P-version Finite Element Research Articles

P-Version FEM for structural acoustics with a posteriori error estimation

We demonstrate the advantages of using p-version finite element approximations for structural-acoustics problems in the mid-to-high frequency regime. We then present a sub-domain-based a posteriori error estimation procedure to quantify the errors in the setting of a 3D interior-acoustics problem with resonances, and give numerical results. Effectivity indices show robust behavior of the error estimator away from the resonant frequencies.

Nonlinear vibrations of simply-supported plates by the p-version finite element method

The p-version, hierarchical finite element method is applied to study geometrically nonlinear periodic vibrations of isotropic simply supported plates, in the elastic regime. Free vibrations are st...

On the use of three-dimensional h- and p-version finite elements in solving vibration response problems

The potential for reliably solving vibration response problems by using h- and p-versions of the finite element (FE) method is investigated. Each FE model is based on full three-dimensional (3-D) displacement-based continuum theory. Special attention is given to the ability to handle thin and/or nearly incompressible elastic and viscoelastic materials. Steady-state time-harmonic problems in the low- and mid-frequency ranges are treated. Convergence studies are performed on plate-like model problems with simply supported boundary conditions, using a series elasto-dynamic Navier solution extended to the viscoelasto-dynamic case by means of the elasto-viscoelastic correspondence principle. The importance of eliminating the discretisation error when using numerical solutions to estimate the frequency-dependent viscoelastic material parameters from experiments is stressed. The superior efficiency of p-enrichments compared to h-refinements for resolving regular elastostatic solutions is observed in elasto/viscoelasto-dynamics as well. Requirements on p-enrichments to avoid shear- and volumetric locking are given. The frequently used polynomial interpolation functions employed are shown to produce a mass contribution to the dynamic stiffness, whose condition number deteriorates with increasing polynomial order. The practical implications and limitations of this observation are outlined. The recommended approach is corroborated using the measured frequency response on a laminate consisting of two aluminium plates and a constrained viscoelastic polymer damping treatment.

Superconvergence of spectral collocation and $p$-version methods in one dimensional problems

Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ∥u (k) ∥L∞ ≤ cM k on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussian points or Lobatto points, the rate of convergence may gain one or two orders of the polynomial degree.

High order finite elements for shells

In this article the p-version finite element method is applied to thin-walled structures. Two different hierarchic element formulations are compared, a shell approach as well as a shell-like, solid formulation. Both approaches are compared for linear elastic and elastoplastic problems. Special emphasis is placed on the efficiency as well as on determining the area of application for both formulations.

Nonlinear vibrations of simply-supported plates by the p-version finite element method

The p-version, hierarchical finite element method is applied to study geometrically nonlinear periodic vibrations of isotropic simply supported plates, in the elastic regime. Free vibrations are studied by the harmonic balance and continuation methods. The nonlinear mode shapes and backbones curves are defined for the first three vibration modes. Resonance curves due to external excitations are also derived, employing the shooting method. The characteristics of the nonlinear motions under harmonic forces are investigated and new results presented.

Automatic p-version mesh generation for curved domains

To achieve the exponential rates of convergence possible with the p-version finite element method requires properly constructed meshes. In the case of piecewise smooth domains, these meshes are characterized by having large curved elements over smooth portions of the domain and geometrically graded curved elements to isolate the edge and vertex singularities that are of interest. This paper presents a procedure under development for the automatic generation of such meshes for general three-dimensional domains defined in solid modeling systems. Two key steps in the procedure are the determination of the singular model edges and vertices, and the creation of geometrically graded elements around those entities. The other key step is the use of general curved element mesh modification procedures to correct any invalid elements created by the curving of mesh entities on the model boundary, which is required to ensure a properly geometric approximation of the domain. Example meshes are included to demonstrate the features of the procedure.

A new hierarchic degenerated shell element for geometrically non-linear analysis of composite laminated square and skew plates

This paper extends the use of the hierarchic degenerated shell element to geometric non-linear analysis of composite laminated skew plates by the p-version of the finite element method. For the geometric non-linear analysis, the total Lagrangian formulation is adopted with moderately large displacement and small strain being accounted for in the sense of von Karman hypothesis. The present model is based on equivalent-single layer laminate theory with the first order shear deformation including a shear correction factor of 5/6. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. A wide variety of linear and non-linear results obtained by the p-version finite element model are presented for the laminated skew plates as well as laminated square plates. A numerical analysis is made to illustrate the influence of the geometric non-linear effect on the transverse deflections and the stresses with respect to width/depth ratio (a/h), skew angle (b), and stacking sequence of layers. The present results are in good agreement with the results in literatures.

J-integral and fatigue life computations in the incremental plasticity analysis of large scale yielding by p-version of F.E.M.

Since the linear elastic fracture analysis has been proved to be insufficient in predicting the failure of strain hardening materials, a number of fracture concepts have been studied which remain applicable in the presence of plasticity near a crack tip. This work thereby presents a new finite element model to predict the elastic-plastic crack-tip field and fatigue life of center-cracked panels(CCP) with ductile fracture under large-scale yielding conditions. Also, this study has been carried out to investigate the path-dependence of J-integral within the plastic zone for elastic-perfectly plastic, bilinear elastic-plastic, and nonlinear elastic-plastic materials. Based on the incremental theory of plasticity, the p-version finite element is employed to account for the accurate values of J-integral, the most dominant fracture parameter, and the shape of plastic zone near a crack tip by using the J-integral method. To predict the fatigue life, the conventional Paris law has been modified by substituting the range of J-value denoted by DJ for DK. The experimental fatigue test is conducted with five CCP specimens to validate the accuracy of the proposed model. It is noted that the relationship between the crack length a and DK in LEFM analysis shows a strong linearity, on the other hand, the nonlinear relationship between a and DJ is detected in EPFM analysis. Therefore, this trend will be depended especially in the case of large scale yielding. The numerical results by the proposed model are compared with the theoretical solutions in literatures, experimental results, and the numerical solutions by the conventional h-version of the finite element method.

P-FEM applied to finite isotropic hyperelastic bodies

In this article the p-version finite element method is applied to finite strain problems. In particular, the behaviour of high order finite elements is studied for an isotropic hyperelastic material in the case of near incompressibility. The question of robustness with respect to distortion and efficiency of anisotropic p-version elements for thin-walled or beam-like structures will be addressed. It will be shown that the p-version finite element approach is a promising method to compute geometrically highly non-linear structures.

Computing maxwell eigenvalues by using higher order edge elements in three dimensions

We use recently proposed hierarchic basis functions and a tetrahedral partitioning to compute Maxwell eigenvalues on a bounded polygonal domain in /spl Ropf//sup 3/, using a p-version finite-element procedure based on edge elements. The problem formulation requires a set of basis functions that are H(curl)-conforming and another compatible set that is H/sup 1/-conforming. In this preliminary study, we employ a uniform order of approximation throughout the domain.

The application of p-version finite-element methods to fracture-dominated problems encountered in engineering practice

This paper details experience applying higher-order ( p-version) finite-element methods to highly specialized and difficult problems in structural life predictions, namely in predictions of aircraft structural fatigue life using crack growth analyses. Crack growth analyses require reliable estimates of the Mode I stress intensity factors (SIFs) for given load, boundary conditions, and geometry. Fortunately, accurate and reliable stresses and stress intensity factors, obtained quickly and efficiently from finite-element simulations performed with higher-order methods, enable basic procedures to be successfully applied to the development of advanced life prediction methods. Many FEA software packages could have been used to obtain engineering data such as SIFs and stresses. However, advanced numerical methods incorporated into higher-order finite-element methods allow the engineering analyst to have confidence in the results. This confidence comes from being able to obtain more accurate and reliable results using convenient error checking and numerical convergence of engineering data such as maximum stress and strain and SIFs, stress contours, and deformations. In addition, greatly shortened analysis schedules are realized through the use of procedures such as the automatic extraction of SIF parameters using the contour integral method (CIM). The versatility of higher-order methods is demonstrated by some examples encountered in engineering practice—multisite damage scenarios and corroded plates.

Performance of the p-version finite element method for limit analysis

In this paper, we investigate the use of the p-version finite element method in carrying out limit analysis using a mathematical programming-based static approach. An important motivation for this study is to overcome the well-known locking behavior—caused by the incompressibility constraint—that may occur in plane strain and 3D problems for such common yield criteria as von Mises’. Quadrilateral elements and plane stress or plane strain structures are specifically considered. Various examples are provided to highlight the robustness and accuracy of the approach.

Two-dimensional viscoelastic vibration by analytic Fourier p-elements

Three new Fourier p-elements of rectangular, skew and trapezoidal shapes are given analytically for plane viscoelastic vibration problems. The natural frequencies of the plane viscoelastic structures with complex Young’s modulus are computed by a complex eigenvalue solver. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly increased. Since trigonometric functions are used as enriching functions instead of polynomials in the proposal elements, the ill-conditioning problems associated with polynomials of higher degree in the traditional p-version finite element method are avoided. The two mapped plane coordinates in the Jacobian are uncoupled for trapezoidal elements whose element matrices can then be integrated analytically. A triangle can easily be divided into three trapezoids. Therefore, any plane viscoelastic problem with polygonal shape can be analyzed by a combination of rectangular and trapezoidal elements. Numerical examples show that the convergence of the present elements is very fast with respect to the number of trigonometric terms. The natural frequencies of several polygonal viscoelastic plates subject to in-plane vibration are presented.

Materially and geometrically nonlinear analysis of laminated anisotropic plates by p-version of FEM

A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with moderately large deflections and small rotations being accounted for in the sense of von Karman hypothesis. The material model is based on the Huber–Mises yield criterion and Prandtl–Reuss flow rule in accordance with the theory of strain hardening yield function, which is generalized for anisotropic materials by introducing the parameters of anisotropy. The model is also based on the equivalent-single layer laminate theory. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. Gauss–Lobatto numerical quadrature is used to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.

Free vibration of skew Mindlin plates by p-version of F.E.M.

Accurate natural frequencies and mode shapes of skew plates with and without cutouts are determined by p-version finite element method using integrals of Legendre polynomials for p=1–14. The hierarchical plate element is formulated based on Mindlin's plate theory including rotatory inertia effects and based on a skew co-ordinate system. Non-dimensional frequency parameter and mode shapes are presented for a range of skew angle ( β), aspect ratio ( a/ b), thickness–width ratio ( h/ b), cutout dimensions and different boundary conditions. The results were verified by comparison with those available in the open literature.

Shape optimization of dental implant designs under oblique loading using the p‐version finite element method

The aim of this study was to introduce design optimization of an endosseous dental implant in order to minimize cortical bone strains at the crest of the implant/bone interface. Linear elastic p-version finite element analysis was used to find the optimal design in four different two-dimensional finite element models of the mandibular bone: model 1 (medium density cancellous bone and 2 mm thickness of cortical bone), model 2 (low density cancellous bone and 2 mm thickness of cortical bone), model 3 (medium density cancellous bone and 1 mm thickness of cortical bone), model 4 (low density cancellous bone and 1 mm thickness of cortical bone). Shape of the implant was evaluated with respect to its length and diameter, as well as the presence or absence of taper in the neck. Implants with maximum length, maximum diameter, and no taper had the lowest crestal strains in models 1, 3, and 4. Taper in the neck of the implant increased the strains in models 1, 3, and 4. In cases of poor quality of bone, increasing the diameter of the implant appeared to be the most effective means of reducing crestal bone strains.

SHAPE OPTIMIZATION OF DENTAL IMPLANT DESIGNS UNDER OBLIQUE LOADING USING THE p-VERSION FINITE ELEMENT METHOD

The aim of this study was to introduce design optimization of an endosseous dental implant in order to minimize cortical bone strains at the crest of the implant/bone interface. Linear elastic p-version finite element analysis was used to find the optimal design in four different two-dimensional finite element models of the mandibular bone: model 1 (medium density cancellous bone and 2 mm thickness of cortical bone), model 2 (low density cancellous bone and 2 mm thickness of cortical bone), model 3 (medium density cancellous bone and 1 mm thickness of cortical bone), model 4 (low density cancellous bone and 1 mm thickness of cortical bone). Shape of the implant was evaluated with respect to its length and diameter, as well as the presence or absence of taper in the neck. Implants with maximum length, maximum diameter, and no taper had the lowest crestal strains in models 1, 3, and 4. Taper in the neck of the implant increased the strains in models 1, 3, and 4. In cases of poor quality of bone, increasing the diameter of the implant appeared to be the most effective means of reducing crestal bone strains.

Multigrid Solver for the Inner Problem in Domain Decomposition Methods forp-FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p-version finite element discretizations of elliptic boundary value problems. The ingredients of such a preconditioner are a preconditioner for the Schur complement, a preconditioner related to the Dirichlet problems in the subdomains, and an extension operator from the boundaries of the subdomains into their interior. In the case of Poisson's equation, we propose a preconditioner for the problems in the subdomains which can be interpreted as the stiffness matrix resulting from an h-version finite element discretization of a degenerate operator. For solving the corresponding systems of finite element equations a multigrid algorithm with a special line smoother is used. We prove that the convergence rate of the multigrid method is independent of the discretization parameter. The proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples.

Eigen‐frequencies in thin elastic 3‐D domains and Reissner–Mindlin plate models

AbstractThe eigen‐frequencies of elastic three‐dimensional thin plates are addressed and compared to the eigen‐frequencies of two‐dimensional Reissner–Mindlin plate models obtained by dimension reduction. The qualitative mathematical analysis is supported by quantitative numerical data obtained by the p‐version finite element method.The mathematical analysis establishes an asymptotic expansion for the eigen‐frequencies in power series of the thickness parameter. Such results are new for orthotropic materials and for the Reissner–Mindlin model. The 3‐D and R–M asymptotics have a common first term but differ in their second terms.Numerical experiments for clamped plates show that for isotropic materials and relatively thin plates the Reissner–Mindlin eigen‐frequencies provide a good approximation to the three‐dimensional eigen‐frequencies. However, for some anisotropic materials this is no longer the case, and relative errors of the order of 30 per cent are obtained even for relatively thin plates. Moreover, we showed that no shear correction factor is known to be optimal in the sense that it provides the best approximation of the R–M eigen‐frequencies to their 3‐D counterparts uniformly (for all relevant thicknesses range). Copyright © 2002 John Wiley & Sons, Ltd.