variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

an iterative penalty method for the finite element solution of the stationary navier stokes equations abstract the objective of this paper is to analyse an iterative procedure for the finite element solution of the stokes and navier stokes stationary problems for the latter case the usual condition on the viscosity and the data that ensures uniqueness is assumed the method is based on the iterative imposition of the incompressibility condition via penalization theoretical and numerical results show that this constraint can be approximated iteratively within the same iterative loop used to deal with the nonlinear term of the equations two particular iterative schemes are analysed namely those based on the picard and newton raphson algorithms

on the inequalities of babuska aziz friedrichs and horgan payne the equivalence between the inequalities of babuska aziz and friedrichs for sufficiently smooth bounded domains in the plane was shown by horgan and payne 30 years ago we prove that this equivalence and the equality between the associated constants is true without any regularity condition on the domain for the horgan payne inequality which is an upper bound of the friedrichs constant for plane star shaped domains in terms of a geometric quantity known as the horgan payne angle we show that it is true for some classes of domains but not for all bounded star shaped domains we prove a weaker inequality that is true in all cases

a penalty finite element method for non newtonian creeping flows summary in this paper we present an iterative penalty finite element method for viscous non newtonian creeping flows the basic idea is solving the equations for the difference between the exact solution and the solution obtained in the last iteration by the penalty method for the case of newtonian flows one can show that for sufficiently small penalty parameters the iterates converge to the incompressible solution the objective of the present work is to show that the iterative penalization can be coupled with the iterative scheme used to deal with the non linearity arising from the constitutive law of non newtonian fluids some numerical experiments are conducted in order to assess the performance of the approach for fluids whose viscosity obeys the power law the use of the penalty method for the finite element solution of incompressible flow problems has attracted the attention of many researchers its popularity is mainly due to the fact that it reduces the number of nodal unknowns of the problem on the other hand if the pure incompressible mixed velocity pressure formulation is adopted the element stiffness matrix of the discrete finite element problem has zero diagonal terms most of the standard finite element direct solvers use the elements of the diagonal as pivots in this case renumbering algorithms have to be modified in order to prevent the appearance of zeroes in the diagonal of the assembled matrix when reducing the equations in the solution process this results in the increase of the bandwidth of the global matrix this problem is circumvented if the incompressibility constraint is penalized the main drawback of the penalty method is the ill conditioning of the stiffness matrix when the penalty parameter is very small for newtonian flows a fairly wide range of values of this parameter are known to yield good results that is the incompressibility equation is sufficiently well approximated and to be easily handled if direct solvers are employed experience shows that the range e 10 6p to i 10 9pc 1 is rec mmended here and below e denotes the penalty number and u the dynamical viscosity of the fluid two questions arise the first is what happens for non newtonian flows in this case the viscosity may vary several orders of magni tude in the fluid domain especially if the physical properties of the material are considered to be thermally sensitive the above rule for choosing the penalty parameter has to be applied using the smallest value of the viscosity in order to avoid ill conditioning which is unknown before the calculation besides that the incompressibility constraint will be excessively relaxed in the high viscosity zones another question to be considered is whether iterative solvers can be safely used or not usually their convergence is very sensitive to the condition number of the stiffness matrix which grows as the penalty parameter decreases when the effect of the non constant viscosity is

site percolation thresholds for archimedean lattices precise thresholds for site percolation on eight archimedean lattices are determined by the hull walk gradient percolation simulation method with the results p c 0 697 043 honeycomb or 6 3 0 807 904 3 12 2 0 747 806 4 6 12 0 729 724 4 8 2 0 579 498 3 4 6 0 621 819 3 4 6 4 0 550 213 3 3 4 2 and 0 550 806 3 2 4 3 4 with errors of about 3 x 10 6 the remaining archimedean lattices are the square 4 4 triangular 3 6 and kagome 3 6 3 6 for which p c is already known exactly or to a high degree of accuracy the numerical result for the 3 12 2 lattice is consistent with the exact value 1 2 sin pi 18 1 2 the values of p c for all 11 archimedean lattices as well as a number of nonuniform lattices are found to be well correlated by a nearly linear function of a generalized scher zallen filling factor this correlation is much more accurate than recently proposed correlations based solely upon coordination number

on polya random walks lattice green functions and the bond percolation threshold based on numerical evidence the authors conjecture a connection between the bond percolation threshold of bravais lattices in three or more dimensions and the value at the origin of a lattice green function related to the probability of return to the origin for a polya random walk

locking spurious mechanisms and pressure divergence in penalty finite element methods for stokes flow problems abstract solution locking spurious mechanisms and pressure divergence exhibited in the stokes flow calculations by the penalty finite element methods are analyzed employing a fourier analysis technique comparison of the approximate discrete operator components with the corresponding continuum ones enables one to explicitly identify the conditions for avoiding locking spurious mechanisms and pressure divergence in terms of discrete operator components which are applicable to either the finite difference or finite element discretizations a synthesis approach is explored to eliminate the solution locking spurious mechanisms and pressure divergence in uniformly reduced integrated four noded elements extensions of the present analysis and synthesis approach to navier stokes equations appear to be straightforward if one deals with a set of consistently linearized operators

on the range of bond percolation thresholds for fully triangulated graphs this paper constructs a sequence of planar periodic fully triangulated lattices which have bond percolation thresholds tending to zero as a consequence we see that bond percolation thresholds of such graphs can range from 0 to 2 sin 18 0 3473 which is the threshold of the triangular lattice rigorous bounds and simulations for other fully triangulated lattices suggest that the threshold for the triangular lattice is the largest for this class of graphs

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

a low order finite element method for three dimensional linear elasticity problems with general meshes abstract the paper is concerned with a low order finite element method namely the staggered cell centered finite element method which has been proposed and analyzed in ong et al 2015 for two dimensional compressible and nearly incompressible linear elasticity problems in this work we extend the results to the three dimensional case and focus on the creating of the meshes in particular from a general primal mesh m we construct a polygonal dual mesh m and its submesh m in a way such that each dual control volume of m corresponds to a primal vertex and is a union macro element of some fixed number of adjacent tetrahedral elements of m the displacement is approximated by piecewise trilinear functions on the subdual mesh m and the pressure by piecewise constant functions on the dual mesh m as for two dimensional case such construction of the meshes and approximation spaces satisfies the macroelement condition which implies stability and convergence of the scheme numerical experiments are carried out to investigate the performance of the proposed method on various mesh types

bond percolation critical probability bounds derived by edge contraction a contraction principle is valid for bond percolation models if gc is obtained from g by contraction of a set of edges then the bond percolation critical probabilities satisfy pc gc or pc g the contraction principle provides relationships between graphs which do not follow from the inclusion principle application of the principle provides the following bounds 0 5000 or pc pentagon lattice or 0 6527 pc kagome lattice or 0 6180 0 3820 or pc dice lattice

universal formulas for percolation thresholds a power law is postulated for both site and bond percolation thresholds the formula is mathit p mathit c mathit p 0 d 1 q 1 mathrm ensuremath mathit a mathit d mathit b where d is the space dimension and q the coordination number all thresholds up to d ensuremath rightarrow ensuremath infty are found to belong to only three universality classes for two classes b 0 for site dilution while b a for bond dilution the remaining class associated with high dimensions is characterized by b 2a 1 for both sites and bonds classes are defined by a set of value for mathit p 0 a deviations from available numerical estimates at d ensuremath le 7 are within ifmmode pm else textpm fi 0 008 and ifmmode pm else textpm fi 0 0004 for high dimensional hypercubic expansions at d ensuremath ge 8 the formula is found to be also valid for ising critical temperatures textcopyright 1996 the american physical society

galerkin methods for the time dependent incompressible elasticity equations abstract we consider semidiscrete and single step fully discrete approximations to the solutions of the time dependent incompressible elasticity equations two methods of dealing with the constraint are analysed a a formulation involving a discrete form of incompressibility condition and b a penalty formulation h1 and l2 error estimates are obtained

site percolation thresholds in all dimensions abstract site percolation thresholds are reproduced in all dimensions for all lattices using a linear combination of two analytic terms one is the well known cayley tree percolation threshold which is believed to be exact at infinite dimension the other one is obtained from a new approach to two dimensional site dilution using a ferromagnetic ising system these two formulas are then combined with weighting factors a and 1 a respectively where a is a fitting parameter which depends only on space dimension it is equal to 0 047 at d 2 to 0 924 at d 6 and 1 at d our results agree with exact numerical estimates for any lattice in any dimension within few percent

some exact results on bond percolation we present some exact results on bond percolation we derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice by l bonds connecting the same adjacent vertices thereby yielding the lattice l this relation is used to calculate the bond percolation threshold on l we show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d 2 but changes it on a one dimensional lattice and quasi one dimensional infinite length strips we also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the n limits of several families of n vertex graphs finally we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as n

convergence of mixed finite element approximations for the shallow arch problem the stability and convergence of mixed finite element methods are investigated for an equilibrium problem for thin shallow elastic arches the problem in its standard form contains two terms corresponding to the contributions from the shear and axial strains with a small parameter lagrange multipliers are introduced to formulate the problem in an alternative mixed form questions of existence and uniqueness of solutions to the standard and mixed problems are addressed it is shown that finite element approximations of the mixed problem are stable and convergent reduced integration formulations are equivalent to a mixed formulation which in general is distinct from the formulation shown to be stable and convergent except when the order of polynomial interpolationt of the arch shape satisfies 1 t min 2 r wherer is the order of polynomial approximation of the unknown variables

enhanced functional evaluation for the finite element penalty method abstract we analyze the impact of using the penalty method on the estimation of a prescribed solution functional or quantity of interest qoi specifically we consider the use of penalty methods to enforce dirichlet boundary constraints focusing our attention on boundary fluxes as qois we propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter we show that the new estimator reduces the error arising from the use of the penalty method from o to o a well posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method errors in the enhanced flux estimator are then controlled using adjoint based techniques several numerical experiments are presented to demonstrate that the enhanced estimator in combination with adjoint error estimation allows one to efficiently control both the discretization and penalty errors in target qois

universal formulas for percolation thresholds ii extension to anisotropic and aperiodic lattices in a recent paper we reported a universal power law for both site and bond percolation thresholds in any bravais lattice with q equivalent nearest neighbors in dimension d we now extend it to three different classes of lattices which are respectively anisotropic lattices without equivalent nearest neighbors non bravais lattices with two atom unit cells and quasicrystals the investigation is focused on d 2 and 3 due to the lack of experimental data at higher dimensions the extension to these lattices requires the substitution of q by an effective non integer value q mathrm eff in the universal law for each of the 17 lattices which constitute our sample we argue for the existence of one q mathrm eff which reproduces both the site and the percolation threshold with a deviation with respect to numerical estimates which does not exceed ensuremath mp 0 01

adaptive dynamic relaxation algorithm for non linear hyperelastic structures part i formulation abstract the formulation of an adaptive dynamic relaxation algorithm with application to non linear hyperelastic structures is presented a complete derivation of the dynamic relaxation method is given and the adaptive scheme used to ensure reliability and improve performance is described finite element equations are derived for the non linear analysis of elastic and hyperelastic solids subject to large deformations the hyperelastic material is assumed to be nearly incompressible and it is modeled using the mooney rivlin material law using a penalty method to enforce incompressibility a very simple and efficient contact algorithm based on solver constraints is developed to enforce frictionless contact conditions

topology optimization by penalty top method in traditional structural topology optimization to the material properties of continuum finite elements of fixed form and coupling are varied to find the optimal topology that satisfies the design problem we develop an alternative fundamental formulation where the design space search is dependent on the coupling and the goal of the topology optimization by penalty top method is to determine the optimal finite element coupling constraints by this approach seemingly disparate topology design problems e g the design of structural supports topology optimization for fluid mechanics problems and topology optimization by the element connectivity parameterization ecp method can be understood as related formulations under a common topology optimization umbrella and more importantly this general framework can be applied to new design problems for example in modern multibody dynamics synthesis the geometric form of finite elements of fixed material properties and interconnectivity are varied to find the optimal topology that satisfies the mechanism design problem the a priori selection of coupling e g by revolute or translational joints severely limits the design space search the top method addresses this limitation in a novel way we develop the methodology and apply the top method to the diverse design problems discussed above

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

certified reduced basis methods for parametrized saddle point problems we present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems first we develop new a posteriori error estimates that unlike earlier approaches provide upper bounds for the errors in the approximations of the primal variable and the lagrange multiplier separately the proposed method is an application of brezzi s theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods second based on an analysis of brezzi s theory we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost finally we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and compared to earlier approaches computationally much more efficient the method is applied to a stokes flow problem in a two dimensional channel with a parametrized rectangular obstacle

penalty finite element method for the navier stokes equations abstract we present an analysis of a penalty formulation of the stationary navier stokes equations for an incompressible fluid subject to restrictions on the viscosity and prescribed body force it is shown that there exists a unique solution to this penalty problem the solution to the penalty problem is shown to converge to the solution of the navier stokes problem as o e where e 0 is the penalty parameter existence uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized navier stokes problem presented here numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases

a capillary free boundary problem governed by the navier stokes equations abstract in this paper an unsteady capillary free boundary in a viscous imcompressible fluid is studied from the numerical point of view the linearized problem is formulated and existence and uniqueness of a solution is proved the linearized problem is discretized by a finite element method in space and a finite difference method in time the numerical scheme is proved to be convergent and stable the method is applied to the computation of free boundaries eigenfrequencies damping factors and vibration modes the influence of the initial conditions the boundary conditions the contact angle the ohnesorge number and the bond number is studied

continuation techniques for a penalty approximation of the navier stokes equations continuation techniques are used to solve a penalty finite element approximation of the navier stokes equations sufficient conditions are given for convergence of the euler newton continuation method in a reynolds parameter to an isolated solution of the finite element problem numerical results are presented for the driven cavity problem the approach is extended to arc length continuation to treat problems with singular points on the continuation path

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

a hybrid finite element method for stokes flow part i formulation and numerical studies abstract a hybrid finite element scheme based on assumed deviatoric fluid stresses in the element and continuous velocity fields at the element boundaries is presented the deviatoric stress and hydrostatic pressure field are subject a priori to the constraints of balance of momenta the advantages of the present scheme are discussed and its versatility is demonstrated through a few numerical examples studies of convergence and stability of the method are included in part ii of the paper

criteria for evaluation of universal formulas for percolation thresholds several universal formulas that predict approximate values for percolation thresholds of all periodic graphs have been proposed in the physics and engineering literature the existing universal formulas have been found to have substantial errors in their predictions for some lattices this paper proposes a set of desirable criteria for universal formulas to satisfy and investigates which criteria are satisfied by two bond threshold formulas and two site threshold formulas most cited in the literature the analysis is limited to lattices in two dimensions

critical probability bounds for two dimensional site percolation models the authors present three techniques for determining rigorous bounds for site percolation critical probabilities of two dimensional lattices a technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that pc d or 0 5020 for the dice lattice d combining this method with kesten s duality result simplifies toth s derivation of the lower bound pc s or 0 5034 for the square lattice s the authors also present a technique for deriving upper bounds for bipartite graphs a technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice h pc h or 0 8079 and pc h or pc s the grouping technique is applied to the dice lattice to find the upper bound pc d or 0 7937

a bond percolation critical probability determination based on the star triangle transformation the bond percolation critical probability of a planar graph with square and triangular faces obtained by inserting a diagonal in every other face of the square lattice is the root of 1 p 6p2 6p3 p5 0 in 0 1 which is approximately 0 404 518 the proof uses the star triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given

biased continuous time random walks between parallel plates the generalized scheme of continuous time random walks in moving fluids a compte phys rev e 55 6821 1997 is applied to particles diffusing between parallel plates whose jumps are biased by a nonhomogeneous longitudinal velocity field we observe that when the statistics governing diffusion is brownian the results are those of taylor dispersion i e enhanced longitudinal diffusion due to the coupling of the transverse diffusion of the solute and the unidirectional velocity field however for l evy flights with infinite mean waiting time we observe an anomalous dispersion approaching ballistic diffusion we interpret this behavior as a consequence of the coupling between the flow and the waiting time statistics

an energy consistent material point method for dynamic finite deformation plasticity energy consistency for the material point method mpm is examined for thermodynamically consistent hyperelastic plastic materials it is shown that mpm can be formulated with implicit three field variational finite element algorithms which dissipate energy and conserve momentum for that class of material models with a consistent mass matrix the resulting overall numerical method inherits the energy dissipative and momentum conserving properties of the mesh solution thus the proposed mpm algorithm satisfies by construction a time discrete form of the second law of thermodynamics properties of the method are illustrated in numerical examples

analysis of hourglass instabilities and control in underintegrated finite element methods belytschko et al 1981 1984 has developed stabilization methods for the treatment of underintegrated fem problems these methods involve the computation of an underintegrated stiffness matrix which is rank deficient and the addition of a stabilization matrix which effectively eliminates the spurious modes an attempt is presently made to give this a priori stabilization method a mathematical means of support attention is also given to an a posteriority stabilization method for hourglass control in which an approximate solution of the underintegrated system is obtained and then subjected to a special projection in order to eliminate the hourglass modes a proof is obtained for the convergence of this stabilized underintegrated approximation to the exact solution of a model problem at almost the same rate as the mesh is refined as the fully integrated solutions

analysis of some mixed elements for the stokes problem abstract in this paper we discuss some mixed finite element methods related to the reduced integration penalty method for solving the stokes problem we prove optimal order error estimates for bilinear constant and biquadratic bilinear velocity pressure finite element solutions the result for the biquadratic bilinear element is new while that for the bilinear constant element improves the convergence analysis of johnson and pitkaranta 1982 in the degenerate case when the penalty parameter is set to be zero our results reduce to some related known results proved in by brezzi and fortin 1991 for the bilinear constant element and bercovier and pironneau 1979 for the biquadratic bilinear element our theoretical results are consistent with the numerical results reported by carey and krishnan 1982 and oden et al 1982

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem