic

an error estimate free and remapping free variational mesh refinement and coarsening method for dissipative solids at finite strains a variational h adaptive finite element formulation is proposed the distinguishing feature of this method is that mesh refinement and coarsening are governed by the same minimization principle characterizing the underlying physical problem hence no error estimates are invoked at any stage of the adaption procedure as a consequence linearity of the problem and a corresponding hilbert space functional framework are not required and the proposed formulation can be applied to highly non linear phenomena the basic strategy is to refine respectively unrefine the spatial discretization locally if such refinement respectively unrefinement results in a sufficiently large reduction respectively sufficiently small increase in the energy this strategy leads to an adaption algorithm having o n complexity local refinement is effected by edge bisection and local unrefinement by the deletion of terminal vertices dissipation is accounted for within a time discretized variational framework resulting in an incremental potential energy in addition the entire hierarchy of successive refinements is stored and the internal state of parent elements is updated so that no mesh transfer operator is required upon unrefinement the versatility and robustness of the resulting variational adaptive finite element formulation is illustrated by means of selected numerical examples

computational two mode asymptotic bifurcation theory combined with hyper dual numbers and applied to plate shell buckling abstract for error free computation of higher order derivatives of a complex mathematical expression composed of elementary functions hyper dual numbers hdns are receiving close attention in computational engineering and sciences differently from conventional finite differences hdns numerically evaluate for instance the exact first order and second order derivatives of the stiffness matrix with respect to nodal degrees of freedom dof in nonlinear finite element fe analysis one of the most promising applications of hdns is the asymptotic structural stability theory which usually requires in conventional series expansions an unrealistically large number of second order derivative of the stiffness matrix to predict the instability behavior of large scale fe models this computational bottleneck may be avoided along with so called classical lyapunov schmidt koiter condensation by a novel idea proposed in the present study the proposed two mode computational asymptotic theory is implementable in engineering practice to algebraically identify snap through and path branching in stability problems one more associated innovation is a graphical solution of two simultaneous equations resulting from the asymptotic expansions this sophisticated solution idea utilizing a popular mathematical tool matlab is widely applicable to quickly visualize the location and number of existing solutions more specifically the present paper formulates a two mode asymptotic bifurcation theory in combination with hdns and presents an examination of the proposed computational theory on stability problems of plates and shells near a precisely computed singular point on the equilibrium path the incremental displacement field is represented in two modes and one mode is the critical eigenvector corresponding to the zero eigenvalue of the singular tangent stiffness matrix the other one is a linear combination of all other non critical eigenvectors and respectively constitute the homogeneous and particular solutions of the linearized equilibrium equations established in the singular point two perturbation parameters and work as weights of and respectively to find possible equilibria around the singular point the nonlinear equilibrium equations are expanded into asymptotic series of and to use hdns to compute 20 coefficients including first order and second order derivatives of the stiffness matrix in polynomials the simultaneous polynomial equations of and are solved by using the matlab graphic operations the obtained results in numerical examples well predict the stability behavior in excellent agreement with solutions available in conventional fe stability analysis the two mode asymptotic bifurcation theory combined with hdns is sufficient to diagnose snap through asymmetric branching unstable and stable symmetric bifurcation in structural stability problems

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

from constant mean curvature hypersurfaces to the gradient theory of phase transitions given a nondegenerate minimal hypersurface in a riemannian manifold we prove that for all e small enough there exists ue a critical point of the allen cahn energy ee u e2 u 2 1 u2 2 whose nodal set converges to as e tends to 0 moreover if is a volume nondegenerate constant mean curvature hypersurface then the same conclusion holds with the function ue being a critical point of ee under some volume constraint

hill top branching its asymptotically expanded and visually solved bifurcation equations abstract hill top branching htb is an exceptional stability problem in which the bifurcation point bp coincides exactly with a limit point lp in such cases snap through and branching instability occur simultaneously then htb is often obscured in snap through behavior and is often overlooked in stability design the compound instability is in fact not well realized in the community of computational mechanics when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined no report of the relevant literature has established diagnosis of path branching or proposed a branch switching procedure furthermore few htb models are available in the literature this report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue there might be a single or two critical eigenvectors of the singular stiffness matrix higher derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper dual numbers hdns the resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as matlab from the locations of existing real roots displayed on a graphic monitor the crossover situation of path branching is visually clear above and below the hill top for the present study three htb examples are computed to verify the proposed algorithm strategy using hdns and graphical tools the obtained numerical results show excellent agreement with analytical and finite element predictions the proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric htb in structural stability problems and function well in computational engineering problems

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

review and unification of methods for computing derivatives of multidisciplinary computational models this paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework this framework hinges on a new equation the unifying chain rule from which all the methods can be derived the computation of derivatives is described as a two step process the evaluation of the partial derivatives and the computation of the total derivatives which are dependent on the partial derivatives finite differences the complex step method and symbolic differentiation are discussed as options for computing the partial derivatives it is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation the direct and adjoint methods and coupled analytic methods for multidisciplinary systems each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches the paper ends with a discussion of current challenges and possib

time like minimal submanifolds as singular limits of nonlinear wave equations abstract we consider the sharp interface limit 0 of the semilinear wave equation u w u 2 0 in r 1 n where u takes values in r k k 1 2 and w is a double well potential if k 1 and vanishes on the unit circle and is positive elsewhere if k 2 for fixed 0 we find some special solutions constructed around minimal surfaces in r n in the general case under some additional assumptions we show that the solutions converge to a radon measure supported on a time like k codimensional minimal submanifold of the minkowski space time this result holds also after the appearance of singularities and enforces the observation made by j neu that this semilinear equation can be regarded as an approximation of the born infeld equation

invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions in this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains the main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of ball in order to guarantee the existence of minimizers based on the introduction of additional argument tensors the so called structural tensors the free energies and the anisotropic stress response functions are represented by scalar valued and tensor valued isotropic tensor functions respectively in order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments we assume an additive structure a variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived where each individual term fulfills a priori the polyconvexity condition the tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented furthermore we propose an extension to orthotropic symmetry

third order maximum principle satisfying dg schemes for convection diffusion problems with anisotropic diffusivity abstract for a class of convection diffusion equations with variable diffusivity we construct third order accurate discontinuous galerkin dg schemes on both one and two dimensional rectangular meshes the dg method with an explicit time stepping can well be applied to nonlinear convection diffusion equations it is shown that under suitable time step restrictions the scaling limiter proposed in liu and yu 2014 23 when coupled with the present dg schemes preserves the solution bounds indicated by the initial data i e the maximum principle while maintaining uniform third order accuracy these schemes can be extended to rectangular meshes in three dimension the crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions this is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages numerical results are presented to validate the numerical methods and demonstrate their effectiveness

robust numerical calculation of tangent moduli at finite strains based on complex step derivative approximation and its application to localization analysis abstract an extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard fem software this method is based on the complex step derivative approximation csda approach the csda is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e g finite difference approximation therefore the csda can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision implementation details of the robust numerical approximation of tangent moduli from stress calculations using the csda are given and their performance is illustrated through representative examples involving finite deformations in addition to that we focus on the determination of material instabilities therefore an accompanying localization analysis is performed where the acoustic tensor is directly computed from the approximation of the moduli it is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation on the other hand the csda approach provides high accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely

numerically generated tangent stiffness matrices using the complex variable derivative method for nonlinear structural analysis accurate tangent stiffness matrices are essential for solving nonlinear structural problems using newton like methods this paper presents a numerical procedure for generating accurate tangent stiffness matrices from the derivative approximation of internal force using the complex variable derivative method cvdm the present procedure is suitable for problems that require complex constitutive relations and or kinematic nonlinearities whose models are difficult to linearize as well as for the incorporation of newton like solution strategies with nonlinear structural analyses

a newton schur alternative to the consistent tangent approach in computational plasticity we introduce a newton schur alternative to the consistent tangent approach in plasticity computations and show that the structure of the consistent tangent approach is preserved in the new newton schur approach however the local constitutive iterations used in the conventional computational plasticity analyses are eliminated when using the newton schur approach two numerical examples are presented which illustrate the efficiencies gained by the new approach

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

effective relaxation for microstructure simulations algorithms and applications for a wide class of problems in continuum mechanics like those involving phase transitions or finite elastoplasticity the governing potentials tend to be not quasiconvex this leads to the occurrence of microstructures of in principle arbitrarily small scale which cannot be resolved by standard discretization schemes their effective macroscopic properties however can efficiently be recovered with relaxation theory the paper introduces the variational framework necessary for the implementation of relaxation algorithms with emphasis on problems with internal variables in a time incremental setting the methods developed are based on numerical approximations to notions of generalized convexification the focus is on the thorough analysis of numerical algorithms and their efficiency in applications to benchmark problems an outlook to time evolution of microstructures within the framework of relaxation theory concludes the paper 2004 published by elsevier b v

a free energy satisfying discontinuous galerkin method for one dimensional poisson nernst planck systems we design an arbitrary order free energy satisfying discontinuous galerkin dg method for solving time dependent poisson nernst planck systems both the semi discrete and fully discrete dg methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions positivity of numerical solutions is enforced by an accuracy preserving limiter in reference to positive cell averages numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation free energy dissipation as well as the preservation of steady states

towards variational constitutive updates for non associative plasticity models at finite strain models based on a volumetric deviatoric split in this paper an enhanced variational constitutive update suitable for a class of non associative plasticity theories at finite strain is proposed in line with classical numerical formulations for plasticity models such as the by now established return mapping algorithm variational constitutive updates represent a numerical method for computing the unknown state variables however in contrast to conventional algorithms variational constitutive updates are fully variational i e all unknown variables follow jointly from minimizing a certain potential in addition to the physical and mathematical elegance of these variational schemes they show several practical advantages as well for instance numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem since mathematically plasticity is a non smooth problem and often it leads to highly singular systems of equations as known from single crystal plasticity a robust implementation is of utmost importance so far variational constitutive updates have been developed for different classes of standard dissipative solids i e solids characterized by associative evolution equations and flow rules in the present paper this framework is extended to a certain class of non associative plasticity models at finite strain all models falling into this class show a volumetric deviatoric split of the helmholtz energy and the yield function typical prototypes are drucker prager or mohr coulomb models playing an important role in soil mechanics the efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples

high order schemes entropy inequalities and nonclassical shocks we are concerned with the approximation of undercompressive regularization sensitive nonclassical solutions of hyperbolic systems of conservation laws by high order accurate conservative and semidiscrete finite difference schemes nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables first we investigate the existence and the properties of entropy conservative schemes a notion due to tadmor math comp 49 1987 pp 91 103 in particular we exhibit a new five point scheme which is third order accurate at least second we study a class of entropy stable and high order accurate schemes satisfying a single cell entropy inequality they are built from any high order entropy conservative scheme by adding to it a mesh independent numerical viscosity which preserves the order of accuracy of the base scheme these schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality these entropy stable schemes exhibit mild oscillations near shocks and interestingly may converge to classical or nonclassical entropy solutions depending on the sign of their dispersion coefficient then based on a third order entropy conservative scheme we propose a general scheme for the numerical computation of nonclassical shocks importantly our scheme satisfies a cell entropy inequality following hayes and lefloch siam j numer anal 35 1998 pp 2169 2194 we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws our results compare favorably with previous analytical and numerical results finally we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete lax wendroff type schemes

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

entropy stable high order discontinuous galerkin methods with suitable quadrature rules for hyperbolic conservation laws abstract it is well known that semi discrete high order discontinuous galerkin dg methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws jiang and shu 1994 39 and symmetric hyperbolic systems hou and liu 2007 36 in any space dimension and for any triangulations however this property holds only for the square entropy and the integrations in the dg methods must be exact it is significantly more difficult to design dg methods to satisfy entropy inequalities for a non square convex entropy and or when the integration is approximated by a numerical quadrature in this paper we develop a unified framework for designing high order dg methods which will satisfy entropy inequalities for any given single convex entropy through suitable numerical quadrature which is specific to this given entropy our framework applies from one dimensional scalar cases all the way to multi dimensional systems of conservation laws for the one dimensional case our numerical quadrature is based on the methodology established in carpenter et al 2014 5 and gassner 2013 19 the main ingredients are summation by parts sbp operators derived from legendre gauss lobatto quadrature the entropy conservative flux within elements and the entropy stable flux at element interfaces we then generalize the scheme to two dimensional triangular meshes by constructing sbp operators on triangles based on a special quadrature rule a local discontinuous galerkin ldg type treatment is also incorporated to achieve the generalization to convection diffusion equations extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable dg methods

a discontinuous galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials abstract we consider a class of time dependent second order partial differential equations governed by a decaying entropy the solution usually corresponds to a density distribution hence positivity non negativity is expected this class of problems covers important cases such as fokker planck type equations and aggregation models which have been studied intensively in the past decades in this paper we design a high order discontinuous galerkin method for such problems if the interaction potential is not involved or the interaction is defined by a smooth kernel our semi discrete scheme admits an entropy inequality on the discrete level furthermore by applying the positivity preserving limiter our fully discretized scheme produces non negative solutions for all cases under a time step constraint our method also applies to two dimensional problems on cartesian meshes numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics

a variational finite volume scheme for wasserstein gradient flows we propose a variational finite volume scheme to approximate the solutions to wasserstein gradient flows the time discretization is based on an implicit linearization of the wasserstein distance expressed thanks to benamou brenier formula whereas space discretization relies on upstream mobility two point flux approximation finite volumes our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level our scheme can be applied to a wide range of energies guarantees non negativity of the discrete solutions as well as decay of the energy we show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem and we prove its convergence in the case of the linear fokker planck equation with positive initial density numerical illustrations show that it is first order accurate in both time and space and robust with respect to both the energy and the initial profile

the computation of the exponential and logarithmic mappings and their first and second linearizations we describe two simple methods for the evaluation of the exponential and logarithmic mappings and their first and second linearizations based on the taylor expansion and the spectral representation we also provide guidelines for switching between those representations on the basis of the size of the argument the first and second linearizations of the exponential and logarithmic mappings provided here are based directly on the exponential formula for the solutions of systems of linear ordinary differential equations this representation does not require the use of perturbation formulae for eigenvalues and eigenvectors our approach leads to workable and straightforward expressions for the first and second linearizations of the exponential and logarithmic mappings regardless of degeneracies in the spectral decomposition of the argument

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed

boundary conditions for the microscopic fene models we consider the microscopic equation of finite extensible nonlinear elasticity fene models for polymeric fluids under a steady flow field it is shown that for the underlying fokker planck type of equations any preassigned distribution on the boundary will become redundant once the nondimensional number text it li frac hb k bt geq 2 where h is the elasticity constant sqrt b is the maximum dumbbell extension t is the temperature and k b is the usual boltzmann constant moreover if the probability density function is regular enough for its trace to be defined on the sphere m sqrt b then the trace is necessarily zero when text it li 2 these results are consistent with our numerical simulations as well as some recent well posedness results by preassuming a zero boundary distribution

on the approximation of the fokker planck equation of the finitely extensible nonlinear elastic dumbbell model i a new weighted formulation and an optimal spectral galerkin algorithm in two dimensions we propose a new weighted weak formulation for the fokker planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well posedness in weighted sobolev spaces we also propose simple and efficient semi implicit time discretization schemes which are unconditionally stable i e the step size of time marching does not depend on the number of the bases used in configurational space we then restrict ourselves to the two dimensional case and construct two fourier jacobi spectral galerkin algorithms which enjoy the following properties i they are unconditionally stable spectrally accurate and of optimal computational complexity ii they conserve the volume and provide accurate approximation to higher order moments of the distribution function and iii they can be easily extended to coupled nonhomogeneous systems numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress

vortex dynamics of the nonlinear wave equation abstract is a complex scalar field defined on 2 1 d dimensional minkowski space it satisfies the nonlinear wave equation nlwe 1 0 we study the solutions which contain time like world lines of zeros with winding numbers 1 or 1 we call these particle like defects of the field vortices we formulate an asymptotic particle field vortex dynamics which derives from the nlwe it is analogous to particle field electrodynamics in 2 1 d in particular the winding number plays the role of charge and gradients of arg play the role of the electromagnetic field

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

numerical computation of algorithmic consistent tangent moduli in large strain computational inelasticity abstract an algorithm for the numerical computation of so called algorithmic or consistent tangent moduli in finite inelasticity is presented these moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation they serve as iteration operators by application of newton type solvers for the iterative solution of non linear initial boundary value problems in finite inelasticity the underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation the algorithmic procedure is material independent and surprisingly simple it is outlined for a lagrangian as well as an eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity

a variational scheme for hyperbolic obstacle problems abstract we consider an obstacle problem for possibly non local wave equations and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme we provide the corresponding numerical implementation and raise some open questions

a polyconvex framework for soft biological tissues adjustment to experimental data the main goal of this contribution is to provide a simple method for constructing transversely isotropic polyconvex functions suitable for the description of biological soft tissues the advantage of our approach is that only a few parameters are necessary to approximate a variety of stress strain curves and to satisfy the condition of a stress free reference configuration a priori in the framework of polyconvexity the proposed polyconvex stored energies are embedded into the concept of structural tensors and the representation theorems for isotropic tensor functions are utilized as an example the medial layer of a human abdominal aorta is investigated modeled by some of the proposed polyconvex functions and compared with experimental data hereby the economic fitting to experimental data and hence the easy handling of the functions is shown

a highly accurate 1st and 2nd order differentiation scheme for hyperelastic material models based on hyper dual numbers abstract in this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models which are derived in terms of the first and second derivatives of a strain energy function this numerical scheme provides a compact model independent framework which means that once the framework is implemented any other hyperelastic material model can be incorporated by solely modifying the energy function the method is based on the numerical calculation of strain energy derivatives using hyper dual numbers and thus referred to as hyper dual step derivative hdsd the hdsd does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values furthermore it enables the calculation of derivatives of arbitrary order this is a great advantage compared to other numerical approaches as e g the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values another alternative the complex step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values but it only provides first derivatives and is thus not able to calculate stresses and moduli at once in this paper representative numerical examples using an anisotropic model are provided showing the performance of the proposed method in detail an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme furthermore examples demonstrate the robustness and simple implementation of the hdsd scheme in finite element software it turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time

variational h adaption in finite deformation elasticity and plasticity we ropose a variational h adaption strategy in which the evolution of the mesh is driven directly by the governing minimum principle this minimum principle is the principle of minimum potential energy in the case of elastostatics and a minimum principle for the incremental static problem of elasto viscoplasticity in particular the mesh is refined locally when the resulting energy or incremental pseudo energy released exceeds a certain threshold value in order to avoid global recomputes we estimate the local energy released by mesh refinement by means of a lower bound obtained by relaxing a local patch of elements this bound can be computed locally which reduces the complexity of the refinement algorithm to o n we also demonstrate how variational h refinement can be combined with variational r refinement to obtain a variational hr refinement algorithm because of the strict variational nature of the h refinement algorithm the resulting meshes are anisotropic and outperform other refinement strategies based on aspect ratio or other purely geometrical measures of mesh quality the versatility and rate of convergence of the resulting approach are illustrated by means of selected numerical examples copyright 2007 john wiley sons ltd

maximum principle satisfying third order discontinuous galerkin schemes for fokker planck equations we design and analyze up to third order accurate discontinuous galerkin dg methods satisfying a strict maximum principle for fokker planck equations a procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution this is achievable by taking advantage of the two parameters in the numerical flux and a novel decomposition of weighted cell averages based on this result a scaling limiter for the dg method with first order euler forward time discretization is proposed to solve the one dimensional fokker planck equations strong stability preserving high order time discretizations will keep the maximum principle it is straightforward to extend the method to two and higher dimensions on rectangular meshes we also show that a modified limiter can preserve the strict maximum principle for dg schemes solving fokker planck equations as a consequence the present schemes preserve steady states numerical tests

implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers abstract in this paper novel implementation schemes for the automatic calculation of internal variables stresses and consistent tangent moduli for incremental variational formulations ivfs describing inelastic material behavior are proposed ivfs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables in the so called multilevel newton raphson method for the inelasticity theory this minimization problem is typically solved by using second derivatives with respect to the internal variables in addition to that to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required compared with classical formulations such as the return mapping method the ivfs are relatively new and their implementation is much less documented furthermore higher order derivatives are required in the algorithms demanding increased implementation efforts therefore even though ivfs are mathematically and physically elegant their application is not standard here novel approaches for the implementation of ivfs using hdns of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy the proposed formulations are quite general and can be applied to a broad range of different constitutive models which means that once the proposed schemes are implemented as a framework any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions these include the helmholtz free energy function the dissipation potential function and additional side constraints such as e g the yield function in the case of plasticity its uncomplicated implementation for associative finite strain elasto plasticity and performance is illustrated by some representative numerical examples

a variational approach for materially stable anisotropic hyperelasticity abstract in this work we propose an anisotropic stored energy function which satisfies a priori the legendre hadamard condition which is strongly related to the material stability of the constitutive equations in the linearized case this condition implies positive wave speeds the legendre hadamard condition plays also an important role for the local existence of solutions in the neighborhood of stationary points we apply the proposed hyperelastic energies to soft tissues and compare the formulation with existing models which have been used for the calculation of medial collateral ligament and arterial walls in our numerical and analytical investigations we discuss the distribution of wave speeds for a sequence of deformation states containing some essential stress strain characteristics of the compared models

kinks and the minimal surface equation in minkowski space abstract the scalar field u is defined on 4 d minkowski space and satisfies the nonlinear wave equation 2 u v u 0 v u has minima at u 0 1 with v 0 in the limit 0 there are solutions in which regions with u 0 or u 1 are separated by layers of thickness about 3 d time like surfaces called kinks we show by singular perturbation analysis that the kink surfaces satisfy a minimal surface equation in minkowski space a kink with spherical time sections collapses to a point but the generic fate of closed kinks is not necessarily collapse we show that a kink with a microstructure of small ripples is stable against collapse this result is based upon an asymptotic description of the coupling between large and small scales of kink geometry the analysis is based upon whitham s averaged lagrangian method

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

non convex potentials and microstructures in finite strain plasticity a mathematical model for a finite strain elastoplastic evolution problem is proposed in which one time step of an implicit time discretization leads to generally non convex minimization problems the elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations the results for a single slip system and von mises plasticity illustrate that finite strain elastoplasticity generates reduced problems with non quasiconvex energy densities and so allows for non attainment of energy minimizers and microstructures

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

the role of a strong confining potential in a nonlinear fokker planck equation abstract we show that solutions of nonlinear nonlocal fokker planck equations in a bounded domain with no flux boundary conditions can be approximated by cauchy problems with increasingly strong confining potentials defined in the whole space two different approaches are analyzed making crucial use of uniform estimates for l 2 energy functionals and free energy or entropy functionals respectively in both cases we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials the free energy approach extends to the case degenerate diffusion

a semi implicit semi lagrangian p adaptive discontinuous galerkin method for the shallow water equations a semi implicit and semi lagrangian discontinuous galerkin method for the shallow water equations is proposed for applications to geophysical scale flows a non conservative formulation of the advection equation is employed in order to achieve a more treatable form of the linear system to be solved at each time step the method is equipped with a simple p adaptivity criterion that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution numerical results show that the method captures well the main features of gravity and inertial gravity waves as well as reproducing correct solutions in nonlinear test cases with analytic solutions the accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high courant numbers and with automatic choice of the local approximation degree

a structure preserving discontinuous galerkin scheme for the fisher kpp equation an implicit euler discontinuous galerkin scheme for the fisher kolmogorov petrovsky piscounov fisher kpp equation for population densities with no flux boundary conditions is suggested and analyzed using an exponential variable transformation the numerical scheme automatically preserves the positivity of the discrete solution a discrete entropy inequality is derived and the exponential time decay of the discrete density to the stable steady state in the l 1 norm is proved if the initial entropy is smaller than the measure of the domain the discrete solution is proved to converge in the l 2 norm to the unique strong solution to the time discrete fisher kpp equation as the mesh size tends to zero numerical experiments in one space dimension illustrate the theoretical results

stabilized non dissipative approximations of euler equations in generalized curvilinear coordinates we discuss stabilization strategies for finite difference approximations of the compressible euler equations in generalized curvilinear coordinates that do not rely on explicit upwinding or filtering of the physical variables our approach rather relies on a skew symmetric like splitting of the convective derivatives that guarantees preservation of kinetic energy in the semi discrete low mach number limit a locally conservative formulation allows efficient implementation and easy incorporation into existing compressible flow solvers the validity of the approach is tested for benchmark flow cases including the propagation of a cylindrical vortex and the head on collision of two vortex dipoles the tests support high accuracy and superior stability over conventional central discretization of the convective derivatives the potential use for dns les of turbulent compressible flows in complex geometries is discussed

on structure preserving discontinuous galerkin methods for hamiltonian partial differential equations energy conservation and multi symplecticity abstract in this paper we present and study discontinuous galerkin dg methods for one dimensional multi symplectic hamiltonian partial differential equations we particularly focus on semi discrete schemes with spatial discretization only and show that the proposed dg methods can simultaneously preserve the multi symplectic structure and energy conservation with a general class of numerical fluxes which includes the well known central and alternating fluxes applications to the wave equation the benjamin bona mahony equation the camassa holm equation the korteweg de vries equation and the nonlinear schrodinger equation are discussed some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods numerically we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes

variational updates for thermomechanically coupled gradient enhanced elastoplasticity implementation based on hyper dual numbers abstract this paper deals with the implementation of thermomechanically coupled gradient enhanced elastoplasticity at finite strains the presented algorithmic formulation heavily relies on the variational structure of the considered initial boundary value problem consequently such a variational structure is elaborated while variational formulations are well known in the case of isothermal plasticity theory thermomechanically coupled gradient enhanced plasticity theory has not been considered before the resulting time continuous variational principle allows the computation of all unknown variables jointly and naturally from the stationarity condition of an incremental potential by discretization of this time continuous potential in time a discrete approximation is obtained which represents the foundation of the algorithmic formulation as a matter of fact stationarity of the respective potential again defines all unknown variables now in a time discrete fashion within this paper the necessary condition associated with stationarity a vanishing first gradient is solved numerically by means of newton s method hence the first as well as the second derivatives of the incremental potential are required they are computed by numerical differentiation based on hyper dual numbers by considering a perturbation with respect to hyper dual numbers the first as well as the second derivatives are computed in an exact manner without introducing any numerical errors although numerical differentiation based on hyper dual numbers is numerically more extensive than real valued perturbations it is shown that the scalability of the resulting parallel finite element implementation is not dominated by this effect for sufficiently large problems

a positivity preserving high order discontinuous galerkin scheme for convection diffusion equations abstract for constructing high order accurate positivity preserving schemes for convection diffusion equations we construct a simple positivity preserving diffusion flux discontinuous galerkin dg schemes with such a positivity preserving diffusion flux are nonlinear schemes which can be regarded as a reduction of the high order positivity preserving dg schemes for compressible navier stokes equations in 1 to scalar diffusion operators in this paper we focus on the local dg method to discuss how to apply such a flux a limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity preserving in the sense that dg schemes with this flux satisfies a weak positivity property together with a positivity preserving limiter high order dg schemes with strong stability preserving time discretizations can be rendered positivity preserving without losing conservation or high order accuracy for convection diffusion problems with periodic boundary conditions or a special class of dirichlet or neumann boundary conditions numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

analysis of thin shells using anisotropic polyconvex energy densities in this contribution the numerical framework for the simulation of anisotropic thin shells is presented on the basis of polyconvex strain energy functions the nonlinear shell theory is based on the reissner mindlin kinematic along with inextensible director vectors for the variational framework we consider a three field variational functional taking into account independent displacements enhanced strains and stress resultants where the latter field is eliminated by the evaluation of some orthogonality conditions the iterative enforcement of the zero normal stress condition at the integration points allows consideration of arbitrary three dimensional constitutive equations due to the fact that we are interested in fiber reinforced materials we consider an additive structure of the energy decoupled in an isotropic part for the matrix and a superposition of transversely isotropic parts for the fiber families for the representation of the anisotropy we use the concept of structural tensors and formulate the strain energy function in terms of principal and mixed invariants of the right cauchy green tensor and the structural tensor in order to guarantee the existence of minimizers we focus on polyconvex strain energy functions the anisotropy effect is documented in several representative examples

convergence of the parabolic ginzburg landau equation to motion by mean curvature for the complex parabolic ginzburg landau equation we prove that asymptotically vorticity evolves according to motion by mean curvature in brakke s weak formulation the only assumption is a natural energy bound on the initial data in some cases we also prove convergence to enhanced motion in the sense of ilmanen

numerical differentiation for local and global tangent operators in computational plasticity in this paper numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators the derivatives of the constitutive equations are approximated by means of difference schemes these derivatives are needed to achieve quadratic convergence in the integration at gauss point level and in the solution of the boundary value problem numerical differentiation is shown to be a simple robust and competitive alternative to analytical derivatives quadratic convergence is maintained provided that adequate schemes and stepsizes are chosen this point is illustrated by means of some numerical examples