moduli spaces of lumps on real projective space harmonic maps that minimise the dirichlet energy in their homotopy classes are known as lumps lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement this has consequences for the behaviour of lumps and their symmetries an interesting feature is that the moduli space of charge three lumps is a 7 dimensional manifold of cohomogeneity one which can be described as a one parameter family of symmetry orbits of d2 symmetric maps in this paper we discuss the charge three moduli spaces of lumps from two perspectives discrete symmetries of lumps and the riemann hurwitz formula we then calculate the metric and find explicit formulas for various geometric quantities we also discuss the implications for lump decay

discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

design parameterization for topology optimization by intersection of an implicit function abstract this paper introduces a new approach to topology optimization where the structural boundary is defined by the intersection of an implicit signed distance function and a cutting surface the cutting surface is discretized using finite element shape function polynomials and the nodal values become the design variables during optimization thus the design parameterization is separated from the analysis discretization which enables a reduction in the number of design variables compared with methods that use element wise variables the proposed method can obtain solutions with smooth boundaries and the parameterization allows solution by standard optimization methods several 2d and 3d examples are used to demonstrate the effectiveness of the method including minimization of compliance and complaint mechanism problems the results show that the method can obtain good solutions to well known problems with smooth clearly defined boundaries and that this can be achieved using significantly fewer design variables compared with element based methods

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

discontinuous galerkin finite element methods for incompressible non linear elasticity abstract a discontinuous galerkin finite element method dgfem for incompressible non linear elasticity is derived one of the limitations of the continuous galerkin finite element method cgfem when applied to incompressible elasticity is that the accuracy of the numerical solution may be adversely affected by the phenomenon known as locking this may prevent the use of a low order polynomial approximation to the displacement on each element we demonstrate using simulations that the dgfem presented here does not suffer from this drawback two further advantages in this setting of this dgfem over cgfems are that i highly anisotropic meshes meshes containing elements with a very high aspect ratio may be used without significantly degrading the accuracy of the solution and ii discontinuities in the elasticity parameters are handled more effectively

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

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

a discontinuous galerkin based immersed boundary method with non homogeneous boundary conditions and its application to elasticity we propose a discontinuous galerkin based immersed boundary method for elasticity problems the resulting numerical scheme does not require boundary fitting meshes and avoids boundary locking by switching the elements intersected by the boundary to a discontinuous galerkin approximation special emphasis is placed on the construction of a method that retains an optimal convergence rate in the presence of non homogeneous essential and natural boundary conditions the role of each one of the approximations introduced is illustrated by analyzing an analog problem in one spatial dimension finally extensive two and three dimensional numerical experiments on linear and nonlinear elasticity problems verify that the proposed method leads to optimal convergence rates under combinations of essential and natural boundary conditions

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

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

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

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

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

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

the geodesic approximation for lump dynamics and coercivity of the hessian for harmonic maps the most fruitful approach to studying low energy soliton dynamics in field theories of bogomol nyi type is the geodesic approximation of manton in the case of vortices and monopoles stuart has obtained rigorous estimates of the errors in this approximation and hence proved that it is valid in the low speed regime his method employs energy estimates which rely on a key coercivity property of the hessian of the energy functional of the theory under consideration in this article we prove an analogous coercivity property for the hessian of the energy functional of a general sigma model with compact kahler domain and target we go on to prove a continuity property for our result and show that for the cp1 model on s2 the hessian fails to be globally coercive in the degree 1 sector we present numerical evidence which suggests that the hessian is globally coercive in a certain equivariance class of the degree n sector for n 2 we also prove that within the geodesic approximation a single cp1 lump movi

stability of classical solutions of two dimensional grassmannian models we show that the only finite action solutions of the two dimensional grassmannian model that are stable under small fluctuations are the anti instanton solutions

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

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

the importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity mixed fems versus nurbs based approximations in the present paper we investigate the capability of finite element methods to correctly reproduce the stability range of finite strain problems in the incompressible regime to this end we develop a numerical scheme obtained combining a stream function formulation with an isogeometric nurbs approach which is able to sharply estimate the stability limits of the continuous problem using such a method we show a pair of benchmark problems on which various well known finite element methods largely fail in approximating the correct stability range

discontinuous galerkin finite element approximation of nonlinear second order elliptic and hyperbolic systems we develop the convergence analysis of discontinuous galerkin finite element approximations to symmetric second order quasi linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in mathbb r d subject to mixed dirichlet neumann boundary conditions optimal order asymptotic bounds are derived on the discretization error in each case without requiring the global lipschitz continuity or uniform monotonicity of the stress tensor instead only local smoothness and a garding inequality are used in the analysis

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

a velocity stress mixed stabilized nodal finite element for elastodynamics analysis and computations with strongly and weakly enforced boundary conditions abstract a new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain our method is based on equal order interpolation discrete spaces for both the velocity or displacement and stress or strain tensor variables the mixed form is derived using either the velocity stress or velocity strain pair of unknowns the latter being instrumental in extensions of the method to nonlinear mechanics the proposed approach works equally well on hexahedral or tetrahedral grids and for this reason it is suitable for time domain engineering applications in complex geometry the peculiarity of the proposed approach is the use of the rate form of the stress update equation which yields a set of governing equations with the structure of a non dissipative space time friedrichs system we complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity or displacement weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes because they do not require dedicated data structures for the imposition of the boundary degrees of freedom but rather they utilize the structure of the variational formulation we also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi domain simulations of heterogeneous media a complete analysis including stability and convergence proofs is included in the case of a space time variational approach a series of computational tests are used to demonstrate and verify the performance of the proposed approach

sigma model solitons and their moduli space metrics we discuss the supersymmetric model and its soliton solutions in 2 1 dimensions we classify supersymmetric maps and derive bogomolny bounds we also give the modified superalgebra and describe the metric on the parameter space of solitons

discontinuous galerkin methods for non linear elasticity this paper presents the formulation and a partial analysis of a class of discontinuous galerkin methods for quasistatic non linear elasticity problems these methods are endowed with several salient features the equations that define the numerical scheme are the euler lagrange equations of a one field variational principle a trait that provides an elegant and simple derivation of the method in consonance with general discontinuous galerkin formulations it is possible within this framework to choose different numerical fluxes numerical evidence suggests the absence of locking at near incompressible conditions in the finite deformations regime when piecewise linear elements are adopted finally a conceivable surprising characteristic is that as demonstrated with numerical examples these methods provide a given accuracy level for a comparable and often lower computational cost than conforming formulations stabilization is occasionally needed for discontinuous galerkin methods in linear elliptic problems in this paper we propose a sufficient condition for the stability of each linearized non linear elastic problem that naturally includes material and geometric parameters the latter needed to account for buckling we then prove that when a similar condition is satisfied by the discrete problem the method provides stable linearized deformed configurations upon the addition of a standard stabilization term we conclude by discussing the complexity of the implementation and propose a computationally efficient approach that avoids looping over both elements and element faces several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous galerkin method within the class copyright 2006 john wiley sons ltd

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

locking free adaptive discontinuous galerkin fem for linear elasticity problems an adaptive discontinuous galerkin finite element method for linear elasticity problems is presented we develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials absence of volume locking furthermore we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes

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