symmetry characterization and measurement errors of elasticity tensors it is often desirable to approximate a full anisotropic tensor given by 21 independent parameters by one with a higher symmetry if one considers measurement errors of an elasticity tensor the standard approaches of finding the best approximation by a higher symmetric tensor do not produce the most likely tensor to find such a tensor i replace the distance metric used in previous studies with one based on probability distribution functions of the errors of the measured quantities in the case of normally distributed errors the most likely tensor with higher symmetries coincides with the closest higher symmetric tensor using a deviation scaled euclidean metric

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

short note volume consistency in a staggered grid lagrangian hydrodynamics scheme staggered grid lagrangian schemes for compressible hydrodynamics involve a choice of how internal energy is advanced in time the options depend on two ways of defining cell volumes an indirect one that guarantees total energy conservation and a direct one that computes the volume from its definition as a function of the cell vertices it is shown that the motion of the vertices can be defined so that the two volume definitions are identical a so modified total energy conserving staggered scheme is applied to the coggeshall adiabatic compression problem and now also entropy is basically exactly conserved for each lagrangian cell and there is increased accuracy for internal energy the overall improvement as the grid is refined is less than what might be expected

the internal consistency stability and accuracy of the discrete compatible formulation of lagrangian hydrodynamics this work explores the somewhat subtle meaning and consequences of the salient properties of the discrete compatible formulation of lagrangian hydrodynamics in particular since this formulation preserves total energy to roundoff error the amount of error in the conservation of total energy cannot be used to gauge the internal consistency of calculations as is often done with the older forms of this algorithm however the compatible formulation utilizes two definitions of zone volume the first is the usual definition whereby the volume of a zone is defined as some prescribed function of the coordinates of the points that define it the second is given as the integration in time of the continuity equation for zone volume as expressed in lagrangian form it is the use of this latter volume in the specific internal energy equation that enables total energy to be exactly conserved these two volume definitions are generally not precisely equal it is the analysis of this difference that forms the first part of this study it is shown that this difference in zone volumes can be used to construct a practical internal consistency measure that not only takes the place of the lack of total energy conservation of the older forms of lagrangian hydrodynamics but is more general in that it can be defined on a single zone basis it can also be used to ascertain the underlying spatial and temporal order of accuracy of any given set of calculations the difference in these two definitions of zone volume may be interpreted as a type of entropy error however this entropy error is found to be significant only when a given calculation becomes numerically unstable otherwise it remains at or far beneath truncation error levels in fact it can be utilized to provide an upper bound on the size of the spatial truncation error for a stable computation it is also shown how this volume difference can be used as an indicator of numerical difficulties since exact local conservation of total energy does not guarantee numerical stability or the quality of any numerical calculation the discrete compatible formulation of lagrangian hydrodynamics utilizes a two level predictor corrector type of time integration scheme a stability analysis both analytical and numerical is given this analysis reveals a novel stability diagram that has not been heretofore published and gives definitive information as to how the stabilizing corrector step should be centered in time

estimating effective elasticity tensors from christoffel equations we consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities these quantities the wavefront slowness and polarization vectors are used in the christoffel equation a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor to obtain an effective tensor of particular symmetry we do not assume its orientation thus the regression using the residuals of the christoffel equation results in a nonlinear optimization problem we find the absolute extremum and to avoid numerical instability of a global search obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression the issue is twofold first finding the closest tensor of particular symmetry without assuming its orientation is challenging second the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information rather it relies on an abstract norm in the space of elasticity tensors to include seismic concepts and statistical information we distinguish between the closest tensor of particular symmetry and the effective one the former is the initial guess to search for the latter

computing the optimal transversely isotropic approximation of a general elastic tensor mathematically 21 stiffnesses arranged in a 6 6 symmetric matrix completely describe the elastic properties of any homogeneous anisotropic medium regardless of symmetry system and orientation however it can be difficult in practice to characterize an anisotropic medium s properties merely from casual inspection of its often experimentally measured stiffness matrix for characterization purposes it is better to decompose a measured stiffness matrix into a stiffness matrix for a canonically oriented transversely isotropic ti medium whose properties can be readily understood plus a generally anisotropic perturbation representing the medium s deviation from perfect symmetry followed by a rotation giving the relationship between the medium s natural coordinate system and the measurement coordinate system to accomplish this decomposition we must find the rotated symmetric medium that best approximates a given stiffness matrix an analytical formula exists for calculating the distance between

an efficient algorithm for computing nearest medium approximations to an arbitrary anisotropic stiffness tensor abstractthe problem of reducing a fully anisotropic triclinic stiffness tensor comprised of 21 distinct components to a higher symmetry form having fewer distinct elements is of interest in many geophysical applications where assuming a particular type of symmetry is often necessary to sufficiently reduce computational complexity to allow practical solutions in addition recent advances in the upscaling of realistically large field scale discrete fracture networks have led to the need for an efficient way to derive a very large number of nearest medium approximations to the triclinic stiffness tensors obtained owing to rotational symmetries and nonlinearity the problem of efficiently finding such approximations is generally nontrivial because optimal orientations are intrinsically nonunique an algorithm proposed by dellinger computes nearest orthotropic and transverse isotropic approximations for a given stiffness tensor using the federov norm as an objective function to iteratively minimize the f

on preservation of symmetry in r z staggered lagrangian schemes in the focus of this work are symmetry preservation conservation of energy and volume and other important properties of staggered lagrangian hydrodynamic schemes in cylindrical r z geometry it is well known that on quadrilateral cells in r z preservation of spherical symmetry perfect satisfaction of the geometrical conservation law gcl and total energy conservation are incompatible even on conforming grids this paper suggests a novel staggered grid approach that preserves symmetry conserves total energy by construction and tries to do its best by diminishing the gcl error to the order of entropy error in particular the forces from an existing volume consistent scheme are corrected so that spherical symmetry is preserved the incorporation of subcell pressure mechanism to reduce spurious grid deformations is described and the relation of the new scheme to popular area weighted and control volume approaches studied

energy preservation and entropy in lagrangian space and time staggered hydrodynamic schemes usual space and time staggered sts leap frog lagrangian hydrodynamic schemes such as von neumann richtmyer s 1950 wilkins 1964 and their variants are widely used for their simplicity and robustness despite their known lack of exact energy conservation since the seminal work of trulio and trigger 1950 and despite the later corrections of burton 1991 it is generally accepted that these schemes cannot be modified to exactly conserve energy while retaining all of the following properties sts stencil with velocities half time centered with respect to positions explicit second order algorithm locally implicit for internal energy and definite positive kinetic energy it is shown here that it is actually possible to modify the usual sts hydrodynamic schemes in order to be exactly energy preserving regardless of the evenness of their time centering assumptions and retaining their simple algorithmic structure burton s conservative scheme 1991 is found as a special case of time centering which cancels the term here designated as incompatible displacements residue in contrast von neumann richtmyer s original centering can be preserved provided this residue is properly corrected these two schemes are the only special cases able to capture isentropic flow with a third order entropy error instead of second order in general the momentum equation is presently obtained by application of a variational principle to an action integral discretized in both space and time the internal energy equation follows from the discrete conservation of total energy entropy production by artificial dissipation is obtained to second order by a prediction correction step on the momentum equation the overall structure of the equations explicit for momentum locally implicit for internal energy remains identical to that of usual sts leap frog schemes though complementary terms are required to correct the effects of time step changes and artificial viscosity updates in deriving these schemes an apparently novel approach of flux in time was introduced to correct numerical residues and ensure energy conservation this method can be applied to essentially any numerical scheme whenever required or desired provided space and time numerical consistency is preserved numerical test cases are presented confirming the conservative character of the new csts schemes down to computer round off errors and showing various improvements compared to the standard von neumann richtmyer and wilkins sts schemes mostly on shock levels shock velocities singularity induced distortions and cfl stability limits

approximation by polynomials in two diffeomorphisms soit 1 k z et soient f et g des diffeomorphismes c k de c dans c ayant des degres opposes alors c f g est dense dans l espace de frechet c k c

polynomial approximation local polynomial convexity and degenerate cr singularities abstract we begin with the following question given a closed disc d c and a complex valued function f c d is the uniform algebra on d generated by z and f equal to c d when f c 1 d this question is complicated by the presence of points in the surface s graph d f that have complex tangents such points are called cr singularities let p s be a cr singularity at which the order of contact of the tangent plane with s is greater than 2 i e a degenerate cr singularity we provide sufficient conditions for s to be locally polynomially convex at the degenerate singularity p this is useful because it is essential to know whether s is locally polynomially convex at a cr singularity in order to answer the initial question to this end we also present a general theorem on the uniform algebra generated by z and f which we use in our investigations this result may be of independent interest because it is applicable even to non smooth complex valued f

formulations of artificial viscosity for multi dimensional shock wave computations in this paper we present a new formulation of the artificial viscosity concept physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated the first important property is that by definition a viscosity must always be dissipative transferring kinetic energy into internal energy and must never act as a false pressure the artificial viscous force should be galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion and remain zero for the latter case these requirements significantly constrain the functional form that the artificial viscous force can have in addition an artificial viscosity should be able to distinguish between shock wave and adiabatic compression and not result in spurious entropy production when only the latter is present it must therefore turn off completely for self similiar motion where only a uniform stretching and or a rigid rotation occurs an additional important but more subtle condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front since the velocity is only discontinuous in the normal direction our principal result is the development of a new formulation of an edge centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions a central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions and to substantially reduce its effect when strong velocity gradients are absent

local polynomial convexity of graphs of functions in several variables abstract in this paper we investigate the locally polynomial convexity of graphs of smooth functions in several variables we also give a sufficient condition for a real analytic function g defined near 0 in c which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on d for all disks d close enough to the origin in c

numerical preservation of symmetry properties of continuum problems we investigate the problem of perfectly preserving a symmetry associated naturally with one coordinate system when calculated in a different coordinate system this allows a much wider range of problems that may be viewed as perturbations of the given symmetry to be investigated we study the problem of preserving cylindrical symmetry in two dimensional cartesian geometry and spherical symmetry in two dimensional cylindrical geometry we show that this can be achieved by a simple modification of the gradient operator used to compute the force in a staggered grid lagrangian hydrodynamics algorithm in the absence of the supposed symmetry we show that the new operator produces almost no change in the results because it is always close to the original gradient operator our technique thus results in a subtle manipulation of the spatial truncation error in favor of the assumed symmetry but only to the extent that it is naturally present in the physical situation this not only extends the range of previous algorithms and the use of new ones for these studies but for spherical or cylindrical calculations it reduces the sensitivity of the results to grid setup with equal angular zoning that has heretofore been necessary with these problems

a nominally second order cell centered lagrangian scheme for simulating elastic plastic flows on two dimensional unstructured grids in this paper we describe a cell centered lagrangian scheme devoted to the numerical simulation of solid dynamics on two dimensional unstructured grids in planar geometry this numerical method utilizes the classical elastic perfectly plastic material model initially proposed by wilkins m l wilkins calculation of elastic plastic flow meth comput phys 1964 in this model the cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state regarding the deviatoric stress its time evolution is governed by a classical constitutive law for isotropic material the plasticity model employs the von mises yield criterion and is implemented by means of the radial return algorithm the numerical scheme relies on a finite volume cell centered method wherein numerical fluxes are expressed in terms of sub cell force the generic form of the sub cell force is obtained by requiring the scheme to satisfy a semi discrete dissipation inequality sub cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node centered solver which results from total energy conservation the nominally second order extension is achieved by developing a two dimensional extension in the lagrangian framework of the generalized riemann problem methodology introduced by ben artzi and falcovitz m ben artzi j falcovitz generalized riemann problems in computational fluid dynamics cambridge monogr appl comput math 2003 finally the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases

the local polynomial hull near a degenerate cr singularity bishop discs revisited let be a smooth real surface in and let be a point at which the tangent plane is a complex line how does one determine whether or not is locally polynomially convex at such a p i e at a cr singularity even when the order of contact of with at p equals 2 no clean characterisation exists difficulties are posed by parabolic points hence we study non parabolic cr singularities we show that the presence or absence of bishop discs around certain non parabolic cr singularities is completely determined by a maslov type index this result subsumes all known facts about bishop discs around order two non parabolic cr singularities sufficient conditions for bishop discs have earlier been investigated at cr singularities having high order of contact with these results relied upon a subharmonicity condition which fails in many simple cases hence we look beyond potential theory and refine certain ideas going back to bishop

3d level set methods for evolving fronts on tetrahedral meshes with adaptive mesh refinement the level set method is commonly used to model dynamically evolving fronts and interfaces in this work we present new methods for evolving fronts with a specified velocity field or in the surface normal direction on 3d unstructured tetrahedral meshes with adaptive mesh refinement amr the level set field is located at the nodes of the tetrahedral cells and is evolved using new upwind discretizations of hamiltonjacobi equations combined with a rungekutta method for temporal integration the level set field is periodically reinitialized to a signed distance function using an iterative approach with a new upwind gradient the details of these level set and reinitialization methods are discussed results from a range of numerical test problems are presented