Chapter 2 - Galerkin Method of Approximation: Irreducible and Mixed Forms

If a naturally straight thin rod, such as a plastic or metal ruler, is subjected to a small compressive thrust applied to its ends, it remains straight. If the thrust is slowly increased beyond a certain critical value, called the buckling load, the rod assumes a configuration, called a buckled state, that is not straight. See Fig. 1.1. This process is called buckling. Depending on the precise mode of loading and the nature of the rod, the transition to a buckled state can be very rapid. If the thrust is further increased, the deflection of the rod from its straight state is likewise increased. If this entire process is repeated, the rod may well buckle into another configuration such as the reflection of the first through a plane of symmetry. The performance of a whole series of such experiments on different rods would lead to the observation that the buckling loads and the nature of buckled states depend upon the material and shape of the rod and upon the manner in which it is supported at its ends. It can also be observed that the results of experiments are highly sensitive to slight deviations of the rod from perfect straightness or of the thrust from perfect symmetry. The study of buckling for different bodies is one of the richest sources of important problems in nonlinear solid mechanics.

Alternative stress-integration schemes for large-deformation problems of solid mechanics

A new approach to parallelization of materially non‐linear problems in solid mechanics is developed. It is based on approximating generalized models of subdomains. The procedure does not retain the same substructuring technique used in a linear version. The convergence proof of the single‐ and multilevel‐domain decomposition algorithms uses the principle of minimum potential energy dissipation and investigated properties of the substructural models. The high efficiency of the approach introduced is shown through the study of several examples. The method developed in this paper for steady creep can be used without modification to solve non‐linear elasticity problems and, at active loading, plasticity problems for bodies of the power‐law strain–stress diagrams. Copyright © 2001 John Wiley & Sons, Ltd.

This paper presents the solution of geometrically nonlinear problems in solid mechanics by the Domain-Boundary Element Method. Because of the Total-Lagrange approach, the arising domain and boundary integrals are evaluated in the undeformed configuration. Therefore, the system matrices remain unchanged during the solution procedure, and their time-consuming computation needs to be performed only once. While the integral equations for axisymmetric finite deformation problems will be derived in detail, the basic ideas of the formulation in two and three dimensions can be found in [1]. The present formulation includes torsional problems with finite deformations, where additional terms arise due to the curvilinear coordinate system. A Newton–Raphson scheme is used to solve the nonlinear set of equations. This involves the solution of a large system of linear equations, which has been a very time-consuming task in former implementations, [1, 2]. In this work, an iterative solver, i.e. the generalized minimum residual method, is used within the Newton–Raphson algorithm, which leads to a significant reduction of the computation time. Finally, numerical examples will be given for axisymmetric and two/three-dimensional problems.

In the present study, the adjuvant activity of flagellin was compared, in the conjugated and mixed forms, against a peptide vaccine from human immunodeficiency virus type 1 (HIV-1) p24-Nef vaccine candidate. Mice were immunized with the HIV-1 p24-Nef peptide with flagellin in both conjugated and mixed forms. Lymphocyte proliferation, CTL activity, and IL-4 and IFN-γ cytokines were evaluated by bromodeoxyuridine, carboxyfluorescein succinimidyl ester and ELISA methods, respectively. At the same time, the frequency of IFN-γ-producing T-lymphocytes, as well as total and isotype-specific antibodies, were assessed by ELISPOT and indirect ELISA, respectively. Our experimental results showed that the immunized mice with the HIV-1 p24-Nef conjugated or mixed forms of flagellin led to increases in the proliferative responses and Th1 cytokine pattern. The conjugated form of vaccine led to increased CTL activity and a Th1 cytokine pattern of immune responses, as well as an IgM isotype of humoral responses in comparison with the mixed form. The flagellin-conjugated vaccine seems to be more potent in increasing vaccine immunogenicity.

For the better management of available groundwater, it is essential to understand the flow behavior within the aquifer systems. The groundwater flow behavior in the complex aquifer system can be studied by solving the governing equations using either analytical or numerical methods. As the analytical solutions are available only for simple idealized cases, numerical methods such as finite difference method (FDM) and finite element method (FEM) are generally used. The Meshless method is a recently developed alternative numerical approach for the solution of existing problem. A variety of Meshless methods like point interpolation method, smooth particle hydrodynamics, Galerkin methods, etc. are under the development for the solution of many solid mechanics and fluid mechanics problems. Meshless method eliminates many of the drawbacks of grid-based methods such as difficulties in meshing and re-meshing as in FDM and FEM which require huge efforts in pre-processing and with Meshless method it translates in the lesser computational time and costs. In this paper, Meshless Local Petrov-Galerkin (MLPG) method with strong form of collocation with exponential/Gaussian radial basis function is used for solving the groundwater flow problem. A computer model in MATLAB has been developed in 1D and 2D for the solution of confined aquifer problems. The developed 1D model is verified with available analytical solution for a hypothetical problem and found to be satisfactory. Further, the 2D model is applied to two case studies and results are compared with the FEM based solutions and found to be satisfactory. The present study shows that the MLPG based Meshless method is very effective in the simulation of groundwater flow problems.

Peter Dronke illuminates a unique literary tradition: the narrative that mixes prose with verse. Highlighting a wide range of text, he defines and explores the creative ways in which mixed forms were used in Europe from antiquity through to the 13th century. Verse with Prose distinguishes some of the most significant use of mixed forms. Dronke looks at the way prose and verse elements function in satirical works, beginning in the 3rd century BC with Menippus. He examines allegorical techniques in the mixed form, paying attention to Boethius' Consolation of Philosophy. His analysis encompasses a feast of medieval saga and romances - ranging from Iceland to Italy - including vernacular works by Marguerite Porete in France and Mechthild in Germany. A number of the medieval Latin texts presented have remained virtually unknown, but emerge here as narratives with unusual and at times brilliant literary qualities. To enable not only specialists but all who love literature to respond to the works discussed, they are quoted on fresh translations, as well as in the originals.

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

Chapter 8 - Basic Equations and Solution Procedure

Purpose – A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities. Design/methodology/approach – A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation. Findings – Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency. Originality/value – Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

Approximate Solutions to Nonlinear Problems of Solid Mechanics by Quasilinearization Method

This commentary on thermodynamics in solid mechanics aims to provide an overview of the main concepts of thermodynamic processes as they apply to, and may be exploited for, studies in nonlinear solid mechanics. We give a descriptive commentary on the (physical) interpretation of these concepts, and relate these where appropriate to behaviour of solids under thermo-mechanical conditions. The motivation is firstly that students of solid mechanics have often had less exposure to thermodynamics than those in other branches of science and engineering, yet there is great value in analytical formulations of material behaviour derived from the principles of thermodynamics. It also sets the contributions in this Theme Issue in context. Along with the deliberately descriptive treatment of thermodynamics, we do outline the main mathematical statements that define the subject, knowing that full details are provided by the authors in their corresponding contributions to this issue. The commentary ends on a lighter note. In order to aid understanding and to stimulate discussion of thermodynamics in solid mechanics, we have invented a number of very basic and completely fictitious materials. These have strange and extreme behaviours that describe certain thermodynamics concepts, such as entropy, in isolation from the complexities of real material behaviour.

Structural and solid mechanics problems can show complex global behavior and, in many cases, without an analytical solution. Numerical methods, such as the Finite Element Method (FEM), are traditionally used to study these systems. The Cellular Automata (CA) is a model based on the idea that the microscopic behavior of a system can be reproduced by smaller systems with simpler local rules. As this approach goes from a microscale to a macroscale, this technique enables the study of microscopic effects in macroscopic objects in a natural way. This paper studies the convergence of a hybrid technique of CA and FEM by developing a computational module capable of simulating solid- and structural-mechanics problems. Four update schemes are studied, evaluating the convergence, with different ordinations and degrees of discretization. Keywords: Finite Elements Methods, Cellular Automata, solid and structural mechanics.

This paper proves the existence and uniqueness of non-negative solutions of the (full) nonlinear configurational probability diffusion equation—which is of Fokker–Planck–Smoluchowski type—of the Doi-Edwards model for polymer melt rheology. This is achieved using the Schauder fixed point theorem and Galerkin's approximation method.

Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: Compressible and incompressible plasticity