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Introduction
1.1 Numerical Modelling
The advances in numerical modelling techniques and computer technology during the past few decades have transformed the analysis and design of many types of engineering structures. Today, numerical simulations as an integral part of Computer-Aided Engineering (CAE) are routinely performed in civil engineering, mechanical engineering, aerospace and other industries.
Many problems in engineering and science are formulated as field problems by mathematical models in terms of field variables, such as displacements, potentials, etc. The mathematical models consist of governing differential equations to describe physical laws (such as equilibrium and compatibility in a stress analysis), material constitutive models and boundary conditions enforced on the problem domains. Classical methods of engineering analysis are often based on analytical solutions to the mathematical model. The analytical solutions are expressed as mathematical functions and can be evaluated at any locations of interest. However, analytical solutions are available only for very simple problems. For many engineering problems, considerable simplifications have to be made in order to apply the classical analysis methods. This often leads to over-conservative designs.
Numerical modelling techniques have been developed to analyse complex engineering problems. In the numerical modelling of a field problem, the mathematical model is solved by a numerical method. A common feature of the numerical methods is the use of mesh discretization to divide a complex problem domain into a set of discrete pieces. Simple formulations are constructed on each piece and assembled together. The mathematical model of a field problem is approximated by a system of algebraic equations with unknowns at a finite number of discrete points. The solution generally converges to the exact solution with the increasing number of discrete points. Numerical methods rely on computers for number crunching.
The finite element method (Bathe, 1996; Cook et al., 2002; Reddy, 2005; Zienkiewicz et al., 2005, to name a few) is probably the most popular and powerful numerical method for the stress analysis of solids and structures. In the standard finite element method, a problem domain is discretized into a set of discrete subdomains called elements. The edges of the elements have to conform to the boundary of the problem domain. The shapes of the elements are limited to a few simple ones. They are triangles and quadrilaterals (i.e. polygons with 3 or 4 edges) in two dimensions, and tetrahedrons, pyramid, wedge and hexahedra (i.e. polyhedra with faces) in three dimensions. These elementary shapes allow the development of simple element solution. Typically, the solution within an element is approximated locally by piecewise polynomials, which leads to extremely efficient procedures (e.g. for numerical integration) suited to number crunching by computers. Element formulations are constructed as algebraic equations by satisfying the governing differential equations in a weak form. Assembling the element equations according to the element connectivity leads to a global system of algebraic equations that can be solved by known solution techniques.
The finite element method is highly versatile and widely applicable in numerical modelling largely due to the simplicity in formulating element equations. Nowadays, many commercial software packages implementing the finite element method are available. Finite element analysis is widely performed in many disciplines of engineering and science. However, the conventional finite element method faces challenges when applied to certain classes of engineering problems. Examples pertinent to the contents of this book include the problems involving dynamic responses of unbounded domains, strain/stress singularities and moving boundaries. In addition, the simple shapes and formulations of finite elements place a heavy burden on mesh generation, which often requires frequent human interventions.
In the numerical model of a structure under an earthquake action, the supporting soil is often simplified as an unbounded domain extending to infinity in comparison with the dimension of the structure. A so-called radiation condition, which states that no energy be radiated from infinity towards the structure, has to be enforced. Direct application of the finite element method would be computationally expensive as the size of the finite elements is limited by the frequency/wavelength of interest. Various techniques such as the viscous boundary, viscous-spring boundary, etc. have been developed for use with finite elements to overcome this limitation. The boundary element method, which satisfies the radiation condition by a fundamental solution, is another attractive alternative in the modelling of unbounded domains.
In the application of linear elastic fracture mechanics to evaluate the propagation of a crack in a brittle material, the stress field around the crack tip is of primary interest. Based on the theory of elasticity, stress/strain singularities exist at the crack tip. The standard finite element analysis of cracks suffers from slow convergence as the polynomial basis functions of standard finite elements do not resemble the singular functions in the strain and stress solutions. The same happens when modelling interface cracks and multi-material corners of composite materials. To achieve accurate solutions, a large number of elements are required around a singularity point. This further increases the burden on mesh generation. Additional post-processing techniques, such as path-independent integrals, are often required to extract the stress intensity factors and other parameters for engineering design.
In engineering practice, structural designs are routinely performed on Computer- Aided Design (CAD) systems. The geometry of a design is commonly described by Non-Uniform Rational B-Splines (NURBS). To perform a finite element analysis, a mesh that conforms to the boundary of the geometric model has to be generated. For computational efficiency considerations, the element size gradation is indispensable. Automatic mesh generation from CAD models has been an active research subject and various meshing techniques have been developed (Watson, 1981; Yerry and Shephard, 1984; Löhner and Parikh, 1988; Blacker and Meyers, 1993, to name a few). Generally speaking, a triangular/tetrahedral mesh is easier to generate automatically than a quadrilateral/hexahedral mesh, while quadrilateral/hexahedral elements are more advantageous in a finite element analysis. Nowadays, an automatic mesh generator is an indispensable feature of any popular commercial finite element software package. However, the generation of a high-quality finite element mesh conforming to the boundary very often requires tedious human interventions in practical engineering computations. The process can be cumbersome and time-consuming. Furthermore, continuous remeshing is highly desirable in some types of analysis. Examples include the modelling of moving boundaries that result from crack propagation or phase transition in matter, adaptive analysis that aims to automatically achieve a specified accuracy in an optimal way, and topology optimization. Robust and efficient remeshing is a challenging task in a finite element analysis.
Various numerical methods have been proposed recently to overcome the above shortcomings of the finite element method in solving practical engineering problems. One major trend is to develop numerical methods that reduce the meshing burden, especially the human efforts and interventions required in the mesh generation. Many novel numerical methods have been proposed. Notable examples include the various meshfree methods (Lucy, 1977; Belytschko et al., 1994; Atluri and Zhu, 1998), extended/generalized finite element methods (Moës et al., 1999), the finite cell method (Parvizian et al., 2007), and isogeometric analysis (Hughes et al., 2005) to name a few. A comprehensive review of the recent rapid developments in this area is a daunting task beyond the scope of this book.
Advances in digital technology have led to the development and increasing popularity of other file formats than NURBS to describe the geometries of objects. Two notable examples are the STL (STereoLithography) format and the digital image.
- The STL format is widely used in 3D printing and rapid prototyping (Bassoli et al., 2007; Rengier et al., 2010). It is now widely supported by CAD systems. The popularity of the STL format is largely attributed to its extreme simplicity: the boundary of a geometric model is represented by a list of unstructured triangular facets. However, this simple format allows the existence of ill-shaped, overlapped or self-intersecting facets and holes on the boundary. These defects, albeit small in size, are incompatible with the finite element method.
- Digital images are increasingly being applied in engineering and science. Various digital imaging technologies, such as X-ray computed tomography (X-ray CT) scans, magnetic resonance imaging (MRI) and ultrasound have become available with a rapid increase in resolution. For example, the meso-structure of a material can be acquired non-destructively by X-ray CT. A digital image is composed of pixels in 2D and voxels in 3D. The geometry is represented by the colour intensity of the pixels/voxels. The algorithms developed for the automatic mesh generation of...
