Stability Analysis of Smoothed Finite Element Methods with Explicit Method for Transient Heat Transfer Problems

In this paper, transient heat transfer problems are analyzed using the smoothed finite element methods (S-FEMs) with explicit time integration. For a numerical method with spatial discretization, the computational cost per time step in the explicit method is less than that in the implicit method, but the time step is much smaller in the explicit analysis than that in the implicit analysis when the same mesh is used. This is because the stability is of essential importance. This work thus studies the stability of S-FEMs, when applied to transient heat transfer problems. Relationships are established between the critical time steps used in S-FEMs with the maximum eigenvalues of the thermal stiffness (conduction) matrix and mass matrix. It is found that the critical time step relates to the “softness” of the model. For example, node-based smoothed finite element method (NS-FEM) is softer than edge-based smoothed finite element method (ES-FEM), which leads to that the critical time step of NS-FEM is larger than that of ES-FEM. Because computing the eigenvalues and condition numbers of the stiffness matrices is very expensive but valuable for stability analysis, we proposed a concise and effective algorithm to estimate the maximum eigenvalue and condition number. Intensive numerical examples show that our scheme for computing the critical time step can work accurately and stably for the explicit method in FEM and S-FEMs.

A Simple Explicit–Implicit Time-Marching Technique for Wave Propagation Analysis

A new explicit–implicit time integration technique is proposed here for wave propagation analysis. In the present formulation, the time integrators of the model are selected at the element level, allowing each element to be considered as explicit or implicit. In the implicit elements, controllable algorithm dissipation is provided, enabling an [Formula: see text]-stable formulation. In the explicit elements, null amplitude decay is considered, enabling maximal critical time-step values. The new methodology renders a very simple and effective time-marching algorithm. Here, just displacement–velocity relations are considered, and no computation of accelerations is required. Moreover, explicit/implicit analyses can be carried out just by the tuning of local effective matrices, inputting or not stiffness matrices into their computations. At the end of the paper, numerical results are presented, illustrating the performance and potentialities of the new method.

Application of Weight Functions in Nonlinear Analysis of Structural Dynamics Problems

This paper presents a weighted residual method with several weight functions for solving differential equation of motion in nonlinear structural dynamics problems. Order of variation of acceleration is assumed to be quadratic in each time step in which polynomial of displacement would contain five unknown coefficients. Five equations are required for determination of these coefficients in each time step. These equations are obtained from initial conditions, satisfying equation of motions at both ends, and weighted residual integration. In this study, four procedures are considered for weight function to be used in the weighted residual integration as; unit weight function, Petrov–Galerkin’s weight function, least square weight function, and collocation weight function. Due to higher order of acceleration in the proposed method, the results indicate better and more accurate responses. Among the tested functions, the unit weighted function method demonstrated to be non-dissipative and its numerical dispersion showed to be clearly less than the common Newmark’s linear acceleration method. Also critical time step duration in stability investigation for weighted function procedure showed to be larger than the critical time step duration obtained by other methods used in the nonlinear structural dynamics problems.