Bernstein–Bézier bases for tetrahedral finite elements

Correction of stiffness and mass matrices utilizing simulated measured modal data

In this work, we choose the points and weights of the Gauss–Jacobi, Gauss–Radau–Jacobi and Gauss–Lobatto–Jacobi quadrature rules that optimize the number of operations for the mass and stiffness matrices of the high‐order finite element method. The procedure is particularly applied to the mass and stiffness matrices using the tensor‐based nodal and modal shape functions given in (Int. J. Numer. Meth. Engng 2007; 71(5):529–563). For square and hexahedron elements, we show that it is possible to use tensor product of the 1D mass and stiffness matrices for the Poisson and elasticity problem. For the triangular and tetrahedron elements, an analogous analysis given in (Int. J. Numer. Meth. Engng 2005; 63(2):1530–1558) was considered for the selection of the optimal points and weights for the stiffness matrix coefficients for triangles and mass and stiffness matrices for tetrahedra. Copyright © 2009 John Wiley & Sons, Ltd.

We propose in this paper an anisotropic, adaptive, finite element algorithm for steady, linear advection-diffusion-reaction problems with strong anisotropic features. The error analysis is based on the dual weighted residual methodology, allowing us to perform “goal-oriented” adaptation of a certain functional $J(u)$ of the solution and derive an “optimal” metric tensor for local mesh adaptation with linear and quadratic finite elements. As a novelty, and to evaluate the weights of the error estimator on unstructured meshes composed of anisotropic triangles, we make use of a patchwise, higher-order interpolation recovery readily extendable to finite elements of arbitrary order. We carry out a number of numerical experiments in two dimensions so as to prove the capabilities of the goal-oriented adaptive method. We compute the convergence rate and the effectivity index for a series of output functionals of the solution. The results show the good performance of the algorithm with linear as well as quadratic ...

Purpose In order to avoid the problem of volumetric locking often encountered when using constant strain tetrahedral finite elements, the purpose of this paper is to present a new composite tetrahedron element which is especially designed for the combined finite-discrete element method (FDEM). Design/methodology/approach A ten-noded composite tetrahedral (COMPTet) finite element, composed of eight four-noded low order tetrahedrons, has been implemented based on Munjiza’s multiplicative decomposition approach. This approach naturally decomposes deformation into translation, rotation, plastic stretches, elastic stretches, volumetric stretches, shear stretches, etc. The problem of volumetric locking is avoided via a selective integration approach that allows for different constitutive components to be evaluated at different integration points. Findings A number of validation cases considering different loading and boundary conditions and different materials for the proposed element are presented. A practical application of the use of the COMPTet finite element is presented by quantitative comparison of numerical model results against simple theoretical estimates and results from acrylic fracturing experiments. All of these examples clearly show the capability of the composite element in eliminating volumetric locking. Originality/value For this tetrahedral element, the combination of “composite” and “low order sub-element” properties are good choices for FDEM dynamic fracture propagation simulations: in order to eliminate the volumetric locking, only the information from the sub-elements of the composite element are needed which is especially convenient for cases where re-meshing is necessary, and the low order sub-elements will enable robust contact interaction algorithms, which maintains both relatively high computational efficiency and accuracy.

<div class="htmlview paragraph">With highly complex structures, it is sometimes desirable to derive a dynamic model of the system from experimental vibration data. This paper presents algorithms for synthesizing the mass and stiffness matrices from experimentally derived modal data in a way which preserves the physical significance of the individual mass and stiffness elements. The synthesizing procedures allow for the incorporation of other mass and stiffness data, whether empirical or based on the analyst's insight. The mass and stiffness matrices are derived for a cantilever beam example and are compared with those obtained using earlier techniques.</div>

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

Finite element vibration analysis of a helically wound tubular and laminated composite material beam

This paper proposes a method to identify mass and stiffness matrices of structures. Element mass and stiffness matrices have mass and stiffness distributions. Therefore, a ratio between two elements in these matrices has a proper value. Element mass and stiffness matrices are divided into several groups considering this relation. A group is represented by a sparse matrix (called group matrix) with only elements in the group matrix. Mass and stiffness matrices are modified group by group. The objective function to be minimized is the sum of each norm of difference between analytical and identified group matrices. A numerical example is given to show the effectiveness of the method.

Development of size-dependent consistent couple stress theory of Timoshenko beams

In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al2O3) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e0a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness matrices are fixed.

Adding damped structure can decrease dynamic stress of blade and avoid blade fatigue failure from forced vibration. Based on the structural feature of long blade with friction damper, the numerical model for dynamic analysis of damped blade in steam turbine has been developed. The blade was described by twisted beam element, the usual space beam element was adopted to analyze the frame of damper, and the slip motion between rubbing surface was modeled by a damping connector. The following matrices which are necessary for finite element analysis were obtained: the stiffness matrix, mass matrix and damping matrix of finite element for blade and damper, the stiffness matrix and damping matrix of damping connector. Then the gross finite element motion equation of the blade was got. Meanwhile, harmonic wave propagation method was adopted to improve calculation efficiency. The comparison of calculation results and experimental data of a 360mm blade shows good agreement. The dynamic characteristic of a last stage long blade in steam turbine with damper was analyzed in detail, its responses with different thickness shroud and gap between shrouds were investigated in detail too, then the optimal structure of damped shroud was obtained, the comparison for response between damped blade and freestanding blade shows the maximum response of blade with optimal damper is 42.4% of that of freestanding blade. At last, a tie wire was added to the damped blade, numerical result shows it can decrease blade response further.

In order to present an efficient, practical technique to determine progressive failure mechanism of structures, modelling of member deterioration by using a spring system is outlined. The procedure uses updates of member stiffness and mass matrices as well as the random load vector in incremental forms. In this procedure, the assembly process produces redistributions of the system stiffness and mass matrices, and the load vector. In the calculation of response spectral values, the original forms remain unchanged. Inversion of the stiffness matrix is calculated by using the Neumann expansion solution in which the original stiffness matrix is inverted only once so that a considerable computation time is saved in the whole calculation process. An incremental solution technique is presented for spectral analyses of both static and dynamic sensitive structures. In the case of dynamic analysis, special attention is paid to estimations of modified natural frequencies and mode shapes of deteriorated structures, which may affect response spectral values considerably. The technique, which is presented in the paper, is attractive in practical applications and can be efficiently used in the reliability calculation as well, and also it can be successfully used to determine a progressive failure mechanism of the structure.

A new strain smoothing method for triangular and tetrahedral finite elements

An efficient multiphysics solid shell based finite element approach for modeling thin sheet metal forming processes