lie bialgebra structures on the twisted heisenberg virasoro algebra abstract in this paper we investigate lie bialgebra structures on the twisted heisenberg virasoro algebra with the determination of certain lie bialgebra structures on the virasoro algebra we determine certain structures on the twisted heisenberg virasoro algebra moreover some general and useful results are obtained with our methods and results we also can easily determine certain structures on some lie algebras related to the twisted heisenberg virasoro algebra

singular dimensions of the n 2 superconformal algebras ii the twisted n 2 algebra we introduce a suitable adapted ordering for the twisted n 2 superconformal algebra i e with mixed boundary conditions for the fermionic fields we show that the ordering kernels for complete verma modules have two elements and the ordering kernels for g closed verma modules just one therefore spaces of singular vectors may be two dimensional for complete verma modules whilst for g closed verma modules they can only be one dimensional we give all singular vectors for the levels 1 and for both complete verma modules and g closed verma modules we also give explicit examples of degenerate cases with two dimensional singular vector spaces in complete verma modules general expressions are conjectured for the relevant terms of all primitive singular vectors i e for the coefficients with respect to the ordering kernel these expressions allow to identify all degenerate cases as well as all g closed singular vectors they also lead to the discovery of subsingular vectors for the twisted n 2 superconformal algebra explicit examples of these subsingular vectors are given for the levels 1 and finally the multiplication rules for singular vector operators are derived using the ordering kernel coefficients this sets the basis for the analysis of the twisted n 2 embedding diagrams

rate dependent projection operators for frictional contact constraints we develop rate dependent regularization approaches for three dimensional frictional contact constraints based on the kelvin and maxwell viscoelastic constitutive models with the present regularization schemes we aim to provide a basis to better model friction and to stabilize the contact analysis while keeping the contact model as simple as possible the key feature of the regularization approaches implemented using an implicit time integrator is that one can recover in the limit the widely used rate independent elastoplastic regularization framework without encountering numerical difficulties intermediate contact tractions are defined in terms of the relative displacement the relative velocity and the regularization parameters the projection operators operate on the intermediate tractions and yield contact tractions that satisfy all the discretized contact constraints the use of projection operators allows a systematic implementation of the present regularization schemes through numerical simulations we observed that the maxwell type regularization effectively avoids convergence problems even for relatively large time step sizes while the kelvin type regularization does not copyright 2003 john wiley sons ltd

the analysis of incompressible hyperelastic bodies by the finite element method abstract a method is developed for the finite element analysis of problems involving incompressible hyperelastic bodies the constitutive relation is based on a class of strain energy functions due to ogden 4 which involve sums of real powers of principal stretches incremental equilibrium equations are derived from a rate form of the principle of virtual work and an additional set of equations which express the condition of incompressibility in an average manner is appended to the equilibrium equations examples of solutions are given and compared either with closed form solutions or with numerical solutions found using conventional approaches

new method for deriving eigenvalue rate with respect to support location the authors present a new method to derive the formulas of eigenvalue rate with respect to in span support location using the generalized variational principle of the rayleigh quotient following the ideas of this note the eigenvalue rate with respect to such variables as location of discrete or continuous in span supports the location of concentrated mass location of elastic support and location of a substructure can be derived without any conceptual difficulties

finite element analysis of anisotropic non linear incompressible elastic solids by a mixed model a mixed finite element method is presented for geometrically and materially non linear analysis of anisotropic incompressible hyperelastic materials an incremental iteractive total lagrangian formulation is adopted the nodal displacements and the hydrostatic pressure are independently interpolated leading to a mixed system of equations with characteristic zero diagonal terms computations are carried out using a three dimensional linear displacement constant pressure element a mixed penalty approximation is then employed to eliminate the pressure variables at the element level the anisotropic material handling capability of the formulation is tested through a number of transversely isotropic problems and the results compared to analytical solutions to demonstrate the applicability of this formulation to model complex anisotropic problems the inflation of a cut toroidal tube with helical fibre orientation is analysed

finite element analysis of laminated composites with contact constraint by extended interior penalty methods in this paper the extended interior penalty methods are applied to solve the bolted joint problems of the laminated composite plates the conventional interior penalty methods have the weak point such that the searches for optimized solution in the neighbourhood of the constraint surfaces are difficult since the values of penalty term in the infeasible domain are infinity to improve the weak point the extended interior penalty formulation is proposed in this work mathematical analysis on the characteristics of the solutions is attempted and the convergence of the penalized solutions is estimated as a numerical example first the case of single bolted joint is solved for a variety of lamination angles of the composite plates the computed results are compared with the previous works and are proved to agree well next the problem of the two bolt holes in series and in parallel are treated in a similar manner it is concluded that the formulations for the class of the problems by the extended interior penalty methods are mathematically sound and numerically efficient

near tip field for diffraction on square lattice by crack the displacement field near a tip of a finite crack due to the diffraction of a wave on a square lattice is studied the finite section method in the theory of toeplitz operators on ell 2 is invoked as the semi infinite crack diffraction problem is shown equivalent to the inversion of a toeplitz operator a truncation of which appears in the finite crack diffraction problem for the same incident wave the existence and uniqueness of the solution in ell 2 for the semi infinite crack problem is established by an application of the well known krein conditions continuum limit of the semi infinite crack diffraction problem is established in a discrete sobolev space a graphical illustration of convergence in the relevant sobolev norm is also included a low frequency asymptotic approximation of the normalized shear force in vertical bonds ahead of the crack tip recovers the classical crack tip singularity displacement of particles in the vicinity of the crack tip a closed form expression of wh

a level set enhanced natural kernel contact algorithm for impact and penetration modeling summary a natural kernel contact nkc algorithm under the framework of the semi lagrangian reproducing kernel particle method semi lagrangian rkpm is proposed to model multi body contact with specific consideration for impact and penetration modeling the nkc algorithm utilizes the interaction of the semi lagrangian kernel functions associated with contacting bodies to serve as the non penetration condition the effects of friction are represented by introducing a layer of the friction like elasto plasticity material between contacting bodies this approach allows the frictional contact conditions and the associated kinematics to be naturally embedded in the semi lagrangian rkpm inter particle force calculation the equivalence in the karush kuhn tucker conditions between the proposed nkc algorithm and the conventional contact kinematic constraints as well as the associated state variable relationships are identified a level set method is further introduced in the nkc algorithm to represent the contact surfaces without pre defined potential contact surfaces the stability analysis performed in this work shows that temporal stability in the semi lagrangian rkpm with nkc algorithms is related to the velocity gradient between contacting bodies the proposed methods have been verified by several benchmark problems and applied to the simulation of impact and penetration processes copyright 2014 john wiley sons ltd

on the convergence factor of uzawa s algorithm abstract in this paper we study convergence properties of uzawa type algorithms for the iterative solution of mixed finite element approximations to two dimensional boundary value problems with equality constraints such as the plane stokes problem or the plane incompressible elasticity problem we present estimates of the convergence factors which are optimal with respect to the discretization parameter h it is shown in particular that for the problems mentioned the convergence factor of the classical uzawa algorithm is independent of h

analytic expressions for singular vectors of the n 2 superconformal algebra using explicit expressions for a class of singular vectors of then 2 untwisted algebra and following the approach of malikov feigin fuchs and kent we show that the analytically extended verma modules contain two linearly independent neutral singular vectors at the same grade we construct this two dimensional space and we identify the singular vectors of the original verma modules we show that in some verma modules these expressions lead to two linearly independent singular vectors which are at the same grade and have the same charge

diffraction on the two dimensional square lattice we solve the thin slit diffraction problem for two dimensional lattice waves more precisely for the discrete helmholtz equation on the semi infinite square lattice with data prescribed on the left boundary the aperture we use lattice green s functions and a discrete sommerfeld outgoing radiation condition to derive the exact solution everywhere in the lattice the solution is a discrete convolution that can be evaluated in closed form for the wave number k 2 for other wave numbers we give a recursive algorithm for computing the convolution kernel

aspects of the formulation and finite element implementation of large strain isotropic elasticity the paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation on the theoretical side an eulerian setting of isotropic elasticity is discussed exclusively in terms of the finger tensor as a strain measure noval aspects are a direct representation of the eulerian elastic moduli in terms of the finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts the isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the finger tensor a constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity on the computational side the implementation of the constitutive model in three possible finite element formulations is discussed after pointing out algorithmic techniques for the treatment of incompressible elasticity several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity

an effective two dimensional frictional contact model for arbitrary curved geometry a finite element model is developed on the basis of a variational formulation of the perturbed lagrange type and the classical coulomb law of friction for the analysis of frictional contact problems in two dimensions the model accounts for all geometric kinematic non linearities associated with large sliding motions as well as arbitrary contact surface curvatures explicit forms for the contact force and tangent stiffness operators and a penalty type format is utilized in the implementation an extensive number of numerical simulations are used to demonstrate the effectiveness and practical usefulness of the model

a paradigm for higher order polygonal elements in finite elasticity using a gradient correction scheme abstract recent studies have demonstrated that polygonal elements possess great potential in the study of nonlinear elastic materials under finite deformations on the one hand these elements are well suited to model complex microstructures e g particulate microstructures and microstructures involving different length scales and incorporating periodic boundary conditions on the other hand polygonal elements are found to be more tolerant to large localized deformations than the standard finite elements and to produce more accurate results in bending and shear with mixed formulations lower order mixed polygonal elements are also shown to be numerically stable on voronoi type meshes without any additional stabilization treatment however polygonal elements generally suffer from persistent consistency errors under mesh refinement with the commonly used numerical integration schemes as a result non convergent finite element results typically occur which severely limit their applications in this work a general gradient correction scheme is adopted that restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field with the correction scheme the recovery of optimal convergence for solutions of displacement based and mixed formulations with both linear and quadratic displacement interpolants is confirmed by numerical studies of several boundary value problems in finite elasticity in addition for mixed polygonal elements the various choices of the pressure field approximations are discussed and their performance on stability and accuracy are numerically investigated we present applications of those elements in physically based examples including a study of filled elastomers with interphasial effect and a qualitative comparison with cavitation experiments for fiber reinforced elastomers

classification of indecomposable modules of the intermediate series over the twisted n 2 superconformal algebra in this paper a classification of indecomposable modules of the intermediate series over the twisted n 2 superconformal algebra is obtained

numerical diffraction by a uniform grid in this report we analyze the influence of a spatial discretization on the propagation of a wave generated by a harmonic point source we consider finite element and finite difference schemes on a uniform grid for the wave equation we study the asymptotic behavior in space of the elementary solution of the associated helmholtz equation this study permits us to define the notions of asymptotic wave fronts a w f and asymptotic amplitude diagrams a a d we point out some results about the anisotropy and the dispersion of numerical scheme which complete those obtained from the more classical plane wave analysis see r m alford r kelly and d m boore accuracy of finite modeling of the acoustic wave equation biophysics 6 1974 pp 834 842 a bamberger g chavent and p lailly etude de schemas numeriques des equations de l elastodynamique lineaire rapport inria no 41 october 1980 t belytschko and r mullen dispersion analysis of finite element semi discretizations of the two dimen

on the convergence of the bilinear velocity constant pressure finite element method in viscous flow abstract in this paper we show that the standard isoparametric q 1 q 0 velocity pressure finite element method for solving viscous flow problems leads to a fully and optimally convergent sequence of approximations if an appropriate quadrangulation of the domain is used

mixed finite element methods of higher order for model contact problems this paper presents mixed finite element methods of higher order for a simplified signorini problem and an idealized frictional problem the discretization is based on a mixed variational formulation proposed by haslinger et al which is extended to higher order finite elements to guarantee the unique existence of the solution of the mixed method a discrete inf sup condition is proven approximation results of the p method of finite elements and some inverse estimates for higher order polynomials are applied numerical results confirm the theoretical findings

galerkin methods for the time dependent incompressible elasticity equations abstract we consider semidiscrete and single step fully discrete approximations to the solutions of the time dependent incompressible elasticity equations two methods of dealing with the constraint are analysed a a formulation involving a discrete form of incompressibility condition and b a penalty formulation h1 and l2 error estimates are obtained

remarks on 4cst elements for incompressible materials abstract three node triangular elements obtained from the intersecting diagonals of quadrilaterals are thoroughly investigated for the penalty method formulation of incompressible problems such as stokes flow navier stokes flow and small deformation rubber elasticity equivalence of the present formulation to the one using 4 node quadrilateral isoparametric elements with selective reduced integration is presented convergence instability phenomenon smoothing of the pressure distribution are studied together with decompositions into triangular elements

enhanced functional evaluation for the finite element penalty method abstract we analyze the impact of using the penalty method on the estimation of a prescribed solution functional or quantity of interest qoi specifically we consider the use of penalty methods to enforce dirichlet boundary constraints focusing our attention on boundary fluxes as qois we propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter we show that the new estimator reduces the error arising from the use of the penalty method from o to o a well posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method errors in the enhanced flux estimator are then controlled using adjoint based techniques several numerical experiments are presented to demonstrate that the enhanced estimator in combination with adjoint error estimation allows one to efficiently control both the discretization and penalty errors in target qois

adaptive dynamic relaxation algorithm for non linear hyperelastic structures part i formulation abstract the formulation of an adaptive dynamic relaxation algorithm with application to non linear hyperelastic structures is presented a complete derivation of the dynamic relaxation method is given and the adaptive scheme used to ensure reliability and improve performance is described finite element equations are derived for the non linear analysis of elastic and hyperelastic solids subject to large deformations the hyperelastic material is assumed to be nearly incompressible and it is modeled using the mooney rivlin material law using a penalty method to enforce incompressibility a very simple and efficient contact algorithm based on solver constraints is developed to enforce frictionless contact conditions

singular dimensions of the n 2 superconformal algebras i verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1 following a method developed for the virasoro algebra by kent we introduce the concept of adapted orderings on superconformal algebras we prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space we apply this method to the topological n 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological verma modules 0 1 2 or 3 depending on the type of verma module and the type of singular vector as a consequence we prove the conjecture of gato rivera and rosado on the possible existing types of topological singular vectors 4 in chiral verma modules and 29 in complete verma modules interestingly we have found two dimensional spaces of singular vectors at level 1 finally by using the topological twists and the spectral flows we also obtain the maximal dimensions of the singular vector spaces for the neveu schwarz n 2 algebra 0 1 or 2 and for the ramond n 2 algebra 0 1 2 or 3

the b bar method and the limitation principles abstract the generalized elastic material provides a reference model to cast in a unitary framework many structural models which are based on nonlinear monotone multivalued relations such as viscoelasticity plasticity and unilateral models the modified forms of the hu washizu and hellinger reissner principles and the displacement based variational formulation are recovered for the generalized elastic material starting from a functional in the complete set of state variables the related limitation principles are derived and their specialization to elasticity and elastoplasticity with mixed hardening are provided it is shown that the interpolating fields for the pressure and the volumetric strain usually adopted in the b bar method lead to a limitation principle accordingly the same elastic and elastoplastic solutions can be obtained by means of an approximate mixed displacement pressure variational principle a second application is concerned with the conditions ensuring the coincidence of the solutions between an approximate two field mixed formulation and the displacement based method numerical examples are provided to show the coincidence of the solutions obtained from different mixed finite element formulations in elasticity or elastoplasticity under the validity of the limitation principles

on the perturbed lagrangian formulation for nearly incompressible and incompressible hyperelasticity a perturbed lagrangian formulation for nearly incompressible and incompressible hyperelasticity is developed by performing a conjugate transformation on the volumetric energy function the formulation is developed without reference to any specific material models or volumetric energy functions solution existence conditions for general nearly incompressible hyperelasticity problems and their discrete counterparts are described a discussion on the convergence of nearly incompressible solutions to an incompressible solution of continuous as well as discrete hyperelasticity problems is given the rate of convergence with respect to the compressibility parameter is shown to be one in several numerical examples

numerical solving of frictionless contact problems in perfectly plastic bodies abstract we consider the frictionless unilateral contact problems of a perfectly plastic solid with a rigid foundation and two perfectly plastic solids under an analogous variational formulation for each one we introduce a discrete approximation via the finite element method and in due course prove convergence also we introduce a converging iterative algorithm to the solution of the discrete problems and present some numerical experiences

penalty finite element models for nonlinear dynamic analysis a simple penalty finite element formulation is presented for the large rotation dynamic analysis of curved beams the analytical formulation is based on a form of reissner s large deformation theory with the effects of transverse shear deformation and the extensibility of the centerline constrained through the use of the penalty method reduced integration is used in evaluating the elemental stiffness arrays and the temporal integration is performed by using newmark s method numerical results are presented to demonstrate the effectiveness of the finite elements developed

finite element methods for constrained problems in elasticity several different variational formulations of boundary value problems with constraints are discussed with particular reference to constrained problems in elasticity special attention is given to exterior penalty methods a discussion of the conditions necessary for penalty methods to provide a basis for stable and convergent finite element methods is given in particular the use of reduced integration is discussed and criteria on the order of reduced integration rules sufficient to produce stable and convergent schemes are described applications of reduced integration penalty methods to incompressible elasticity problems and contact problems are described

the adapted ordering method for lie algebras and superalgebras and their generalizations in 1998 the adapted ordering method was developed for the representation theory of the superconformal algebras in two dimensions it allows us to determine maximal dimensions for a given type of space of singular vectors to identify all singular vectors by only a few coefficients to spot subsingular vectors and to set the basis for constructing embedding diagrams in this paper we present the adapted ordering method for general lie algebras and superalgebras and their generalizations provided they can be triangulated we also review briefly the results obtained for the virasoro algebra and for the n 2 and ramond n 1 superconformal algebras

a finite element solution method for contact problems with friction a new finite element solution method for the analysis of frictional contact problems is presented the contact problem is solved by imposing geometric constraints on the pseudo equilibrium configuration defined as a configuration at which the compatibility conditions are violated the algorithm does not require any a priori knowledge of the pairs of contactor nodes or segments the contact condition of sticking slipping rolling or tension release is determined from the relative magnitudes of the normal and tangential global nodal forces contact iterations are in general found to converge within one or two iterations the analysis method is applied to selected problems to illustrate the applicability of the solution procedure

a penalty finite element method based on the euler implicit explicit scheme for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations where the time discretization is based on the euler implicit explicit scheme with some implicit linear terms and an explicit nonlinear term and the finite element spatial discretization is based on the p 1b p 1 element pair which satisfies the discrete inf sup condition this method allows us to separate the computation of the velocity from the computation of the pressure with a larger time step size dt so that the numerical velocity u e h n and the pressure p e h n are easily computed an optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter e the time step size dt and the mesh size h satisfy the following stability conditions ec 1 1 dt k 1 1 and h 2 b 1 dt respectively for some positive constants c 1 k 1 and b 1 finally some numerical tests to confirm the theoretical results of the penalty finite element method are provided

a stabilised mixed meshfree method for incompressible media application to linear elasticity and stokes flow abstract this paper addresses some problems of modelling incompressible media using meshfree methods with a stabilised mixed formulation the aims of this paper are twofold to avoid the volumetric locking that occurs at the incompressible limit and to circumvent the inf sup condition while approximating both displacement and pressure with the same set of nodes and shape functions we employ a modified polynomial pressure projection stabilisation method to stabilise and filter spurious pressure modes where the gradient is corrected as proposed by duan et al 2014 the gradient correction is used to counteract the lack of variational consistency when numerical integration of the rational shape functions is performed a threat that could undermine the stabilisation effort the resulting novel method passes the mixed patch tests and is demonstrated to be effective and optimally accurate in linear elasticity and stokes flow test cases

accuracy of some finite element models for arch problems abstract this paper presents a theoretical accuracy study of some finite element models for thin arches as is well known the selection of finite elements for curved members is quite a delicate problem we obtain order estimates of errors of finite element solutions by means of the perturbation theory of mixed models and the technique of asymptotic expansion in particular we theoretically show that certain finite element models may suffer from the so called locking phenomenon numerical results are also given to be compared with the theoretical error estimates

an analysis of metal forming processes using large deformation elastic plastic formulations a concise survey of the literature related to the large deformation elasto plasticity problems including unilateral contact and friction is presented together with an extension of the friction law for large deformation analysis starting from the principle of virtual work the so called total lagrangian and updated lagrangian formulations are derived based on some fundamental assumptions in linearizing the nonlinear equations by introducing the zaremba jaumann co rotational increment to the cauchy stress tensor the classical prandtl reuss equations are generalized for describing the elastic plastic material behavior to allow a proper consideration of the contact conditions in the incremental analysis a general friction law with an associated isotropic coulomb sliding rule is obtained by the similarity between dry friction and plasticity finite element discretizations and approximations are applied to the resulting formulation of the updated lagrangian approach four example problems are solved to test the formulations developed in this paper the emphasis is made toward the numerical accuracy of the finite element solutions

on the fundamental solutions for the difference helmholtz operator in this paper we construct the difference approximation of the helmholtz operator on a square grid and we analyze asymptotic properties of the difference operator fundamental solutions we show that for a sommerfeld type asymptotic condition there exists a fundamental solution that is determined in a unique way

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

on the structure of verma modules over virasoro and neveu schwarz algebras in the paper we present a different proof of the theorem of b l feigin and d b fuchs about the structure of verma modules over virasoro algebra we state some new results about the structure of verma modules over neveu schwarz the proof has two advantages first it is simpler in the most interesting cases for example in the so called minimal models second it can be generalized for neveu schwarz algebra for some class of verma modules

analysis of hourglass instabilities and control in underintegrated finite element methods belytschko et al 1981 1984 has developed stabilization methods for the treatment of underintegrated fem problems these methods involve the computation of an underintegrated stiffness matrix which is rank deficient and the addition of a stabilization matrix which effectively eliminates the spurious modes an attempt is presently made to give this a priori stabilization method a mathematical means of support attention is also given to an a posteriority stabilization method for hourglass control in which an approximate solution of the underintegrated system is obtained and then subjected to a special projection in order to eliminate the hourglass modes a proof is obtained for the convergence of this stabilized underintegrated approximation to the exact solution of a model problem at almost the same rate as the mesh is refined as the fully integrated solutions

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem

diffraction of waves on triangular lattice by a semi infinite rigid constraint and crack abstract an exact solution is provided for a discrete analog of each of the two sommerfeld diffraction problems using a triangular lattice model with nearest neighbor interactions deforming in the anti plane direction the discrete helmholtz equation with time harmonic data prescribed on semi infinite row s of lattice sites is solved using the discrete wiener hopf method an asymptotic expression of the exact solution given in the form of a contour integral has been obtained in far field using the standard approximation of the diffraction integrals based on the method of stationary phase in case of the rigid constraint diffraction the displacement of particles on the semi infinite row complementing the constrained lattice sites as well as on the adjacent row is presented in closed form as a discrete convolution for the crack diffraction problem the length of both types of slant bonds on the semi infinite row complementing the crack as well as the crack opening displacement is given in a similar form but in terms of the fourier coefficients of the associated wiener hopf kernel which are not available in closed form the displacement field associated with the scattered waves at sites far from the defect tip as well as few sites near the tip is compared graphically with that of a numerical solution on a finite grid both discrete sommerfeld problems are naturally relevant to their continuous counterparts involving the traditional helmholtz equation that model the diffraction of electromagnetic and acoustic waves by a semi infinite screen based on 7 point discretization on a triangular grid