redundant cordic methods with a constant scale factor for sine and cosine computation proposes two redundant cordic coordinate rotation digital computer methods with a constant scale factor for sine and cosine computation called the double rotation method and the correcting rotation method in both methods the cordic is accelerated by the use of a redundant binary number representation as in the previously proposed redundant cordic in the proposed methods since the number of rotation extensions performed for each angle is a constant the scale factor is a constant independent of the operand hence one does not need to calculate the scale factor during the computation and can make a more efficient sine and cosine generator than that based on the previous redundant cordic

on the partial differential equations of electrostatic mems devices iii refined touchdown behavior abstract this paper is a continuation of n ghoussoub y guo on the partial differential equations of electrostatic mems devices stationary case siam j math anal 38 2007 1423 1449 and n ghoussoub y guo on the partial differential equations of electrostatic mems devices ii dynamic case nodea nonlinear differential equations appl 2008 in press where we analyzed nonlinear parabolic problem u t u f x 1 u 2 on a bounded domain of r n with dirichlet boundary conditions this equation models a simple electrostatic micro electromechanical system mems device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at 1 here u is modeled to describe dynamic deflection of the elastic membrane when a voltage represented here by is applied the membrane deflects towards the ground plate and a snap through touchdown must occur when it exceeds a certain critical value pull in voltage creating a so called pull in instability which greatly affects the design of many devices in an effort to achieve better mems design the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f x in this work some a priori estimates of touchdown behavior are established based on which the refined touchdown profiles are obtained by adapting self similar method and center manifold analysis applying various analytical and numerical techniques some properties of touchdown set such as compactness location and shape are also discussed for different classes of varying permittivity profiles

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

cordic vectoring with arbitrary target value the computation of additional functions in the cordic module increases its flexibility we consider here the extension of the vectoring mode angle calculation so that the vector is rotated until one of the coordinates for instance y attains a target value t in contrast to the value 0 as in standard vectoring the main problem in the algorithm is that the modulus of the vector is scaled in each cordic iteration so that a direct comparison of y j with t does not assure convergence we present a scheme that overcomes this and in which the implementation consists of a standard cordic module plus a module to determine the direction of rotation this improves over a previous proposal in which more complex iterations are introduced as part of the cordic algorithm moreover an error analysis is performed to determine the datapath width required for convergence since this width is large we consider also the characteristics of the algorithm for a narrower datapath

projection schemes for one point quadrature shell elements abstract a new projection scheme to improve the performance of one point quadrature shell elements is presented the projection is performed elementwise and enforces invariance of the internal power to rigid body motion and drill rotation the projection is especially suitable for 6 degree of freedom per node explicit programs and can be easily implemented as an integral part of the subroutine that calculates the internal forces on the element level problems that could not be solved accurately with one point quadrature elements such as the twisted beam problem show excellent results even for very coarse meshes when the projection is used

the blow up rate of solutions of semilinear heat equations on caracterise le comportement asymptotique de la solution de l equation de la chaleur semilineaire u t u exp u dans o t u k sur o t u x 0 x k pour xe k 0 dans un voisinage d un point d explosion quand t approche du temps d explosion fini t

self similar solutions for a quenching problem with spatially dependent nonlinearity abstract this paper is devoted to the study of the self similar solutions for a semilinear parabolic equation with spatially dependent nonlinearity which arises in the model of micro electro mechanical system we first provide a result on the non existence of slow orbit for a certain range of parameters next we prove the existence of backward solutions with the desired polynomial growth condition at infinity to the associated equation by a fixed point argument then we give a detailed analysis of the behavior of global solutions at the origin finally as an application of the above results we prove a uniqueness theorem

bisector and zero macrospin co rotational systems for shell elements a principal issue in any co rotational approach for large displacement analysis of plates and shells is associated with the specific choice of the local reference system in relation to the current deformed element configuration previous approaches utilised local co rotational systems which are invariant to nodal ordering a characteristic that is deemed desirable on several fronts however the associated definitions of the local reference system suffered from a range of shortcomings including undue complexity dependence on the local element formulation and possibly an asymmetric tangent stiffness matrix in this paper new definitions of the local co rotational system are proposed for quadrilateral and triangular shell elements which achieve the invariance characteristic to the nodal ordering in a relatively simple manner and address the aforementioned shortcomings the proposed definitions utilise only the nodal coordinates in the deformed configuration where two alternative definitions namely bisector and zero macrospin definitions are presented for each of quadrilateral and triangular finite elements in each case the co rotational transformations linking the local and global element entities are presented highlighting the simplicity of the proposed approach several numerical examples are finally presented to demonstrate the effectiveness and relative accuracy of the alternative definitions proposed for the local co rotational system

computing functions cos sup 1 and sin sup 1 using cordic an extension of the cordic coordinate rotation digital computer algorithm that makes it possible to compute the functions cos sup 1 sin sup 1 square root 1 t sup 2 sinh sup 1 cosh sup 1 and square root 1 t sup 2 is presented the algorithms are suitable for vlsi implementation and require only a slight modification of the original cordic algorithm

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

a co rotational quasi conforming 4 node resultant shell element for large deformation elasto plastic analysis abstract a co rotational quasi conforming formulation of a 4 node stress resultant shell element is presented for non linear analysis of plate and shell structures the tangent stiffness matrix in this quasi conforming formulation is explicitly integrated this makes the element computationally efficient in incremental non linear analysis it includes drilling degrees of freedom which improves membrane behavior and allows the modeling of stiffened plates and shells it is also free of shear locking behavior the formulation of the geometrical stiffness is derived using the full definition of green strain tensor the inclusion of the bending moment and transverse shear resultant forces in the geometric stiffness allows effective analysis of stability problems of moderately thick plates and shells the stresses are accurately taken at the nodal points without extrapolation the plasticity is traced by applying the von mises yield condition and prandtl reuss flow rule to discrete points through the thickness the multi layered approach is based on equally spaced stations including extreme fibers a modified trapezoidal rule is used for the numerical integration of the constitutive relation in the plasticity part numerous tests are carried out for the non linear validation of present 4 node shell element and the results are in good agreement with references

a formulation of the mitc4 shell element for finite strain elasto plastic analysis abstract a finite strain elasto plastic formulation is developed for the mitc4 shell element the developed formulation is based on lee s multiplicative decomposition of the deformation gradient and on the hyperelastic expression of von mises flow rule expressed in terms of hencky s strain tensor the formulation incorporates thickness stretching degrees of freedom that are condensed at the element level by imposing the classical in layer plane stress condition a symmetric and consistent tangent stiffness matrix is also developed

a unified formulation of small strain corotational finite elements i theory this paper presents a unified theoretical framework for the corotational cr formulation of finite elements in geometrically nonlinear structural analysis the key assumptions behind cr are i strains from a corotated configuration are small while ii the magnitude of rotations from a base configuration is not restricted following a historical outline the basic steps of the element independent cr formulation are presented the element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms it is shown that this framework permits the derivation of a set of cr variants through selective simplifications this set includes some previously used by other investigators the different variants are compared with respect to a set of desirable qualities including self equilibrium in the deformed configuration tangent stiffness consistency invariance symmetrizability and element independence we discuss the main benefits of the cr formulation as well as its modeling limitations

isogeometric kirchhoff love shell formulation for elasto plasticity abstract an isogeometric thin shell formulation allowing for large strain plastic deformation is presented a stress based approach is adopted which means that the constitutive equations are evaluated at different integration points through the thickness allowing the use of general 3d material models the plane stress constraint is satisfied by iteratively updating the thickness stretch at the integration points the deformation of the shell structure is completely described by the deformation of its midsurface and furthermore the formulation is rotation free which means that the discrete shell model involves only three degrees of freedom several numerical benchmark examples with comparison to fully 3d solid simulations confirm the accuracy and efficiency of the proposed formulation

large elasto plastic finite element analysis of solids and shells with the enhanced assumed strain concept abstract a finite element simulation of arbitrarily large elasto plastic deformations of three dimensional and shell structures is described two numerical aspects originally developed by j c simo have been combined i e the multiplicative large strain plasticity and the enhanced assumed strain finite element concept the constitutive model is worked out in its algorithmic version applying projection methods and consistent linearization the enhanced strain field is added to the conventional hexahedral displacement elements avoiding volumetric locking in the plastic range in addition to solid structures the three dimensional material model without any modifications is also applied to thin walled shell structures here an extended shell formulation a so called seven parameter model is used in which the thickness strain is augmented by a linear term the applicability of the entire formulation is demonstrated by several numerical examples

two fold normalized square root schur rls adaptive filter a square root schur rls adaptive filter with two fold input and residual normalization is presented the algorithm has several attractive features such as a fully systolic structure based on elementary hyperbolic plane rotations all internal variables are bounded in the unit interval and fully utilize it in successive stages due to an inherent autoscaling property of the algorithm this work extends the previous work of the author in the area of schur rls adaptive filtering

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

a geometrically non linear quasi conforming nine node quadrilateral degenerated solid shell element in this paper a geometrically non linear quasi conforming nine node quadrilateral degenerated solid shell element has been constructed from generalization of a linear quasi conforming nine node quadrilateral degenerated solid shell element 10 based upon the non linear quasi conforming finite element method first the geometrical description and displacement model of the nine node quadrilateral degenerated solid shell element is given then the non linear quasi conforming finite element method is discussed and assumed strains and weighed functions are given last analysis of numerical examples for a clamped square plate a circular arch a spherical shell two cylindrical shells the patch test and a simply supported square plate is carried out the results are compared with exact and other finite element solutions the results indicate that the shell element in the paper is accurate and efficient

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

redundant and on line cordic application to matrix triangularization and svd several modifications to the cordic method of computing angles and performing rotations are presented 1 the use of redundant carry free addition instead of a conventional carry propagate one 2 a representation of angles in a decomposed form to reduce area and communication bandwidth 3 the use of on line addition left to right digit serial addition to replace shifters by delays and 4 the use of online multiplication square root and division to compute scaling factors and perform the scaling operations the modifications improve the speed and the area of cordic implementations the proposed scheme uses efficiently floating point representations the application of the modified cordic method to matrix triangularization by givens rotations and to the computation of the singular value decomposition svd are discussed

a unified and division free cordic argument reduction method with unlimited convergence domain including inverse hyperbolic functions one of the main problems of the cordic algorithm is the limited convergence domain in which the functions can be calculated two different approaches can be employed to overcome this constraint first an argument reduction method and second an expansion of the cordic convergence domain while the first approach requires significant processing overhead due to the need for divisions especially for tanh sup 1 the second technique achieves an increased but still limited convergence domain only in this brief contribution we present a unified division free argument reduction method and a regular pipeline array architecture for floating point or fixed point implementations which results in savings of computation time in contrast to previous methods we avoid extra cordic arithmetic for realization of argument reduction

parallel singular value decomposition of complex matrices using multidimensional cordic algorithms the singular value decomposition svd of complex matrices is computed in a highly parallel fashion on a square array of processors using kogbetliantz s analog of jacobi s eigenvalue decomposition method to gain further speed new algorithms for the basic svd operations are proposed and their implementation as specialized processors is presented the algorithms are 3 d and 4 d extensions of the cordic algorithm for plane rotations when these extensions are used in concert with an additive decomposition of 2 spl times 2 complex matrices which enhances parallelism and with low resolution rotations early on in the svd process which reduce operation count a fivefold speedup can be achieved over the fastest alternative approach

the quantization effects of the cordic algorithm a detailed analysis of the quantization error encountered in the cordic coordinate rotation digital computer algorithm is presented two types of quantization error are examined an approximation error due to the quantized representation of rotation angles and a rounding error due to the finite precision representation in both fixed point and floating point arithmetic tight error bounds for these two types of error are derived the rounding error due to a scaling normalization operation in the cordic algorithm is also discussed an expression for overall quantization error is derived and several simulation examples are presented

on quenching for some parabolic problems abstract the universal quenching of a porous medium equation is established we also obtain some universal quenching properties for some semilinear parabolic problems and dichotomy properties for some general one dimensional problems

dynamics for a complex valued heat equation with an inverse nonlinearity abstract we study the cauchy problem for a parabolic system which is derived from a complex valued heat equation with an inverse nonlinearity first we provide some criteria for the global existence of solutions then we consider the case when the initial data are asymptotically constants and obtain that depending on the asymptotic limits the solution quenches at space infinity or exists globally in time

discrete basis and computation of elementary functions we give necessary and sufficient conditions in order that the infinite product or sum of the terms of a positive decreasing sequence generates the reals in a given interval

constant factor redundant cordic for angle calculation and rotation a constant factor redundant cordic cfr cordic scheme where the scale factor is kept constant while an angle for plane rotations is computed is developed the direction of rotation is determined from an estimate of the sign and convergence is assured by suitably placed correcting iterations the number of iterations in the cordic rotation unit is reduced by about 25 by expressing the direction of the rotation in radix 2 and radix 4 and conversion to conventional representation is done on the fly the performance of cfr cordic is estimated and compared with that of previously proposed schemes it is found to provide an execution time similar to that of redundant cordic with a variable scaling factor with a significant saving in area

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

on a highly robust spurious mode filtering method for uniformly reduced integrated shell elements summary in structural elements that allow shear deformation uniformly reduced integration is used to avoid shear and membrane locking the spurious modes of an underintegrated stiffness matrix can be filtered using perturbation methods although finite elements with perturbed stiffness show very good results the displacements still conceal a component of spurious modes albeit with very small magnitude depending on the perturbation factor to desensitize the numerical results completely from the perturbation factor a projection of the applied force is proposed to complement the perturbation of the stiffness such that spurious modes are not excited thus providing robustness to the filtering of spurious modes the proposed methodology is general in nature but is applied in this paper to a 9 node shell element with uniformly reduced integration as an illustration the perturbation of the stiffness matrix and the projection of the force vector are performed at the element level avoiding expensive local global transformations as a consequence of the force projection the perturbation factor for the stiffness can be arbitrarily small and the element yields accurate results for a wide range of shell thickness

quenching rate for a nonlocal problem arising in the micro electro mechanical system abstract in this paper we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro electro mechanical system this question is equivalent to the stabilization of the solution to the transformed problem in self similar variables first some a priori estimates are provided in order to construct a lyapunov function due to the lack of time monotonicity property we then derive some very useful and challenging estimates by a delicate analysis finally with this lyapunov function we prove that the quenching rate is self similar which is the same as the problem without the nonlocal term except the constant limit depends on the solution itself

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

incomplete quenching in a system of heat equations coupled at the boundary in this paper we find a possible continuation for quenching solutions to a system of heat equations coupled at the boundary condition this system exhibits simultaneous and non simultaneous quenching for non simultaneous quenching our continuation is a solution of a parabolic problem with neumann boundary conditions we also give some results for simultaneous quenching and present some numerical experiments that suggest that the approximations are not uniformly bounded in this case

on a hyperbolic equation arising in electrostatic mems abstract we consider the initial boundary value problem of a damped wave equation with singular nonlinearity which describes an electrostatic micro electro mechanical system mems device the results of the pull in voltage being the critical threshold for global existence and quenching are obtained if the applied voltage then the equation admits a unique global small solution that exponentially converges to the minimal steady state while large solution may quench in finite time if then any solution quenches in finite time finally in the sense of the viscosity dominated limit the asymptotic relation of solutions between the hyperbolic equation and the parabolic one is investigated also the related error estimates in arbitrary order are derived

a unified co rotational framework for solids shells and beams abstract the paper describes a unified framework for applying the co rotational method to the analysis of solids shells and beams the general method stems from an unusual application of the technique which involves solid elements the proposed framework allows a formulation that is simpler than many of the earlier procedures and in addition gives a direct indication of the terms in the tangent stiffness which may be ignored

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

quenching behavior of solutions to heat equations with coupled boundary singularities abstract this paper deals with finite time quenching for one dimensional heat equations coupled via negative singular boundary sources in the more general product form of two component powers u p 1 v q 1 and u p 2 v q 2 differently from the generally considered there are two additional singular factors namely u p 1 and v q 2 for the boundary sources of u and v respectively we at first propose a criterion to distinguish simultaneous and non simultaneous quenching and then determine all possible blow up rates in the model three for simultaneous quenching and two for non simultaneous quenching under different nominations this shows real contributions of the two additional singular factors to the quenching behavior of solutions

nonlinear postbuckling analysis of plates and shells by four noded strain element this paper deals with the nonlinear postbuckling analysis of plates and shells by the finite element method the four noded c strain element for nonlinear plate anaysis developed by the authors is first extended to the nonlinear shell analysis and then used for the nonlinear postbuckling analysis of plates and shallow shells the element tangent stiffness matrix presented here is given explicitly i e without any numerical integration consequently this c strain element is much more computationally economical and efficient than the widely used elements obtained from numerical integration the efficiency and accuracy of the present four noded strain element for postbuckling analysis are demonstrated by five numerical examples of various plates and shells

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

error analysis and reduction for angle calculation using the cordic algorithm in this paper we consider the errors appearing in angle computations with the cordic algorithm circular and hyperbolic coordinate systems using fixed point arithmetic we include errors arising not only from the finite number of iterations and the finite width of the data path but also from the finite number of bits of the input we show that this last contribution is significant when both operands are small and that the error is acceptable only if an input normalization stage is included making unsatisfactory other previous proposals to reduce the error we propose a method based on the prescaling of the input operands and a modified cordic recurrence and show that it is a suitable alternative to the input normalization with a smaller hardware cost this solution can also be used in pipelined architectures with redundant carry save arithmetic

numerical quenching for the semilinear heat equation with a singular absorption in this paper we study the numerical approximation for the heat equation with a singular absorption we prove that the numerical quenching rate coincides with the continuous one we also see that the quenching time and the quenching set converge to the continuous one in fact under some restriction on the initial data the numerical quenching coincides with the continuous one finally we give some numerical results to illustrate our analysis

optimal solid shells for non linear analyses of multilayer composites ii dynamics abstract we are presenting a simple low order solid shell element formulation having only displacement degrees of freedom dofs i e without rotational dofs that has an optimal number of parameters to pass the patch tests and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio with the dynamics referred to a fixed inertial frame the elements can be used to analyze multilayer shell structures undergoing large overall motion the formulation of this element is based on the mixed hu washizu variational principle leading to a novel enhancing strain tensor enhanced assumed strain eas method that renders the computation particularly efficient with improved in plane and out of plane bending behavior poisson thickness locking especially in refined analyses of composite structures involving a large number of high aspect ratio layers the energy momentum conserving algorithm in the context of current solid shell element is presented we discuss the eas formulation based on the displacement gradient and its complexity compared to formulation on the green lagrange strain shear locking and curvature thickness locking are treated using the assumed natural strain ans method the element has an optimal combination of the ans method and the minimal number of eas parameters required to pass the plate bending patch test numerical examples involving dynamic analyses with conservation of energy and momentum of multilayer shell structures having a large range of element aspect ratios are presented several implicit direct integration methods with without numerical dissipation are used and compared in terms of the accuracy stability and cost in multilayer shell structure finally we note that the topic in this paper is a fitting dedication to professor ekkehard ramm who has made important pioneering contributions in this research direction

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

square rooting algorithms for high speed digital circuits two binary algorithms for the square rooting of a number or of a sum of two numbers are presented they are based on the classical nonrestoring method the main difference lies in the replacement of subtractions and additions by a parallel reduction f three summands which may be positive and negative to two summands to eliminate the carry propagation two of three summands form the successive partial remainder their most significant bit triples sometimes together with a sign bit of the earlier partial remainder are used to determine digits 1 0 1 of a redundant square root notation these digits are transformed during the square rooting process into the conventional notation square root bits which are next used in further square rooting steps to form the third reduced summands

a stabilized 9 node non linear shell element a non linear 9 node stress resultant shell finite element with six degrees of freedom per node is formulated the material non linearity is based on an implicit integration scheme using the von mises yield criterion and linear isotropic bardening the small strain geometric non linearity is formulated using the polar decomposition theorem of continuum mechanics via a corotational updated lagrangian method which represents finite rotations with accuracy reduced integration is used to remove locking and calculate the stresses at their optimal stress accuracy points a practical procedure is employed to stabilize the troublesome spurious zero energy modes a number of tests covering the non linear material and geometry ranges and buckling show the good performance of the new element

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

on the quenching behavior of the solution of a semilinear parabolic equation where 0 e 0 0 6 u0 0 there exists an e b 0 such that u quenches at finite time t if e e b we say that a is a quenching point for u if there exists a sequence x t with x a and t r t such that u x t 1 as n co throughout this paper we assume that u quenches at finite time t and that u0 is smooth and satisfies

stress resultant plasticity for shells revisited in this work we revisit the stress resultant elastoplastic geometrically exact shell finite element formulation that is based on the ilyushin shapiro two surface yield function with isotropic and kinematic hardening the main focus is on implicit projection algorithms for computation of updated values of internal variables for stress resultant shell elastoplasticity four different algorithms are derived and compared three of them yield practically identical final results yet they differ considerably in computational efficiency and implementation complexity since they solve different sets of equations and they use different procedures that choose active yield surfaces one algorithm does not provide acceptable accuracy it turns out that the most simple and straightforward algorithm performs surprisingly well and efficiently several numerical examples are presented to illustrate the ilyushin shapiro stress resultant shell formulation and the numerical performance of the presented integration algorithms