Polynomial Shape Functions Research Articles

Natural frequencies, modeshapes and modal interactions for strings vibrating against an obstacle: Relevance to Sitar and Veena

We study the vibration characteristics of a string with a smooth unilateral obstacle placed at one of the ends similar to the strings in musical instruments like sitar and veena. In particular, we explore the correlation between the string vibrations and some unique sound characteristics of these instruments like less inharmonicity in the frequencies, a large number of overtones and the presence of both frequency and amplitude modulations. At the obstacle, we have a moving boundary due to the wrapping of the string and an appropriate scaling of the spatial variable leads to a fixed boundary at the cost of introducing nonlinearity in the governing equation. Reduced order system of equations has been obtained by assuming a functional form for the string displacement which satisfies all the boundary conditions and gives the free length of the string in terms of the modal coordinates. To study the natural frequencies and mode-shapes, the nonlinear governing equation is linearized about the static configuration. The natural frequencies have been found to be harmonic and they depend on the shape of the obstacle through the effective free length of the string. Expressions have been obtained for the time-varying mode-shapes as well as the variation of the nodal points. Modal interactions due to coupling have been studied which show the appearance of higher overtones as well as amplitude modulations in our theoretical model akin to the experimental observations. All the obtained results have been verified with an alternate formulation based on the assumed mode method with polynomial shape functions.

Application of the lumped mass technique in dynamic analysis of a flexible L-shaped structure under moving loads

This paper presents a lumped-mass approach for the approximate determination of the dynamic response of a flexible L-shaped structure with tip mass subjected to a moving load. The load in the form of a force of a constant magnitude moving across the top beam of the L-shaped frame with a constant speed is considered. A two degrees of freedom model is formed, which allows to avoid complex analytic postulations and finite element formulations. The incorporation of the moving load influence on the frame structure is achieved through moving force flexibility influence coefficients. By using the assumed-modes method with polynomial shape functions, the validation of the approach is given.

Superconvergence Analysis for the Stable Conforming Rectangular Mixed Finite Elements for the Linear Elasticity Problem

In this paper, we consider the linear elasticity problem based on the Hellinger-Reissner variational principle. An O(h 2 ) order superclose property for the stress and displacement and a global superconvergence result of the displacement are established by employing a Clement interpolation, an integral identity and appropriate postprocessing techniques. In this paper, we consider the mixed finite element (for short MFE) approximation of a stress-displacement system derived from the Hellinger-Reissner variational principle for the linear elasticity problem. As is known to all, the MFE methods require that the pair of finite element spaces satisfying the B-B condition. Although there are a number of well-known stable MFEs for the analogous problems involving vector fields and scalar fields (1), the combination of the symmetry and continuity conditions of the stress field is a substantial additional difficulty. On the other hand, a lot of efforts, dating back four decades, have been devoted to develop stable MFEs for the linear elasticity problem, but no stable MFE scheme with polynomial shape functions are yielded. Not until the year 2002, were there some development in this direction. In (2), a sufficient condition was given and then a family of stable MFEs were constructed with respect to arbitrary triangular meshes, with 24 stress and 6 displacement degrees of freedom for the lowest order element, and an optimal order error estimate was obtained. An analogous family of conforming MFEs based on rectangular meshes were proposed in (3), involving 45 stress and 12 displacement degrees of freedom for the lowest order element. Two nonconforming triangular elements were presented in (4) with 12 degrees of freedom for the stress and 3 degrees of freedom for the displacement. Although many stable elements have been constructed for this problem, they involve too much degrees of freedom. Recently, some more simple elements have been developed. In (5), a group of nonconforming rectangular elements were introduced, with the convergence order of

STATIC ANALYSIS OF C-S SHORT CYLINDRICAL SHELL UNDER INTERNAL LIQUID PRESSURE USING POLYNOMIAL SERIES SHAPE FUNCTION

The static analysis of C-S short cylindrical shell under internal liquid pressure is presented. Pasternak’s equation was adopted as the governing differential equation for cylindrical shell. By satisfying the boundary conditions of the C-S short cylindrical shell in the general polynomial series shape function, a particular shape function for the shell was obtained. This shape function was substituted into the total potential energy functional of the Ritz method, and by minimizing the functional, the unknown coefficient of the particular polynomial shape function was obtained. Bending moments, shear forces and deflections of the shell were determined, and used in plotting graphs for cases with a range of aspect ratios, 1 ≤ L/r ≤ 4. For case 1, the maximum deflection was 8.65*10-4metres, maximum rotation was 3.06*10- 3radians, maximum bending moment was -886.45KNm and maximum shear force was -5316.869KN. For case 2, the maximum deflection was 2.18*10-4metres, maximum rotation was 7.74*10-4radians, maximum bending moment was 223.813KNm and maximum shear force was -1342.878KN. For case 3, the maximum deflection was 9.71*10-5metres, maximum rotation was 3.44*10-4radians, maximum bending moment was -99.463KNm and maximum shear force was -596.779KN. For case 4,the maximum deflection was 5.48*10-5metres, maximum rotation was 1.94*10- 4radians, maximum bending moment was 56.097KNm and maximum shear force was -336.584KN. It was observed that as the aspect ratio increases from 1 to 4, the deflections, bending moments and shear forces decreases, and the shell tends to behave like long cylindrical shell.

A 2D local discontinuous Galerkin method for contaminant transport in channel bends

In this work, a third-order local discontinuous Galerkin (LDG) method is applied to the numerical integration of a two-dimensional, depth-integrated mathematical model describing the flow hydrodynamics and the transport of a passive contaminant in open-channel bends.The mathematical model is described in Begnudelli et al. (2010) [14] and treats the main physical aspects of the flow in curved channels (including bottom shear, momentum dispersion, scalar dispersion and turbulent diffusion) with a homogeneous degree of complexity and exhaustiveness. The numerical integration is performed using the scheme presented in Caleffi and Valiani (2012) [27], which is extended in this work to allow the treatment of diffusion terms. The capability of the model to correctly and efficiently treat computational domains with curved boundaries while preserving the exact well-balancing property is of fundamental importance for the correct simulation of the free-surface flow in laboratory flumes and real channels.Three test cases are used to validate the model. The first case consists of a problem, with an analytical solution, related to the tracer convection–diffusion in a uniform flow. This test is selected to highlight the excellent resolution obtainable using an LDG method. The other test cases, consisting of comparisons between numerical results and laboratory data, are selected to verify the capability of the model to reproduce real world phenomena. Both steady and unsteady tracer dispersion are taken into account.The results show the potentiality of the high-order accuracy models when applied to engineering problems that are characterized by inherently complex physical phenomena. The results also confirm the possibility of using extremely coarse grids, which leads to a high computational efficiency, in these real-world applications. Moreover, the good agreement between the experiments and simulations is a further confirmation of the suitability of two-dimensional depth-averaged models for the study of the convection–dispersion of momentum and pollutants in curved channels and rivers.Finally, the reconstruction of the solution by polynomial shape functions, which are typical of the LDG schemes, allows the streamline curvature, which is necessary to evaluate the diffusion coefficients, to be computed in a self-consistent manner, without the use of arbitrary reconstructions.

Static and instability analysis of moderately thick viscoelastic plates using a fully discretized nonlinear finite strip formulation

A linear finite strip plate element based on the first order shear deformation theory is considered for the analysis of viscoelastic plates. The plate end conditions are considered to be simply supported and the polynomial shape functions are used to evaluate the deflection of plates in transverse direction. The mechanical properties of the material are considered to be linear viscoelastic by expressing the relaxation modulus in terms of Prony series. The time history of maximum deflection of viscoelastic plates subjected to loading and unloading is calculated. In addition, a nonlinear procedure is used to calculate the changing of in-plane critical load of plate with time. Moreover, the effect of thickness and the interaction of biaxial loading on critical load of plate are studied.

Evaluation of Singular and Nearly Singular Integrals in the BEM with Exact Geometrical Representation

The geometries of many problems of practical interest are created from circular or elliptic arcs. Arc boundary elements can represent these boundaries exactly, and consequently, errors caused by representing such geometries using polynomial shape functions can be removed. To fully utilize the geometry of circular boundary, the non-singular boundary integral equations (BIEs) and a general nonlinear transformation technique available for arc elements are introduced to remove or damp out the singular or nearly singular properties of the integral kernels. Several benchmark 2D elastostatic problems demonstrate that the present algorithm can effectively handle singular and nearly singular integrals occurring in the boundary element method (BEM) for boundary layer effect and thin-walled structural problems. Owing to the employment of exact geometrical representation, only a small number of elements need to be divided along the boundary and high accuracy can be achieved without increasing other more computational efforts.

A novel hybrid finite element model for modeling anisotropic composites

In this paper we proposed a novel hybrid Trefffz-type finite element method to simulate mechanical behavior of anisotropic composite materials based on its associated Green's functions. The hybrid method is formulated based on two independent assumptions: intra-element field covering the element domain and inter-element frame field along the element boundary. A modified functional, which satisfies the governing equation, boundary and continuity conditions between elements, is proposed to derive the element stiffness equation. In this work, the foundational solutions of anisotropic materials in terms of the elegant and powerful Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. Several numerical examples are presented to demonstrate the performance of the new method by comparing with the analytical or numerical results from literatures and ABAQUS. Numerical results show that the proposed method is accurate and efficient in modeling anisotropic composites materials in contrast to traditional FEM, and it is potential to be further extended to analyze composite laminates easily.

Novel basis functions for the partition of unity boundary element method for Helmholtz problems

SUMMARYThe BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid‐element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain. Copyright © 2012 John Wiley & Sons, Ltd.

A new hybrid finite element approach for plane piezoelectricity with defects

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green’s functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii’s formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.

Enrichment of a rational polynomial family of shape functions with regularity Ck0, k=0,2,4…

PurposeThe purpose of this paper is to investigate the approximation performance of a family of piecewise rational polynomial shape functions, which are enriched by a set of monomials of order p to obtain high order approximations. To numerically demonstrate the features of the enriched approximation some examples on the mechanical elastic response and free‐vibration of axisymmetric plates and shells are carried out.Design/methodology/approachThe global approximation is based on a particular family of weight function, which is defined on the parametric domain of the element, ξ∈[−1,1], resulting in shape functions with compact support, which have regularity C0k,k=0,2,4… in the global domain Σ. The PU shape functions are enriched by a set of monomials of order p to obtain high order approximation spaces.FindingsBased on the numerical results of elastic axisymmetric plates and shells, it is demonstrated that the proposed methodology produces satisfactory results in terms of keeping the ill‐conditioning of the system of equations under accepted levels. Comparisons are established between linear and Hermitian shape functions showing similar results. The observed results for the free‐vibration problem of plates and shells show the potential of the proposed approximation space.Research limitations/implicationsIn this paper the formulation is limited to the modeling of axisymmetric plate and shell problems. However, it can be applied to model other problems where the high regularity of the approximation is required.Originality/valueThe paper presents an alternative approach to construct partition of unity shape functions based on a particular family of weight function.

Development, Validation and Comparison of Higher Order Finite Element Approaches to Compute the Propagation of Lamb Waves Efficiently

When considering structural health monitoring (SHM) applications efficient and powerful numerical methods are required to predict the behavior of ultrasonic guided waves and to design SHM systems. The existing commercial explicit finite element analysis tools based on standard linear displacement elements quickly reach their limits when applied to ultrasonic waves in thin plates, so called Lamb waves. It is known that the required temporal and spatial resolution causes enormous computational costs. One resort to overcome this problem is the application of special finite elements utilizing higher order polynomial shape functions (p> 2). The current paper is focused on the development of such higher order finite elements and the verification of their accuracy and performance. In the paper we compare and evaluate the capabilities of spectral finite elements,p-version finite elements and isogeometric finite elements. Their advantages and disadvantages with respect to ultrasonic wave propagation problems are discussed and their properties are demonstrated by solving a benchmark problem. Higher order finite elements with varying polynomial degrees in longitudinal and transversal direction (anisotropic ansatz space) are investigated and convergence studies are performed. The results of the convergence studies are summarized and a guideline to estimate the optimal discretization is prepared. If the required accuracy is known, the proposed guideline provides a helpful means to determine both the element size and the polynomial degree template for a given model.

An enriched finite element for crack opening and rebar slip in reinforced concrete members

Object of the paper is the simulation of reinforced concrete bars behaviour, accounting for crack opening and concrete-rebar slippage. A macro beam element with a single uniform reinforcement is studied in details in the uniaxial case. Distinct constitutive hypotheses are formulated for the materials. The CEB-FIP Model Code 90 rules the behaviour of the materials interface that is assumed to be fully dissipative. Steel is supposed to behave elastoplastically with hardening. Crack opening in the concrete matrix is introduced by means of a strong discontinuity approach (SDA). All the relevant equations of the problem are variationally derived from a mixed energy functional. Two enhancements of the enriched kinematics, based on polynomial or exponential shape functions, respectively, are compared with the usual SDA enhancement. As an alternative approach, high-order interpolation of the displacement field based on B-splines, both for steel and concrete, is proposed. These functions appear to be adequate in reproducing rapidly varying fields, like the stress gradients occurring in the shear lag problem near the boundaries or where slips and/or cracks occur. Their use allow to use few macro-element instead of the very dense meshing required in those areas by the traditional FE interpolations.

A frame element model for the analysis of reinforced concrete structures under shear and bending

Modeling the response of reinforced concrete structures under combined shear and normal forces involves handling the anisotropic behavior that takes place in the post-cracked and ultimate ranges. Current 2D (plane-stress and plane-strain) and 3D (solid) non-linear finite element formulations capable of modeling such response are too costly and not suitable for daily engineering practice. In this paper a new developed frame element model which combines an efficient fully coupled in-plane shear–normal sectional model with a beam–column element, suitable for structural analysis applications is described and verified with tests results. Both normal and shear stresses and strains are involved in the sectional equilibrium and compatibility equations, respectively. The shear and vertical strain distributions are assumed to be composed by a series of polynomial shape functions; each one is affected by a constant whose value is internally calculated according to the materials state. The non-linear analysis of the frame structure is carried out using the Generalized Matrix Formulation, which is a force-based approach with “exact” interpolation of forces along the element and implicitly accounts for the shear deformation of the element. The model has been verified using the results of selected well documented tests, in which the influence of the level of shear stresses on the structure response is experimentally evidenced. Good agreement has been obtained between the experimental and the theoretical results provided by the model, showing its capability to reproduce displacements, stresses and strains in the concrete and in the reinforcements, and different types of failure, with more accuracy than previous models.

Dynamics behavior of rotating bladed discs: A finite element formulation for the study of second and higher order harmonics

Annular finite elements for the computation of second and higher order harmonics modes of bladed rotating discs are developed. The elements take into account gyroscopic effect and stiffening due to centrifugal and thermal stresses (the latter not present in arrays of blades). The displacement field is expressed by a truncated Fourier series along the angle and by polynomial shape functions in the radial direction. This paper is the generalization of a previous study limited to zero- and first-order harmonics and deals only with second and higher order harmonics modes that are uncoupled from the modes involving the behavior of the rotor as a whole. Several cases have been studied to verify the accuracy of the disc and array of blades elements.

Analytical integrations for the approximation of 3D hyperbolic scalar boundary integral equations

The present note deals with 3D hyperbolic scalar problems modeled by integral equations [1]. It aims at showing analytical integrations for collocation in time [2] as well as “variational” [3] approximation schemes. Numerical solution is achieved adopting polynomial shape functions of arbitrary degree (in space and time) on a trapezoidal flat tiling of a polygonal domain. Analytical integrations are performed both in space and time for Lebesgue integrals, working in a local coordinate system. For singular integrals, both a limit to the boundary as well as the finite part of Hadamard approach have been pursued. Outcomes and computational issues are presented. Extension to elastodynamics will be the subject of forthcoming publications.

Vibration analysis of multi-span plates using finite strip-elements

This paper discusses the use of the finite strip-element method for the vibration analysis of multi-span, thick plate structures. The method uses a combination of polynomial and trigonometric shape functions that enables all boundary conditions to be correctly treated. The global equations are derived using standard finite element procedures, and accurate solutions are obtained with the generation and solution of few equations.

Analytical integrations in 3D BEM for elliptic problems: Evaluation and implementation

AbstractThe present publication deals with 3D elliptic boundary value problems (potential, Stokes, elasticity) in the framework of linear, isotropic, and homogeneous materials. Numerical approximation of the unique solution is achieved by 3D boundary element methods (BEMs). Adopting polynomial test and shape functions of arbitrary degree on flat triangular discretizations, the closed form of integrals that are involved in the 3D BEMs is proposed and discussed. Analyses are performed for all operators (single layer, double layer, hypersingular). The Lebesgue integrals are solved working in a local coordinate system. For singular integrals, both a limit to the boundary as well as the finite part of Hadamard (Lectures on Cauchy's Problem in Linear Partial Differential Equations. Yale University Press: New Haven, CT, U.S.A., 1923) approach have been considered. Copyright © 2010 John Wiley & Sons, Ltd.

A boundary meshless method with shape functions computed from the PDE

This paper presents a new meshless method using high degree polynomial shape functions. These shape functions are approximated solutions of the partial differential equation (PDE) and the discretization concerns only the boundary. If the domain is split into several subdomains, one has also to discretize the interfaces. To get a true meshless integration-free method, the boundary and interface conditions are accounted by collocation procedures. It is well known that a pure collocation technique induces numerical instabilities. That is why the collocation will be coupled with the least-squares method. The numerical technique will be applied to various second order PDE's in 2 D domains. Because there is no integration and the number of shape functions does not increase very much with the degree, high degree polynomials can be considered without a huge computational cost. As for instance the p-version of finite elements or some well established meshless methods, the present method permits to get very accurate solutions.

A discontinuous enrichment method for the efficient solution of plate vibration problems in the medium‐frequency regime

AbstractA discontinuous enrichment method (DEM) is presented for the efficient discretization of plate vibration problems in the medium‐frequency regime. This method enriches the polynomial shape functions of the classical finite element discretization with free‐space solutions of the biharmonic operator governing the elastic vibrations of an infinite Kirchhoff plate. These free‐space solutions, which represent flexural waves and decaying modes, are discontinuous across the element interfaces. For this reason, two different and carefully constructed Lagrange multiplier approximations are introduced along the element edges to enforce a weak continuity of the transversal displacement and its normal derivative, and discrete Lagrange multipliers are introduced at the element corners to enforce there a weak continuity of the transversal displacement. The proposed DEM is illustrated with the solution of sample plate vibration problems with different types of harmonic loading in the medium‐frequency regime, away from and close to resonance. In all cases, its performance is found to be significantly superior to that of the classical higher‐order finite element method. Copyright © 2010 John Wiley & Sons, Ltd.