linear stability of propagating phase boundaries in capillary fluids abstract dynamical phase changes occurring along planar traveling waves in capillary fluids are considered their multi dimensional stability which is encoded in the spectrum of a third order variational differential operator is addressed by two approaches on one hand an original energy method is used to show that these diffuse interfaces are weakly stable more precisely the spectrum of the involved operator is found to coincide with the imaginary axis on the other hand an evans function technique reveals a bifurcation phenomenon about zero the interpretation of which still needs some clarification it is the counterpart of the surface waves exhibited in an earlier work for the corresponding sharp interfaces

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

the determination of canonical forms for lattice quadrature rules abstract lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s dimensional unit hypercube us 0 1 s for a given lattice rule say ql a set of points l the integration lattice regularly spaced in all of rs is generated by a finite number of rational vectors the abscissa set for ql is then p ql l us it is known that p ql is a finite abelian group under addition modulo the integer lattice zs and that ql f may be written in the form of a nonrepetitive multiple sum q l f 1 n 1 n m j 1 1 n 1 j m 1 n m f j 1 n 1 z 1 j m n m z m known as a canonical form in which denotes addition modulo zs in this form zi zs m is called the rank and n1 n2 nm are called the invariants of ql and n i 1 n i for i 1 2 m 1 the rank and invariants are uniquely determined for a given lattice rule in this paper we provide a construction of a canonical form for a lattice rule ql given a generator set for the lattice l we then show how the rank and invariants of ql may be determined directly from the generators of the dual lattice l

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

existence and stability of viscoelastic shock profiles we investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by antman and malek madani we establish that the resulting equations fall into the class of symmetrizable hyperbolic parabolic systems hence spectral stability implies linearized and nonlinear stability with sharp rates of decay the new contributions are treatment of the compressible case formulation of a rigorous nonlinear stability theory including verification of stability of small amplitude lax shocks and the systematic incorporation in our investigations of numerical evans function computations determining stability of large amplitude and nonclassical type shock profiles

dynamical stability of phase transitions in the p system with viscosity capillarity we show for the p system with viscosity capillarity that all heteroclinic phase transitions lying sufficiently close to the maxwell line are linearly dynamically stable under localized perturbations independent of the form of the flux on the other hand we show that nearby homoclinic phase transitions involving two or more changes in phase are exponentially unstable these results together imply that stationary and slowly moving phase transitions are dynamically stable if and only if they are monotone in analogy with results obtained by carr gurtin and slemrod in the variational setting arch rational mech anal 86 1984 pp 317 351 this research is significant as the first proof of stability of a strong viscous shock wave for systems our analysis uses the technical framework developed in r gardner and k zumbrun comm pure appl math 51 1998 pp 797 855 k zumbrun and p howard indiana univ math j 47 1998 pp 741 871 together with a nonstandard energy estimate

convex entropy hopf bifurcation and viscous and inviscid shock stability we consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy specifically for a system possessing a convex entropy in particular for the equations of gas dynamics with a convex equation of state we ask 1 can inviscid instability occur 2 can viscous instability not detected by inviscid theory occur 3 can there occur the necessarily viscous effect of hopf bifurcation or galloping instability and perhaps most important from a practical point of view 4 as shock amplitude is increased from the stable weak amplitude limit can there occur a first transition from viscous stability to instability that is not detected by inviscid theory we show that 1 does occur for strictly hyperbolic genuinely nonlinear gas dynamics with certain convex equations of state while 2 and 3 do occur for an artifically constructed system with convex viscosity compatible entropy we do not know of an example for which 4 occurs leaving this as a key open question in viscous shock theory related to the principal eigenvalue property of sturm liouville and related operators in analogy with and partly proceeding close to the analysis of smith on non uniqueness of the riemann problem we obtain convenient criteria for shock in stability in the form of necessary and sufficient conditions on the equation of state

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

symmetrization of dissipative dispersive traveling waves for systems of conservation laws abstract this work addresses existence of traveling waves for systems of conservation laws through an hamiltonian formalism a related hamilton jacobi equation is given and studied our proofs rely on the construction of a cost functional and require a smallness condition for the dispersion with respect to diffusion some numerical applications are also presented concerning a model phase transition problem

conditional stability of unstable viscous shock waves in compressible gas dynamics and mhd extending our previous work in the strictly parabolic case we show that a linearly unstable lax type viscous shock solution of a general quasilinear hyperbolic parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 3 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by mascia and zumbrun the main new difficulty in the hyperbolic parabolic case is to construct an invariant manifold in the absence of parabolic smoothing

a study of incremental iterative strategies for non linear analyses the paper describes a study of incremental iterative solution techniques for geometrically non linear analyses the solution methods documented are based on a modified newton raphson approach meaning that the tangent stiffness matrix is computed at the commencement of each load step but is then held constant throughout the equilibrium iterations a consistent mathematical notation is employed in the description of the iterative and load incrementation strategies enabling the simple inclusion of several solution options in a computer program the iterative strategies investigated are iteration at constant load iteration at constant displacement iteration at constant arc length iteration at constant external work iteration at minimum unbalanced displacement norm iteration at minimum unbalanced force norm and iteration at constant weighted response the load incrementation schemes investigated include strategies based on the number of iterations required to achieve convergence in the previous load step strategies based on the current stiffness parameter and a strategy based on a parabolic approximation to the load deflection response criteria for detecting when the applied external load increment should reverse sign are described a challenging example of a circular arch exhibiting snap through load limit point behaviour and snap back displacement limit point behaviour is solved using several different iterative and load incrementation strategies the performance of the solution schemes is evaluated and conclusions are drawn

nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case abstract extending previous results of oh zumbrun and johnson zumbrun we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions key to our analysis is a nonlinear cancellation estimate observed by johnson and zumbrun along with a detailed understanding of the whitham averaged system the latter motivates a careful analysis of the bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave allowing us to derive the sharpened low frequency estimates needed to close the nonlinear iteration arguments

variationally consistent state determination algorithms for nonlinear mixed beam finite elements abstract this paper presents and evaluates several variationally consistent state determination algorithms for geometric and material nonlinear beam finite elements based on the hellinger reissner hr assumed stress variational principle the specifically targeted element type is derived from the two field displacement and generalized stress form of this principle momentum balance and strain compatibility are expressed in a weak form and the constitutive equations are satisfied directly at each integration point i e section along the element length a specific formulation of one element of this type is presented in a companion paper interelement compatibility is not enforced in the dual stress resultant variables for this element and thus various state determination algorithms can be formulated at the local i e element and section levels four state determination algorithms referred to as the l l l n n l and n n procedures are considered the first symbol in these names indicates the element level and the second denotes the section level furthermore the symbol n indicates that local nonlinear iteration is performed prior to returning to the higher level whereas the symbol l indicates that only the linearized equations are satisfied at the corresponding local level the l l algorithm corresponds to conventional newton iteration on both the element and section levels whereas the n n algorithm entails the iterative satisfaction of the nonlinear element compatibility and section constitutive equations for each global iteration the performance of these algorithms is evaluated and compared it is found that the algorithms in which the local nonlinear strain displacement compatibility conditions are solved iteratively at the element level for every global iteration lead to fewer global iterations at the expense of iterations at the element level in particular the n n algorithm is overall the most efficient of the four algorithms for the problems tested also the n n algorithm has substantially smaller element storage requirements the n n algorithm is expected to be particularly advantageous for cases in which a large number of elements remain elastic and inelasticity is concentrated in a few elements

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

a stability index for detonation waves in majda s model for reacting flow abstract using evans function techniques we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in siam j math anal 32 2001 929 commun pure appl math 51 7 1998 797 yielding useful necessary conditions for stability here we carry out the analysis in the context of the majda model a simplified model for reacting flow the method is extended to the full navier stokes equations of reacting flow in g lyng one dimensional stability of detonation waves doctoral thesis indiana university 2002 g lyng k zumbrun stability of detonation waves preprint 2003 the resulting stability condition is satisfied for all nondegenerate i e spatially exponentially decaying weak and strong detonations of the majda model in agreement with numerical experiments of siam j sci statist comput 7u 1986 1059 and analytical results of commun math phys 204 3 1999 551 commun math phys 202 3 1999 547 for a related model of majda and rosales we discuss also the role in the znd limit of degenerate subalgebraically decaying weak detonation and for a modified bump type ignition function deflagration profiles as discussed in siam j math anal 24 1993 968 siam j appl math 55 1995 175 for the full equations

the uniqueness and stability of the solution of the riemann problem of a system of conservation laws of mixed type we establish the uniqueness and stability of the similarity solution of the riemann problem for a 2 2 system of conservation laws of mixed type with initial data separated by the elliptic region which satisfies the viscosity capillarity travelling wave admissibility criterion

stability of periodic solutions of conservation laws with viscosity analysis of the evans function nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena including homoclinic pulse type and periodic solutions along with the standard heteroclinic shock or front type solutions here we investigate stability of periodic traveling waves within the abstract evans function framework established by r a gardner our main result is to derive a useful stability index analogous to that developed by gardner and zumbrun in the traveling front or pulse context giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period t as the traveling wave moreover we show that the periodic stability index has an interpretation analogous to that of the traveling front or pulse index in terms of well posedness of an associated riemann problem for an inviscid medium now to be interpreted as allowing a wider class of measure valued solutions or alternatively in terms of existence and nonsingularity of a local mass map from perturbation mass to potential time asymptotic t periodic states a closely related calculation yields also a complementary long wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period nt n we augment these analytical results with numerical investigations analogous to those carried out by brin in the traveling front or pulse case approximating the spectrum of the linearized operator about the wave

a mixed finite element for three dimensional nonlinear analysis of steel frames abstract this paper develops and demonstrates a mixed beam finite element that can represent the inelastic three dimensional stability behavior of frames composed of open walled cross section members the kinematics of deformation of the element include finite rotation and warping of the cross section due to torsion the material inelasticity is based on a two space model that includes the effect of shear stresses due to uniform torsion in addition to normal stresses due to axial force biaxial bending and bimoment initial geometric imperfections and residual stresses are accommodated the formulation is based on a two field displacement and generalized stress hellinger reissner hr variational principle the interpolation of the generalized stresses i e cross section stress resultants along the element length is based on the geometrically exact nonlinear governing differential equations of equilibrium this leads to significant improvements in the accuracy over conventional displacement based elements in problems involving highly nonlinear variations in the curvature along the member lengths due to geometric nonlinearity and or distributed plasticity effects several numerical examples involving two and three dimensional elastic and inelastic stability behavior are presented to illustrate the capabilities of the formulation

corotational mixed finite element formulation for thin walled beams with generic cross section the corotational technique is adopted here for the analysis of three dimensional beams the technique exploits the technology that applies to a two noded element a coordinate system which continuously translates and rotates with the element in this way the rigid body motion is separated out from the deformational motion in this paper a mixed formulation are adopted for the derivation of the local element tangent stiffness matrix and nodal forces the mixed finite element formulation is based on an incremental form of the two field hellinger reissner variational principle to permit elasto plastic material behavior the local beam kinematics is based on a low order nonlinear strain expression using bernoulli assumption the present formulation captures both the saint venant and warping torsional effects of thin walled open cross sections shape functions that satisfy the nonlinear local equilibrium equations are selected for the interpolation of the stress resultants in particular for the torsional forces and the twist rotation degree of freedom a family of hyperbolic interpolation functions is adopted in lieu of conventional polynomials governing equations are expressed in a weak form and the constitutive equations are enforced at each integration cross section along the element length a consistent state determination algorithm is proposed this local element together with the corotational framework can be used to analyze the nonlinear buckling and postbuckling of thin walled beams with generic cross section the present corotational mixed element solution is compared against the results obtained from a corotational displacement based model having the same beam kinematics and corotational framework the superiority of the mixed formulation is clearly demonstrated

stability of undercompressive shock profiles abstract using a simplified pointwise iteration scheme we establish nonlinear phase asymptotic orbital stability of large amplitude lax undercompressive overcompressive and mixed under overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations u 0 x e 0 1 x 3 2 in c 0 e 0 sufficiently small under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile this completes the program initiated by zumbrun and howard in k zumbrun p howard pointwise semigroup methods and stability of viscous shock waves indiana univ math j 47 4 1998 741 871 extending to the general undercompressive case results obtained for lax and overcompressive shock profiles in a szepessy z xin nonlinear stability of viscous shock waves arch ration mech anal 122 1993 53 103 t p liu pointwise convergence to shock waves for viscous conservation laws comm pure appl math 50 11 1997 1113 1182 k zumbrun p howard pointwise semigroup methods and stability of viscous shock waves indiana univ math j 47 4 1998 741 871 k zumbrun refined wave tracking and nonlinear stability of viscous lax shocks methods appl anal 7 2000 747 768 m r raoofi l 1 asymptotic behavior of perturbed viscous shock profiles thesis indiana univ 2004 c mascia k zumbrun pointwise green s function bounds and stability of relaxation shocks indiana univ math j 51 4 2002 773 904 c mascia k zumbrun stability of small amplitude shock profiles of symmetric hyperbolic parabolic systems comm pure appl math 57 7 2004 841 876 c mascia k zumbrun pointwise green s function bounds for shock profiles with degenerate viscosity arch ration mech anal 169 3 2003 177 263 c mascia k zumbrun stability of large amplitude shock profiles of hyperbolic parabolic systems arch ration mech anal 172 1 2004 93 131 c mascia k zumbrun stability of large amplitude shock profiles of general relaxation systems siam j math anal in press and for special weakly coupled respectively scalar diffusive dispersive undercompressive profiles in t p liu k zumbrun nonlinear stability of an undercompressive shock for complex burgers equation comm math phys 168 1 1995 163 186 t p liu k zumbrun on nonlinear stability of general undercompressive viscous shock waves comm math phys 174 2 1995 319 345 respectively p howard k zumbrun pointwise estimates for dispersive diffusive shock waves arch ration mech anal 155 2000 85 169 in particular together with spectral results of k zumbrun dynamical stability of phase transitions in the p system with viscosity capillarity siam j appl math 60 2000 1913 1924 our results yield nonlinear stability of large amplitude undercompressive phase transitional profiles near equilibrium of slemrod s model m slemrod admissibility criteria for propagating phase boundaries in a van der waals fluid arch ration mech anal 81 4 1983 301 315 for van der waal gas dynamics or elasticity with viscosity capillarity

asymptotic behavior of solutions to a system of viscous conservation laws related to a phase transition problem we study the asymptotic behavior of solutions for a system of viscous conservation laws with discontinuous initial data we discuss mainly the case where the system without the viscosity term is of hyperbolic elliptic mixed type this problem is related to a phase transition problem we study the initial value problem and show the decay rates of solutions to piecewise constant states where two phases coexist the modification necessary for the hyperbolic case is also discussed

a geometrically exact rod model incorporating shear and torsion warping deformation abstract a fully nonlinear three dimensional rod model is developed that incorporates transverse shear and torsion warping deformation the geometric setting is that of a constrained body model with configuration space modeled on r3 so 3 r a differentiable manifold the proposed model incorporates the classical notion of bi moment and bi shear in a fully nonlinear geometrically exact context explicit properly invariant constitutive equations that generalize those of the linearized theory are developed the underlying variational formulation of the model is discussed and computational procedures employing a galerkin projection are addressed numerical examples are presented that illustrate the performance of the formulation

a vanishing viscosity approach on the dynamics of phase transitions in van der waals fluids abstract the existence of solutions for the riemann problem of a mixed type system is established by a vanishing similarity viscosity approach the solutions that we construct are also admissible by the travelling wave criterion derived from the most common form of viscosity the assumption needed includes the constitutive function of van der waals fluids

stability of periodic solutions of conservation laws with viscosity pointwise bounds on the green function we investigate stability of periodic traveling wave solutions of systems of conservation laws with viscosity within the abstract evans function framework established by r a gardner our main result generalizing the work of zumbrun and howard in the traveling front or pulse setting is to establish sharp pointwise bounds on the green function for the linearized evolution equations provided that an appropriate evans function condition applies to the linearized operator about the wave this condition is equivalent to a spectral stability criterion introduced by schneider in the context of periodic reaction diffusion waves an immediate consequence is that strong spectral stability in the sense of schneider implies linearized l1 l p asymptotic stability for all p 1 on the other hand we show that the strict version of schneider s condition generically fails in the conservation law setting leading to complicated new metastable behavior reminiscent of that seen for degenerate neutrally stable families in the traveling front or pulse case our results apply also to the reaction diffusion setting sharpening at the linearized level results obtained by schneider using weighted norm and bloch decomposition methods

lattice methods for multiple integration theory error analysis and examples this paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension the concept of a lattice rule is introduced and error bounds are developed in terms of the dual lattice examples of lattice rules discussed in the paper are the rectangle rule the number theoretic good lattice methods of korobov and others the body centred cubic rule and a generalization of the latter to a family w nr the w nr rules appear to have interesting extrapolation properties both for fixed r and for r n

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

on the shock wave spectrum for isentropic gas dynamics with capillarity abstract we consider the stability problem for shock layers in slemrod s model of an isentropic gas with capillarity we show that these traveling waves are monotone in the weak capillarity case and become highly oscillatory as the capillarity strength increases using a spectral energy estimate we prove that small amplitude monotone shocks are spectrally stable we also show that monotone shocks have no unstable real spectrum regardless of amplitude this implies that any instabilities of these monotone traveling waves if they exist must occur through a hopf like bifurcation where one or more conjugate pairs of eigenvalues cross the imaginary axis we then conduct a systematic numerical evans function study which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths in particular we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities

a unified approach for variationally consistent finite elements in elastoplasticity abstract a general variationally consistent approach for elastoplastic finite element analysis is presented a weak formulation of the finite step i e in terms of finite increments elastoplastic boundary value problem is obtained by enforcing the stationarity of a mixed functional of hu washizu type a peculiarity of the present formulation is that the constitutive equations are expressed in weak form also all fields are then independently discretized in terms of so called generalized variables the result is a set of non linear algebraic equations a subset of which can be interpreted as the backward difference time integrated constitutive law of the finite element an iterative procedure where all calculations are at the element and not at the gauss point level resting on newton raphson scheme is proposed for the solution of the governing equations it is shown how the customary displacement approach and also some other mixed formulations in elastoplasticity can be regarded as special cases of the present formulation

frame elements with mixed formulation for singular section response mixed formulations of frame elements offer significant advantages over more traditional displacement formulations particularly under large cyclic inelastic deformations including the effects of shear this paper complements the recent proposal of a consistent variational basis for the mixed formulation of frame elements by supplying the explicit definition of the stress field over the cross section the paper also addresses the numerical stability of the element state determination algorithm in the presence of an ill conditioned or even singular section stiffness matrix the proposed algorithm is based on the eigendecomposition of the section stiffness matrix and uses the sherman morrison woodbury formula and the moore penrose pseudoinverse to avoid the inversion of ill conditioned matrices in the element state determination in the extreme case of uniform tension or uniform flexure the section flexibility matrix is split into an elastic and a plastic component before eigendecomposition with the proposed method the inelastic response of the element under multiple perfectly plastic hinges can be successfully traced numerical examples demonstrate the capabilities of the approach copyright 2008 john wiley sons ltd

the representation of lattice quadrature rules as multiple sums we provide a classification of lattice rules applying elementary group theory we assign to each s dimensional lattice rule a rank m and a set of positive integer invariants n sub 1 n sub 2 n sub s the number nu q of abscissas required by the rule is the product n sub 1 n sub 2 n sub s and the rule may be expressed in a canonical form with m independent summations under this classification an n point number theoretic rule in the sense of korobov and conroy is a rank m 1 rule having invariants n 1 1 1 and the product trapezoidal rule using n sup s points in rank m s rule having invariants n n n besides providing a canonical form we give some of the properties of copy rules and of projections into lower dimensions

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

worst case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the hermite and smith normal forms of an integer matrix an o s 5 m s 2 algorithm for computing the canonical structure of a finite abelian group represented by an integer matrix of size s this is the smith normal form of the matrix is presented moreover an o s 3 m s 2 algorithm for computing the hermite normal form of an integer matrix of size s is given the upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by kannan and bachem in siam j comput 8 1979 pp 499 507 and chou and collins in siam j comput 11 1982 pp 687 708

an application of diophantine approximation to the construction of rank 1 lattice quadrature rules lattice quadrature rules were introduced by frolov 1977 sloan 1985 and sloan and kachoyan 1987 they are quasi monte carlo rules for the approximation of integrals over the unit cube in r s and are generalizations of number theoretic rules introduced by korobov 1959 and hlawka 1962 themselves generalizations in a sense of rectangle rules for approximating one dimensional integrals and trapezoidal rules for periodic integrands error bounds for rank 1 rules are known for a variety of classes of integrands for periodic integrands with unit period in each variable these bounds are conveniently characterized by the figure of merit p which was originally introduced in the context of number theoretic rules the problem of finding good rules of order n that is having n nodes then becomes that of finding rules with large values of p this paper presents a new approach based on the theory of simultaneous diophantine approximation which uses a generalized continued fraction algorithm to construct rank 1 rules of high order

a sturm liouville theorem for quadratic operator pencils abstract we establish a sturm liouville theorem for quadratic operator pencils with matrix valued potentials counting their unstable real roots with applications to stability of waves such pencils arise for example in reduction of eigenvalue systems to higher order scalar problems