Variational methods for selective mass scaling

A new variational method for selective mass scaling is proposed. It is based on a new penalized Hamilton’s principle where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices. In this framework new mass matrices with desired properties can be constructed. It is demonstrated how usage of these non-diagonal mass matrices decreases the maximum frequency of the discretized system and allows for larger steps in explicit time integration. At the same time the lowest eigenfrequencies in the range of interest and global structural response are not significantly changed. Results of numerical experiments for two-dimensional and three-dimensional problems are discussed.     

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

A variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated. Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B-splines of high continuity and Bézier elements with a standard C⁰ continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures.     

Selective mass scaling for explicit finite element analyses

Due to their inherent lack of convergence problems explicit finite element techniques are widely used for analysing non‐linear mechanical processes. In many such processes the energy content in the high frequency domain is small. By focusing an artificial mass scaling on this domain, the critical time step may be increased substantially without significantly affecting the low frequency behaviour. This is what we refer to as selective mass scaling. Two methods for selective mass scaling are introduced in this work. The proposed methods are based on non‐diagonal mass matrices that scale down the eigenfrequencies of the system. The applicability of the methods is illustrated in two example models where the critical time step is increased by up to 30 times its original size. Copyright © 2005 John Wiley & Sons, Ltd.     

Selective mass scaling for thin walled structures modeled with tri-linear solid elements

A method for selective mass scaling in explicit finite element analyses of thin walled structures, modeled with solid elements, is introduced. The method aims at increasing the critical time step without significantly altering the dynamical response of the system. The proposed method is based on the exclusion of certain rigid body motions from the applied mass scaling by filtering the local velocity field.     

Large-step explicit time integration via mass matrix tailoring

A method for tailoring mass matrices that allows large time-step explicit transient analysis is presented. It is shown that the accuracy of the present tailored mass matrix preserves the low-frequency contents while effectively replacing the unwanted higher mesh frequencies by a user-desired cutoff frequency. The proposed mass tailoring methods are applicable to elemental, substructural as well as global systems, requiring no modifications of finite element generation routines. It becomes most computationally attractive when used in conjunction with partitioned formulation as the number of higher (or lower) modes to be filtered out (or retained) are significantly reduced. Numerical experiments with the proposed method demonstrate that they are effective in filtering out higher modes in bars, beams, plain stress, and plate bending problems while preserving the dominant low-frequency contents.     

Lumped mass finite element implementation of continuum theories with micro‐inertia

Continuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro‐structured materials. Such terms, often called micro‐inertia, are convenient and straightforward extensions of classical continuum theories. Furthermore, the critical time step is usually increased via the inclusion of micro‐inertia. However, the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro‐inertia terms. In this paper, we will develop a solution algorithm based on Neumann expansions by which this disadvantage is avoided altogether. The micro‐inertia terms are translated into modifications of the residual force vector, so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained. The numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail. Numerical examples are used to illustrate the various aspects of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

Selective mass scaling for distorted solid‐shell elements in explicit dynamics: optimal scaling factor and stable time step estimate

The use of solid‐shell elements in explicit dynamics has been so far limited by the small critical time step resulting from the small thickness of these elements in comparison with the in‐plane dimensions. To reduce the element highest eigenfrequency in inertia dominated problems, the selective mass scaling approach previously proposed in [G. Cocchetti, M. Pagani and U. Perego, Comp. & Struct. 2013; 127:39‐52.] for parallelepiped elements is here reformulated for distorted solid‐shell elements. The two following objectives are achieved: the critical time step is governed by the smallest element in‐plane dimension and not anymore by the thickness; the mass matrix remains diagonal after the selective mass scaling. The proposed approach makes reference to one Gauss point, trilinear brick element, for which the maximum eigenfrequency can be computed analytically. For this element, it is shown that the proposed mass scaling can be interpreted as a geometric thickness scaling, obtaining in this way a simple criterion for the definition of the optimal mass scaling factor. A strategy for the effective computation of the element maximum eigenfrequency is also proposed. The considered mass scaling preserves the element translational inertia, while it modifies the rotational one, leading to errors in the kinetic energy when the motion rotational component is dominant. The error has been rigorously assessed for an individual element, and a simple formula for its estimate has been derived. Numerical tests, both in small and large displacements and rotations, using a state‐of‐the‐art solid‐shell element taken from the literature, confirm the effectiveness and accuracy of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

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

A 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 multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

One of the main computational issues with explicit dynamics simulations is the significant reduction of the critical time step as the spatial resolution of the finite element mesh increases. In this work, a selective mass scaling approach is presented that can significantly reduce the computational cost in explicit dynamic simulations, while maintaining accuracy. The proposed method is based on a multiscale decomposition approach that separates the dynamics of the system into low (coarse scales) and high frequencies (fine scales). Here, the critical time step is increased by selectively applying mass scaling on the fine scale component only. In problems where the response is dominated by the coarse (low frequency) scales, significant increases in the stable time step can be realized. In this work, we use the proper orthogonal decomposition (POD) method to build the coarse scale space. The main idea behind POD is to obtain an optimal low‐dimensional orthogonal basis for representing an ensemble of high‐dimensional data. In our proposed method, the POD space is generated with snapshots of the solution obtained from early times of the full‐scale simulation. The example problems addressed in this work show significant improvements in computational time, without heavily compromising the accuracy of the results. Copyright © 2013 John Wiley & Sons, Ltd.     

On generation of lumped mass matrices in partition of unity based methods

The partition of unity based methods, such as the extended finite element method and the numerical manifold method, are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials. In the dynamic analysis of cracked bodies using an explicit time integration algorithm, as a result, huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions. Observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force, we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses, where we inversely utilize the ‘from local to global’ skill. The proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix, but has more excellent dynamic property than the latter. Meanwhile, the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes — a very excellent dynamic property. Copyright © 2017 John Wiley & Sons, Ltd.     

Hybrid‐mixed discretization of elasto‐dynamic contact problems using consistent singular mass matrices

An alternative spatial semi‐discretization of dynamic contact based on a modified Hamilton's principle is proposed. The modified Hamilton's principle uses displacement, velocity and momentum as variables, which allows their independent spatial discretization. Along with a local static condensation for velocity and momentum, it leads to an approach with a hybrid‐mixed consistent mass matrix. An attractive feature of such a formulation is the possibility to construct hybrid singular mass matrices with zero components at those nodes where contact is collocated. This improves numerical stability of the semi‐discrete problem: the differential index of the underlying differential‐algebraic system is reduced from 3 to 1, and spurious oscillations in the contact pressure, which are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Results of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis.Copyright © 2013 John Wiley & Sons, Ltd.     

A semi-implicit fractional step finite element method for viscous incompressible flows

This paper describes a new semi-implicit finite element algorithm for time-dependent viscous incompressible flows. The algorithm is of a general type and can handle both low and high Reynolds number flows, although the emphasis is on convection dominated flows. An explicit three-step method is used for the convection term and an implicit trapezoid method for the diffusion term. The consistent mass matrix is only used in the momentum phase of the fractional step algorithm while the lumped mass matrix is used in the pressure phase and in the pressure Poisson equation. An accuracy and stability analysis of the algorithm is provided for the pure convection equation. Two different types of boundary conditions for the end-of-step velocity of the fractional step algorithm have been investigated. Numerical tests for the lid-driven cavity at Re=1 and Re=7500 and flow past a circular cylinder at Re=100 are presented to demonstrate the usefulness of the method.     

Free vibration analysis of arches using curved beam elements

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.     

Inverse mass matrix via the method of localized lagrange multipliers

An efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces and constructs the free‐free inverse mass matrix using the standard FEM procedure. The various boundary conditions are realized by the the method of localized Lagrange multipliers. In particular, the present paper constructs the kernel inverse matrix by using the standard FEM elemental mass matrices. It is shown that the accuracy of the present inverse mass matrix is almost identical to that of a conventional consistent mass matrix or a combination of lumped and consistent mass matrices. Numerical experiments with the proposed inverse mass matrix are conducted to validate its effectiveness when applied to vibration analysis of bars, beams, and plain stress problems.     

Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4‐node tetrahedral and 8‐node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three‐dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.     

On a numerical solution of the supersonic panel flutter eigenproblem

An automated digital computer procedure is presented in this paper which enables efficient solution of the eigenvalue problem associated with the supersonic panel flutter phenomena. The step-by-step incremental solution procedure is based on an inverse iteration technique which effectively utilizes solution results from the previous step in determining such results during the current solution step. Also, the computations are limited to the determination of a few specific roots only, which are expected to contain the flutter mode, and this is achieved at each step without having to compute any other root. The structural discretization achieved by the finite element method yields highly banded stiffness, mass, and aerodynamic matrices; the aerodynamic matrix evaluated by the linearized piston theory is real but unsymmetric in nature. The solution algorithm presented in this paper fully exploits the banded form of the associated matrices, and the resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be most efficient and economical when compared to existing procedures of such analysis. Numerical results are presented for a two-dimensional panel flutter problem.     

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

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.     

Customization of reciprocal mass matrices via log-det heuristic

Customization of finite elements for low-dispersion error through grid dispersion analysis requires a symbolic expansion of a determinant of a representative dynamic stiffness matrix. Such an expansion turns out to be a bottleneck for many practical cases with the size of the representative matrix greater than eight or ten even if the modern computer algebra systems are applied. In this contribution, we propose an alternative approach for low-dispersion customization that avoids explicit determinant expansion. This approach reduces the customization problem to a series of quadratic programming problems and consist of two main steps. First, the customization problem is reformulated as a rank minimization problem for the representative dynamic stiffness matrix evaluated at several discrete pairs of wavenumbers and frequencies. Second, the rank minimization problem is solved approximately via log-det heuristic. Examples for customization of reciprocal mass matrices illustrate capabilities of the proposed approach.     

A faster and accurate explicit algorithm for quasi‐harmonic dynamic problems

It is known that the explicit time integration is conditionally stable. The very small time step leads to increase of computational time dramatically. In this paper, a mass‐redistributed method is formulated in different numerical schemes to simulate transient quasi‐harmonic problems. The essential idea of the mass‐redistributed method is to shift the integration points away from the Gauss locations in the computation of mass matrix for achieving a much larger stable time increment in the explicit method. For the first time, it is found that the stability of explicit method in transient quasi‐harmonic problems is proportional to the softened effect of discretized model with mass‐redistributed method. With adjustment of integration points in the mass matrix, the stability of transient models is improved significantly. Numerical experiments including 1D, 2D and 3D problems with regular and irregular mesh have demonstrated the superior performance of the proposed mass‐redistributed method with the combination of smoothed finite element method in terms of accuracy as well as stability. Copyright © 2016 John Wiley & Sons, Ltd.