Mass Matrix Formulation Research Articles

A Unified Meshfree Path to Arbitrary Order Hermite Finite Elements for Euler-Bernoulli Beams

The fourth-order governing equation of Euler-Bernoulli beams necessitates the employment of C1 Hermite shape functions in finite element analysis. However, unlike the C0 finite element shape functions that can be easily formulated as Lagrangian polynomials in a unified manner, the Hermite shape functions of Euler-Bernoulli beam elements are usually constructed case by case. In this work, a simple and unified meshfree path is proposed to develop arbitrary order Hermite shape functions of Euler-Bernoulli beam elements. This approach is realized by proving that the Hermite reproducing kernel meshfree shape functions degenerate to the interpolatory Hermite finite element shape functions under a proper choice of support sizes. The proposed approach provides an explicit formalism for Hermite finite element shape functions, which enables an easy construction of arbitrary order Hermite beam elements. Accordingly, in addition to the conventional cubic Hermite beam elements, explicit shape functions, mass and stiffness matrices for quintic and septic Hermite beam elements are presented. Meanwhile, the frequency errors and general higher order mass matrix formulation for these Hermite beam elements are elaborated as well. These theoretical results are consistently demonstrated by numerical examples with respect to both static and free vibration analysis.

Template mass and stiffness for vibration analysis and damage detection of plane structures

This paper investigates he vibration properties of template plane elements in both themes of optimum mass matrix formulation and damage identification. To find the optimum mass matrices for the elements, the classical and unconventional mass formulations for the plane problems are explored. The optimum mass is obtained by the template approach, along with an optimization procedure, based on the perturbation theory and sensitivity analysis. The efficiencies of the proposed optimum mass are investigated through numerical examples for some plane structures. Those pairs of the optimum mass and stiffness, which yield more accurate results in the vibration analysis, are utilized for damage identification. A novel damage localization index based on the in-plane transverse vibrations is proposed and applied successfully in a new two-stage damage identification procedure for planar structures. By using Test Analysis-Model correlation (TAM) and the element damage equations; authors' algorithm overcomes the challenges of limitations in measuring all the degrees of freedoms. Furthermore, the proposed algorithm has the potential to diagnose the false alarms and prevents the ill-condition problem in damage identification. The abilities of the suggested method are demonstrated through 11 different damage scenarios and the presence of noise.

A Mid-Node Mass Lumping Scheme for Accurate Structural Vibration Analysis with Serendipity Finite Elements

A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.

A Meshfree Higher Order Mass Matrix Formulation for Structural Vibration Analysis

An accurate meshfree formulation with higher order mass matrix is proposed for the structural vibration analysis with particular reference to the 1D rod and 2D membrane problems. Unlike the finite element analysis with an explicit mass matrix, the mass matrix of Galerkin meshfree formulation usually does not have an explicit expression due to the rational nature of meshfree shape functions. In order to develop a meshfree higher order mass matrix, a frequency error measure is derived by using the entries of general symmetric stiffness and mass matrices. The frequency error is then expressed as a series expansion of the nodal distance, in which the coefficients of each term are related to the meshfree stiffness and mass matrices. It is theoretically proved that the constant coefficient in the frequency error vanishes identically provided with the linear completeness condition, which does not rely on any specific form of the shape functions. Furthermore, a meshfree higher order mass matrix is developed through a linear combination of the consistent and lumped mass matrices, in which the optimal mass combination coefficient is attained via eliminating the lower order error terms. In particular, the proposed higher order mass matrix with Galerkin meshfree formulation achieves a fourth-order accuracy when the moving least squares or reproducing kernel (RK) meshfree approximation with linear basis function is employed; nonetheless, the conventional meshfree method only gives a second-order accuracy for the frequency computation. In the multidimensional formulation, the optimal mass combination coefficient is a function of the wave propagation angle so that the proposed accurate meshfree method is applicable to the computation of frequencies associated with any wave propagation direction. The superconvergence of the proposed meshfree higher order mass matrix formulation is validated via numerical examples.

ACCURATE COMPUTATION ON DYNAMIC SIFS USING IMPROVED XFEM

Compared to the standard extended finite element method (XFEM), the improved XFEM overcomes, in theory, the linear dependence and the ill-conditioning issues, and improves in magnitude of orders the efficiency of linear system solve. In particular, in dynamic crack propagation problems, thanks to the exclusion of the extra dofs on crack tip enriched nodes, the new method eliminates the issue of energy inconsistency or the inconservitive energy transfer caused by dof dynamics, and provides more accurate dynamic stress intensity factors (DSIFs) with much less numerical oscillation. To the best of our knowledge, numerical solution on DSIFs for crack propagation under dynamic loading remains engineeringly unsatisfied. In this paper, the extra-dof free XFEM is extended to implicitly dynamic crack propagation problems-still a remaining difficulty for the current XFEMs. The implicit Newmark algorithm is used for time discretization. A dynamic interaction integral method is employed for DSIFs for both stationary and moving cracks under dynamic loading. Compared with the interaction integral method for static cracks, the dynamic method considers the effects of crack growth speed and inertia terms. The paper investigates in detail the influences of element size, mass matrix formulation, time step size, crack tip enriched zone size, inertia term, crack growth speed, and J-domain mesh/cell sizes of the interaction integral. Numerical tests show satisified accuracy and efficiency of the new method for dynamic crack problems. In particular on the challenging benchmark problem Mode I semi-infinite stationary and then moving crack, the new improved XFEM provides the best DSIFs up to the time of publication.

Dynamic analysis of laminated composite skew plates with cut-out

The aim of the present paper deals with free vibration analysis of laminated composite skew plates with single and multiple cut-outs. For complete understanding of the dynamic behavior of laminated skew plates with cut-out a numerical analysis has been carried out by developing a computer code in FOTRAN. Special attention is drawn on the formulation of mass matrix by considering effect of rotary inertia. The results obtained by the finite element formulation using nine noded isoparametric plate bending element are validated by comparing the results from relevant published literature. Few new results on laminated skew plates with cut-out have been presented.

Free vibration analysis of laminated composite skew plates with cut-out

The present paper deals with free vibration analysis of laminated composite skew plates with and without cut-outs. The experimental investigation along with numerical simulation has been presented in this paper for complete understanding of the dynamic behaviour of laminated skew plates with cut-out. Glass fibre-reinforced laminated composite plates have been prepared by resin infusion process using vacuum bagging technique in the laboratory for experimental analysis. The experimental studies have been carried out on skew composite plates with varying size of cut-out placed at the centre. The numerical analysis has been carried out by developing a computer code in MATLAB. Special attention is drawn on the formulation of mass matrix by considering effect of rotary inertia. The results obtained by the finite element formulation using nine-noded isoparametric plate-bending elements are validated by comparing the results from relevant published literature. The numerical and experimental data are then compared for experimental verification of present investigation. The consistency of mode shapes between experimental and numerical investigations is checked by using modal assurance criteria.

A quadrature-based superconvergent isogeometric frequency analysis with macro-integration cells and quadratic splines

A quadrature-based superconvergent isogeometric method is presented for efficient frequency analysis with macro-integration cells and quadratic splines. The present development particularly refers to the isogeometric analysis of scalar wave equations. The single element, two-element and three-element macro-integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy, in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix. The 1D quadrature rules with various macro-integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix. Consequently, 3-point, 5-point and 7-point quadrature rules with identical precision are established for the single element, two-element and three-element macro-integration cells, respectively. By construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1D case. The 2D and 3D superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1D integration algorithms. It is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem, which needs special treatments for the multidimensional higher order mass matrix formulation. The proposed approach is featured by its simplicity for numerical implementation and efficiency using macro-integration cells. Numerical examples confirm the efficacy of the present methodology.

A unified quadrature-based superconvergent finite element formulation for eigenvalue computation of wave equations

A simple and unified finite element formulation is presented for superconvergent eigenvalue computation of wave equations ranging from 1D to 3D. In this framework, a general method based upon the so called $$\alpha $$ź mass matrix formulation is first proposed to effectively construct 1D higher order mass matrices for arbitrary order elements. The finite elements discussed herein refer to the Lagrangian type of Lobatto elements that take the Lobatto points as nodes. Subsequently a set of quadrature rules that exactly integrate the 1D higher order mass matrices are rationally derived, which are termed as the superconvergent quadrature rules. More importantly, in 2D and 3D cases, it is found that the employment of these quadrature rules via tensor product simultaneously for the mass and stiffness matrix integrations of Lobatto elements produces a unified superconvergent formulation for the eigenvalue or frequency computation without wave propagation direction dependence, which usually is a critical issue for the multidimensional higher order mass matrix formulation. Consequently the proposed approach is capable of computing arbitrary frequencies in a superconvergent fashion. Meanwhile, numerical implementation of the proposed method for multidimensional problems is trivial. The effectiveness of the proposed methodology is systematically demonstrated by a series of numerical examples. Numerical results revealed that a superconvergence with $$2(p+1)\hbox {th}$$2(p+1)th order of frequency accuracy is achieved by the present unified formulation for the $$p\hbox {th}$$pth order Lobatto element.

Modal analysis of laminates by a mixed assumed-strain finite element model

Fibre-reinforced plates and shells are finding an increasing interest in engineering applications; in most cases dynamic phenomena need to be taken into account. Consequently, effective and robust computational tools are sought in order to provide reliable results for the analysis of such structural models. In this paper the mixed assumed-strain laminated plate element, previously used for static analyses, has been extended to the dynamic realm. This model is derived within the framework of the so-called First-order Shear Deformation Theory (FSDT). What is peculiar in this assumed-strain finite element is that in-plane strain components are modeled directly; the corresponding stress components are deduced via constitutive law. By enforcing the equilibrium equations for each lamina, and taking continuity requirements into account, the out-of-plane shear stresses are computed and, finally, constitutive law provides the corresponding strains. The resulting global strain field depends only on a fixed number of parameters, regardless of the total number of layers. Since the proposed element is not locking-prone, even in the thin plate limit, and provides an accurate description of inter-laminar stresses, an extension to the dynamic range seems to be particularly attractive. The same kinematic assumptions will lead to the formulation of a consistent mass matrix. The element, developed in this way, has been extensively tested for several symmetric lamination sequences; comparison with available analytical solutions and with numerical results obtained by refined 3-D models are also presented.

An exact solution to compute the band gap in phononic crystals

In this paper, an alpha finite element method (αFEM) is formulated to study the propagation of elastic waves in periodic crystals. Using the simple triangular elements, the αFEM with optimal α value can always provide the exact solutions in the prediction of the band gap of phononic crystals. In addition, the upper and lower solutions are always observed in the computation of the band gap of phononic crystals with the adjustment of α value. The performance of αFEM is compared with standard finite element method (FEM) with different types of mass matrix formulation. The numerical examples have strongly demonstrated that the αFEM has given more accuracy prediction in the analysis of phononic crystals compared with standard FEM.

A superconvergent isogeometric formulation for eigenvalue computation of three dimensional wave equation

A superconvergent isogeometric formulation is presented to compute the eigenvalues for three dimensional wave equation. This three dimensional superconvergent isogeometric formulation is characterized by a higher order mass matrix formulation with particular reference to the quadratic basis functions. The three dimensional higher order mass matrix is built upon an optimal combination of the reduced bandwidth mass matrix and the consistent mass matrix. The frequency error associated with the isogeometric discretization of three dimensional wave equation is derived in detail. In particular, the optimal mass combination parameter for higher order mass matrix is devised as a function of the two spatial wave propagation angles, which enables that arbitrary frequency corresponding to a given wave propagation direction can be computed in a superconvergent way. Two extra orders of accuracy, i.e., 6th order of accuracy, are attained by the proposed higher order mass matrix than the consistent mass matrix for the frequency computation of three dimensional wave equation. The dispersion property of the present three dimensional higher order mass matrix formulation is examined as well. The accuracy of the proposed three dimensional superconvergent isogeometric formulation is testified by several numerical examples.

SPEEDING BEYOND FDTD, PERFORATED FINITE ELEMENT TIME DOMAIN METHOD FOR 3D ELECTROMAGNETICS

A three-dimensional (3D) time domain approach can be particularly valuable for the analysis of many different types of practical structures. In this regard, the finite difference time domain (FDTD) method is a popular technique, being used successfully to analyze the electromagnetic properties of many structures, including a range of optical or photonic devices. This FDTD method offers several major advantages: a minimum level of calculation is required for each of the cells into which the structure is divided, as well as data parallelism and explicit and easy implementation: however, the use of a cuboid grid makes the method very resource intensive for large simulations, especially those in 3D. Although the finite element (FE) approach is superior for the discretization of two-dimensional (2D) and 3D structures, most of the FE-based time domain approaches reported so far suffer from limitations due to the implicit or iterative form or the mass matrix formulation, for example. This paper presents an FE based time domain technique for 3D structures which uses a unique perforated mesh system. It calculated the numerical dispersion characteristics for the FDTD and the proposed method and compared. This paper finally discusses how to utilize the improved numerical dispersion characteristics of the proposed method to increase the simulation speed beyond the FDTD 3D method by using Intel micro-processors.

Spatial force-based non-prismatic beam element for static and dynamic analyses of circular flexure hinges in compliant mechanisms

Abstract This paper presents a spatial force-based non-prismatic beam element modeling approach for circular flexure hinges, which is capable of accurate and efficient modeling of both static and dynamic characteristics of flexure-based compliant mechanisms. The spatial force-based non-prismatic beam element is an improvement and extension of the existing beam elements by considering shear and torsional effects of circular flexure hinge. The new consistent mass matrix formulation was derived using the Unit Load method. A parallel-guided compliant mechanism was taken as an example to validate the efficiency and accuracy of the proposed approach through static and modal analyses with fewer non-prismatic beam elements, and the results are well agreed with ANSYS simulation results with massive 3D solid or 2D shell elements. The developed approach is also available to model arbitrarily shaped flexure hinges in compliant mechanisms.

A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parameter vector

A mass-matrix formulation of the fluctuation splitting schemes for solving compressible, unsteady flows is proposed. This formulation is consistent with the conservative linearisation based on parameter vector and allows to extend to unsteady flows the ‘invariance under similarity transformations’ property that had been shown to hold for the steady version of the schemes. Second-order time accuracy is achieved using a Petrov–Galerkin finite element interpretation of the fluctuation splitting schemes. The approach may however be readily applicable to all other time-accurate fluctuation splitting formulations that have been so far proposed in the literature. Applications of the proposed methodology to two- and three-dimensional, inviscid and viscous compressible flows are reported and discussed in the paper.

A study of transient response of initially stressed composite sandwich folded plates

In this paper a C° four node flat facetted shell element with drilling rotational degrees of freedom based on a first order shear deformation shell theory is developed to study the transient response of initially stressed composite sandwich folded plates. The new shell element contains three translations, two rotations of the normals about the shell mid-plane, and one drilling rotational degree of freedom per node. A consistent mass matrix formulation is employed to evaluate the total kinetic energy of the system. A generic validation study is carried out to evaluate the performance of the present finite element formulation. New results are presented for the transient analysis of initially stressed composite sandwich folded plate structures.

Dynamic response of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid

In this study, a method of analysis is presented for investigating the dynamic response behavior of moderately thick plates (Mindlin plates) resting on arbitrarily orthotropic two parameter foundation and partially in contact with a quiescent fluid on its other side. A mixed-type finite element formulation is derived for the Mindlin plate–arbitrarily orthotropic Pasternak foundation interaction by applying the Gâteaux differential. A four nodded isoparametric C0 class element is adopted and at each node eight degrees of freedom are assigned. The rotary inertia effect is taken into account through the consistent mass matrix formulation. For calculation of the fluid–structure interaction effects, the boundary element method is adopted. The fluid is assumed to be ideal, i.e., inviscid, incompressible, and its motion is irrotational. The infinite frequency limit condition is applied on the fluid's free surface using a modified fundamental solution implicitly satisfying the appropriate free surface boundary condition. It is assumed that the plate–elastic orthotropic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. The fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To assess the influence of the arbitrarily orthotropic Pasternak foundation and fluid on the dynamic response behavior of moderately thick plates, the natural frequencies and associated mode shapes are presented for rectangular and circular plates with various boundary conditions.

Toward a higher order unsteady finite volume solver based on reproducing kernel methods

During the last decades, research efforts are headed to develop high order methods on CFD and CAA to reach most industrial applications (complex geometries) which need, in most cases, unstructured grids. Today, higher-order methods dealing with unstructured grids remain in infancy state and they are still far from the maturity of structured grids-based methods when solving unsteady cases. From this point of view, the development of higher order methods for unstructured grids become indispensable. The finite volume method seems to be a good candidate, but unfortunately it is difficult to achieve space flux derivation schemes with very high order of accuracy for unsteady cases. In this paper we propose, a high order finite volume method based on Moving Least Squares approximations for unstructured grids that is able to reach an arbitrary order of accuracy on unsteady cases. In order to ensure high orders of accuracy, two strategies were explored independently: (1) a zero-mean variables reconstruction to enforce the mean order at the time derivative and (2) a pseudo mass matrix formulation to preserve the residuals order.

Triangular Spectral Element simulation of two-dimensional elastic wave propagation using unstructured triangular grids

SUMMARY A Triangular Spectral Element Method (TSEM) is presented to simulate elastic wave propagation using an unstructured triangulation of the physical domain. TSEM makes use of a variational formulation of elastodynamics based on unstructured straight-sided triangles that allow enhanced flexibility when dealing with complex geometries and velocity structures. The displacement field is expanded into a high-order polynomial spectral approximation over each triangular subdomain. Continuity between the subdomains of the triangulation is enforced using a multidimensional Lagrangian interpolation built on a set of Fekete points of the triangle. High-order accuracy is achieved by resorting to an analytical computation of the associated internal product and bilinear forms leading to a non-diagonal mass matrix formulation. Therefore, the time stepping involves the solution of a sparse linear algebraic system even in the explicit case. In this paper the accuracy and the geometrical flexibility of the TSEM is explored. Comparison with classical spectral elements on quadrangular grids shows similar results in terms of accuracy and stability even for long simulations. Surface and interface waves are shown to be accurately modelled even in the case of complex topography with the TSEM. Numerical results are presented for 2-D canonical examples as well as more specific problems, such as 2-D elastic wave scattering by a cylinder embedded in an homogeneous half-space. They all illustrate the enhanced geometrical flexibility of the TSEM. This clearly suggests the need of further investigations in computational seismology specifically targeted towards efficient implementations of the TSEM both in the time and the frequency domain.

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

The existing statically exact beam finite element (FE) based on the first order shear deformation theory (FSDT) is used to study the geometric nonlinear effects on static and dynamic responses in isotropic, composite and functionally graded material (FGM) beams. In this beam element, the exact solution of the static part of the governing differential equations is used to construct the interpolating polynomials for the element stiffness and mass matrix formulation. The rotary inertia is also taken into account while formulating the mass matrix. Consequently, the stiffness matrix is statically exact and the mass matrix is much more accurate compared to any other existing FEs. These two aspects make the element devoid of shear locking and an efficient instrument for analysing wave propagation problems, wherein refined mesh is essential. A total Lagrangian formulation of the linear beam element is employed for large displacement and large rotation analysis. The superconvergent property of the beam element is shown by comparing the convergence rate of this beam element to the reduced integrated FSDT beam element formed using the linear shape functions. Numerical examples deal with the static and wave propagation problems to highlight the nonlinear effects. In the static case, both Von-Karman strain tensor and Green’s strain tensor is used, whereas, for the wave propagation studies only the Von-Karman strains are used. Power-law variations of the material property distribution is used for the FGM beam. Both high and low frequency pulse loading is employed to bring out the effect of nonlinearity on the transient response.