Mass Matrix Formulation Research Articles

Transient response of composite sandwich plates

The transient response of orthotropic, layered composite sandwich plates is investigated by using two new C 0 four and nine node finite element formulations of a refined form of Reddy's higher-order theory. This refined third order theory accounts for parabolic variation of the transverse shear stresses, and requires no shear correction factors. The assumed strain approach is employed to model both thin and thick plates without any major defects like shear locking and parasitic spurious zero energy modes. A consistent mass matrix formulation is adopted. The Newmark direct integration scheme is used to solve the governing equilibrium equations. The parametric effects of plate aspect ratio, length to thickness ratio, boundary conditions and lamination scheme on the transient response are investigated. The present results are in very close agreement with earlier published results in the literature and can serve as a benchmark for future investigators.

Free vibration analysis of Reissner plates by mixed finite element

In this study, free vibration analysis of Reissner plates on Pasternak foundation is carried out by mixed finite element method based on the Gateaux differential. New boundary conditions are established for plates on Pasternak foundation. This method is developed and applied to numerous problems by Akoz and his co-workers. In dynamic analysis, the problem reduces to the solution of a standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and bending and torsional moments, transverse shear forces, rotations and displacements are the basic unknowns. The element performance is assessed by comparison with numerical examples known from literature. Validity limits of Kirchhoff plate theory is tested by dynamic analysis. Shear locking effects are tested as far as h/2a =10 -6 and it is observed that REC32 is free from shear locking.

A fractional-step Padé–Galerkin model for dam-break flow simulation

A Padé–Galerkin solution for dam-break simulations is proposed. The fractional time-step integration of Saint-Venant equations is obtained by a three-stage Padé implicit approximation, whereas the spatial discretization is obtained by the standard Galerkin finite-element method. The model, unconditionally stable, becomes dissipative with the formulation of the unknown quantity mass matrix in lumped form. Results, in terms of water profiles and depth hydrographs, following dam-break phenomena, are investigated in one- and two-dimensional problems.

Particle-in-Cell Simulation of Electrical Gas Discharges

A fluid particle-in-cell (PIC) model is proposed for the numerical solution of the continuity equation of electrons and ions in transient electrical gas discharges. The reactions occurring in a gaseous discharge, such as ionization of neutral molecules, electron attachment, and recombination between electrons and ions, are implemented through the variation of the mass of the computational particles used in the simulation. Two different forms of interpolation of the gain/loss rates from the grid to the computational particles are suggested, depending on the reaction type. The PIC model is first applied to the problem of an idealized electron avalanche in a non-attaching gas. This problem possesses an analytical solution where the electron density grows exponentially in time as it propagates, but keeps the square-wave form of the initial electron distribution. This problem is used to validate the optimum interpolation of the gain/loss rate and to analyze the effect of the mass matrix formulation of the PIC model. Then, a more realistic model is applied to simulate the propagation of a Trichel pulse between a sphere and a plate. In this case, the continuity equation for electrons and positive and negative ions, coupled to the Poisson equation, has been solved. This second test has proved the ability of the present numerical method to deal with those discharges dominated by the space charge effect. The results of the PIC simulation are compared with those obtained from the application of a flux-corrected transport method.

Discretization errors associated with reproducing kernel methods: one-dimensional domains

The reproducing kernel particle method (RKPM) is a discretization technique for partial differential equations that uses the method of weighted residuals, classical reproducing kernel theory and modified kernels to produce either “mesh-free” or “mesh-full” methods. Although RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quantified. In order to address the numerical performance of RKPM, von Neumann analysis is performed for semi-discretizations of three model one-dimensional PDEs. The von Neumann analyses results are used to examine the global and asymptotic behavior of the semi-discretizations. The model PDEs considered for this analysis include the parabolic and hyperbolic (first- and second-order wave) equations. Numerical diffusivity for the former and phase speed for the latter are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero. Group speed is also presented for the hyperbolic problems. Excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter. In contrast, the row-sum lumped mass matrix formulation severely degraded performance. The asymptotic analysis indicates that very good rates of convergence (O( x 6)–O( x 8)) are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter and quadrature rule.

Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

It is well known that isotropic, nondispersive continuous hyperbolic problems become dispersive and anisotropic upon discretization. The purpose of this paper is to conduct a dispersion analysis of the nondissipative numerical approximations to plane wave motions in isotropic elastic solids. The discrete formulations considered are: an explicit, second-order accurate finite difference scheme, a consistent mass matrix formulation with linear quadrilateral elements and the corresponding lumped mass matrix formulation. Dispersion relation is derived for each of these formulations. In the context of the finite difference scheme, expressions for group velocity for both the shear and longitudinal waves are derived and the effect of using meshes of unequal size in x and y directions is studied. Results from numerical experiments confirming the predictions of analysis are also presented.

A mass-matrix formulation for three-dimensional dynamic fracture mechanics

In this paper, the dual reciprocity boundary element method is developed for the analysis of three-dimensional elastodynamic fracture mechanics mixed-mode problems. The dual boundary element method is used to calculate the unknowns of boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method. Dynamic stress intensity factors are determined by the crack opening displacement (COD) in the time domain. Several numerical examples are presented to demonstrate the accuracy of the method.

Flexibility-based linear dynamic analysis of complex structures with curved-3D members

A flexibility-based formulation of a new mass matrix for the dynamic analysis of spatial frames consisting of curved elements with variable cross-sections is presented. The main characteristic of such formulations is the exact equilibrium of forces at any interior point, with no additional hypotheses about the distribution of displacements, strains or stresses. Accordingly, the derived element mass matrix takes into account the exact stiffness and mass distribution throughout each element. In validation tests, results obtained with this method are compared with those obtained by other numerical or analytical formulations, showing the accuracy of the proposed method. The comparison of experimental results for a multispan arch bridge subjected to a dynamic load with those achieved by means of the proposed method are finally included to illustrate its efficiency in the treatment of complex structures. © 1998 John Wiley & Sons, Ltd.

Hermitian lumped mass matrix formulation for flexural wave propagation

Hermitian finite difference operators are employed to formulate a new diagonal mass matrix for solving flexural wave propagation problems. The dynamic equilibrium for moments is considered. The consistency of the formulation is evident because these operators explicitly present second-order convergence. Numerical results indicate that the velocity dispersion is very close to the consistent mass matrix. © 1998 John Wiley & Sons, Ltd.

Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements

A quantitative study of dispersion in Chebyshev spectral finite element solutions to the one- and two-dimensional scalar wave equations is presented. The spectral elements employ central time differencing and three mass matrix treatments: consistent, row-summed and diagonal-scaled. The one-dimensional axisymmetric wave equation is also formulated and solved with Chebyshev spectral elements. The computationally efficient, row-summed mass matrix formulation is shown to exhibit minimal dispersion. One- and two-dimensional examples highlight the effects of dispersion.

FREE VIBRATION ANALYSIS OF KIRCHHOFF PLATES RESTING ON ELASTIC FOUNDATION BY MIXED FINITE ELEMENT FORMULATION BASED ON GÂTEAUX DIFFERENTIAL

The main objective of the present work is to give the systematic way for derivation of Kirchhoff plate-elastic foundation interaction by mixed-type formulation using the Gâteaux differential instead of well-known variational principles of Hellinger–Reissner and Hu–Washizu. Foundation is a Pasternak foundation, and as a special case if shear layer is neglected, it converges to Winkler foundation in the formulation. Uniform variation of the thickness of the plate is also included into the mixed finite element formulation of the plate element PLTVE4 which is an isoparametric C0 class conforming element discretization. In the dynamic analysis, the problem reduces to solution of the standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and at each node transverse displacement two bending and one torsional moment is the basic unknowns. Proper geometric and dynamic boundary conditions corresponding to the plate and the foundation is given by the functional. Performance of the element for bending and free vibration analysis is verified with a good accuracy on the numerical examples and analytical solutions present in the literature. © 1997 by John Wiley & Sons, Ltd.

Mass matrix formulation of the FLIP particle-in-cell method

A refinement of FLIP [ Brackbill and Ruppel, J. Comput. Phys. 65, 314 (1986) ] is described which uses a mass matrix formulation to achieve greater accuracy and less numerical diffusion over the previous version. Without the refinement, there is a significant dissipation of energy in modeling elastic vibrations of a solid. Moreover, in modeling an initial flow discontinuity there are sub-grid-scale oscillations in the particle velocity field in the neighborhood of the discontinuity. These difficulties are eliminated using the mass matrix. In addition, the mass matrix formulation conserves kinetic energy, linear and angular momentum, and is Galilean invariant.

New concepts for finite-element mass matrix formulations

Improved finite-element formulations are presented for dynamic analysis. The improvements are based on modifications of the finite-element mass-matrix and are of high importance, since no additional computational effort is required, compared with the necessary effort for the standard, consistent mass-matrix formulation. Formulations are presented for the one-dimensional bar, two-dimensional membrane, and the pure bending beam element. Free vibration analysis and forced vibration results are presented employing new mass matrices. In general, the frequency errors and stress errors are at least three times smaller than the corresponding consistent or lumped-mass formulation errors. Accurate model data can be achieved for all the modes of relatively coarse meshes.

Multilayered Finite Element Formulation for Vibration and Stability Analysis of Plates

A finite element formulation for multilayered and thick plates, based on a multilayered theory, which was previously applied to a static analysis is extended to include vibration and stability analyses. First, a mass matrix formulation is presented which is consistent with the stiffness coefficient derivation, namely a virtual work expression is obtained which is then evaluated through numerical differentiation. This is followed by a more general discussion of a geometric stiffness derivation which is seen within the concept of linear elastic stability. The geometric stiffness itself, in this context is the different between the linear stiffness and the tangent stiffness where the tangent stiffness is evaluated, within the general nonlinear theory, usually at a unit load stress state. In both cases vibration and stability analysis, the usual generalized eigenvalue problem follows. In some applications for layered beams and plates the vibration and stability analysis presented is compared with exact solutions.

Structural design sensitivity analysis with generalized global stiffness and mass matrices

First- and second-order derivatives of measures of structural response with respect to design variables are calculated using the generalized global stiffness and mass matrix formulation of structural equations. The adjoint variable method is extended for first-order design sensitivity calculation and a mixed direct differentiationadjoint variable method is employed to calculate second derivatives of static structural response measures. The variational or virtual work form of the structural equations is shown to be well suited for design sensitivity analysis with generalized global stiffness and mass matrices, avoiding the requirement for explicit reduction of the matrices. The method is illustrated using a simple structure with a multipoint constraint.

Frequency analysis of thick orthotropic plates on elastic foundation using a high precision triangular plate bending element

AbstractA high precision triangular thick orthotropic plate bending element on an elastic foundation is developed for the free vibration analysis of thick plates on elastic foundation. The element has three nodes with twelve degrees‐of‐freedom per node, and takes into account the shear deformation and rotatory inertia. The accuracy of the element is established by comparison of the natural frequencies of certain thick and thin plates, determined from a consistent mass matrix formulation, with available results.

Lumped mass matrices with non−zero inertia for general shell and axisymmetric shell elements

AbstractA diagonal lumped mas formulation with non−zero inertia terms is presented for Ahmad's general shell element. The effect of co−ordinate transformation of the mas matrix is demonstrated. Due to arbitray co−ordinate transformation on nodal variables, the diagonal lumped mas matrix becomes a banded matrix of half bandwidth three. It is shown that with some approximations this matrix can be made diagnoal without effecting the results appreciably. Numerical examples are presented to illustrate the accuracy of the formulation. Similar lumped mass matrix formulation is also given for axisymmetric shell elements.