a schur complement formulation for solving free boundary stefan problems of phase change a schur complement formulation that utilizes a linear iterative solver is derived to solve a free boundary stefan problem describing steady state phase change via the isotherm newton approach which employs newton s method to simultaneously and efficiently solve for both interface and field equations this formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire jacobian matrix for a two dimensional sample problem that discretizes the field equations using a galerkin finite element method and employs a deforming grid approach to represent the melt solid interface all methods demonstrate quadratic convergence for sufficiently accurate newton solves but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size of these two approaches the schur formulation proves to be more robust converging with significantly smaller krylov subspaces than those required to solve the global system of equations further improvement of performance are made through approximations and preconditioning of the schur complement problem hence the new schur formulation shows promise as an affordable robust and scalable method to solve free boundary stefan problems such models are employed to study a wide array of applications including casting welding glass forming planetary mantle and glacier dynamics thermal energy storage food processing cryosurgery metallurgical solidification and crystal growth

conservative rezoning remapping for general quadrilateral meshes a classic problem in lagrangian numerical hydrodynamics is the conservative transfer of quantities from an old distorted mesh to a new mesh the same problem arises whenever the mesh is changed as for example in adaptive mesh techniques this transfer of information is an interpolation process which is frequently called rezoning or remapping the general problem of conservative rezoning from one arbitrary mesh to another may be formulated as follows m sub k integral integral integral sub v sub k rho r dv that is we compute the mass m sub k of each cell of the new mesh by integrating the known density distribution in the old mesh over the cell volume v sub k a direct intregation is generally prohibitive we show however that is is possible to convert this integral to a surface integral by the appropriate use of the divergence theorem thus greatly reducing the complexity of the problem for two dimensional generall quadrilateral meshes the resulting method is exact and particularly simple

finite element simulation of dynamic wetting flows as an interface formation process a mathematically challenging model of dynamic wetting as a process of interface formation has been for the first time fully incorporated into a numerical code based on the finite element method and applied as a test case to the problem of capillary rise the motivation for this work comes from the fact that as discovered experimentally more than a decade ago the key variable in dynamic wetting flows the dynamic contact angle depends not just on the velocity of the three phase contact line but on the entire flow field geometry hence to describe this effect it becomes necessary to use the mathematical model that has this dependence as its integral part a new physical effect termed the hydrodynamic resist to dynamic wetting is discovered where the influence of the capillary s radius on the dynamic contact angle and hence on the global flow is computed the capabilities of the numerical framework are then demonstrated by comparing the results to experiments on the unsteady capillary rise where excellent agreement is obtained practical recommendations on the spatial resolution required by the numerical scheme for a given set of non dimensional similarity parameters are provided and a comparison to asymptotic results available in limiting cases confirms that the code is converging to the correct solution the appendix gives a user friendly step by step guide specifying the entire implementation and allowing the reader to easily reproduce all presented results including the benchmark calculations

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

on the influence of boundary conditions poisson s ratio and material non linearity on the optimal shape abstract design of a boundary in a material and in a structure is in principle an identical problem however for structures we mostly assume the external loads to be given while in the design for material moduli the loads are evaluated from forced displacements and the loads are therefore design dependent in the present study we shall focus on the influence of these different load assumptions also the influence of poisson s ratio and of material non linearity will be shown based on sensitivity analysis we also obtain general information the objective may be the stiffest design the strongest design or just a design of uniform energy density along the shape in an energy formulation it was proven earlier that these three objectives have the same solution at least within the limits of geometrical constraints including the parametrization without involving stress strain fields this proof holds good for 3d problems for power law non linear elasticity and for anisotropic elasticity with this background the results of a number of influence studies are presented

optimal bounds on the effective bulk modulus of polycrystals the classical problem of bounding the effective bulk modulus of isotropic polycrystalline aggregates built from a single anisotropic phase is studied for three dimensional composites we find that the voigt and reuss bounds are each attained by multi layered laminate geometries and hence are optimal for two dimensional composites the reuss lower bound is proved to be optimal while a new optimal upper bound is derived

multiple points of sample paths of markov processes 15 in 7 kahane treated symmetric stable levy processes iand obtained some zero probability and positive probability results the probability 1 result in the case k 2 was then proved by testard 16 in this paper we study this problem both positive probability and probability 1 for two large classes of markov processes namely diffusions driven by stochastic differential equations and levy processes with positive lower indices we first prove in theorem 2 1 a result for a general time homogeneous markov process with the transition density functions satisfying certain explicit conditions the existence of the k multiple points of x e is then ensured by the positivity of h0 e the hausdorff 01 measure of e where 0 0 01 1 and 0 is a

stress invariance in planar cosserat elasticity we extend the clm theorem cherkaev lurie milton proc r soc lond a 1992 438 519 529 to planar linear elastic materials of cosserat micropolar type that is those having microrotation as an additional degree of freedom besides the two in plane displacements more specifically it is shown that in the first planar problem of such media with smoothly varying material properties a shift in three out of four compliances is possible without changing the stress field the fourth coefficient represents a connection between the couple stress tensor and the torsion tensor while the other three represent compliances relating traction stress vector with the strains same shift holds for a locally anisotropic material whereby the shift tensor is seen to be a multiple of a rotation by a right angle a null lagrangian formulation is set up on this basis these results are obtained for materials with smooth properties with natural implications being drawn for their macroscopically effective moduli also it is shown that no shift is possible in case of a pseudo continuum a material admitting couple stresses but restricted to have the same connection between rotation and displacement gradient as in classical elasticity finally we establish that there is no shift in the second planar problem which represents a micropolar generalization of the classical out of plane elasticity

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

an isochoric domain deformation method for computing steady free surface flows with conserved volumes abstract the domain deformation method has been applied successfully to steady state free surface flows where the volume of the flow domain is unknown v f de almeida gas liquid counterflow through constricted passages ph d thesis university of minnesota minneapolis mn 1995 p a sackinger p r schunk r r rao a newton raphson pseudo solid domain mapping technique for free and moving boundary problems a finite element implementation j comput phys 125 1996 83 103 l c musson two layer slot coating ph d thesis university of minnesota minneapolis mn 2001 however this method does not handle effectively problems where the volume of the flow domain is known a priori this work extends the original domain deformation method to a new isochoric domain deformation method to account for the volume conservation like in the original domain deformation method the unknown shape of the flow domain is mapped onto a reference domain by using the equations of an elastic pseudo solid the difference with the original method is that this pseudo solid is considered incompressible because of the incompressibility the pseudo pressure of the mapping appears as a lagrange multiplier in the equations and it is determined only up to an arbitrary uniform datum by analyzing the coupled fluid flow mapping problem we show that in the finite element setting such pressure datum can be specified by replacing one continuity equation in the fluid domain

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

generalized plane deformation of electromagnetic thermoelastic solids i correspondence and invariance shifts this two part paper extends the recent results in plane elasticity to generalized plane deformation of electromagnetic thermoelastic materials these include the correspondences between various phy

effective properties of composite materials containing voids recent results of theoretical and practical importance prove that the two dimensional in plane effective average young s modulus for an isotropic elastic material containing voids is independent of the poisson s ratio of the matrix material this result is true regardless of the shape and morphology of the voids so long as isotropy is maintained the present work uses this proof to obtain explicit analytical forms for the effective young s modulus property forms which simplify greatly because of this characteristic in some cases the optimal morphology for the voids can be identified giving the shapes of the voids at fixed volume that maximize the effective young s modulus in the two dimensional situation recognizing that two dimensional isotropy is a subset of three dimensional transversely isotropic media it is shown in this more general case that three of the five properties are independent of poisson s ratio leaving only two that depend upon it for three dimensionally isotropic composite media containing voids it is shown that a somewhat comparable situation exists whereby the three dimensional young s modulus is insensitive to variations in poisson s ratio v m over the range 0 v m although the same is not true for negative values of v m this further extends the practical usefulness of the two dimensional result to three dimensional conditions for realistic values of v m

more on stress invariance conditions for the traction boundary value problem of plane linear elasticity abstract cherchaev 1992 gave a sufficient condition for a change in the elastic tensor field to preserve the stress state of a plane linearly elastic solid subject to prescribed traction forces on its boundary this condition turned out to have a number of fruitful applications in the mechanics of composite materials and was later extended by dundurs and markenstoff 1993 the present work addresses and answers the question whether there is a yet more general condition we show that i if the change in the elastic tensor field is required to be hyperelastic the extended condition of dundurs and markenscoff is not only sufficient but also necessary ii if the change in the elastic tensor field is relaxed so as to be elastic a more general necessary and sufficient condition exists in proving these two conclusions several orthogonal decompositions are constructed for third and fourth order tensors presenting index permutation symmetries such decompositions are probably also useful for solving other problems in mechanics

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

a new stabilization of adaptive step trapezoid rule based on finite difference interrupts the adaptive step trapezoid rule tr is a generally effective numerical integrator but it is prone to ringing instability and solution stall we introduce a new stabilization algorithm based on finite difference interrupts fdi that reduces ringing and prevents stall unlike previously reported stabilization schemes our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion tr with fdi is at least as stable and accurate as existing methods when solving the prototypical scalar problem we demonstrate that it has better stability accuracy and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids though we demonstrate tr with fdi in the context of a finite element method of lines it is applicable to any tr based algorithm

stress and couple stress invariance in non centrosymmetric micropolar planar elasticity the stress invariance problem for a chiral non centrosymmetric micropolar material model is explored in two different planar problems the in plane and the anti plane problems this material model grasps direct coupling between the cauchy type and cosserat type or micropolar effects in hooke s law an identical strategy of invariance is set for both problems leading to a remarkable similarity in their results for both problems the planar components of stress and couple stress undergo strong invariance while their out of plane counterparts can only attain weak invariance which restricts all compliance moduli transformations to a linear type as an application when heterogeneous composite materials are subjected to weak invariance their effective volume averaged compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material independently of the microstructure geometry and phase distribution these analytical results constitute a valuable means to validate computational procedures that handle this particular type of material model

composite three dimensional grids generated by elliptic systems a technique is presented for constructing a boundary conforming grid throughout a general threedimensional flow region as a composite of subregion grids each subregion grid is generated numerically by solving a nonlinear system of elliptic equations the boundary values represent nodal points in a quasi twodimensional grid that covers the curved surface bounding the subregion and are generated numerically by a modified elliptic system a primary feature of the technique is that the composite three dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions sample results are given for a wing body combination

discretization of three dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles abstract a new boundary fitted technique to describe free surface and moving boundary problems is presented we have extended the 2d elliptic grid generator developed by dimakopoulos and tsamopoulos 2003 19 and further advanced by chatzidai et al 2009 18 to 3d geometries the set of equations arises from the fulfillment of the variational principles established by brackbill and saltzman 1982 21 and refined by christodoulou and scriven 1992 22 these account for both smoothness and orthogonality of the grid lines of tessellated physical domains the elliptic grid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2d quasi elliptic grid methodologies the capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces the sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters subsequently the mesh equations are coupled with the navier stokes equations in order to reveal the full potential of the proposed methodology in free surface flows more specifically the problem of gas assisted injection in ducts of circular and square cross sections is examined where the fluid domain experiences extreme deformations finally the flow mesh solver is used to calculate the equilibrium shapes of static menisci in capillary tubes

cavities vis a vis rigid inclusions elastic moduli of materials with polygonal inclusions in this paper we explore the correspondence between rigid inclusions and cavities dundurs 1989 j appl mech 56 786 790 as applied to the effective elastic moduli of materials with polygonal rigid inclusions and cavities in the analysis we use a complex variable method of elasticity and a conformal transformation to solve for the stress field due to a single rigid inclusion then we use a far field approach to obtain the effective elastic constants of composites with a dilute concentration of rigid polygonal inclusions by employing the dundurs correspondence we can automatically obtain the result for the effective elastic moduli of materials with cavities finally we use effective medium theories to predict the elastic moduli of materials containing a finite concentration of inclusions

new results in the theory of elasticity for two dimensional composites we bring together and discuss a number of exact relationships in two dimensional or plane elasticity that are useful in studying the effective elastic constants and stress fields in two dimensional composite materials the first of these dates back to michell 1899 and states that the stresses induced by applied tractions are independent of the elastic constants in a two dimensional material containing holes the second involves the use of dundurs constants which for a composite consisting of two isotropic elastic phases reduce the dependence of stresses on the elastic constants from three independent dimensionless parameters to two it is shown that these two results are closely related to a recently proven theorem by cherkaev lurie and milton which we use to show that the effective young s modulus of a sheet containing holes is independent of the poisson s ratio of the matrix material we also show that the elastic moduli of a composite can be found exactly if the shear moduli of the components are all equal a previously known result we illustrate these results with computer simulations where appropriate finally we conjecture on generalizations to multicomponent composite materials and to situations where the bonding between the phases is not perfect

sur la saucisse de wiener et les points multiples du mouvement brownien on considere b t t 0 un mouvement brownien a valeurs dans l espace euclidien a d dimensions r d d 2 issu de o on s interesse a la probabilite que b penetre avant un instant donne t dans une petite boule de l espace

harmonic analysis of additive levy processes let x1 xn denote n independent d dimensional levy processes and consider the n parameter random field mathfrak x t x 1 t 1 cdots x n t n first we demonstrate that for all nonrandom borel sets f subseteq bf r d the minkowski sum mathfrak x bf r n oplus f of the range mathfrak x bf r n of mathfrak x with f can have positive d dimensional lebesgue measure if and only if a certain capacity of f is positive this improves our earlier joint effort with yuquan zhong by removing a certain condition of symmetry in khoshnevisan et al ann probab 31 2 1097 1141 2003 moreover we show that under mild regularity conditions our necessary and sufficient condition can be recast in terms of one potential densities this rests on developing results in classical non probabilistic harmonic analysis that might be of independent interest as was shown in khoshnevisan et al ann probab 31 2 1097 1141 2003 the potential theory of the type studied here has a large number of consequences in the theory of levy processes presently we highlight a few new consequences

generalized plane deformation of electromagnetic thermoelastic solids ii further results on invariance shifts and reduced dependences the validity of the invariance shifts on electromagnetic thermoelastic moduli is rigorously proven to be true for a multiply connected domain with piecewise continuously varying moduli and or with various spring layer and constrained spring layer interfaces these results are further applied to establish the reduced dependence relations of the effective moduli upon the material parameters particularly for a homogeneous matrix with parallel infinitely long cylindrical surfaces including parallel tunnels of holes of various shapes and microcracks in generalized plane strain deformation all the stress electric displacement and magnetic induction fields under prescribed stress electric displacement and magnetic induction boundary conditions and all the normalized differences between the effective compliance and the matrix compliance depend upon only the completely symmetric parts of the moduli tensors for example the in plane minor symmetric elastic compliance s ijkl s jikl s ijlk and the piezoelectric tensor p kij p kji may at most contain nine and six independent components respectively while the completely symmetric parts of s ijkl and p kij contain at most five and four completely symmetric parts

assessing the quality of curvilinear coordinate meshes by decomposing the jacobian matrix an algebraic decomposition of the jacobian matrix j x i email protected j which relates physical and computational variables is presented this invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation cell orthogonality cell volume and cell aspect ratio this decompostion can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm this is worked out in detail for laplace s equation in nonconservative and conservative forms an analysis by g h k 1 of the mesh contribution to truncation error for the full potential code tair is given in abbreviated form the variables introduced here and their derivatives are also natural lagrange multipliers for adaptive mesh algorithms based on a variational principle

interface design of neutral elastic inclusions abstract the paper studies the design of the neutral elastic inclusions that do not disturb the prescribed uniform stress field in the surrounding elastic body such neutral inclusion does not exist if a perfectly bonded interface between inclusion and elastic body is assumed the design method presented here is based on the model of imperfect interface characterized by the interface parameters the basic equations for interface design of a single neutral elastic inclusion in plane and anti plane deformations are given and the corresponding interface parameters are discussed for several typical inclusion shapes in particular the design of a neutral inclusion under uniaxial or equal biaxial tension is studied with a restriction imposed on the interface parameters that facilitates implementation of the designed interface the results obtained in the paper affirm the feasibility of designing a neutral elastic inclusion in many typical cases by means of the imperfect interface

predictability and stopping on lattices of sets as a first step in the development of a general theory of set indexed martingales we define predictability on a general space with respect to a filtration indexed by a lattice of sets we prove a characterization of the predictable algebra in terms of adapted and left continuous processes without any form of topology for the index set we then define a stopping set and show that it is a natural generalization of the stopping time in particular the predictable algebra can be characterized by various stochastic intervals generated by stopping sets

algebraic grid generation three methods are described for transforming grids in bounded two and three dimensional physical domains into a uniform grid in a rectangular computational domain the methods are based on mathematical interpolation functions and do not require the solution of differential equations or the use of complex variables they are simply referred to as algebraic methods and are called transfinite interpolation the multisurface method and the two boundary technique the primary advantage of the methods is that they provide explicit control of physical grid shape and physical grid spacing secondly they require relatively few computations consequently the application of inter active computer graphics in conjunction with the methods is advocated for rapid generation of grids the basic mathematical structure of each method is described and particular attention is given to surface representation parameter variable generation and control function generation physical boundary topology and grid derivative requirements are presented

random walks and intersection local time with each random walk on z 2 we associate a functional related to the number of steps which the walk spends in sites occupied at least k times we show that if our random walk is in the domain of attraction of a stable process of order greater than 2 2k 2 2k 1 then our functional converges to the intersection local time of the limiting process

a quasi elliptic transformation for moving boundary problems with large anisotropic deformations we have developed a quasi elliptic set of equations for generating a discretization mesh that optimally conforms to an entire domain that undergoes large deformations in primarily one direction we have applied this method to the axisymmetric problem of the transient displacement of a viscous liquid by a high pressure gas the liquid initially fills completely a tube the diameter of which may be constant or change smoothly suddenly or periodically along its finite length key ingredients for the success of the proposed transformation are limiting the orthogonality requirements on the mesh and employing an improved node distribution function along the deforming boundary the retained orthogonal term along with the penalty method for the imposition of the boundary conditions overcome the inherent restrictions of a conformal transformation producing meshes of high quality this term also eliminates the discontinuous slopes of the coordinate lines that are normal to the free surface these usually arise due to the harmonic transformation around highly deforming surfaces the mathematical anisotropy in the mesh generating equations directly corresponds to the physical one when the initially straight and normal to the axis of symmetry air fluid interface at the tube entrance develops mostly along the axial direction generating a long open bubble moreover the generalized node distribution imposes an optimal discretization of the bubble surface itself combining this scheme with the mixed finite element method produces a powerful tool with extended robustness and accuracy in order to reduce the computational cost the resulting set of equations is solved with a non linear gauss seidel technique and a variable time step specific applications of the proposed scheme are presented

application of a variational method for generating adaptive grids the application of variational methods for generating adaptive grids is not as straightforward as one is led to believe proper scaling suitable weight functions and appropriate clustering on boundaries must be employed to obtain a satisfactory grid this work which is based on the framework developed by brackbill and saltzman provides simple methods for determining scaling and investigates possible options for selecting the weight function and clustering points on the boundaries the concepts developed here are applied to two two dimensional problems a model problem based on burger s equation which contains two length scales and transonic flow past airfoils using the euler equations

tracking birth of vortex in flows in order to avoid undesired effects from vortices in many industrial processes it is important to know the set of operating parameters at which the flow does not have recirculation the map of these conditions in the parameter space is called vortex free operating window here we propose an efficient way to construct such window automatically without expensively checking every possible flow states the proposed technique is based on tracking a path in the parameter space at which the local kinematic condition at a stagnation point for vortex birth is satisfied this multiparameter continuation is performed by solving an augmented navier stokes system in the augmented system the birth condition and the governing equations was represented in galerkin s finite element context we used the proposed method in two important coating flows with free surfaces single layer slot coating and forward roll coating

local times of fractional brownian sheets let b0h b0h t t i mathbb r n be a real valued fractional brownian sheet consider the n d gaussian random field bh defined by bh t bh1 t bhd t t i mathbb r n where bh1 bhd are independent copies of b0h in this paper the existence and joint continuity of the local times of bh are established

measure of the multiple self intersection set of a markov process let x t t 0 be a markov process in rm with homogeneous transition density p t x y for a closed bounded set b rm x is said to have a self intersection of order r 2 in b if there are distinct points t1 tr such that x t1 b and x tj x t1 for j 2 r the focus of this work is the hausdorff measure suitably defined of the set of such r tuples the main result is that under general conditions on p t x y as well as the specific condition there is a measure function m t defined in terms of the integral above such that the corresponding hausdorff measure of self intersection set is positive with positive probability the results are applied to levy and diffusion processes and are shown to extend recent results in this area

stress invariance and redundant moduli in three dimensional elasticity a three dimensional framework is established for generating invariant stress configurations and associated shifts in the elastic compliance under these shifts the stress throughout an elastic body is unaltered while the compatibility equations for the strain are automatically satisfied the types of invariant stress fields and translations of the compliance identified here generalize the re9 19 2008sults of cherkaev lurie and milton clm for planar elasticity the key to the classification is the partitioning of the fourth order compliance tensor into symmetric and antisymmetric components the clm theorem and its generalization are closely linked to the six dimensional antisymmetric part of the compliance and several examples are given of stress invariance under shifts of these elements of the compliance tensor

flows in forward deformable roll coating gaps roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a continuous flexible substrate of the two rolls that make a forward roll coating gap one is often covered by a layer of deformable elastomer thin films can be obtained without the risk of clashing two hard rolls liquid carried into the converging side of the gap can develop high enough pressure to deform the resilient cover which changes the gap geometry and thus alters the velocity and pressure fields the complete understanding of the flow in this situation is vital to the optimization of this widely used coating method however this elastohydrodynamic action is not well understood the situation is similar to what is called the soft elastohydrodynamic lubrication regime soft ehl however the range of minimum distance between the rotating rolls roll speed and therefore flow rate through the gap in the roll coating process is one to three orders of magnitude larger than the typical values reported in previous work on soft ehl earlier works on deformable roll coating analyzed the action with both the lubrication approximation and the full navier stokes solution and different one dimensional models of roll cover deformation in order to test the accuracy of the past approaches and to evaluate the relationship between the empirical constant used in the one dimensional model to the relevant physical parameters a complete two dimensional formulation has to be employed for both the liquid flow and the solid deformation in this work the flow between a rigid and a deformable rotating roll was examined by solving the complete navier stokes system coupled with a non linear plane strain model of the roll cover deformation the approximate and computationally cheaper approach is evaluated in which the compliant wall is represented by an array of radially oriented hookean springs the equation system was solved by the galerkin finite element method the resulting set of non linear algebraic equations of the fully coupled problem was solved by newton s method with initialization by pseudo arc length continuation as parameters were varied results show how roll deformation affects the total flow rate and forces on the rolls and illustrate how a deformable roll can be used to obtain thin coated layers with much less sensitivity to roll runout than those obtained with rigid rolls

an exact correspondence between plane piezoelectricity and generalized plane strain in elasticity we consider an anisotropic body bounded by a cylindrical surface which is infinitely long in the axial direction suppose the body is loaded in such a way that the field variables do not vary along the generators an exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity in particular we show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa the assertation is true for rectilinearly as well as for cylindrically anisotropic solids the equivalence is found for the most general anisotropic case which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic triclinic elastic solid a few degenerate systems are also identified in addition these correspondences can be extended to inhomogeneous media applied to composite materials or polycrystalline aggregates they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors and vice versa we also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by cherkaev lurie and milton in illustration we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity lekhnitskii9s complex potential approach together with the conformal mapping technique are employed general solutions of the fields inside the inhomogeneity and the matrix are obtained the results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

exact transient full field analysis of a finite crack subjected to dynamic anti plane concentrated loadings in anisotropic materials in this study the elastodynamic full field response of a finite crack in an anisotropic material subjected to a dynamic anti plane concentrated loading with heaviside function time dependence is investigated a linear coordinate transformation is introduced to simplify the problem the linear coordinate transformation reduces the anisotropic finite crack problem to an equivalent isotropic problem an alternative methodology different from the conventional superposition method is developed to construct the reflected and diffracted wave fields the transient solutions are determined by superposition of two proposed fundamental solutions in the laplace transform domain the fundamental solutions to be used are the problems for applying exponentially distributed traction and displacement on the crack faces and along the crack tip line in the laplace transform domain respectively exact analytical transient solutions for dynamic shear stresses displacement and stress intensity factor are obtained by using the cagniard de hoop method of laplace inversion and are expressed in explicitly compact formulations the solutions have accounted for the contributions of all diffracted waves generated from two crack tips numerical results for the time history of shear stresses and stress intensity factors during the transient process are calculated based on analytical solutions and are discussed in detail the transient solutions of stresses have been shown to approach the corresponding static values after the first eight waves have passed the field point the dynamic stress intensity factor will reach a maximum value when the incident wave arrives at the crack tip and remain constant before the first diffracted wave generated from the other crack tip arrives and then will oscillate near the static value a simple explicit expression of the dynamic overshoot for stress intensity factors is derived as a function of the location for applied loadings the crack length and material constants

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

discretization of free surface flows and other moving boundary problems a system of elliptic partial differential equations and boundary conditions has been developed for generating boundary fitted element discretizations of two dimensional free and moving boundary problems terms in the differential equations are scaled for dimensional homogeneity and adjustable weighting of orthogonality smoothness and concentration of the coordinate mesh they govern grid points become finite element nodes mapped isoparametrically or subparametrically from a simple or patched computational domain concentration terms contain control functions and parameters that influence node spacing along each coordinate independently overall control is by patchwise parameters and functions successful selection of these to follow deforming flow regions is straightforward and is illustrated by analysis of steady and transient slide coating flows

holder conditions for the local times and the hausdorff measure of the level sets of gaussian random fields let y t t n be a real valued strongly locally nondeterministic gaussian random field with stationary increments and y 0 0 consider the n d gaussian random field defined by where x 1 x d are independent copies of y the local and global holder conditions in the set variable for the local time of x t are established and the exact hausdorff measure of the level set x 1 x is evaluated

invariance of stresses under a change in elastic compliances a recent discovery by cherkaev lurie and milton shows that in plane elasticity involving a material with varying compliances it is possible to keep the stress state the same by compensating for a constant shift in the bulk compliance with the opposite shift in the shear compliance the present article demonstrates that depending on the connectivity of the domain and some side conditions the stress state may also be invariant upon a linear shift in the elastic compliances discontinuous compliances such as encountered in composite materials and various interface conditions as well as global compatibility are discussed in detail

marangoni benard convection with a deformable free surface computations of marangoni convection are usually performed in two or three dimensional domains with rigid boundaries in two dimensions allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure we describe a finite element technique for calculating bifurcations that arise due to thermal gradients in a two dimensional domain with a deformable free surface the fluid is assumed to be newtonian to conform to the boussinesq approximation and to have a surface tension that varies linearly with temperature an orthogonal mapping from the physical domain to a reference domain is employed which is determined as the solution to a pair of elliptic partial differential equations the mapping equations and the equilibrium equations for the velocity pressure and temperature fields and their appropriate nonlinear boundary conditions are discretized using the finite element method and solved simultaneously by newton iteration contact angles other than 90 degrees are shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner the loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to and less than 90 degrees

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

a newton raphson pseudo solid domain mapping technique for free and moving boundary problems an implicit pseudo solid domain mapping technique is described that facilitates finite element analysis of free and moving boundary problems the technique is based on an implicit full newton strategy free of restrictions on mesh structure this leads to many advantages over existing domain mapping techniques the fully coupled approach using newton s method is particularly effective for problems with strong coupling between the internal bulk physics and the governing physics at unknown free boundary locations it is also useful when the distinguishing conditions which constrain the free boundary shape provide only an implicit dependence on the boundary location unstructured meshes allow for efficient resolution of internal and boundary layers and other regions of strong local variations in the solution and they also reduce the amount of user interaction required to define a problem since the meshes may be generated automatically the technique is readily applied to steady or transient problems in complex geometries of two and three dimensions examples are shown that include free and moving boundary problems from solidification and capillary hydrodynamics

coordinate system control adaptive meshes the discussion of the formulation and properties of the adaptive mesh algorithm is as complete as space permits as should be clear from the discussion adaptive gridding is most useful for singular perturbation problems where localized regions with large gradients develop spontaneously often choosing the correct weight function is not trivial and sometimes it is appropriate to cause the mesh to respond to structure in several dependent variables simultaneously

generation of orthogonal and nearly orthogonal coordinates with gridcontrol near boundaries a numerical procedure is presented for the generation of boundary fitted curvilinear coordinates with controllable mesh spacing and orthogonality the technique is based on a straightforward two step procedure involving the solution of poisson s equation the method is capable of generating orthogonal grids with partial control of the mesh spacing or nearly orthogonal grids with strict control of the mesh spacing the inhomogeneous terms in the poisson equations are automatically adjusted during the grid generation the technique is applicable to two dimensional simply connected regions typical of airfoils cascades diffusers and inlets

on the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations we present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions this method belongs to the class of arbitrary lagrangian eulerian ale methods for solving moving boundary problems its tools are either a separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or b domain decomposition and separate mappings of each subdomain using suitable coordinate systems the latter is shown to be more versatile and general in both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by dimakopoulos and tsamopoulos y dimakopoulos j a tsamopoulos a quasi elliptic transformation for moving boundary problems with large anisotropic deformations j comput phys 192 2003 494 522 we shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi elliptic set of pdes fail to produce an acceptable mesh and accurate solutions in such geometries furthermore in contrast to other methods appropriate boundary conditions and constraints such as orthogonality of specific mesh lines and prespecified node distributions on them can be easily implemented along a specific part of the domain or its boundary hence no attractive terms at specific corners or singular points are needed to increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time a local mesh refinement technique has been incorporated as well the method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations in the first one the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined while in the second one the linear and non linear dynamics of two bubbles in a viscous medium are determined in an acoustic field the large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein the governing equations are solved using the finite element galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid these are coupled with an implicit euler method for time integration or with arnoldi s algorithm for normal mode analysis

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

random fields associated with multiple points of the brownian motion abstract let xt be the brownian motion in r d the random set t1 tn z xtl xtn z in r d n is empty a s except in the following cases a n 1 d 1 2 b d 2 n 2 3 c d 3 n 2 in each of these cases a family of random measures m concentrated on is constructed takes values in a certain class of measures on r d measures m characterize the time space location of self intersections for brownian paths if n d 1 then m dt dz dz nz dt where n2 is the local time at z in the case n 2 the set can be identified with the set of brownian loops the measure m explodes on the diagonal t1 t2 and to study small loops a random distribution which regularizes m is constructed

self intersection local time critical exponent large deviations and laws of the iterated logarithm if t is renormalized self intersection local time for planar brownian motion we characterize when ee 1 is finite or infinite in terms of the best constant of a gagliardo nirenberg inequality we prove large deviation estimates for 1 and 1 we establish lim sup and lim inf laws of the iterated logarithm for t as t