Consistent Mass Matrices Research Articles

Dynamic analysis of crack problems in functionally graded materials using a new graded singular finite element

In this research a recently developed graded singular element is used for studying dynamic crack problems in linear elastic isotropic functionally graded materials (FGMs) with spatially varying elastic parameters. The formulation for the graded singular element is obtained by analyzing the crack tip stress field using the Westergaard stress function method. Shape functions are extracted by presenting expressions for displacements near the crack tip in terms of the complex functions with unknown coefficients. The stiffness and consistent mass matrices for the singular element are then determined based on the principle of minimum potential energy and kinetic energy of the element, respectively. The global stiffness and mass matrices are then generated by assembling matrices for the singular and adjacent regular elements by considering the displacement continuity at their common boundary. The results from the current method are compared with existing solutions to evaluate the accuracy of the technique. A set of simulations are performed using this element for analyzing dynamic crack problems in FGMs whose elastic properties vary normal to the crack line. Compared with traditional elements, the proposed element includes higher order terms in addition to the singular term, which leads to accurate description of the stress field near the crack tip. It also significantly reduces the calculation time needed in comparison with using very fine mesh of traditional elements. With the use of the singular element, no post processing technique is required for calculating fracture parameters such as stress intensity factors, energy release rate, and mode mixity.

A cubic spline layerwise time domain spectral FE for guided wave simulation in laminated composite plate structures with physically modeled active piezoelectric sensors

A novel theoretical framework is presented enabling enhanced simulation of guided waves generated by piezoelectric actuators in laminated composite plates. A new multi-field multi-physics layerwise theory is formulated for laminated plates with piezoelectric actuators and sensors which captures symmetric and anti-symmetric stress waves. Third-order Hermite splines are employed in the approximation of displacements and electric potential through the thickness. The piezoelectric actuators and sensors are physically modeled though coupled electromechanical governing equations. An explicit time-domain spectral finite element is formulated entailing displacement and electric degrees of freedom at the nodes which are further collocated with Gauss–Lobatto–Legendre integration points. The stiffness, consistent semi-diagonal mass, piezoelectric and electric permittivity matrices are calculated. Furthermore, an explicit time integration scheme is presented for the calculation of the coupled transient electromechanical response including the free sensor response. Numerical guided wave predictions of the developed multi-node time domain spectral finite elements are correlated with well established numerical tools and with experiments conducted on laminated carbon/epoxy plates with active piezoceramic sensor networks. Important effects introduced by the physical presence of the active actuator/sensor system on guided wave propagation, such as wave reflections, mode conversions and sensor signal attenuation, are successfully captured by the developed finite element and correlated and evaluated with numerical and experimental results.

Consistent matrices for steel framed structures with semi-rigid connections accounting for shear deformation and rotary inertia effects

Estimation of vibration characteristics and thus the seismic loads acted on steel framed structures are influenced by the presence of semi-rigid connections and accurate modeling of shear deformations and rotary inertia effects. This paper presents a finite element model that takes into account all these effects in order to calculate consistent stiffness and mass matrices. The formulation of the element utilizes three-fields Hu-Washizu-Barr principle, where the need for displacement shape function approximation is eliminated through the use of force-based approach. The proposed model does not require extra discretization to capture localized connection response. An accurate shear correction coefficient for I-shaped steel sections is implemented to represent shear deformation and rotary inertia along steel beams and columns. Numerical examples on single member, portal frame and multi-story steel framed structures verify the accuracy and robustness of the proposed element with and without semi-rigid connections.

B-spline based finite element method in one-dimensional discontinuous elastic wave propagation

The B-spline variant of the finite element method (FEM) is tested in one-dimensional discontinuous elastic wave propagation. The B-spline based FEM (called Isogeometric analysis IGA) uses spline functions as testing and shape functions in the Galerkin continuous content. Here, the accuracy of stress distribution and spurious oscillations of the B-spline based FEM are studied in numerical modeling of one-dimensional propagation of stress discontinuities in a bar, where the analytical solution is known. For time integration, the Newmark method, implicit form of the generalized-α method, the central difference method and the predictor/multi-corrector method are tested and compared. The use of lumped and consistent mass matrices in explicit time integration is discussed. Due to accuracy, the consistent mass matrix is preferred in explicit time integration in IGA.

Buckling and free vibration of shear-flexible sandwich beams using a couple-stress-based finite element

In this work, we create a framework for linear buckling and free vibration analyses of sandwich beams using a microstructure-dependent Timoshenko beam model founded on the modified couple-stress theory. The stiffness parameters of a structural web-core sandwich panel are determined by unit cell analysis. An extension to homogeneous cores is also carried out. By employing the exact general solution to the governing equations of the beam, an accurate approximate finite element stiffness matrix is formulated. Furthermore, the static shape functions are used to derive consistent linear geometric stiffness and mass matrices. A convergence study shows that the approximate finite element has good accuracy although the hyperbolic terms of the exact general solution have been expanded into only relatively low-order polynomial series. Results from examples show that the microstructure-dependent beam can predict critical buckling loads and natural frequencies with very good accuracy when compared to more sophisticated finite element models. Unlike the classical Timoshenko beam model, the microstructure-dependent model yields accurate results also when the sandwich assembly is transversely flexible and the bending stiffness of the faces non-negligible.

Generalized Finite Element Method in linear and nonlinear structural dynamic analyses

PurposeThe purpose of this study is to evaluate the accuracy and numerical stability of the Generalized Finite Element Method (GFEM) for solving structural dynamic problems.Design/methodology/approachThe GFEM is a numerical method based on the Partition of Unity (PU) concept. The method can be understood as an extension of the conventional Finite Element Method (FEM) for which the local approximation provided by the shape functions can be improved by means of enrichment functions. Polynomial enrichment functions are hereby used combined with an implicit time-stepping integration technique for improving the dynamical response of the models. Both consistent and lumped mass matrices techniques are tested. The method accuracy and stability are investigated through linear and nonlinear elastic problems.FindingsThe results indicate that the adopted strategies can provide stable and accurate responses for GFEM in dynamic analyses. Furthermore, the mass lumping technique provided remarkable reductions of the system of equation condition number, therefore leading to more stable numerical models.Originality/valueThe evaluated features of GFEM models for implicit time-stepping integration schemes represent new information of great deal of interest regarding linear and nonlinear dynamic analyses using such a method.

Stability analysis of generalized mass formulation in dynamic heat transfer

ABSTRACTIn this article, the generalized mass formulation is developed in an explicit analysis of transient transport problems. It has been well known that the time step is typically smaller in explicit analysis than in implicit analysis when the same size mesh is used. Further, the over-stiffness of conventional finite-element model may result in poor accuracy with linear triangular or tetrahedral elements. In order to improve the computational efficiency and numerical accuracy, this article proposes a generalized mass formulation by matching the mass matrix to the smoothed stiffness matrix using linear triangular elements in 2-D problems. The proposed mass matrix can be obtained by simply shifting the integration points from the conventional locations. Without loss of generality, several 2-D examples, including conduction, convection, and radiation heat transfer problems, are presented to demonstrate that the generalized mass formulation allows a larger time step in explicit analysis compared with the lumped and consistent mass matrices. In addition, it is found that the maximum allowable time step is proportional to the softened effect of the discretized model in an explicit analysis.

Accelerated Approach for the Band Structures Calculation of Phononic Crystals by Finite Element Method

We present here a fast and easily realized computational approach based on the finite element methods with consistent and lumped mass matrices (CM-FEM and LM-FEM, respectively), and the Bloch’s theorem, to calculate the elastic band structures of phononic crystals. Two improvements, the adjustment of the introduction of Bloch’s theorem as well as weighting treatment of consistent and lumped mass matrices, are performed. Numerical simulations show that convergence speed is accelerated obviously. Furthermore, the method is verified by analytical solutions in specified homogeneous cases. It is concluded that compared with CM-FEM or LM-FEM, the present method gives higher precision results with sparser mesh and takes less time.

Stochastic Damped Free Vibration Analysis of Composite Sandwich Plates

In this paper, stochastic damped free vibration analysis of composite sandwich plates has been carried out using a nine node Heterosis plate bending element based on a first order shear deformation theory with a-priori shear correction factors. The plate bending element contains one transverse displacement and two rotations of the normals about the plate's mid-plane. Selective reduced integration scheme is adopted to integrate terms associated with the stiffness matrix formulations. Both lumped and consistent mass matrices are considered in the analysis. The accuracy and reliability of the present finite element formulation is verified with previously published results in the literature. New results are presented which are beneficial for designers of composite structures.

Efficient coupled polynomial interpolation scheme for out-of-plane free vibration analysis of curved beams

The performance of two curved beam finite element models based on coupled polynomial displacement fields is investigated for out-of-plane vibration of arches. These two-noded beam models employ curvilinear strain definitions and have three degrees of freedom per node namely, out-of-plane translation (v), out-of-plane bending rotation (θz) and torsion rotation (θs). The coupled polynomial interpolation fields are derived independently for Timoshenko and Euler–Bernoulli beam elements using the force-moment equilibrium equations. Numerical performance of these elements for constrained and unconstrained arches is compared with the conventional curved beam models which are based on independent polynomial fields. The formulation is shown to be free from any spurious constraints in the limit of ‘flexureless torsion’ and ‘torsionless flexure’ and hence devoid of flexure and torsion locking. The resulting stiffness and consistent mass matrices generated from the coupled displacement models show excellent convergence of natural frequencies in locking regimes. The accuracy of the shear flexibility added to the elements is also demonstrated. The coupled polynomial models are shown to perform consistently over a wide range of flexure-to-shear (EI/GA) and flexure-to-torsion (EI/GJ) stiffness ratios and are inherently devoid of flexure, torsion and shear locking phenomena.

Free vibration analysis of soft-core sandwich beams by the novel weak form quadrature element method

The free vibration of soft-core sandwich beams is analyzed by using the weak form quadrature element method. An N-node novel weak form quadrature sandwich beam element is established based on the extended high-order sandwich panel theory and the differential quadrature rule. Gauss quadrature is used to obtain the element stiffness and consistent mass matrices numerically. Detailed formulations are given. Convergence study shows the fast rate of convergence. The validity and the capability of the novel beam element are examined through comparison with analytic results based on the same theory and two-dimensional finite element data. It is shown that one proposed beam element can yield very accurate frequencies with a relatively small number of nodal points. New data of soft-core sandwich beams with boundary conditions other than the simply supported one, not available in literature, are provided. These data may be referential with which other researchers can compare their results during developing new numerical methods.

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

In this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods–such as the spectral element method based on Lagrange polynomials through Gauss–Lobatto–Legendre points or the iso-geometric analysis using non-uniform rational B-splines–can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

A NURBS‐based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach

SummaryA numerical model to deal with nonlinear elastodynamics involving large rotations within the framework of the finite element based on NURBS (Non‐Uniform Rational B‐Spline) basis is presented. A comprehensive kinematical description using a corotational approach and an orthogonal tensor given by the exact polar decomposition is adopted. The state equation is written in terms of corotational variables according to the hypoelastic theory, relating the Jaumann derivative of the Cauchy stress to the Eulerian strain rate.The generalized‐α method (Gα) method and Generalized Energy‐Momentum Method with an additional parameter (GEMM+ξ) are employed in order to obtain a stable and controllable dissipative time‐stepping scheme with algorithmic conservative properties for nonlinear dynamic analyses.The main contribution is to show that the energy–momentum conservation properties and numerical stability may be improved once a NURBS‐based FEM in the spatial discretization is used. Also it is shown that high continuity can postpone the numerical instability when GEMM+ξ with consistent mass is employed; likewise, increasing the continuity class yields a decrease in the numerical dissipation. A parametric study is carried out in order to show the stability and energy budget in terms of several properties such as continuity class, spectral radius and lumped as well as consistent mass matrices. Copyright © 2015 John Wiley & Sons, Ltd.

Superconvergent isogeometric free vibration analysis of Euler–Bernoulli beams and Kirchhoff plates with new higher order mass matrices

A superconvergent isogeometric free vibration analysis is presented for Euler–Bernoulli beams and Kirchhoff plates. This method is featured by new higher order mass matrices. For the 1D Euler–Bernoulli beam problem, it is shown that a new higher order mass matrix can be directly established by optimizing a reduced bandwidth mass matrix. The reduced bandwidth mass matrix is designed based upon the consistent mass matrix and it contains adjustable parameters to be determined via maximizing the order of accuracy of the vibration frequency. As a result, 4th and 6th orders of accuracy are observed for the proposed quadratic and cubic higher order mass matrices, while their corresponding consistent mass matrices are 2nd and 4th order accurate, respectively. Thus the higher order beam mass matrices have more superior frequency accuracy simultaneously with smaller bandwidth compared with their corresponding consistent mass matrices. While for the 2D Kirchhoff plate problem, in order to compute arbitrary frequency in a superconvergent fashion, a mixed mass matrix is formulated through a linear combination of the consistent mass matrix and the reduced bandwidth mass matrix. Then by introducing the wave propagation angle, the higher order plate mass matrix can be rationally derived from the mixed mass matrix. It is proved that the optimal combination parameter for higher order mass matrix depends on the wave propagation angle. Consequently a particular higher order mass matrix can always be set up for a superconvergent computation of arbitrary vibration frequency. It is shown that the quadratic and cubic higher order mass matrices possess 4th and 6th orders of accuracy, which are two orders higher than those of the consistent plate mass matrices. The construction of higher order mass matrices and the analytical results for vibration frequencies are systematically demonstrated by a set of numerical examples.

Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling

SummaryClassical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non‐diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element‐wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed. Copyright © 2014 John Wiley & Sons, Ltd.

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

SummaryWe demonstrate the potential of collocation methods for efficient higher‐order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra‐weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher‐order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher‐order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. Copyright © 2014 John Wiley & Sons, Ltd.

Mass Matrix Templates: General Description and 1D Examples

This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The main focus is on adjusting the mass matrix. Two well known discretization methods, described in FEM textbooks since the late 1960s, lead to diagonally lumped and consistent mass matrices, respectively. Those models are sufficient to cover many engineering applications but for some problems they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes. The paper provides a general overview of the topic, and illustrates it with one-dimensional structural elements: bars and beams.

Numerical quadrature and operator splitting in finite element methods for cardiac electrophysiology

We study the numerical accuracy and computational efficiency of alternative formulations of the finite element solution procedure for the monodomain equations of cardiac electrophysiology, focusing on the interaction of spatial quadrature implementations with operator splitting and examining both nodal and Gauss quadrature methods and implementations that mix nodal storage of state variables with Gauss quadrature. We evaluate the performance of all possible combinations of 'lumped' approximations of consistent capacitance and mass matrices. Most generally, we find that quadrature schemes and lumped approximations that produce decoupled nodal ionic equations allow for the greatest computational efficiency, this being afforded through the use of asynchronous adaptive time-stepping of the ionic state variable ODEs. We identify two lumped approximation schemes that exhibit superior accuracy, rivaling that of the most expensive variationally consistent implementations. Finally, we illustrate some of the physiological consequences of discretization error in electrophysiological simulation relevant to cardiac arrhythmia and fibrillation. These results suggest caution with the use of semi-automated free-form tetrahedral and hexahedral meshing algorithms available in most commercially available meshing software, which produce nonuniform meshes having a large distribution of element sizes.

Grid dispersion analysis of plane square biquadratic serendipity finite elements in transient elastodynamics

SUMMARYThe spatial discretisation of a continuum by the FEM introduces dispersion errors to the numerical solution of stress wave propagation. Errors of phase and group velocities and the scatter of wave propagation are induced. When these propagating phenomena are modelled by the FEM, the speed of a single harmonic wave depends on its frequency. Parasitic effects do not exist in an ‘ideal’ unbounded continuum. With higher order finite elements, there are optical modes in the spectrum resulting in spurious oscillations of stress and velocity distributions near the sharp wavefronts. In principle, the dispersion errors and the spurious oscillations cannot be removed from these solutions but can be suppressed to a certain extent. For reliable numerical solutions by the FEM, the dispersion errors should be analysed and determined. The dispersion properties of a plane square biquadratic serendipity finite element are examined and compared with a bilinear one. Dispersion analysis is carried out for the consistent and lumped mass matrices. A dispersive ‘improved’ lumped mass matrix for biquadratic serendipity elements is proposed. The paper closes with the recommendation of a choice of permissible dimensionless wavelengths for bilinear and biquadratic serendipity finite element meshes. Copyright © 2013 John Wiley & Sons, Ltd.

Novel higher order mass matrices for isogeometric structural vibration analysis

A set of novel higher order mass matrices are presented for isogeometric analysis of structural vibrations using NURBS. The proposed method for the construction of higher order mass matrices contains two essential steps. Firstly based upon the standard consistent mass matrix a special reduced bandwidth mass matrix is designed. This reduced bandwidth mass matrix meets the requirement of mass conservation while simultaneously preserves the same order of frequency accuracy as the corresponding consistent mass matrix. Subsequently a mixed mass matrix is formulated through a linear combination of the reduced bandwidth mass matrix and the consistent mass matrix. The desired higher order mass matrix is then deduced from the mixed mass matrix by optimizing the linear combination parameter to achieve the most favorable order of accuracy. Both quadratic and cubic formulations are discussed in detail and it is shown that with regard to the vibration frequency, the proposed higher order mass matrices have 6th and 8th orders of accuracy in contrast to the 4th and 6th orders of accuracy associated with the quadratic and cubic consistent mass matrices. A generalization to two dimensional higher order mass matrix is further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices. A series of benchmark examples congruously demonstrate that the proposed higher order mass matrices are capable of achieving the theoretically derived optimal accuracy orders for structural vibration analysis.