			Home About us Contact

Mass Matrix (mass + matrix)

Distribution by Scientific Domains

Engineering	83%
3 Other Domains	17%

Kinds of Mass Matrix

consistent mass matrix

lumped mass matrix

Selected Abstracts

Constraints from F and D supersymmetry breaking in general supergravity theories

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 7-9 2008
M. Gomez-Reino
Abstract We study the conditions under which a generic supergravity model involving chiral and vector multiplets can admit vacua with spontaneously broken supersymmetry and realistic cosmological constant. We find that the existence of such viable vacua implies some constraints involving the curvature tensor of the scalar geometry and the charge and mass matrices of the vector fields, and also that the vector of F and D auxiliary fields defining the Goldstino direction is constrained to lie within a certain domain. We illustrate the relevance of these results through some examples and also discuss the implications of our general results on the dynamics of moduli fields in string models. This contribution is based on [1,3]. [source]

Nonlinear transient dynamic analysis by explicit finite element with iterative consistent mass matrix

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2009
Shen Rong Wu
Abstract Various mass matrices in the explicit finite element analyses of nonlinear transient dynamic problems are investigated. The matrices are obtained as a linear combination of lumped and consistent mass matrices. An iterative procedure to calculate the inverse of the consistent and the mixed mass matrices in the framework of explicit finite element method is presented. The convergence of the iterative procedure is proved. The inverse of the consistent and mixed mass matrices is approximated by the iteration and is used to compare the results from the lumped mass matrix. For the impact of a structural component and a vehicle, some difference in the results by using coarse mesh is observed. For the component using fine mesh, no significant difference is found. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A rational approach to mass matrix diagonalization in two-dimensional elastodynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004
E. A. Paraskevopoulos
Abstract A variationally consistent methodology is presented, which yields diagonal mass matrices in two-dimensional elastodynamic problems. The proposed approach avoids ad hoc procedures and applies to arbitrary quadrilateral and triangular finite elements. As a starting point, a modified variational principle in elastodynamics is used. The time derivatives of displacements, the velocities, and the momentum type variables are assumed as independent variables and are approximated using piecewise linear or constant functions and combinations of piecewise constant polynomials and Dirac distributions. It is proved that the proposed methodology ensures consistency and stability. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Free vibration analysis of arches using curved beam elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2003
Jong-Shyong Wu
Abstract The natural frequencies and mode shapes for the radial (in-plane) bending vibrations of the uniform circular arches were investigated by means of the finite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two-node six-degree-of-freedom arch element, the elemental stiffness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia effect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall stiffness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the finite element method based on the conventional straight beam element and good agreements were achieved. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Spectral-element simulations of wave propagation in porous media

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2008
Christina Morency
SUMMARY We present a derivation of the equations describing wave propagation in porous media based upon an averaging technique which accommodates the transition from the microscopic to the macroscopic scale. We demonstrate that the governing macroscopic equations determined by Biot remain valid for media with gradients in porosity. In such media, the well-known expression for the change in porosity, or the change in the fluid content of the pores, acquires two extra terms involving the porosity gradient. One fundamental result of Biot's theory is the prediction of a second compressional wave, often referred to as ,type II' or ,Biot's slow compressional wave', in addition to the classical fast compressional and shear waves. We present a numerical implementation of the Biot equations for 2-D problems based upon the spectral-element method (SEM) that clearly illustrates the existence of these three types of waves as well as their interactions at discontinuities. As in the elastic and acoustic cases, poroelastic wave propagation based upon the SEM involves a diagonal mass matrix, which leads to explicit time integration schemes that are well suited to simulations on parallel computers. Effects associated with physical dispersion and attenuation and frequency-dependent viscous resistance are accommodated based upon a memory variable approach. We perform various benchmarks involving poroelastic wave propagation and acoustic,poroelastic and poroelastic,poroelastic discontinuities, and we discuss the boundary conditions used to deal with these discontinuities based upon domain decomposition. We show potential applications of the method related to wave propagation in compacted sediments, as one encounters in the petroleum industry, and to detect the seismic signature of buried landmines and unexploded ordnance. [source]

Spectral-element simulations of global seismic wave propagation,II.

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2002
Three-dimensional models, oceans, rotation, self-gravitation
Summary We simulate global seismic wave propagation based upon a spectral-element method. We include the full complexity of 3-D Earth models, i.e. lateral variations in compressional-wave velocity, shear-wave velocity and density, a 3-D crustal model, ellipticity, as well as topography and bathymetry. We also include the effects of the oceans, rotation and self-gravitation in the context of the Cowling approximation. For the oceans we introduce a formulation based upon an equivalent load in which the oceans do not need to be meshed explicitly. Some of these effects, which are often considered negligible in global seismology, can in fact play a significant role for certain source,receiver configurations. Anisotropy and attenuation, which were introduced and validated in a previous paper, are also incorporated in this study. The complex phenomena that are taken into account are introduced in such a way that we preserve the main advantages of the spectral-element method, which are an exactly diagonal mass matrix and very high computational efficiency on parallel computers. For self-gravitation and the oceans we benchmark spectral-element synthetic seismograms against normal-mode synthetics for the spherically symmetric reference model PREM. The two methods are in excellent agreement for all body- and surface-wave arrivals with periods greater than about 20 s in the case of self-gravitation and 25 s in the case of the oceans. At long periods the effect of gravity on multiorbit surface waves up to R4 is correctly reproduced. We subsequently present results of simulations for two real earthquakes in fully 3-D Earth models for which the fit to the data is significantly improved compared with classical normal-mode calculations based upon PREM. For example, we show that for trans-Pacific paths the Rayleigh wave can arrive more than a minute earlier than in PREM, and that the Love wave is much shorter in duration. [source]

Nonlinear transient dynamic analysis by explicit finite element with iterative consistent mass matrix

Kinematic and dynamic analysis of open-loop mechanical systems using non-linear recursive formulation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2006
Yunn-Lin Hwang
Abstract In this paper, a non-linear recursive formulation is developed for kinematic and dynamic analysis of open-loop mechanical systems. The non-linear equations of motion are developed for deformable links that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial co-ordinates as well as the assumed displacement field, and these time invariant quantities represent the dynamic coupling between the rigid-body modes and elastic deformations. A new recursive formulation is presented for solving equations of motion for open-loop chains consisting of interconnected rigid and deformable open-loop mechanical systems. This formulation is expressed by the recursive relationships and the generalized non-linear equations for deformable mechanical systems to obtain a large system of loosely coupled equations of motion. The main processor program consists of three main modules: constraint module, mass module and force module. The constraint module is used to numerically evaluate the relationship between the absolute and joint accelerations. The mass module is used to numerically evaluate the system mass matrix as well as the non-linear Coriolis and centrifugal forces associated with the absolute, joint and elastic co-ordinates. Simultaneously, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic co-ordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The solution techniques used in this investigation yield a much smaller operations count and can more efficiently implement in any computer. The algorithms and solutions presented in this paper are illustrated by using an industrial robotic manipulator system. The numerical results using this formulation are also presented and discussed in this paper. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Diagonalization procedure for scaled boundary finite element method in modeling semi-infinite reservoir with uniform cross-section

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2009
S. M. Li
Abstract To improve the ability of the scaled boundary finite element method (SBFEM) in the dynamic analysis of dam,reservoir interaction problems in the time domain, a diagonalization procedure was proposed, in which the SBFEM was used to model the reservoir with uniform cross-section. First, SBFEM formulations in the full matrix form in the frequency and time domains were outlined to describe the semi-infinite reservoir. No sediments and the reservoir bottom absorption were considered. Second, a generalized eigenproblem consisting of coefficient matrices of the SBFEM was constructed and analyzed to obtain corresponding eigenvalues and eigenvectors. Finally, using these eigenvalues and eigenvectors to normalize the SBFEM formulations yielded diagonal SBFEM formulations. A diagonal dynamic stiffness matrix and a diagonal dynamic mass matrix were derived. An efficient method was presented to evaluate them. In this method, no Riccati equation and Lyapunov equations needed solving and no Schur decomposition was required, which resulted in great computational costs saving. The correctness and efficiency of the diagonalization procedure were verified by numerical examples in the frequency and time domains, but the diagonalization procedure is only applicable for the SBFEM formulation whose scaling center is located at infinity. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Peter Betsch
Abstract In the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unit-length constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy,momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy,momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Efficient explicit time stepping for the eXtended Finite Element Method (X-FEM)

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2006
T. Menouillard
Abstract This paper focuses on the introduction of a lumped mass matrix for enriched elements, which enables one to use a pure explicit formulation in X-FEM applications. A proof of stability for the 1D and 2D cases is given. We show that if one uses this technique, the critical time step does not tend to zero as the support of the discontinuity reaches the boundaries of the elements. We also show that the X-FEM element's critical time step is of the same order as that of the corresponding element without extended degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Application of piece-wise linear weight functions for 2D 8-node quadrilateral element in contact problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004
Chouping Luo
Abstract The present study is a continuation of our previous work with the aim to reduce problems caused by standard higher order elements in contact problems. The difficulties can be attributed to the inherent property of the Galerkin method which gives uneven distributions of nodal forces resulting in oscillating contact pressures. The proposed remedy is use of piece-wise linear weight functions. The methods to establish stiffness and/or mass matrix for 8-node quadrilateral element in 2D are presented, i.e. the condensing and direct procedures. The energy and nodal displacement error norms are also checked to establish the convergence ratio. Interpretation of calculated contact pressures is discussed. Two new 2D 8-node quadrilateral elements, QUAD8C and QUAD8D, are derived and tested in many examples, which show their good performance in contact problems. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Free vibration analysis of arches using curved beam elements

Non-local dispersive model for wave propagation in heterogeneous media: multi-dimensional case

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002
Jacob Fish
Abstract Three non-dispersive models in multi-dimensions have been developed. The first model consists of a leading-order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non-local model, contains a fourth-order spatial derivative and thus requires C1 continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C0 continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Free vibrations of shear-flexible and compressible arches by FEM

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2001
Przemyslaw Litewka
Abstract The purpose of this paper is to analyse free vibrations of arches with influence of shear and axial forces taken into account. Arches with various depth of cross-section and various types of supports are considered. In the calculations, the curved finite element elaborated by the authors is adopted. It is the plane two-node, six-degree-of-freedom arch element with constant curvature. Its application to the static analysis yields the exact results, coinciding with the analytical ones. This feature results from the use of the exact shape functions in derivation of the element stiffness matrix. In the free vibration analysis the consistent mass matrix is used. It is obtained on the base of the same functions. Their coefficients contain the influences of shear flexibility and compressibility of the arch. The numerical results are compared with the results obtained for the simple diagonal mass matrix representing the lumped mass model. The natural frequencies are also compared with the ones for the continuous arches for which the analytically determined frequencies are known. The advantage of the paper is a thorough analysis of selected examples, where the influences of shear forces, axial forces as well as the rotary and tangential inertia on the natural frequencies are examined. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Projection and partitioned solution for two-phase flow problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2005
Andrea Comerlati
Abstract Multiphase flow through porous media is a highly nonlinear process that can be solved numerically with the aid of finite elements (FE) in space and finite differences (FD) in time. For an accurate solution much refined FE grids are generally required with the major computational effort consisting of the resolution to the nonlinearity frequently obtained with the classical Picard linearization approach. The efficiency of the repeated solution to the linear systems within each individual time step represents the key to improve the performance of a multiphase flow simulator. The present paper discusses the performance of the projection solvers (GMRES with restart, TFQMR, and BiCGSTAB) for two global schemes based on a different nodal ordering of the unknowns (ORD1 and ORD2) and a scheme (SPLIT) based on the straightforward inversion of the lumped mass matrix which allows for the preliminary elimination and substitution of the unknown saturations. It is shown that SPLIT is between two and three time faster than ORD1 and ORD2, irrespective of the solver used. Copyright © 2005 John Wiley & Sons, Ltd. [source]

A high-order mass-lumping procedure for B-spline collocation method with application to incompressible flow simulations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2003
O. Botella
Abstract This paper presents new developments of the staggered spline collocation method for cost-effective solution to the incompressible Navier,Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high-accuracy of the B-spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high-order generalization of the mass-lumping technique of the finite-element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi-consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier,Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd. [source]

An approximate projection method for incompressible flow

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2002
David E. Stevens
This paper presents an approximate projection method for incompressible flows. This method is derived from Galerkin orthogonality conditions using equal-order piecewise linear elements for both velocity and pressure, hereafter Q1Q1. By combining an approximate projection for the velocities with a variational discretization of the continuum pressure Poisson equation, one eliminates the need to filter either the velocity or pressure fields as is often needed with equal-order element formulations. This variational approach extends to multiple types of elements; examples and results for triangular and quadrilateral elements are provided. This method is related to the method of Almgren et al. (SIAM J. Sci. Comput. 2000; 22: 1139,1159) and the PISO method of Issa (J. Comput. Phys. 1985; 62: 40,65). These methods use a combination of two elliptic solves, one to reduce the divergence of the velocities and another to approximate the pressure Poisson equation. Both Q1Q1 and the method of Almgren et al. solve the second Poisson equation with a weak error tolerance to achieve more computational efficiency. A Fourier analysis of Q1Q1 shows that a consistent mass matrix has a positive effect on both accuracy and mass conservation. A numerical comparison with the widely used Q1Q0 (piecewise linear velocities, piecewise constant pressures) on a periodic test case with an analytic solution verifies this analysis. Q1Q1 is shown to have comparable accuracy as Q1Q0 and good agreement with experiment for flow over an isolated cubic obstacle and dispersion of a point source in its wake. Copyright © 2002 John Wiley & Sons, Ltd. [source]

LS-DYNA and the 8:1 differentially heated cavity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2002
Mark A. Christon
Abstract This paper presents results computed using LS-DYNA's new incompressible flow solver for a differentially heated cavity with an 8:1 aspect ratio at a slightly super-critical Rayleigh number. Three Galerkin-based solution methods are applied to the 8:1 thermal cavity on a sequence of four grids. The solution methods include an explicit time-integration algorithm and two second-order projection methods,one semi-implicit and the other fully implicit. A series of ad hoc modifications to the basic Galerkin finite element method are shown to result in degraded solution quality with the most serious effects introduced by row-sum lumping the mass matrix. The inferior accuracy of a lumped mass matrix relative to a consistent mass matrix is demonstrated with the explicit algorithm which fails to obtain a transient solution on the coarsest grid and exhibits a general trend to under-predict oscillation amplitudes. The best results are obtained with semi-implicit and fully implicit second-order projection methods where the fully implicit method is used in conjunction with a ,smart' time integrator. Copyright © 2002 John Wiley & Sons, Ltd. [source]

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2009
Zheng-Jian Bai
Abstract In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive-definite matrices with the mass matrix being diagonal and the damping and stiffness matrices being tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newton-type algorithm for the optimization problem, which improves a pre-existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method. Copyright © 2008 John Wiley & Sons, Ltd. [source]