STRONG CONVERGENCE OF IMPLICIT ITERATIVE ALGORITHMS FOR STRICTLY PSEUDO-CONTRACTIVE MAPPINGS

The class of strictly pseudo-contractive mappings is known to have more powerful applications than the class of nonexpansive mappings in solving nonlinear equations such as inverse and equilibrium problems. Motivated by the potency of the class of strictly pseudo-contractive mappings, a generalized viscosity implicit algorithm is constructed for finding their fixed points in the framework of Banach spaces. The strong convergence of the newly constructed sequence to a fixed point of a strictly pseudo-contractive mapping is obtained under some mild conditions on the parameters and the fixed point is shown to solve some variational inequality problems. An example is given to illustrate the convergence analysis of the newly constructed generalized viscosity implicit algorithm for the class of strictly pseudo-contractive mappings. The example also shows that the algorithm and the conditions which are imposed on the parameters are not just optical illusion.

An Implicit Algorithm for Finding a Fixed Point of a Q -Nonexpansive Mapping in Locally Convex Spaces

In this paper, the notion of the q -duality mappings in locally convex spaces is introduced. An implicit method for finding a fixed point of a Q -nonexpansive mapping is provided. Finally, the convergence of the proposed implicit scheme is investigated. Some examples in order to illustrate of the main results are presented.

Fully Implicit Stress Update Algorithm for Distortion−Based Anisotropic Hardening with Cross−Loading Effect: Comparative Algorithmic Study and Application to Large−Size Forming Problem

A fully implicit stress integration algorithm is developed for the distortional hardening model, namely the e−HAH model, capable of simulating cross−hardening/softening under orthogonal loading path changes. The implicit algorithm solves a complete set of residuals as nonlinear functions of stress, a microstructure deviator, and plastic state variables of the constitutive model, and provides a consistent tangent modulus. The number of residuals is set to be 20 or 14 for the continuum or shell elements, respectively. Comprehensive comparison programs are presented regarding the predictive accuracy and stability with different numerical algorithms, strain increments, material properties, and loading conditions. The flow stress and r−value evolutions under reverse/cross−loading conditions prove that the algorithm is robust and accurate, even with large strain increments. By contrast, the cutting−plane method and partially implicit Euler backward method, which are characterized by a reduced number of residuals, result in unstable responses under abrupt loading path changes. Finally, the algorithm is implemented into the finite element modeling of large−size, S−rail forming and the springback for two automotive steel sheets, which is often solved by a hybrid dynamic explicit–implicit scheme. The fully implicit algorithm performs well for the whole simulation with the solely static implicit scheme.

TSDT theory for free vibration of functionally graded plates with various material properties

In this work, an implicit algorithm is used for analyzing the free dynamic behavior of Functionally Graded Material (FGM) plates. The Third order Shear Deformation Theory (TSDT) is used to develop the proposed model. In this contribution, the formulation is written without any homogenization technique as the rule of mixture. The Hamilton principle is used to establish the resulting equations of motion. For spatial discretization based on Finite Element Method (FEM), a quadratic element with four and eight nodes is adopted using seven degrees of freedom per node. An implicit algorithm is used for solving the obtained problem. To study the accuracy and the performance of the proposed approach, we present comparisons with literature and laminate composite modeling results for vibration natural frequencies. Otherwise, we examine the influence of the exponent of the volume fraction which reacts the plates "P-FGM" and "S-FGM". In addition, we study the influence of the thickness on "E-FGM" plates.

An implicit algorithm for computing the minimal Gersgorin set

In this paper we present a new algorithm for the computation of the minimal Gersgorin set that can be considered an extension of the results from [5]. While the general approach to calculation of the boundary of the minimal Gersgorin set is kept, the core numerical calculation is changed. Namely, the problem is formulated in such a way that the eigenvalue computations are replaced by LU decompositions, allowing the algorithm to be used for larger matrices more efficiently. To illustrate the benefits, we compare both algorithms on several test matrices.

Vibration of composite thin-walled beams with variable open cross section by a high order implicit algorithm

In this work, we study the forced nonlinear vibrations with large amplitude and large torsion of composite thinwalled beams with open variable cross sections under external dynamic loads using a high order implicit algorithm. The used algorithm is based on the temporal and spatial discretizations, the homtopoy transformation, Taylor series expansion and the continuation technique. A 3D beam element with two nodes and seven degrees of freedom per node is adopted. The obtained results are compared with those computed by the industriel Abaqus code.