discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

on the exact mutual reactance of a line source array a hilbert transform methodology the mutual reactance between two elements of a lineal array is formulated using fourier transform autocorrelation convolution and hilbert transform concepts with the invocation of the hilbert transform being the novel contribution of this paper it is shown herein that the mutual reactance of an array can be determined from the radiation pattern if one considers the entire time averaged power delivered to both the visible and the nonvisible far field regions power to the nonvisible region corresponds to stored energy in the poynting volume stored energy corresponds to reactive power which is in quadrature with the nonvisible time averaged power and quadrature suggests the invocation of the hilbert transform the end result of the hilbert transform process is a straightforward closed form analytical expression for the mutual reactance since the closed form mutual resistance has been determined previously both closed form expressions can be combined to provide an elegant but easy to use formulation for the mutual impedance for a given current distribution the results are exact and validated against data obtained numerically cases associated with the half wave dipole generalized dipole distribution and the uniform distribution are provided herein as examples of the robustness of the presented theory

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

a relationship between total radiated power of difference patterns and sum patterns using variance beamwidth for a given current distribution along a line source the relationship between the line source s total radiated power and the radiated power of a line source whose current distribution is the derivative of the original current distribution is established herein using the notion of variance beamwidth current distributions of specific interest are symmetrical and antisymmetrical distributions that produce sum and difference patterns respectively an example is provided by using the triangular distribution as the symmetrical current distribution and the bipolar pulse as the antisymmetrical distribution error estimates are also provided that highlight the limitations of using parseval s identity to approximate the total radiated power of a line source this limitation is exemplified by using the bipolar pulse and uniform distributions as a case study numerical data are plotted and assessed to substantiate all theoretical claims

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

spectral distribution of product of pseudorandom matrices formed from binary block codes let a 1 1 na n and b 1 1 nb n be two matrices whose rows are drawn i i d from the codewords of the binary codes ca and cb of length n and dual distances d a and d b respectively under the mapping 0 1 and 1 1 it is proven that as n with ya n na 0 and yb n nb 0 fixed the empirical spectral distribution of the matrix a b na nb resembles a universal distribution closely related to the distribution function of the free multiplicative convolution of two members of the marchenko pastur family of densities in the sense of the levy distance if the asymptotic dual distances of the underlying binary codes are large enough moreover an explicit upper bound on the levy distance of the two distributions in terms of ya yb d a and d b is given under mild conditions the upper bound is strengthened to the kolmogorov distance of the underlying distributions numerical studies on the empirical spectral distribution of the product of random matrices from bch and gold codes are provided which verify the validity of this result

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity analytical estimates this is the second of two papers in which we motivate introduce and analyze a new type of strategy for the stabilization of discontinuous galerkin dg methods in nonlinear elasticity problems the foremost goal behind it is to enhance the robustness of the method without deteriorating the accuracy of the resulting solutions its distinctive property is that for nonlinear elastic problems the stabilization term is solution dependent and hence it is termed an adaptive stabilization mechanism the key contribution of this paper is the construction of a stabilization strategy for which the method is perfectly stable since the stabilization parameters can be explicitly computed as part of the numerical solution this is accomplished through the main result of this paper which consists of a theorem that provides lower bounds for the size of the stabilization parameters numerical examples confirm the guaranteed stability of the resulting method however they also show that the computed lower bounds overestimate the minimum amount of stabilization needed negatively affecting the approximation properties of the method these results underscore the fact that a better understanding of the stabilization mechanisms is needed in order to construct a method that is both robust and efficient

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

exact gain beamwidth product of a line source radiator the exact gain beamwidth product for a line source is formulated using autocorrelation functions the product is valid for any current distribution for any line source length the proposed theory is straightforward to implement and several closed form results are obtained for many different classical current distributions i e uniform cosine cosine squared triangular and generalized dipole asymptotic results for the electrically long line source are provided and reveal a product that tends toward a constant for three of the distributions i e cosine cosine squared and triangular thus confirming the inverse relationship between the gain and the beamwidth additional validation data are given for any line source length thus confirming the robustness of the proposed theory

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

a general theory to determine the exact radiated power directivity and radiation resistance of a line source radiator instead of integrating the power pattern function directly and discounting the effect of the element factor to find the total radiated power of a line source we show herein that an exact analytical expression for the radiated power can be obtained by singular knowledge of the source s current distribution the expression is found by using the fourier principles associated with convolution integrals autocorrelation integrals parseval s identity and the helmholtz operator in one dimension knowledge of the exact radiated power is used to find the expressions for directivity and radiation resistance furthermore we show that the efficiency of the line source is explicitly a function of the relative proportion of two autocorrelation integrals this newly developed theory is validated by considering large and small argument approximations in which closed form results are readily known additional validation is obtained by comparing the exact result with the data obtained from numerical integration of the power pattern function examples associated with the half wave dipole cosine distribution cosine squared distribution generalized dipole triangular distribution and the uniform distribution are provided other than the half wave and generalized dipole solutions we believe that the expressions obtained in this investigation have never been reported in the open literature this is particularly true for the uniform line source in which the classical asymptotic result for the directivity i e 2l lambda is shown to be deficient

variance as a proxy for line source beamwidth the concept of variance is invoked as a means to quantify robustly the spread of a radiation power pattern a precise definition for variance is given in the context of heisenberg s uncertainty principle a methodology is provided to quantify the variance exactly and in closed form using autocorrelation and convolution integrals several examples are provided to demonstrate both the power of the variance definition and the straightforwardness of the mathematical process all mathematical results provided are new and unreported in the open literature additionally a new heisenberg product is formulated that is a relevant addition to the theory of line source radiation data obtained from numerical integration are provided to validate the closed form analytical results derived herein

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

spectral distribution of random matrices from binary linear block codes let c be a binary linear block code of length n dimension k and minimum hamming distance d over gf 2 n let d denote the minimum hamming distance of the dual code of c over gf 2 n let e gf 2 n 1 1 n be the component wise mapping e vi 1 vi for v v1 v2 vn gf 2 n finally for p n let mmb c be a p n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of c under e it is shown that for d large enough and y p n 0 1 fixed as n the empirical spectral distribution of the gram matrix of 1 n mmb c resembles that of a random i i d rademacher matrix i e the marchenko pastur distribution moreover an explicit asymptotic uniform bound on the distance of the empirical spectral distribution of the gram matrix of 1 n mmb c to the marchenko pastur distribution as a function of y and d is presented

the exact radiated power and directivity of a scanning colineal array this paper addresses the fundamental and practical problem of colineal phased array electromagnetic radiation by formulating a general and robust theory that is applicable to any number of elements scanning angle array weights element distribution interelement spacing and element length the theory does not rest on any assumptions and does not invoke any approximations such as discounting the impact of the element factor the results from the theory are exact further support of the theory is provided by showing how the mutual resistance between elements is determined exactly for any specified element length and distribution examples are provided by considering both uniform and chebyshev arrays with current distributions across individual elements that are either uniform or those associated with a wire dipole antenna of any length validation is accomplished by an independent numerical integration of the poynting radiation integral setting aside imperfections associated with numerical round off errors we claim that the agreement between data sets is exact

discontinuous galerkin finite element approximation of nonlinear second order elliptic and hyperbolic systems we develop the convergence analysis of discontinuous galerkin finite element approximations to symmetric second order quasi linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in mathbb r d subject to mixed dirichlet neumann boundary conditions optimal order asymptotic bounds are derived on the discretization error in each case without requiring the global lipschitz continuity or uniform monotonicity of the stress tensor instead only local smoothness and a garding inequality are used in the analysis

random matrices from linear codes and wigner s semicircle law in this paper we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral distribution converges to wigner s semicircle law as the length of the codes goes to infinity one such condition is that the dual distance of the codes is at least five this is analogous to previous work on the empirical spectral distribution of similar matrices obtained in this fashion that converges to the marchenko pastur law

the exact radiated power and directivity of a scanning line source radiator the theory for computing the closed form exact radiated power of a scanning electric line source is provided for any scan angle including zenith nadir and complex valued angles of radiation and for any current distribution of any length including both electrically short and electrically long line sources knowledge of the radiated power pattern is not needed the theory is couched in terms of three straightforward mathematical operations an autocorrelation of the current distribution a modified helmholtz operation and a convolution integral examples associated with the uniform triangular half wave dipole cosine and cosine squared distributions are provided to showcase the theory validation of the theory is established by comparing data obtained from the examples with data obtained via numerical integration of the power pattern the agreement between data sets is effectively perfect

on a question of babadi and tarokh in a series of remarkable papers babadi and tarokh proved the randomness of matrices and product of two matrices arising from binary linear block codes with respect to the empirical spectral distribution provided that their dual distances are sufficiently large however numerical experiments conducted by babadi and tarokh revealed that gold codes which have a dual distance of 5 also possess such a randomness property hence the interesting question was raised as to whether or not the stringent requirement of large dual distances can be relaxed in the theorems in order to explain the mysterious randomness of gold sequences in this paper we improve the results of babadi and tarokh on several fronts and provide an affirmative answer to this question

a general discontinuous galerkin method for finite hyperelasticity formulation and numerical applications a discontinuous galerkin formulation of the boundary value problem of finite deformation elasticity is presented the primary purpose is to establish a discontinuous galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the dg concept can show its superiority compared to the continuous formulation the method is based on a general hu washizu de veubeke functional allowing for displacement and stress discontinuities in the domain interior it is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation the consistency and linearized stability of the method in the non linear range as well as its convergence rate are proven an implementation in three dimensions is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner in order to demonstrate the versatility accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided copyright 2006 john wiley sons ltd

a re examination of the fundamental limits on the radiation q of electrically small antennas an exact method which is more straightforward than those previously published is derived for the calculation of the minimum radiation q of a general antenna this expression agrees with the previously published and widely cited approximate expression in the extreme lower limit of electrical size however for the upper end of the range of electrical size which is considered electrically small the exact expression given here is significantly different from the approximate expression this result has implications on both the bandwidth and efficiency limitations of antennas which fall into this category

locking free adaptive discontinuous galerkin fem for linear elasticity problems an adaptive discontinuous galerkin finite element method for linear elasticity problems is presented we develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials absence of volume locking furthermore we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes