Lumped Mass Matrix Research Articles

A high-order generalized Finite Element Method for multiscale structural dynamics and wave propagation

This paper introduces a high-order multiscale Generalized/eXtended Finite Element Method (GFEM) tailored for the solution of structural dynamics and wave propagation problems exhibiting fine-scale and/or localized solution features such as singularities and discontinuities. The proposed method uses an explicit time-marching scheme together with a block-diagonal lumped mass matrix applicable to arbitrary patch approximation spaces and adopts shape functions computed numerically on-the-fly through the solution of local initial/boundary value problems. Numerical results in one-, two-, and three-dimensions for problems exhibiting singularities in the spatial gradient show that the proposed GFEM is able to capture relevant features of the response using structural-scale meshes that are much coarser than those required by Direct Generalized Finite Element Analyses (DGFEAs) of comparable accuracy. In addition, a detailed study of the critical time step size of the method is presented and compared against the one for DGFEA discretizations that provide similar levels of accuracy. It shows that the proposed method has considerably looser time step size restrictions than available DGFEAs.

Explicit edge-based smoothed numerical manifold method for transient dynamic modeling of two-dimensional stationary cracks

Modeling and prediction of dynamic behaviors of cracked bodies are vital to the safety evaluation of structures. The numerical manifold method (NMM) has achieved considerable success in crack analysis as it provides a powerful representation of complex discontinuities. Strain smoothing has been incorporated into enriched methods such as NMM to improve their accuracy, efficiency, and tolerance to mesh distortion. However, conventional strain smoothing cannot reproduce the large spatial variance in the strain surrounding crack tips. Consequently, the accuracy is reduced when it is applied to crack-tip enrichment bases. High-fidelity representation of crack-tip fields can be achieved by crack-tip enrichment. Therefore, it is desirable to combine compatible strain for the enriched areas with edge-based smoothed strain for the remaining areas. When explicit time integration is employed, it is beneficial to use a lumped mass matrix because it provides a larger critical time step and a straightforward solution of the linear algebraic equation. A local-approximation-based mass lumping strategy is used to obtain a block-diagonal mass matrix with small blocks. Finally, the proposed methodology is applied to various static and dynamic crack analyses, and highly accurate and stable results are obtained.

Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions.

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

Linear Transient Dynamic Analysis of Plates With and Without Cutout

The present work exhibits the analysis of a plate with and without cutout considering the dynamic analysis in a transient zone with its application in the civil, structural, and aerospace industry. For this, first-order shear deformable C0 continuous Finite Element (FE) is implemented to examine transient response of plate. The governing ODE of motion’s is integrated using central difference explicit time integration scheme. A lumped mass matrix is obtained by a special diagonlization scheme and the total mass of the element is conserved by inducing effective rotary inertia terms. A FE program is developed based on derived FE formulation to investigate the effect of dynamic loading on such plates. The analysis exhibits the results in terms of displacement, forces, and bending moment of the plate for various aspect ratios and different parameters. The results are shown in terms of tables and graphs for smaller to larger cutout in plates with its different aspect ratio considering transient analysis. The results reveal that transient dynamics analysis shows a considerable difference in the behavior of plates with and without cutout in terms of time period, amplitude, and bending moment than that obtained from static analysis.

A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

This paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

Material point method simulations using an approximate full mass matrix inverse

All material point method (MPM) codes approximate the full mass matrix with a lumped mass matrix. Because this approach causes dissipation, most MPM simulations rely on so-called FLIP methods to limit dissipation. Recent work to deal with noise caused by those FLIP methods derived the XPIC method (for extended particle in cell method) that filters null space noise from particle velocities using a projection operator. This paper shows that the XPIC projection operator is equivalent to doing grid calculations using an asymptotic expansion of the full mass matrix inverse. From that insight, we derived FMPM(k) for full mass matrix MPM of order k (where the mass matrix inverse is expanded to k terms). Compared to prior MPM algorithms, FMPM changes the methods used to update particle velocities, positions, stresses, and strains. Several examples show that FMPM is more stable and accurate, has less dissipation than XPIC, and with high enough k has less dissipation than FLIP methods. One challenge in MPM is imposing velocity conditions on the grid and those challenges are amplified in FMPM. This paper describes a new moving-wall approach that improves grid boundary conditions and is beneficial to both FMPM and prior MPM. Finally, the use of FMPM for multimaterial mode MPM, with explicit cracks, and with affine extrapolation options is each discussed.

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.

Finite element method based computational time reversal in elastodynamics: Refocusing, reconstruction and its numerical sensitivity

In this paper, we study the accuracy and robustness of the computational time reversal approach based on the explicit finite element method for application in nondestructive testing in solids. The main goal of this paper is to find a methodology for correct and accurate reconstruction of the original source time history. For numerical modeling of frontal (forward) and reverse (backward) problems of elastic wave propagation, we use the finite element method and explicit time integration with the lumped mass matrix. The suggested methodology is applicable in each finite element open source or commercial software. A special attention is paid to prescription of boundary conditions/loading for the reverse problem for accurate reconstruction of time history of the original source. For evaluation of the reconstruction quality, we suggest certain cost functions. Based on several numerical tests, we show effects of prescription of boundary conditions/loading in time reversal, effect of mesh size and time step size, an unknown obstacle, a number of sources, and environmental disturbance (noise) on the correctness of reconstruction of the original source.

A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures

This paper presents a new C1-type beam spectral element method, which aims to ensure the continuity of the deflection and its first derivative at the junction of the adjacent spectral elements. The proposed method is based on the Euler-Bernoulli beam theory. The main novelty of the method is that two different sets of interpolation functions are adopted for the axial displacement and deflection in order to meet the C1 continuous requirement. By removing the rotational degrees of freedom of the internal nodes, a lumped mass matrix can be formed in this spectral element method. Static and dynamic analysis of various beams and frame structures are performed to examine the validity of the proposed spectral beam element model. The numerical results show that the proposed method is accurate and effective. A single element for each component of the structure is able to provide good results.

Three-Dimensional Numerical Model for Seismic Analysis of Structures

Response Spectrum Analysis (RSA) and modal combination techniques are widely used to estimate the peak response of structures subjected to earthquake vibrations. This paper presents the numerical modeling and seismic analysis of structures. The response spectra based on Eurocode 8 is implemented for a five storey moment-resisting 3D structure to estimate peak response to earthquake-induced vibration. Eigenvector analysis is carried out to determine undamped free vibration mode shapes and frequencies. The lumped mass matrix, stiffness matrix of the full 3D structure is derived and natural periods, modal shapes, modal participation factors, and effective modal mass ratios are calculated. The structural response in different modes is combined to estimate the total dynamic response. The Square Root of Sum of Squares (SRSS), Complete Quadratic Combination (CQC), and Absolute Sum (AbsSum) rules are discussed and compared. The results of the SRSS and CQC rule for the examined building are almost identical as all cross-correlation coefficients are approximately equal to zero, and the responses of the individual modes are completely uncorrelated. CQC rule is appropriate when the natural frequencies are close to each other. AbsSum method shows an upper bound estimate of the total response since it assumes that all modal peaks occur at the same time. The simulation for numerical modeling and response spectrum analysis (RSA) is implemented in MATLAB® software.

3D simulations of solidification with liquid phase flow

The work discusses issues related to 3D coagulation simulations including liquid phase flow. The approach chosen by the authors is based on the numerical solution of the enthalpy solidification formula and the Navier-Stokes equation, in which the buoyancy force approximation was added. Both of these equations are discretized in space using Finite Element Method, and in both cases time integration utilizes the Theta scheme, which is a generalized form of Euler Backward approach. The presence of the Navier-Stokes equation in the model forces the use of special techniques to ensure the convergence of the solution. From a number of available methods, the method of adding stabilizing elements in the FEM formulation was chosen, i.e. the stabilization formulas PSPG/SUPG (Pressure Stabilized Petrov Galerkin/Streamline Upwind Stabilized Petrov Galerkin). The occurrence of large temperature gradients, however, made it necessary to use the so-called lumped mass matrix in the energy equation. The work presents the results obtained using a model implemented by the authors in the form of a computer program written in C++ programming language. The use of this language and appropriate libraries made it possible to obtain sufficient performance to perform 3D solidification simulations.

An analysis of implicit time integration schemes for wave propagations

The objective of this paper is to investigate the optimal use of some time integration schemes for the solution of transient wave propagation problems. We study the accuracy characteristics of the trapezoidal rule and the ρ∞-Bathe scheme considering various parameter sets (ρ∞,γ,CFL) with both consistent and lumped mass matrices. The ρ∞-Bathe scheme includes also the standard-, β1/β2-Bathe methods, the Newmark method and the trapezoidal rule. The study of the numerical dispersion shows that in the case of the consistent mass matrix, the ρ∞-Bathe scheme with a proper setting of (ρ∞,γ) and standard Bathe scheme provide similar dispersion errors and outperform the trapezoidal rule. The optimal CFL number of the ρ∞-Bathe scheme is about 25% larger than for the standard Bathe scheme. In addition, we show that using a lumped mass matrix and proper values of ρ∞<0, γ and CFL in the ρ∞-Bathe scheme, more accurate solutions can be obtained in some analyses.

A general mass lumping scheme for the variants of the extended finite element method

SummaryA new strategy for the mass matrix lumping of enriched elements for explicit transient analysis is presented. It is shown that to satisfy the kinetic energy conservation, the use of zero or negative masses for enriched degrees of freedom of lumped mass matrix may be necessary. For a completely cracked element, by lumping the mass of each side of the interface into the finite element nodes located at the same side and assigning zero masses to the enriched degrees of freedom, the kinetic energy for rigid body translations is conserved without transferring spurious energy across the interface. The time integration is performed by adopting an explicit‐implicit technique, where the regular and enriched degrees of freedom are treated explicitly and implicitly, respectively. The proposed method can be viewed as a general mass lumping scheme for the variants of the extended finite element methods because it can be used irrespective of the enrichment method. It also preserves the optimal critical time step of an intact finite element by treating the enriched degrees of freedom implicitly. The accuracy and efficiency of the proposed mass matrix are validated with several benchmark examples.

A fast mesh deformation method for fluid–structure interaction problems

Previous work by the author [1] has developed a very simple moving mesh method, called parabolic extension, allowing for very large deformation in static problems and to very large time steps for time-dependent problems. In this paper, we significantly accelerate the previous method by using a forward Euler scheme with a lumped mass matrix to solve the parabolic extension and we applied it for simulation in fluid-structure interaction. The use of a lumped mass matrix in the forward scheme leads to an explicit algorithm (no linear system to solve) which make the new technique presented here very fast and less computational cost. Numerical benchmarks including complex boundary with irregular deformation are carried out. The results demonstrate that the new method generates the deformed mesh very fast and with high quality even in large deformations, especially concerning the boundary meshes.

A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems

Evaluating inertia forces is a central and complicated task for dynamic analysis of flexible multibody systems (FMS) involving large displacements and rotations. In a number of co-rotational approaches, cubic interpolations are adopted to formulate both inertia and internal forces, where the inertia term may be complicated in derivation and several Gauss points are needed in the numerical integration. The paper presents a high-precision co-rotational formulation of 3D beam elements for dynamic analysis of FMS. The method of switching the rotation vector and its complement is adopted to avoid the singularity problem in spatial finite rotations. In contrast to the traditional co-rotational formulation, the governing equations in the paper are formulated based on the principle of virtual power, without requiring the variation of rotation matrix. In the framework of the proposed co-rotational formulation, the stiffness matrix of small-deformation beam element can be used directly, including the Euler–Bernoulli and Timoshenko–Reissner beam elements. More essentially, the inertia terms of the beam element are formulated by discretizing the beam element into three lumped masses at the left and right end nodes, and an auxiliary node at the middle point of the beam element, resulting in a lumped mass matrix. The equivalence ensures that the total mass is completely accurate and the moment of inertia is high-precision if the beam element undergoes small elastic deformation in the local coordinate system. In the local coordinate system of co-rotational formulations, small elastic deformation can be guaranteed so that the inertia forces are formulated analytically without needing the Gauss integration. Finally, four numerical examples are considered to evaluate the accuracy of the formulation against to the geometrically exact beam theory and the previous co-rotational beam formulations. The proposed method can give high-precision numerical results from the comparison, even if using a small number of elements.

Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

The conditioning of implicit Runge–Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler (the simplest implicit RK method) and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution where the system is solved as a whole and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into a number of smaller systems using the Jordan normal form of the RK matrix and then solve them successively.For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that the conditioning of an implicit RK method behaves like that of the implicit Euler method. If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite.The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Illustrative numerical examples are given.

Explicit time integration with lumped mass matrix for enriched finite elements solution of time domain wave problems

We present a partition of unity finite element method for wave propagation problems in the time domain using an explicit time integration scheme. Plane wave enrichment functions are introduced at the finite elements nodes which allows for a coarse mesh at low order polynomial shape functions even at high wavenumbers. The initial condition is formulated as a Galerkin approximation in the enriched function space. We also show the possibility of lumping the mass matrix which is approximated as a block diagonal system. The proposed method, with and without lumping, is validated using three test cases and compared to an implicit time integration approach. The stability of the proposed approach against different factors such as the choice of wavenumber for the enrichment functions, the spatial discretization, the distortions in mesh elements or the timestep size, is tested in the numerical studies. The method performance is measured for the solution accuracy and the CPU processing times. The results show significant advantages for the proposed lumping approach which outperforms other considered approaches in terms of stability. Furthermore, the resulting block diagonal system only requires a fraction of the CPU time needed to solve the full system associated with the non-lumped approaches.

A mass lumping scheme for the 10-node tetrahedral element

In the dynamic analysis with explicit time integration scheme, a lumped mass matrix (LMM) is always preferable than the consistent mass matrix (CMM), since the solving of large scale simultaneous algebraic equations can be avoided if a LMM is employed. In this study, a mass lumping scheme which is suitable for the 10-node tetrahedral element is proposed. A series of numerical examples show that the natural frequencies assessed from the proposed LMM are very close to those from CMM. In addition, the dynamic responses of structure predicted by the proposed LMM are in very good agreement with those from CMM if an implicit time integration scheme is adopted. More importantly, the proposed LMM is less sensitive to mesh distortion than the CMM. The proposed LMM can supersede the CMM at any cases.

Mass lumping techniques in the spectral element method: On the equivalence of the row-sum, nodal quadrature, and diagonal scaling methods

In the context of wave propagation analysis, the spectral element method (SEM) in conjunction with a diagonal mass matrix is often the method of choice. Therefore, it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results. To this end, we compare the performance of three established methods including the row-sum method, the nodal quadrature method, and the diagonal scaling method. The theoretical analysis of these methods reveals a close connection between them. Under certain conditions, that are discussed in detail in this article, we are able to show a direct equivalence between these three approaches. In this regard, the attainable accuracy of the numerical integration of the mass matrix plays an important role. By means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix SEM in comparison to a formulation based on the consistent mass matrix.

Critical assessment of different mass lumping schemes for higher order serendipity finite elements

An efficient and robust finite element-based transient analysis of structures is important in many engineering applications. In this context, a diagonal or lumped mass matrix is an essential prerequisite. In the last decades, several methods to construct such a mass matrix have been proposed and therefore a comprehensive and unbiased evaluation of these approaches is needed. In the present article, we consequently investigate established mass lumping schemes, such as the row-sum method and the diagonal scaling method, as well as the recently proposed manifold-based method. Furthermore, the latter approach is rigorously extended to higher order serendipity finite elements in two and three dimensions. Note that the manifold-based method and the diagonal scaling method are very general approaches, in the sense that they are applicable to arbitrary nodal-based shape functions with arbitrary polynomial orders, while guaranteeing the positive-definiteness of the lumped mass matrix. With respect to the row-sum method the positivity is not ensured and negative or zero diagonal components might be computed. Consequently, a detailed analysis of the influence and performance of these mass lumping schemes on the numerical results is conducted. To this end, several dynamic benchmark models (including modal, harmonic, and transient analyses) are selected, showing that only suboptimal rates of convergence are attained if serendipity finite elements based on a diagonal mass matrix are employed.