In this paper, first, the von Karman Nonlinear theory of plate is used to present the differential equations of large deformation of thin plates in terms of the in-plane forces and out-of-plane displacement and moment sum. Then, the Galerkin integrated formulation of problem is presented. The independent variable of this equation includes the displacement u, v, w and the moment sum M. As a basic step of the Galerkin method all variables are independent in terms of the basic functions which are given in area coordinate system and the generalized coordinates. The integrated equations for each problem are solved by Newton-Raphson to drive the generalized coordinates. Several examples are solved including isocel and right-angled triangular plates under uniform distributed load and compared with results obtained by other researchers.

This paper introduces a computational strategy to collaboratively develop the Galerkin Finite Volume Method (GFVM) as one of the most straightforward and efficient explicit numerical methods to solve structural problems encountering material Nonlinearity in a small limited area, while the remainder of the domain represents a linear elastic behavior. In this regard, the Element Free Galerkin method (EFG), which is remarkably robust and accurate, but presumably more expensive, has locally been employed as a Nonlinear sub-model to cover the shortcomings of the GFVM in the elastoplastic analysis. Since the formulations of these two methods are fundamentally different, the iterative zonal coupling has been accomplished using overlapping Multi-Grid (MG) patches with a non-matching interface and Iterative Global/Local (IGL) approach. The main property of such an algorithm is its non-intrusiveness, which means the complex Nonlinear EFG solver is locally utilized over an elastic global GFVM without any geometric modification. This method is verified and investigated with available analytical and numerical solutions which gave quiet promising results showing the robustness and accuracy of the method. The Moving Least-Square approximation (MLS) has widely been applied on transfer level due to the non-conforming interface at the patch edges, and easily allows us to attach complex geometries with different mesh patterns. The new type of Quasi-Newtonian accelerator is adopted on the global material constitutive matrices and its convergence property and accuracy is compared with dynamic Aitken accelerators for two-dimensional problems in MATLAB. Finally, various accelerator types and mapping strategies are also concerned in the examination.

Herein, we construct an explicit modal numerical solver based on the spec-tral Petrov–Galerkin method via a specific combination of shifted Cheby-shev polynomial basis for handling the Nonlinear time-fractional Burger-type partial differential equation in the Caputo sense. The process reduces the problem to a Nonlinear system of algebraic equations. Solving this alge-braic equation system will yield the approximate solution’s unknown coef-ficients. Many relevant properties of Chebyshev polynomials are reported, some connection and linearization formulas are reported and proved, and all elements of the obtained matrices are evaluated neatly. Also, conver-gence and error analyses are established. Various illustrative examples demonstrate the applicability and accuracy of the proposed method and depict the absolute and estimated error figures. Besides, the current ap-proach’s high efficiency is proved by comparing it with other techniques in the literature.

The development of mathematical models for describing the dynamic behaviours of fluid conveying pipes, micro-pipes and nanotubes under the influence of some thermo-mechanical parameters results intoNonlinear equations that are very difficult to solve analytically. In cases where the exact analytical solutionsare presented either in implicit or explicit forms, high skills and rigorous mathematical analyses wereemployed. It is noted that such solutions do not provide general exact solutions. Inevitably, comparativelysimple, flexible yet accurate and practicable solutions are required for the analyses of these structures. Therefore, in this study, approximate analytical solutions are provided to the Nonlinear equations arising inflow-induced vibration of pipes, micro-pipes and nanotubes using Galerkin-Newton-Harmonic Method(GNHM). The developed approximate analytical solutions are shown to be valid for both small and largeamplitude oscillations. The accuracies and explicitness of these solutions were examined in limiting casesto establish the suitability of the method.

The analysis of beam deformation on elastic foundation is very important in engineering applications, and studies of beams on linear elastic foundations are abundant and accurate.   For practical problems, it is always demanded that the influence of the Nonlinear effect of the foundation on the analytical methods and results must be considered. This study treats the static bending problem of an elastic beam resting on the Nonlinear elastic foundation by solving the Nonlinear differential equations with the Galerkin method, converting the Nonlinear differential equations to a system of Nonlinear algebraic equations for approximate solutions.   The Nonlinear equations are solved with a series expansion of the deflection satisfying the boundary conditions, and coefficients of the series are obtained with usual techniques including the iterative method. The accuracy of the approximate solution with the Galerkin method is verified through examples from earlier studies.   The procedure and results show that the Galerkin method is effective in solving static Nonlinear differential equations in addition to the Nonlinear vibrations with the extended Galerkin method in earlier studies.

In this paper, the Nonlinear torsional vibrations and internal resonances of nanorods are investigated by considering the surface energy effects. For this purpose, Hamilton’s principle is implemented to derive the Nonlinear governing equation of motion based on the von-Kármán relations. Hamilton's principle includes the strain energy and the kinetic energy of the nanorod surface and bulk. The strain and kinetic energies of the nanorod bulk are obtained using the classical theory of elasticity, and those of the nanorod surface are obtained using the surface elasticity theory. The surface energy parameters, including the surface density and the surface Lame constants, are included in the equations by the surface elasticity theory. Then, the multi-mode Galerkin method is used to convert the partial differential equation of motion to an ordinary differential equation. The Multiple-scale method is employed to solve the governing equations of motion for fixed-free and fixed-fixed end conditions. To investigate the technique presented in this paper, circular nanorods made of aluminum and silicon have been used. The effect of surface energy parameters on the torsional frequencies of nanorods is investigated for different values of length, radius, frequency number, and amplitude of the Nonlinear vibrations. In addition, the cases in which internal resonances occur are reported, and some numerical data are given. The results obtained in this research may be helpful for the better design of nanoelectromechanical devices such as nano-bearings and rotary servo motors.

In this paper, a comparison of weak-form Galerkin and least-squares finite element models of Timoshenko beam theory with the von Kármán strains is presented. Computational characteristics of the two models and the influence of the polynomial orders used on the relative accuracies of the two models are discussed. The degree of approximation functions used varied from linear to the 5th order. In the linear analysis, numerical results of beam bending under different types of boundary conditions are presented along with exact solutions to investigate the degree of shear locking in the newly developed mixed finite element models. In the Nonlinear analysis, convergences of Nonlinear finite element solutions of newly developed mixed finite element models are presented along with those of existing traditional model to compare the performance.