solving time dependent engineering problems with multiquadrics abstract this paper deals with the solution of time dependent problems the multiquadric radial basis function method is formulated with a new approach for transient problems one and two dimensional problems are considered the forward difference and the crank nicolson time marching schemes for parabolic cases are considered the central difference integration method of the newmark family is considered for hyperbolic problems the method proves its accuracy in four numerical examples

a new simple third order shear deformation theory of plates an improved simple third order shear deformation theory for the analysis of shear flexible plates is presented in this paper this new plate theory is composed of three parts the simple third order kinematics of displacements reduced from the higher order displacement field derived previously by the author a system of 10th order differential equilibrium equations in terms of the three generalized displacements of bending plates five boundary conditions at each edge of plate boundaries although the resulting displacement field is the same as that proposed by murthy the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature the applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate a bending beam and two bending plates to which the 3 d elasticity solutions are available and excellent agreements are achieved even for the torsion of a plate with square cross section as well the local effects of stresses at plate boundaries can be characterized accurately these analytical solutions clearly show that the simple third order shear deformation theory developed in this work indeed gives better results than the first order shear deformation theories and other simple higher order shear deformation theories since the present third order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations but also a group of consistent boundary conditions associated with the differential equations the present simple third order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories and improved results could be obtained from the present simple third order shear deformable theories of plates

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

on choosing a radial basis function and a shape parameter when solving a convective pde on a sphere radial basis function rbf approximations have been used for some time to interpolate data on a sphere as well as on many other types of domains their ability to solve to spectral accuracy convection type pdes over a sphere has been demonstrated only very recently in such applications there are two main choices that have to be made i which type of radial function to use and ii what value to choose for their shape parameter denoted by e and with flat basis functions stretched out in the radial direction corresponding to e 0 the recent rbf qr algorithm has made it practical to compute stably also for small values of e results from solving a convective type pde on a sphere are compared here for many choices of radial functions over the complete range of e values from very large down to the limit of e 0 the results are analyzed with a methodology that has similarities to the customary fourier analysis in equispaced 1 d periodic settings in particular we find that high accuracy can be maintained also over very long time integrations we furthermore gain insights into why rbfs sometimes offer higher accuracy than spherical harmonics since the latter arise as an often non optimal special case of the former anticipated future application areas for rbf based methods in spherical geometries include weather and climate modeling

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

meshless implementation of arbitrary 3d shell structures based on a modified first order shear deformation theory abstract a meshless implementation of arbitrary 3d shell structures based on a modified first order shear deformation theory fsdt is investigated in this paper the fsdt is improved in this manuscript in order to assume a parabolic distribution of the shear strain to correct the constant shear stress in the mindlin reissner theory and to get closer to a realistic distribution of the shear strain through the thickness the radial point interpolation method is used in the construction of shape functions based on arbitrarily distributed nodes of general spatial shell geometry the convergence of the proposed model is compared to other well known formulations found in the literature considering isotropic and functionally graded spatial shell structures the results obtained using the proposed meshless method demonstrate that the improved fsdt is very successful compared to closed form solutions and finite element results using different shell theories the effect of the material distribution on the deflections and stresses is also analyzed

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

augmented levy michell equations for flexural plates abstract a new sixth order isotropic elastic and flexural plate theory is presented based on the fourth order equations of levy 1877 and michell 1900 the lack of surface loading in these levy michell equations is overcome by including the dominant flexural component of dougall s 1904 solution for a point load on the surface of a three dimensional isotropic elastic layer the augmented plate equations are arranged to fit the form of a consistent sixth order system of isotropic plate equations which include the well known work of reissner and static equations of mindlin numerical applications to two three dimensional problems with analytical solutions show good comparative results

finite element vibration analysis of composite beams based on higher order beam theory abstract this paper presents a new finite element formulation for the free vibration analysis of composite beams based on the third order beam theory this work also studies the influence of the mass components resulting from higher order displacements on the frequencies of flexural vibration by using hamilton s principle the variational consistent equation of motion in matrix form corresponding to the third order shear deformation theory is derived the resulting mass matrices are decomposed into three parts i e the usual part including the rotary inertia corresponding to first order theory the part resulting from higher order displacement and the part resulting from the coupling between the different components of the axial displacement the numerical examples show that the higher order and coupling mass matrices have a significant influence on the frequencies of high mode flexural vibration the present element formulation for composite beams can be easily extended to composite plates and shells

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

structured kernel dictionary learning with correlation constraint for object recognition in this paper we propose a new discriminative non linear dictionary learning approach called correlation constrained structured kernel ksvd for object recognition the objective function for dictionary learning contains a reconstructive term and a discriminative term in the reconstructive term signals are implicitly non linearly mapped into a space where a structured kernel dictionary each sub dictionary of which lies in the span of the mapped signals from the corresponding class is established in the discriminative term by analyzing the classification mechanism the correlation constraint is proposed in kernel form constraining the correlations between different discriminative codes and restricting the coefficient vectors to be transformed into a feature space where the features are highly correlated inner class and nearly independent between classes the objective function is optimized by the proposed structured kernel ksvd during the classification stage the specific form of the discriminative feature is needless to be known while the inner product of the discriminative feature with kernel matrix embedded is available and is suitable for a linear svm classifier experimental results demonstrate that the proposed approach outperforms many state of the art dictionary learning approaches for face scene and synthetic aperture radar vehicle target recognition

the numerical solution of plane potential problems by improved boundary integral equation methods abstract a modified boundary integral bie method which facilitates accurate solution of laplacian boundary value problems is presented this method is designed specifically for treatment of problems in which singularities occur on the interface between two regions with different physical properties and is illustrated by application to two physical problems analytic expressions for the integrals arising in the piecew se linear and piecewise quadratic bie approximations are also presented these analytical expressions afford an appreciable reduction in computational time when compared with previously employed quadrature formulae

computing eigenmodes ofelliptic operators using radial basis functions radial basis function rbf approximations have been successfully used to solve boundary value problems numerically we show that rbfs can also be used to compute eigenmodes of elliptic operators particular attention is given to the laplacian operator in two dimensions including techniques to avoid degradation of the solution near the boundaries for regions with corner singularities special functions must be added to the basis to maintain good convergence numerical results compare favorably to basic finite element methods

a refined two dimensional theory for thick cylindrical shells abstract a higher order two dimensional theory for thick cylindrical shells is presented in this paper the shell equations given here do not only incorporate the effect of transverse shear deformations but also account for the initial curvature as well as the radial stress the proposed theory presents a very good approximation for the shell constitutive equations and the nonlinear distributions of the in plane stresses across the thickness of the shell the latter is very important in the thick shells analysis the formulation is based on 1 1 assumed out of plane stress components which satisfy the given traction boundary conditions 2 2 three dimensional elasticity equations with an integral form of the equilibrium equations and 3 3 stress resultants and stress couples acting on the middle surface of the shell average displacements along the normal at a point on the middle surface and average rotations of the normal the proposed shell equations can be conveniently used in the finite element analysis an application of this theory to the finite element analysis of circular arches is given in this paper a more convenient form of the proposed shell equations for finite element analysis and its application to cylindrical shells will be presented in a follow up paper

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

analysis of cracked plate using higher order shear deformation theory asymptotic crack tip fields and xiga implementation abstract in this work an extended isogeometric analysis xiga is used for the analysis of through thickness crack in a homogeneous and isotropic plate in isogeometric analysis iga non uniform rational b splines nurbs are used as a basis function the plate kinematics is modelled by reddy s higher order shear deformation theory hsdt the c 1 continuity requirement of hsdt can be easily fulfilled by the nurbs basis functions in order to obtain the plate fracture parameters moment intensity factors the expressions of crack tip fields auxiliary fields are derived using separation of variables and eigen function approach a new expression for moment intensity factors is developed using auxiliary fields solution crack tip fields and interaction integral approach several cracked plate problems are solved by xiga using hsdt the results obtained by hsdt based xiga hsdt xiga are compared with the fsdt based xiga fsdt xiga and literature solutions