mixed perfectly matched layers for direct transient analysis in 2d elastic heterogeneous media abstract we are concerned with elastic waves arising in plane strain problems in an elastic semi infinite arbitrarily heterogeneous medium specifically we discuss the development of a new mixed displacement stress formulation for forward elastic wave simulations in perfectly matched layer pml truncated heterogeneous media to date most pml formulations split the displacement and stress fields resulting in non physical components for each field in this work we favor unsplit schemes primarily for the relative ease by which the resulting forms can be incorporated into existing codes the ease by which the resulting semi discrete forms can be integrated in time and the ease by which they can be used in adjoint formulations arising in inverse problems contrary to most past and current developments we start by following classical lines and apply complex coordinate stretching to the governing equations in the frequency domain while retaining both displacements and stress quantities as unknowns with the aid of auxiliary variables the resulting mixed form is rendered second order in time thereby allowing the use of standard time integration schemes we report on numerical simulations demonstrating the stability and efficacy of the approach

new approach to study splines by blossoming method and application to the construction of a bivariate c 1 quartic quasi interpolant this work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions splines which uses the blossoming approach some existing results in the literature are reformulated such as the smoothness conditions between polynomials of a spline by using the affinity property of the blossom some definitions of sub splines are proposed which can be very useful in the study and the construction of splines such as macro elements or quasi interpolants as an application of the proposed results a c 1 quartic spline quasi interpolant with optimal approximation order is defined without using any mask for smoothness or b spline basis numerical results are presented and compared with other methods given in the literature

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

crossed products by endomorphisms of c_0 x algebras abstract in the first part of the paper we develop a theory of crossed products of a c algebra a by an arbitrary not necessarily extendible endomorphism a a we consider relative crossed products c a j where j is an ideal in a and describe up to morita rieffel equivalence all gauge invariant ideals in c a j and give six term exact sequences determining their k theory we also obtain certain criteria implying that all ideals in c a j are gauge invariant and that c a j is purely infinite in the second part we consider a situation where a is a c 0 x algebra and is such that f a f a a a f c 0 x where is an endomorphism of c 0 x pictorially speaking is a mixture of a topological dynamical system x dual to c 0 x and a continuous field of homomorphisms x between the fibers a x x x of the corresponding c bundle for systems described above we establish efficient conditions for the uniqueness property gauge invariance of all ideals and pure infiniteness of c a j we apply these results to the case when x prim a is a hausdorff space in particular if the associated c bundle is trivial we obtain formulas for k groups of all ideals in c a j in this way we constitute a large class of crossed products whose ideal structure and k theory is completely described in terms of x x x x y where y is a closed subset of x

on smooth multivariate spline functions in this paper the dimensions of bivariate spline spaces with simple cross cut grid partitions are determined and expressions of their basis functions are given consequently the closures of these spaces over all partitions of the same type can be determined a somewhat more detailed study on bivariate splines with rectangular grid partitions is included the results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions

mixed spectral finite elements for the linear elasticity system in unbounded domains in this paper we present a mixed formulation of a spectral element approximation of the linear elasticity system after studying the main features of this approach we construct perfectly matched layers pmls for modeling unbounded domains then algorithmic issues are discussed and numerical results are given

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

the generic dimension of the space of c 1 splines of degree d 8 on tetrahedral decompositions this paper considers the linear space of globally differentiable piecewise polynomial functions defined on a three dimensional polyhedral domain which has been partitioned into tetrahedra combining bernstein bezier methods and combinatorial and geometric techniques from rigidity theory this paper gives an explicit expression for the generic dimension of this space for sufficiently large polynomial degrees d geq 8 this is the first general dimension statement of its kind

triangle buildings and actions of type iii1 q2 abstract we study certain group actions on triangle buildings and their boundaries and some von neumann algebras which can be constructed from them in particular for buildings of order q 3 certain natural actions on the boundary are hyperfinite of type iii 1 q 2

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

a two and a half dimensional displacement based pml for elastodynamic wave propagation summary this paper presents a perfectly matched layer pml technique for the numerical simulation of three dimensional linear elastodynamic problems where the geometry is invariant in the longitudinal direction examples include transportation infrastructure dams lifelines and alluvial valleys for longitudinally invariant geometries a computationally efficient two and a half dimensional 2 5d approach can be applied where the fourier transform from the longitudinal coordinate to the wavenumber domain allows for the representation of the three dimensional radiated wave field on a two dimensional mesh in this 2 5d framework the equilibrium equations of a pml continuum are formulated in a weak form for an isotropic elastodynamic medium and discretized using a galerkin approach the 2 5d pml methodology is validated by computing the green s displacements of a homogeneous halfspace demonstrating that the 2 5d pml absorbs all propagating waves for different angles of incidence furthermore the dynamic stiffness of a rigid strip foundation and the efficiency of a vibration isolating screen are computed the examples demonstrate that the pml methodology is computationally efficient especially when only the response of the structure or the near field response is of interest copyright 2011 john wiley sons ltd

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

explicit finite element perfectly matched layer for transient three dimensional elastic waves the use of a perfectly matched layer pml model is an efficient approach toward the bounded domain modelling of wave propagation on unbounded domains this paper formulates a three dimensional pml for elastic waves by building upon previous work by the author and implements it in a displacement based finite element setting the novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration thus making it practicable for use in large scale three dimensional dynamic analyses an efficient method of calculating the strain terms in the pml is developed in order to take advantage of the lack of the overhead of solving equations at each time step the pml formulation is studied and validated first for a semi infinite bar and then for the classical soil structure interaction problems of a square flexible footing on a i half space ii layer on a half space and iii layer on a rigid base numerical results for these problems demonstrate that the pml models produce highly accurate results with small bounded domains and at low computational cost and that these models are long time stable with critical time step sizes similar to those of corresponding fully elastic models copyright 2008 john wiley sons ltd

a c 2 trivariate macroelement based on the worsey farin split of a tetrahedron a c 2 trivariate macroelement is constructed based on the worsey farin split of a tetrahedron into twelve subtetrahedra the element uses supersplines of degree 9 and provides optimal order approximation of smooth functions

perfectly matched layers for transient elastodynamics of unbounded domains one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies in a recent work computer methods in applied mechanics and engineering 2003 192 1337 1375 the authors presented inter alia time harmonic governing equations of pmls for anti plane and for plane strain motion of visco elastic media this paper presents a corresponding time domain displacement based governing equations of these pmls and b displacement based finite element implementations of these equations suitable for direct transient analysis the finite element implementation of the anti plane pml is found to be symmetric whereas that of the plane strain pml is not numerical results are presented for the anti plane motion of a semi infinite layer on a rigid base and for the classical soil structure interaction problems of a rigid strip footing on i a half plane ii a layer on a half plane and iii a layer on a rigid base these results demonstrate the high accuracy achievable by pml models even with small bounded domains

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

upper and lower bounds on the dimension of multivariate spline spaces we give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangu lations in mathbb r k the bounds are optimal in a certain sense

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

simple purely infinite c algebras and n filling actions let n be a positive integer we introduce a concept which we call the n filling property for an action of a group on a separable unital c algebra a if a c is a commutative unital c algebra and the action is induced by a group of homeomorphisms of then the n filling property reduces to a weak version of hyperbolicity the n filling property is used to prove that certain crossed product c algebras are purely infinite and simple a variety of group actions on boundaries of symmetric spaces and buildings have the n filling property an explicit example is the action of sln z on the projective n space

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

a higher order perfectly matched layer formulation for finite difference time domain seismic wave modeling we have developed a higher order perfectly matched layer pml formulation to improve the absorption performance for finite difference time domain seismic modeling first we outlined a new unsplit correction approach which allowed for traditional first order pmls to be added directly to existing codes in a straightforward manner then using this framework we constructed a pml formulation that can be used to construct higher order pmls of arbitrary order the greater number of degrees of freedom associated with the higher order pml allow for enhanced flexibility of the pml stretching functions thus potentially facilitating enhanced absorption performance we found that the new approach can offer increased elastodynamic absorption particularly for evanescent waves we also discovered that the extra degrees of freedom associated with the higher order pml required careful optimization if enhanced absorption was to be achieved furthermore these extra degrees of freedom increased the computational requirements in comparison with first order schemes we reached our formulations using one compact equation thus increasing the ease of implementation additionally the formulations are based on a recursive integration approach that reduce pml memory requirements and do not require special consideration for corner regions we tested the new formulations to determine their ability to absorb body waves and surface waves we also tested standard staggered grid stencils and rotated staggered grid stencils

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

bivariate spline spaces and minimal determining sets abstract the paper describes bivariate spline spaces and minimal determining sets shows applications of these concepts presents two new triangular finite elements describes public domain software for the analysis of spline spaces and concludes with a concise history of these subjects the software described in this paper is available on http www math utah edu alfeld mds that page also contains links to colored versions of the figures in this paper

optimizing the perfectly matched layer the berenger perfectly matched layer or pml 4 3 is used in computational electromagnetism as a sponge layer to terminate finite element approximations of scattering problems an infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction and the waves decay exponentially in magnitude into the layer this property is lost when the layer is truncated since the truncation boundary generates a reflected wave discretization of the differential equations further changes the reflection coefficient of the layer in this talk we shall describe the pml in the simple case of a two dimensional electromagnetic problem the first part of this abstract is expository we shall show how to derive the layer using a change of variable approach 5 9 6 after this derivation we describe some new analysis to show the effects of truncating and discretizing the pml 9 we end by showing how to compute an optimal layer and make some remarks about pml for the full maxwell system

topological full groups of one sided shifts of finite type we explore the topological full group g of an essentially principal etale groupoid g on a cantor set when g is minimal we show that g and its certain normal subgroup is a complete invariant for the isomorphism class of the etale groupoid g furthermore when g is either almost finite or purely infinite the commutator subgroup d g is shown to be simple the etale groupoid g arising from a onesided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid for such g g is thought of as a generalization of the higman thompson group we prove that g is of type f and so in particular it is finitely presented this gives us a new infinite family of finitely presented infinite simple groups also the abelianization of g is calculated and described in terms of the homology groups of g

on structure of bivariate spline space of lower degree over arbitrary triangulation the structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property a new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces the upper bound of the dimension of spline spaces over triangulation given by l l schumaker is slightly improved via the new vertex coding method the structure of multivariate spline spaces s 2 1 and s 3 1 over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by luo and wang a kind of sufficient conditions on judging non singularity of the s 2 1 and s 3 1 spaces over arbitrary triangulation is given which only depends on the topological property of the triangulation from the sufficient conditions a triangulation strategy is presented at the end of the paper the strategy ensures that the constructed triangulation is non singular or generic for s 2 1 and s 3 1

studies of fe pml for exterior problems of time harmonic elastic waves the equation of an absorbing layer for time harmonic elastic waves based on the perfectly matched layer pml concept is formulated in a manner that is easily implemented in finite element software in the proposed approach the layer is viewed as an anisotropic material with continuously varying complex material properties the effect of the pml parameters on its discrete representation is investigated through dispersion analysis guidelines for proper selection of the pml parameters are presented the formulation with the parameters selected according to the guidelines proposed performs well in computation

perfectly matched layers for elastic waves in cylindrical and spherical coordinates the perfectly matched layer pml for elastic waves in cylindrical and spherical coordinates is developed using an improved scheme of complex coordinates as is known for electromagnetic waves berenger s original pml scheme does not apply to cylindrical and spherical coordinates the straightforward extension of the complex coordinates for elastic waves to cylindrical and spherical coordinates requires extra unknowns for time domain solutions wasting computer memory and computation time the main idea of the improved scheme in this work is the use of integrated complex variables it is shown that for three dimensional cylindrical and spherical coordinates this improved pml scheme requires no more unknowns than in cartesian coordinates the number of unknowns can be further reduced through the use of symmetry in the partial differential equations the pml scheme allows an arbitrary inhomogeneity in the medium and is suitable for numerical solutions of wave equations by finite difference finite element

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

a local lagrange interpolation method based on c 1 cubic splines on freudenthal partitions a trivariate lagrange interpolation method based on c 1 cubic splines is described the splines are defined over a special refinement of the freudenthal partition of a cube partition the interpolating splines are uniquely determined by data values but no derivatives are needed the interpolation method is local and stable provides optimal order approximation and has linear complexity

an efficient time domain perfectly matched layers formulation for elastodynamics on spherical domains summary many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain one widely used approach for truncating the infinite domain is the so called method of perfectly matched layers pmls most existing pml formulations are developed for finite difference methods based on the first order velocity stress form of the elasticity equations and they are not straight forward to implement using standard finite element methods fems on unstructured meshes some of the problems with these formulations include the application of boundary conditions in half space problems and in the treatment of edges and or corners for time domain problems several pml formulations which do work with fems have been proposed although most of them still have some of these problems and or they require a large number of auxiliary nodal history memory variables in this work we develop a new pml formulation for time domain elastodynamics on a spherical domain which reduces to a two dimensional formulation under the assumption of axisymmetry our formulation is well suited for implementation using fems where it requires lower memory than existing formulations and it allows for natural application of boundary conditions we solve example problems on two dimensional and three dimensional domains using a high order discontinuous galerkin dg discretization on unstructured meshes and explicit time stepping we also study an approach for stabilization of the discrete equations and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation copyright 2014 john wiley sons ltd

an efficient finite element time domain formulation for the elastic second order wave equation a non split complex frequency shifted convolutional pml the perfectly matched layer pml technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first order system in velocity and stress in attenuating non grazing bulk and surface waves this paper develops a novel convolutional pml formulation based on the second order wave equation with displacements as the only unknowns to annihilate spurious reflections from near grazing waves the derived variational form allows for the use of e g finite element and the spectral element methods as spatial discretization schemes a recursive convolution update scheme of second order accuracy is employed such that highly stable effective time integration with the newmark beta implicit and explicit with mass lumping method is achieved the implementation requires minor modifications of existing displacement based finite element software and the stability and efficiency of the proposed formulation is verified by relevant two dimensional benchmarks that accommodate bulk and surface waves copyright 2011 john wiley sons ltd

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

the scaled boundary finite element method alias consistent infinitesimal finite element cell method for elastodynamics the scaled boundary finite element method alias the consistent infinitesimal finite element cell method is developed starting from the governing equations of linear elastodynamics only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one no fundamental solution is necessary and thus no singular integrals must be evaluated general anisotropic material is analysed without any increase in computational effort boundary conditions on free and fixed surfaces and on interfaces between different materials are enforced exactly without any discretization this method is exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions for a bounded medium symmetric static stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption a stress singularity is represented very accurately as the condition on the boundary in the vicinity of the point of singularity is satisfied without spatial discretization

c1 quintic interpolation over triangles two explicit representations two explicit representation of a c1 quintic interpolant over triangles are given these representations are derived by generalization of coons methods and bernstein bezier methods respectively

lumped mass finite element implementation of continuum theories with micro inertia summary continuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro structured materials such terms often called micro inertia are convenient and straightforward extensions of classical continuum theories furthermore the critical time step is usually increased via the inclusion of micro inertia however the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro inertia terms in this paper we will develop a solution algorithm based on neumann expansions by which this disadvantage is avoided altogether the micro inertia terms are translated into modifications of the residual force vector so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained the numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail numerical examples are used to illustrate the various aspects of the algorithm copyright 2013 john wiley sons ltd