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

A cracked element is formulated using the two-filed Hellinger–Reissner functional.
 Due to utilization of the linear elastic fracture mechanics, only geometrical nonlinearities can be considered for the cracked element. A clear step-by-step algorithm for the element state determination is also presented. The element flexibility matrix is derived in a basic coordinate system. Co-rotational approach is used to transform the element stiffness matrix and the resisting force vector from the basic system to the global one.
 The suggested element is applicable for static and dynamic analysis, as well as, the stress intensity factor calculation, and also inverse crack detection. Various numerical problems verify accuracy of the proposed element for linear and nonlinear structural analysis.

Using co-rotational method for cracked frame analysis

Analytical solution for free vibration of flexible 2D rectangular tanks

In the last sixty years‎, ‎the Dynamic Relaxation methods have evolved significantly‎. ‎These explicit and iterative procedures are frequently used to solve the linear or nonlinear response of governing equations resulted from structural analyses‎. ‎In the first part of this study‎, ‎the common DR formulations are reviewed‎. ‎Mathematical bases and also physical concepts of these solvers are explained briefly‎. ‎All the DR parameters‎, ‎i.e‎. ‎fictitious mass‎, ‎fictitious damping‎, ‎fictitious time step and initial guess are described‎, ‎as well‎. ‎Furthermore‎, ‎solutions of structural problems along with kinetic and viscous damping formulations are discussed‎. ‎Analyses of the existing studies and suggestions for future research trends are presented‎. ‎In the second part‎, ‎the applications of Dynamic Relaxation method in engineering practices are reviewed.

https://ijnao.um.ac.ir/article_24586_0c73509c02360988751d269ab28e51fa.pdf

The state of the art in Dynamic Relaxation methods for structural mechanics Part 1: Formulations

This paper is devoted to a literature review for various treatments of dynamic analysis of concrete gravity and arch dams. The dynamic analysis of dams can be carried out by using analytical, semi-analytical, and numerical approaches, which may include nonlinear effects or ignore them. This kind of analysis can be conducted in the time or frequency domain. Different parameters significantly affect the dynamic behavior of dams, such as dam-water interaction, water compressibility, mass and damping of the foundation, sediments at the reservoir bottom, free surface waves, the infinite length of the foundation rock, reservoir and so on. As the analysis methods of dams developed, these parameters were gradually included in the process. Consequently, their capability enhanced in a more realistic investigation of the dams' behavior. This paper aims to illustrate these achievements by briefly reviewing the available published articles. Corresponding researches are categorizing based on the analysis techniques and considering or neglecting the dam-water interaction, sediments, and free surface motions. Assessing the current studies and suggesting future research are carried out. Expressing positive or negative aspects of the existing investigations are also offered.

A Literature Review on Dynamic Analysis of Concrete Gravity and Arch Dams

In this paper, a plane quadrilateral element with rotational degrees of freedom is developed. Present formulation is based on a hybrid functional with independent boundary displacement and internal optimum strain field. All the optimality constraints, including being rotational invariant, omitting the parasitic shear error and satisfying Fliepa’s pure bending test, are considered. Moreover, the static equilibrium equations are satisfied in this scheme. Authors’ element has only four nodes and twelve degrees of freedom. For the boundary displacement field, Alman’s second-order displacement function is employed. The validities of the proposed element are demonstrated by eleven numerical examples: thick curved beam, thin cantilever beam, Cooke’s skew beam, thin curved beam, cantilever beam with distortion parameter, high-order patch test, cantilever beam with five and four irregular mesh, Mc Neal’s thin cantilever beam and cantilever shear wall with and without openings. When utilizing the coarse and irregular meshes, numerical tests show the high accuracy, rapid convergence and robustness of the suggested element. Less sensitivity to distortion is another property of the new element.

https://ceij.ut.ac.ir/article_64091_d72f776b552d365f2db23a2bf13913af.pdf

Mohammad Rezaiee-Pajand

Papers

Using co-rotational method for cracked frame analysis

Analytical solution for free vibration of flexible 2D rectangular tanks

The state of the art in Dynamic Relaxation methods for structural mechanics Part 1: Formulations

A Literature Review on Dynamic Analysis of Concrete Gravity and Arch Dams

A Hybrid Stress Plane Element with Strain Field