A Comparison of Large Deflection Analysis of Bending Plates by Dynamic Relaxation

In this paper, various dynamic relaxation methods are investigated for geometric nonlinear analysis of bending plates. Sixteen wellknown algorithms are employed. Dynamic relaxation fictitious parameters are the mass matrix, the damping matrix and the time step. The difference between the mentioned tactics is how to implement these parameters. To compare the efficiency of these strategies, several bending plates’ problems with large deflections are solved. Based on the number of iterations and analysis time, the scores of the different schemes are calculated. These scores determine the ranking of each technique. The numerical results indicate the appropriate efficiency of Underwood and Rezaiee-Pajand & Alamatian processes for the nonlinear analysis of bending plates

On the Geometrically Nonlinear Analysis of Composite Axisymmetric Shells

Composite axisymmetric shells have numerous applications; many researchers have taken advantage of the general shell element or the semi-analytical formulation to analyze these structures. The present study is devoted to the nonlinear analysis of composite axisymmetric shells by using a 1D three nodded axisymmetric shell element. Both low and higher-order shear deformations are included in the formulation. The displacement field is considered to be nonlinear function of the nodal rotations. This assumption eliminates the restriction of small rotations between two successive increments. Both Total Lagrangian Formulation and Generalized Displacement Control Method are employed for analyzing the shells. Several numerical tests are performed to corroborate the accuracy and efficiency of the suggested approach

Evaluation of iterative methods for solving nonlinear scalar equations

This study is aimed at performing a comprehensive numerical evalua-tion of the iterative solution techniques without memory for solving non-linear scalar equations with simple real roots, in order to specify the most efficient and applicable methods for practical purposes. In this regard, the capabilities of the methods for applicable purposes are be evaluated, in which the ability of the methods to solve different types of nonlinear equations is be studied. First, 26 different iterative methods with the best performance are reviewed. These methods are selected based on performing more than 46000 analyses on 166 different available nonlinear solvers. For the easier application of the techniques, consistent mathematical notation is employed to present reviewed approaches. After presenting the diverse methodologies suggested for solving nonlinear equations, the performances of the reviewed methods are evaluated by solving 28 different nonlinear equations. The utilized test functions, which are selected from the re-viewed research works, are solved by all schemes and by assuming different initial guesses. To select the initial guesses, endpoints of five neighboring intervals with different sizes around the root of test functions are used. Therefore, each problem is solved by ten different starting points. In order to calculate novel computational efficiency indices and rank them accu-rately, the results of the obtained solutions are used. These data include the number of iterations, number of function evaluations, and convergence times. In addition, the successful runs for each process are used to rank the evaluated schemes. Although, in general, the choice of the method de-pends on the problem in practice, but in practical applications, especially in engineering, changing the solution method for different problems is not feasible all the time, and accordingly, the findings of the present study can be used as a guide to specify the fastest and most appropriate solution technique for solving nonlinear problems

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium’s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C² and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper we have shown these results for planar systems, in this paper we cover general n-dimensional systems

A New Triangular Hybrid Displacement Function Element for Static and Free Vibration Analyses of Mindlin-Reissner Plate

A new 3-node triangular hybrid displacement function Mindlin- Reissner plate element is developed. Firstly, the modified variational functional of complementary energy for Mindlin-Reissner plate, which is eventually expressed by a so-called displacement function F, is proposed. Secondly, the locking-free formulae of Timoshenko’s beam theory are chosen as the deflection, rotation, and shear strain along each element boundary. Thirdly, seven fundamental analytical solutions of the displacement function F are selected as the trial functions for the assumed resultant fields, so that the assumed resultant fields satisfy all governing equations in advance. Finally, the element stiffness matrix of the new element, denoted by HDF-P3-7β, is derived from the modified principle of complementary energy. Together with the diagonal inertia matrix of the 3-node triangular isoparametric element, the proposed element is also successfully generalized to the free vibration problems. Numerical results show that the proposed element exhibits overall remarkable performance in all benchmark problems, especially in the free vibration analyses

Review on computational methods for Lyapunov functions

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 di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic 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