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Introduction: Background and Motivation

1.1 What This Book Is All About

The book introduces linear and nonlinear structural analysis through a combination of of mesh generation, solid mechanics and a new finite element methodology called c-type finite element method (Ray, 1999, 2003, 2004, 2005, 2007, 2008). The ultimate objective is to present the largest possible (curved) beam, plate and shell elements undergoing extremely large displacement and rotation, and to apply these to solve standard industrial problems. Any finite element method is only as strong as its weakest link. In other words, the book is not just about unification of mesh generation and finite element methodology but it strives to serve as a reference for budding researchers, engineers, analysts, upper division and graduate students and teachers by demonstrating what various interdisciplinary machinery has to be accurately harnessed to devise a solid and conducive theoretical framework upon which to build a robust, reliable and efficient numerical methodology for linear and nonlinear static and dynamic analysis of beams, plates and shells. As indicated, the principal goal of the book is to produce the largest possible arbitrary shaped elements (a) defined and restricted solely by the requirements of geometry, material, loading and support conditions, (b) avoiding computational problems such as locking in the conventional finite element methods and (c) presenting new, accurate and explicit expressions for resolution of the symmetry issue of the tangent operator for beams, plates and shells in areas of extreme nonlinearity. The 'mega-sized' elements may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis, and a reduced engineering manpower requirement for the final quality assurance. For example, the explicit algebraic and symmetric expressions of the tangent operator, as presented in the book, are an absolute necessity for computational cost efficiency, especially in repetitive calculations that are commonly associated with nonlinear problems. It must be recognized that the requirements for numerical convergence should be purely adaptive and subservient to the main delineating factors already mentioned. However, this strategy of computer generation of mega-elements of arbitrary shape, as it turns out, takes its toll on the analyst. Firstly, only accurate theoretical formulation can be used for the underlying continuum or solid mechanics principles without unnecessary 'short-circuiting' by proliferation of ad hoc numerical manipulations. Secondly, it demands that the applicable finite element method be devised to successfully accept computer generated elements with arbitrarily distorted shapes, with edges (or faces) consisting of up to truly 3D curved boundaries (or surfaces) with natural twist and bend (e.g. for shell elements). Thirdly, for these 'hyperelements' with conformity, the finite element method must be able to accommodate effortlessly and naturally C0, C1 or C2 inter-element continuity on demand.

1.2 A Brief Historical Perspective

Every meaningful structural analysis is an exercise in abstraction about a structural system in the real world, so just as with any other natural or man-made phenomenon, the viability and safety of the structural system is intricately associated with the methodology underpinning such abstraction. More specifically, for a structural system, abstractions lie in the geometric modeling of its material body, its relevant support conditions and its imposed loadings, and finally its material properties; we associate the sum total of these abstractions with a structural theory. Moreover, each of these abstractions defines the extent to which a particular structural theory can efficiently and logically predict and control the response of a structural system. To paraphrase Einstein's incisive comment to Heisenberg that led to the latter's discovery of the uncertainty principle, every theory, like a mirror or a horse's blinder, filters and determines what we can see of the real world. Naturally, lest we miss out on important real-world phenomena, as structural analysts, we have to critically evaluate structural theories, propounded in both the distant and the recent past, so that we can be successful in the ultimate goal of our exercise in abstraction, namely, the prediction and control of structural response to external stimuli. Translated into actual methodology for solid bodies, these abstractions reduce to two fundamentally complementary disciplines: solid mechanics and numerical analysis - each determining and harnessing the strength of the other.

1.2.1 Operational Mechanics

For hundreds of years, even before the digital age, the basic theoretical premises of linear and nonlinear solid mechanics involving the study of the deformation of a body, transmission of force through it and the characterization of its material properties, have been well established, and they were perfected during the twentieth century (Eringen, 1962; Green and Zerna, 1968; Ogden, 1997; Malvern, 1969). However, without the computational power of modern computers, the forms of the various equations in solid mechanics, while accurately describing real world problems, were not of much use for finding numerical solutions for material bodies of very complicated geometry and intricate support conditions, subjected to complex loading systems. Thus, before the computer era, solid mechanics had no unifying numerical methodology, and so structural problems were solved on a case-by-case basis using a variety of different analytical methods.

1.2.2 Conventional Finite Element Methods

With the coming of the digital era, numerical methods became much more dominant. Over 60 years ago, an energy-based methodology called the finite element method (Turner, et al., 1956) made its triumphant entry into the realm of numerical structural analysis, its theory having been established earlier (Courant, 1943). Its chief advance was to choose triangular elements with a complete set of basis functions following Pascal's triangle, but soon after, the application of similar basis functions to quadrilateral elements proved to be a poor choice because arbitrarily curtailed and incomplete polynomials resulted in interpolation problems requiring various ad hoc numerical artifacts such as under integration and reduced integration, to rectify numerous locking issues that resulted. Both the conventional finite element methods - h-type with Lagrangian and Hermite basis functions (Zienkiewicz and Taylor, 2000) and p-type with Legendre basis functions (Szabo and Babuska, 1991), - suffered from these ill-conceived ideas.

While the general theoretical formulation of the displacement finite element method of analysis is based on rigorous variational method, the practice of finite element method is another story - a harrowing experience because, in an ad-hoc manner, it tried in vain to tear apart the fundamental and inalienable dictum of the Rayleigh-Ritz-Galerkin method that says: ". method is a 'package deal', and neither requires nor permits the user to make independent decisions about different parts of the problem" (Strang and Fix, 1973, p. 33). Based on the most common construction mechanism of global basis or Ritz functions from appropriate mapping of the chosen elemental or local basis functions, the ultimate success or failure of a practiced or computer-implemented finite element method essentially depends on the choice of the local basis functions (also known as the elemental shape functions), the adequacy of the mapping functions and the evaluations of the integrals. It is no wonder then that for more than 30 years, the primary focus of all notable fundamental researches in the field of finite element analysis has been on devising the "best" shape functions and associated mapping functions. It is instructive to briefly review the interpolation failure and concomitant patchy, ad hoc remedies with the ill side effects and to present some extremely important conceptual problems that remained unsolved by the conventional finite element methods, insofar as interpolation is concerned.

h-type Methodology: Hermitian shape functions, C1 and C2, are used for situations where flexural strain exists such as in the case of beam. The nodes in these cases are what can be considered as multiple nodes; that is, apart from the values, the slopes and/or curvatures are also taken as degrees of freedom at a physical location. The simplest element in the family has two end nodes with two degrees of freedom each for a total of four degrees of freedom. A cubic Hermite is used for this. The inter-element continuity that can be imposed is that the slope or the first derivatives of the functions are continuous, that is, C1 continuity. To speed up convergence where the solution is smooth or has curved flexural elements, one can use a quintic Hermite with curvature or bending moment continuity, that is, C2 continuity. Note that because of the inclusion of the derivative components as degrees of freedom, the interpolation loses its barycentric nature or convexity. The isoparametric elements are obtained following the assumption and procedure that the functional representation of the deformational behavior is employed in representation of the element geometry. In...