【正文】 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 Conditions Boundary, 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. IDEAS軟件在 CAE中的應(yīng)用 (文獻(xiàn)翻譯 ) 23 SIMULATION Discrete System Equations Based 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 IDEAS軟件在 CAE中的應(yīng)用 (文獻(xiàn)翻譯 ) 24 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 problems in Chapter 12. The third approach is based on the Taylor series, which led to the formation of the traditional Finite Difference Method (FDM). The fourth approach is based on the control of conservation laws on each finite volume (elements) in the domain. The Finite Volume Method (FVM) is established using this approach. Another approach is by integral representation, used in some mesh free methods [Liu, 2021]. Engineering practice has so far shown that the first two approaches are most often used for solids and structures, and the other two approaches are often used for fluid flow simulation. However, the FEM has also been used to develop mercial packages for fluid flow and heat transfer problems, and FDM can be used for solids and structures. It may be mentioned without going into detail that the mathematical foundation of all these three approaches is the residual method. An appropriate choice of the test and trial functions in the residual method can lead to the FEM, FDM or FVM formulation. This book first focuses on the formulation of finite element equations for the mechanics of solids and structures based on energy principles. FEM formulations for heat transfer problems are then described, so as to demonstrate how the weighted residual method can be used for derivingFEMequations. This will provide the basic knowledge and key approaches into the FEM for dealing with other physical problems. Equation Solvers After the putational model has been created, it is then fed to a solver to solve the discretized system, simultaneous equations for the field variables at the nodes of the mesh. This is the most puter hardware demanding process. Different software packages use different algorithms depending upon the physical phenomenon to be simulated. There are two very important considerations when choosing algorithms for solving a system of equations: one is the storage required, and another is the CPU (Central Processing Unit) time needed. There are two main types of method for solving simultaneous equations: direct methods and iterative methods. Commonly used direct methods include the Gauss elimination method and the LU deposition method. Those methods work well for relatively small equation systems. Direct methods operate on fully assembled system equations, and therefore demand larger storage space. It can also be coded in such a way that the assembling of the equations is done only for those elements involved in the current stage of equation solving. This can reduce the requirements on storage significantly. IDEAS軟件在 CAE中的應(yīng)用 (文獻(xiàn)翻譯 ) 25 Iterative methods include the Gauss–Jacobi method, the Gauss–Deidel method, the SOR method, generalized conjugate residual methods, the line relaxation method, and so on. These methods work well for relatively larger systems. Iterative methods are often coded in such a way as to avoid full assembly of the system matrices in order to save significantly on the storage. The performance in terms of the rate of convergence of these methods is usually very problemdependent. In using iterative methods, preconditioning plays a very important role in accelerating the convergence process. For nonlinear problems, another iterative loop is needed. The nonlinear equation has to be properly formulated into a linear equation in the iteration. For timedependent problems, time stepping is also required, . first solving for the solution at an initial time (or it could be prescribed by the analyst), then using this solution to march forward for the solution at the next time step, and so on until the solution at the desired time is obtained. There are two main approaches to time stepping: the implicit and explicit approaches. Implicit approaches are usually more stable numerically but less efficient putationally than explicit approaches. Moreover, contact algorithms can be developed more easily using explicit methods. Details on these issues will be giv