This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Structural dynamics: VonGyörgy Vértes. Amsterdam: Elsevier Science Publishers 1985. 320 S., US $72.25

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them.
Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expan- sion, linear programming, linear matrix inequalities, collocation methods, al- gebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.

https://www.aimsciences.org/article/exportPdf?id=ee7e4111-9a13-4e44-95c8-14c5557278fc

Review on computational methods for Lyapunov functions

Analysis of clustered tensegrity structures using a modified dynamic relaxation algorithm

Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements

This paper addresses the modified Dynamic Relaxation algorithm, called mdDR by minimizing displacement error between two successive iterations. In the mdDR method, new relationships for fictitious mass and damping are presented. The results obtained from linear and nonlinear structural analysis, either by finite element or finite difference techniques; demonstrate the potential ability of the proposed scheme compared to the conventional DR algorithm. It is shown that the mdDR improves the convergence rate of Dynamic Relaxation method without any additional calculations, so that, the cost and
computational time are decreased. Simplicity, high efficiency and automatic operations are the main merits of the proposed technique.

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The dynamic relaxation method using new formulation for fictitious mass and damping

Free vibration analysis of functionally graded hybrid matrix/fiber nanocomposite conical shells using multiscale method

A new method of fictitious viscous damping determination for the dynamic relaxation method

New implicit higher-order accuracy (IHOA) time integration, which uses constant time step, is presented for dynamic analysis. By defining weighted factors, current displacement and velocity are assumed to be functions of the accelerations in the several previous time steps. Then, the optimum weighted factors are determined so that displacement and velocity errors in the Taylor series are minimized. The IHOA is conditionally stable and by using the Routh–Hurwitz criterion, permissible time step is obtained. For numerical verification of the proposed technique, some linear and nonlinear dynamic systems from finite element and finite difference will be analyzed and the results are compared with common implicit methods such as Newmark- β and Wilson- θ approach. Further, the dynamic relaxation method is applied to solve equivalent static systems, obtained in implicit algorithms. Results show that the proposed time integration is stable for common time steps and exhibits improved accuracy. The highly accurate, ...

Mohammad Rezaiee-Pajand

Papers

The dynamic relaxation method using new formulation for fictitious mass and damping

Free vibration analysis of functionally graded hybrid matrix/fiber nanocomposite conical shells using multiscale method

A new method of fictitious viscous damping determination for the dynamic relaxation method

Implicit Higher-Order Accuracy Method for Numerical Integration in Dynamic Analysis

Nonlinear analysis of FG-sandwich plates and shells