Fundamental time-domain approaches for variable time delay

The partial differential equation that characterizes variable time delay is analyzed. Analytical solution methods are applicable when endpoint conditions and variable delay time are known in advance; two such methods are given. When variable time delay is but one part of a larger system, however, the simulation of the larger system on a computer can be facilitated by the use of finite difference approximations of the variable delay element. A class of variable delay approximations is developed from which several commonly used approximations follow as special cases. A particular approximation having truncation error of the order of (Δx)⁴ is found to be near optimum. Simulation results are presented.     

Application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations

A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the C^∞ multiquadric (MQ) radial basis function (RBF) expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than 5 · 10⁻⁷.     

Curved finite element methods for the solution of singular integral equations on surfaces in R3

We have shown in [1]that the singular integral equation (1.2) on a closed surface Γ of R³ admits a unique solution q and is variational and coercive in the Hilbert space $\text{H}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$(Γ). In this paper, with the help of curved finite elements, we introduce an approximate surface Γ_h, and an approximate problem on Γ_h, whose solution is q_h. Then we study the error of approximation |q ? q_h| in some Hubert spaces and also the associated error |u ? u_h| of the potential.     

Exponential-time approximation of weighted set cover

The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponential-time algorithms for problems of that kind. The goal is getting the time complexity still of order O(cn), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation.In fact, we describe a kind of approximation schemes—trade-offs between approximation factor and the time complexity. We use general transformations from exponential-time exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate r, one can transform any O^∗(cn)-time¹ algorithm for Set Cover into a (1+lnr)-approximation algorithm running in time O^∗(cn/r). We believe that results of that kind extend the applicability of exact algorithms for NP-hard problems.     

Time step integration algorithms with predetermined coefficients for second order equations

In this paper, the effect of using the predetermined coefficients in constructing time step integration algorithms suitable for linear second order differential equations based on the weighted residual method is investigated. The second order equations are manipulated directly. The displacement approximation is assumed to be in a form of polynomial in the time domain and some of the coefficients can be predetermined from the known initial conditions. The algorithms are constructed so that the approximate solutions are equivalent to the solutions given by the transformed first order equations. If there are m predetermined coefficients (in addition to the two initial conditions) and r unknown coefficients in the displacement approximation, it is shown that the formulation is consistent with a minimum order of accuracy m+r. The maximum order of accuracy achievable is m+2r. This can be related to the Padé approximations for the second order equations. Unconditionally stable algorithms equivalent to the generalized Padé approximations for the second order equations are presented. The order of accuracy is 2r−1 or 2r and it is required that m+1⩽r. The corresponding weighting parameters, weighting functions and additional weighting parameters for the Padé and generalized Padé approximations are given explicitly.     

The effect of numerical integration in finite element approximations to degenerate evolution equations

This paper studies some effects of numerical integration on the rate of convergence of finite element approximations to a degenerate initial, boundary value problem. It is shown that if the quadrature scheme and the finite element space satisfy certain accuracy and consistency conditions then the approximation converges to the true solution at an optimal rate in theW ₂ ¹ norm. When a specific choice of starting data is made, convergence is also optimal inL ². Partially supported by grant MCS-8202025 from the National Science Foundation.     

Finite element approximation of some indefinite elliptic problems

We consider the finite element approximation of some indefinite Neumann problems in a domain of IR^N. From the Fredholm Alternative this kind of problem admits a solution if and only if the right hand term has zero mean value with respect to a measure whose density m is the solution of a homogeneous adjoint problem. The first step consists in the construction of piecewise linear finite element approximations m_h of m, showing their optimal rate of convergence both in energy and L^p norms. The functions m_h are then shown to be crucial in testing admissible data for the Neumann problem and also in its numerical resolution (actually, the standard Galerkin approximation may not be solvable without suitable perturbations of the data).     

A practical approximation algorithm for the LMS line estimator

The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are widely used because of their lack of sensitivity to outlying data points. The least median-of-squares (LMS) regression line estimator is among the best known robust estimators. Given a set of n points in the plane, it is defined to be the line that minimizes the median squared residual or, more generally, the line that minimizes the residual of any given quantile q, where 0<q?1. This problem is equivalent to finding the strip defined by two parallel lines of minimum vertical separation that encloses at least half of the points.The best known exact algorithm for this problem runs in O(n²) time. We consider two types of approximations, a residual approximation, which approximates the vertical height of the strip to within a given error bound ε_r?0, and a quantile approximation, which approximates the fraction of points that lie within the strip to within a given error bound ε_q?0. We present two randomized approximation algorithms for the LMS line estimator. The first is a conceptually simple quantile approximation algorithm, which given fixed q and ε_q>0 runs in O(nlogn) time. The second is a practical algorithm, which can solve both types of approximation problems or be used as an exact algorithm. We prove that when used as a quantile approximation, this algorithm's expected running time is . We present empirical evidence that the latter algorithm is quite efficient for a wide variety of input distributions, even when used as an exact algorithm.     

The solution of multi-parameter systems of equations with application to problems in nonlinear elasticity

Systems of nonlinear equations governed by more than one parameter are discussed with particular attention to bifurcation behaviour. The procedure adopted is to add to the original system of n equations (m − 1) further equations, in the case of m parameters, and to seek solution curves in R^n+m to this augmented system. Two types of additional equations are considered: one describes a piecewise linear path in the space of parameters, and the second constrains the solution curve to be a locus of singular points. These ideas are all subsequently applied to the systems of equations arising from finite element approximations of boundary value problems in nonlinear elasticity. The behaviour of a nonlinear elastic thick-walled cylinder subjected to internal pressure and axial extension is discussed.     

The accuracy of iterated JWBK approximations for Coulomb radial wave functions

Iterated JWBK approximations (IJWBK) can be used for the calculation of Coulomb radial wave functions, for non-negative energies and for values of the radial coordinate, r, such that the functions are oscillatory. Burgess (1963) has given analytical expressions for the first-order IJWBK amplitudes and phases. The second-order theory is discussed. Accurate Coulomb functions are computed using power-series expansions and are used to determine the errors in the zero-, first- and second-order IJWBK approximations. Values of r_l [δ] are given, such that for r > r_l [δ] the errors in the first-order approximation are less than δ, where δ = 10^-3 or 10^-4     

Self-adaptive finite elements in fracture mechanics

A new finite element capability which permits the analyst to vary the order of polynomial approximation over each finite element is discussed with reference to its potential for application to stress intensity factor computations in linear elastic fracture mechanics. Computational experiments, in which polynomial orders ranging from 1 to 8 were used, indicated strong and monotonie convergence of the strain energy release rate even for very coarse finite element meshes as the order p of the approximating polynomial was increased. Pointwise convergence of stresses was achieved by averaging approximations of different polynomial orders. The strong and monotonie convergence of K_I factors with respect to increasing p provides a new method for computing stress intensity factors. The main advantage of this method is that the accuracy of approximation can be established without mesh refinement or the use of special procedures.     

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C⁰ finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C¹ finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.     

Coupling strategy for matching different methods in solving singularity problems

For solving elliptic boundary value problems with singularities, we have proposed the combined methods consisting of the Ritz-Galerkin method using singular (or analytic) basic functions for one part,S ₂, of the solution domainS, where there exist singular points, and the finite element method for the remaining partS ₁ ofS, where the solution is smooth enough. In this paper, general approaches using additional integrals are presented to match different numerical methods along their common boundary Г₀. Errors and stability analyses are provided for such a general coupling strategy. These analyses are important because they form a theoretical basis for a number of combinations between the Ritz-Galerkin and finite element methods addressed in [7], and because they can lead to new combinations of other methods, such as the combined methods of the Ritz-Galerkin and finite difference methods. Moreover, the analyses in this paper can be applied or extended to solve general elliptic boundary value problems with angular singularities, interface singularity or unbounded domain.     

Optimal segmented approximations

Segmented approximations with free knots for a given continuous function are discussed in a general form. This form includes any kind of continuous approximating segments, and a great variety of error criteria, such as anyL _s -norm, or the maximum norm of the pointwise error. It is shown that a solution to the problem exists under very general conditions, and that one solution at least must have continuous pointwise error modulus. Finally, algorithms for the iterative reduction of the approximation error are proposed, which lead to good, but not necessarily best approximations.     

A singular ES-FEM for plastic fracture mechanics

The stress and strain fields around the crack tip for power hardening material, which are singular as r approaches zero, are crucial to fracture and fatigue of structures. To simulate effectively the strain and stress around the crack tip, we develop a seven-node singular element which has a displacement field containing the HRR term and the second order term. The novel singular element is formulated based on the edge-based smoothed finite element method (ES-FEM). With one layer of these singular elements around the crack tip, the ES-FEM works very well for simulating plasticity around the crack tip based on the small strain formulation. Two examples are presented with detailed comparison with other methods. It is found that the results of the presented singular ES-FEM are closer to reference solution, which demonstrates the applicability and the effectiveness of our method for the plastic field around the crack tip.     

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L²-norm.     

A Spectral Element Framework for Option Pricing Under General Exponential Lévy Processes

We derive a spectral element framework to compute the price of vanilla derivatives when the dynamic of the underlying follows a general exponential Lévy process. The representation of the solution with Legendre polynomials allows one to naturally approximate the convolution integral with high order quadratures. The method is spectrally accurate in space for the solution and the greeks, and third order accurate in time. The spectral element framework does not require an approximation of the Lévy measure nor the lower truncation of the convolution integral as commonly seen in finite difference approximations. We show that the spectral element method is ten times faster than Fast Fourier Transform methods for the same accuracy at strike, and two hundred times faster if one reconstructs the greeks from the solution obtained by FFT. We use the SEM approximation to derive the $\Delta $ Δ and $\Gamma $ Γ in a variance gamma model, for which there is no closed form solution.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.