【文章內(nèi)容簡介】 ux in thermal analysis, the electrical charge in electrical analysis, and so on. The FEM is a numerical method seeking an approximated solution of the distribution of field variables in the problem domain that is difficult to obtain analytically. It is done by dividing the problem domain into several elements, as shown in Figures and . Known physical laws are then applied to each small element, each of which usually has a very simple geometry. Figure shows the finite element approximation for a onedimensional case schematically. A continuous function of an unknown field variable is approximated using piecewise linear functions in each subdomain, called an element formed by nodes. The unknowns are then the discrete values of the field variable at the nodes. Next, proper principles are followed to establish equations for the elements, after which the elements are ‘tied’ to one another. This process leads to a set of linear algebraic simultaneous equations for the entire system that can be solved easily to yield the required field variable. This book aims to bring across the various concepts, methods and principles used in the formulation of FE equations in a simple to understand manner. Worked examples and case studies using the well known mercial software package ABAQUS will be discussed, and effective techniques and procedures will be highlighted. IDEAS軟件在 CAE中的應(yīng)用 (文獻翻譯 ) 18 PHYSICAL PROBLEMS IN ENGINEERING There are numerous physical engineering problems in a particular system. As mentioned earlier, although the FEM was initially used for stress analysis, many other physical problems can be solved using the FEM. Mathematical models of the FEM have been formulated for the many physical phenomena in engineering systems. Common physical problems solved using the standard FEM include: ? Mechanics for solids and structures. ? Heat transfer. ? Acoustics. ? Fluid mechanics. ? Others. This book first focuses on the formulation of finite element equations for the mechanics of solids and structures, since that is what the FEM was initially designed for. FEM formulations for heat transfer problems are then described. The conceptual understanding of the methodology of the FEM is the most important, as the application of the FEM to all other physical problems utilizes similar concepts. Computer modelling using the FEM consists of the major steps discussed in the next section. IDEAS軟件在 CAE中的應(yīng)用 (文獻翻譯 ) 19 COMPUTATIONAL MODELLING USING THE FEM The behaviour of a phenomenon in a system depends upon the geometry or domain of the system, the property of the material or medium, and the boundary, initial and loading conditions. For an engineering system, the geometry or domain can be very plex. Further, the boundary and initial conditions can also be plicated. It is therefore, in general, very difficult to solve the governing differential equation via analytical means. In practice, most of the problems are solved using numerical methods. Among these, the methods of domain discretization championed by the FEM are the most popular, due to its practicality and versatility. The procedure of putational modelling using the FEM broadly consists of four steps: ? Modelling of the geometry. ? Meshing (discretization). ? Specification of material property. ? Specification of boundary, initial and loading conditions. Modelling of the Geometry Real structures, ponents or domains are in general very plex, and have to be IDEAS軟件在 CAE中的應(yīng)用 (文獻翻譯 ) 20 reduced to a manageable geometry. Curved parts of the geometry and its boundary can be modeled using curves and curved surfaces. However, it should be noted that the geometry is eventually represented by a collection of elements, and the curves and curved surfaces are approximated by piecewise straight lines or flat surfaces, if linear elements are used. Figure shows an example of a curved boundary represented by the straight lines of the edges of triangular elements. The accuracy of representation of the curved parts is controlled by the number of elements used. It is obvious that with more elements, the representation of the curved parts by straight edges would be smoother and more accurate. Unfortunately, the more elements, the longer the putational time that is required. Hence, due to the constraints on putational hardware and software, it is always necessary to limit the number of elements. As such, promises are usually made in order to decide on an optimum number of elements used. As a result, fine details of the geometry need to be modelled only if very accurate results are required for those regions. The analysts have to interpret the results of the simulation with these geometric approximations in mind. Depending on the software used, there are many ways to create a proper geometry in the puter for the FE mesh. Points can be created simply by keying in the coordinates. Lines and curves can be created by connecting the points or nodes. Surfaces can be created by connecting, rotating or translating the existing lines or curves。 and solids can be created by connecting, rotating or translating the existing surfaces. Points, lines and curves, surfaces and solids can be translated, rotated or reflected to form new ones. Graphic interfaces are often used to help in the creation and manipulation of the geometrical objects. There are numerous Computer Aided Design (CAD) software packages used for engineering design which can produce files containing the geometry of the designed engineering system. These files can usually be read in by modelling software packages, which can significantly save time when creating the geometry of the models. However, in many cases, plex objects read directly from a CAD file may need to be modified and simplified before performing meshing or discretization.