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

Ever since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the ‘consistent mass matrix’ has been a relevant issue. Applied to the time‐dependent incompressible Navier–Stokes equations, it virtually demands the use of implicit time integration methods in which full ‘velocity–pressure coupling’ is also inherent. The high cost of such (high‐quality) FEM calculations led to the development of simpler but ad hoc methods in which the ‘lumped’ mass matrix is employed and the velocity and pressure are uncoupled to the maximum extent possible. Resulting computer codes were less expensive to use but suffered a significant loss of accuracy, caused by lumping the mass when the flow was advection‐dominated and accurate transport of ‘information’ was important. In the second part of this paper we re‐introduce the consistent mass matrix into some semi‐implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized.

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.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

Numerical investigation of ES-FEM with various mass re-distribution for acoustic problems

An ultra-accurate isogeometric dynamic analysis is presented. The key ingredient of the proposed methodology is the development of isogeometric higher order mass matrix. A new one-step method is proposed for the construction of higher order mass matrix. In this approach, an adjustable mass matrix is formulated through introducing a set of mass parameters into the consistent mass matrix under the element mass conservation condition. Then the semi-discrete frequency derived from the free vibration equation with the adjustable mass matrix is served as a measure to optimize the mass parameters. In 1D analysis, it turns out that the present one-step method can perfectly recover the existing reduced bandwidth mass matrix and the higher order mass matrix by choosing different mass parameters. However, the employment of the proposed one-step method to the 2D membrane problem yields a remarkable gain of solution accuracy compared with the higher order mass matrix generated by the original two-step method. Subsequently a full-discrete isogeometric transient analysis algorithm is presented by using the Newmark time integration scheme and the higher order mass matrix. The full-discrete frequency is derived to assess the accuracy of space-time discretization. Finally a set of numerical examples are presented to evaluate the accuracy of the proposed method, which show that very favorable solution accuracy is achieved by the present dynamic isogeometric analysis with higher order mass formulation compared with that obtained from the standard consistent mass approach.

Cavitation effects play an important role in the UNDEX loading of a structure. For far-field UNDEX, the structural loading is affected by the formation of local and bulk cavitation regions, and the pressure pulses resulting from the closure of the cavitation regions. A common approach to numerically modeling cavitation in far-field underwater explosions is Cavitating Acoustic Finite Elements (CAFE) and more recently Cavitating Acoustic Spectral Elements (CASE). Treatment of cavitation in this manner causes spurious pressure oscillations which must be treated by a numerical damping scheme. The focus of this paper is to investigate the severity of these oscillations on the structural response and a possible improvement to CAFE, based on the original Boris and Book Flux-Corrected Transport algorithm on structured meshes [6], to limit oscillations without the energy loss associated with the current damping schemes.

In this study, a theoretical analysis is presented for estimating the in-plane geometric nonlinear elastic stability behavior of steel members with tapered elements under dynamic loads. Beam-column approach is adopted for modeling the structural members as beam-column elements. The formulation is based on the Eulerian description taking into consideration the influence of axial force on bending stiffness. The changes in member chord length due to axial deformation and flexural bowing are also considered. In the dynamic analysis, the system mass properties have been represented using both lumped and consistent mass matrices. The consistent mass matrix is derived in three components: translational, rotational, and axial inertia. The formulation of the mass matrices in local and global coordinate systems of tapered members, which incorporates geometric nonlinearity, has been presented. A parametric study is conducted to examine the effects of number of tapered elements, tapering the prismatic members, time step size, and tapering ratio. The dynamic responses of tapered members have been shown to be significantly affected by these parameters.

This chapter presents extended forms of Hamilton's Principle and the phase space Hamilton's principle based on the extended Lagrangian and Hamiltonian methods. Hamilton's Principle has already been treated in the context of traditional Lagrangian and Hamiltonian mechanics. Noether's theorem, a method for using symmetries of the extended Lagrangian to identify quantities that are conserved during the motion of the system, is also presented here. Noether's theorem is a powerful technique for discovering conserved quantities in complex Lagrangian systems. The chapter presents the basics of the method in the simple context of Lagrangian systems with a finite number of degrees of freedom. Both the traditional and the extended Hamilton's Principles are an application of the calculus of variations to mechanics. The traditional Hamilton's Principle uses the coordinate parametric method, while the extended Hamilton's Principle uses the general parametric method.

On integral principles for nonholonomic systems

The procedure describes the derivation of boundary integral equations for surface acoustic waves propagating under periodic metal strip gratings with piezoelectric films. It takes into account the electrical and mechanical perturbations, including the effects of mass loading caused by the gratings with an arbitrary shape. First, an integral equation is derived with line integrals on the boundaries within one period. This derivation is based on Hamilton's principle and uses Lagrange's method of multipliers to alleviate the continuous conditions of the displacement and the electric potential on the boundaries. Second, boundary integral equations corresponding to each substrate, piezoelectric film, metal strip, and free space region are obtained from the integral equation using the Rayleigh-Ritz method for admissible functions. With this procedure, it is not necessary to make any assumptions for separation of the boundary conditions between two neighboring regions. Consequently, we clarify the theoretical basis for the analytical procedure using boundary integral equations for longitudinal LSAW modes.

Static and dynamic analysis of wind turbine blades using the finite element method

The solution of mass matrix is one of the important parts for dynamic analysis of finite element method (FEM). In general FEM procedure, the numerical integration of consistent mass matrix needs to carry out the same operation as the stiffness matrix, which includes the coordinate mapping and computing of Jacobian matrix. There has been proposed smoothed finite element method for evaluating stiffness matrix to avoid the coordinate mapping and computing of Jacobian matrix in the numerical integration. In this work, a novel integration scheme is proposed to calculate the consistent mass matrix, in which a symbolic integration is implemented by combining indefinite integral with Gauss divergence theorem. Then, the novel integration scheme of consistent mass matrix is incorporated with the smoothing strain technique for free and forced vibration analysis. The accuracy and the convergence properties of the present method are investigated by several numerical examples. It can be concluded from the numerical results that the present method is robust and stability for dynamic analysis.

In The present work the method of Higher Order Finite Strip, based on Auxiliary Nodular Line (ANL) technique, was utilized to analyze the forced vibration of the Simply Supported Box Girder Bridges with cross intermediate diaphragms, Also to find out the dynamic response under moving vehicles loads by means of Newmark's method which falls within the Implicit Time Integration methods. The Bridge Mass Matrix and the Consistent Mass Matrix was used for both bending and in plane actions. While the Guyan Reduction method was used for reducing the Bridge System Matrix through omitting the Freedom Degrees belonging to the Auxiliary Nodular Line for both bending and in plane actions, In order to enable the Higher Order Finite Strip method for analyzing the box girder with cross intermediate diaphragms. vehicle model were used in the analysis is the Moving Force (MF) where the vehicle represents a no. of moving force with a value equal to the weight of the vehicle wheel.

A study of the isoparametric Timoshenko beam element with reduced integration is presented which has four nodes and two degrees of freedom per node. The element stiffness and mass matrices are derived from the same cubic polynomials for the total lateral displacement and the bending slope. The performance of the reduced integration Timoshenko beam element is studied by calculating the natural frequencies of three cantilever beams with different slenderness ratios. It is shown that the rate of convergence of the isoparametric Timoshenko beam element is significantly improved by the use of reduced integration especially for a thin beam.

On the adoption of velocity variable and grid system for fluid flow computation in curvilinear coordinates