【正文】 g system. These features can be deleted, ignored or simplified, though this may not be true in some cases, where a fine geometrical change can give rise to a significant difference in the simulation results.An example of having sufficient knowledge and engineering judgment is in the simplification required by the mathematical modelling. For example, a plate has three dimensions geometrically. The plate in the plate theory of mechanics is represented mathematically only in two dimensions (the reason for this will be elaborated in Chapter 2). Therefore, the geometry of a ‘mechanics’ plate is a twodimensional flat surface. Plate elements will be used in meshing these surfaces. A similar situation can be found in shells. A physical beam has also three dimensions. The beam in the beam theory of mechanics is represented mathematically only in one dimension, therefore the geometry of a ‘mechanics’ beam is a onedimensional straight line. Beam elements have to be used to mesh the lines in models. This is also true for truss structures. MeshingMeshing is performed to discretize the geometry created into small pieces called elements or cells. Why do we discretize? The rational behind this can be explained in a very straightforward and logical manner. We can expect the solution for an engineering problem to be very plex, and varies in a way that is very unpredictable using functions across the whole domain of the problem. If the problem domain can be divided (meshed) into small elements or cells using a set of grids or nodes, the solution within an element can be approximated very easily using simple functions such as polynomials. The solutions for all of the elements thus form the solution for the whole problem domain.How does it work? Proper theories are needed for discretizing the governing differential equations based on the discretized domains. The theories used are different from problem to problem, and will be covered in detail later in this book for various types of problems. But before that, we need to generate a mesh for the problem domain.Mesh generation is a very important task of the preprocess. It can be a very time consuming task to the analyst, and usually an experienced analyst will produce a more credible mesh for a plex problem. The domain has to be meshed properly into elements of specific shapes such as triangles and quadrilaterals. Information, such as element connectivity, must be created during the meshing for use later in the formation of the FEM equations. It is ideal to have an entirely automated mesh generator, but unfortunately this is currently not available in the market. A semiautomatic preprocessor is available for most mercial application software packages. There are also packages designed mainly for meshing. Such packages can generate files of a mesh, which can be read by other modelling and simulation packages.Triangulation is the most flexible and wellestablished way in which to create meshes with triangular elements. It can be made almost fully automated for twodimensional (2D) planes, and even threedimensional (3D) spaces. Therefore, it is monly available in most of the preprocessors. The additional advantage of using triangles is the flexibility of modelling plex geometry and its boundaries. The disadvantage is that the accuracy of the simulation results based on triangular elements is often lower than that obtained using quadrilateral elements. Quadrilateral element meshes, however, are more difficulty to generate in an automated manner. Some examples of meshes are given in Figures –. Property of Material or MediumMany engineering systems consist of more than one material. Property of materials can be defined either for a group of elements or each individual element, if needed. For different phenomena to be simulated, different sets of material properties are required. For example, Young’s modulus and shear modulus are required for the stress analysis of solids and structures, whereas the thermal conductivity coefficient will be required for a thermal analysis. Inputting of a material’s properties into a preprocessor is usually straightforward。 all the analyst needs to do is key in the data on material properties and specify either to which region of the geometry or which elements the data applies. However, obtaining these properties is not always easy. There are mercially available material databases to choose from, but experiments are usually required to accurately determine the property of materials to be used in the system. This, however, is outside the scope of this book, and here we assume that the material property is known. Boundary, Initial and Loading ConditionsBoundary, initial and loading conditions play a decisive role in solving the simulation. Inputting these conditions is usually done easily using mercial preprocessors, and it is often interfaced with graphics. Users can specify these conditions either to the geometrical identities (points, lines or curves, surfaces, and solids) or to the elements or grids. Again, to accurately simulate these conditions for actual engineering systems requires experience, knowledge and proper engineering judgments. The boundary, initial and loading conditions are different from problem to problem, and will be covered in detail in subsequent chapters. SIMULATION Discrete System EquationsBased on the mesh generated, a set of discrete simultaneous system equations can be formulated using existing approaches. There are a few types of approach for establishing the simultaneous equations. The first is based on energy principles, such as Hamilton’s principle (Chapter 3), the minimum potential energy principle, and so on. The traditional Finite Element Method (FEM) is established on these principles. The second approach is the weighted residual method, which is also often used for establishing FEM equations for many physical problems and will be demonstrated for heat transfer probl