【正文】 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(xiàn)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(xiàn)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(xiàn)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。這是因?yàn)橹虚g節(jié)點(diǎn)的過(guò)度偏移會(huì)導(dǎo)致單元應(yīng)力場(chǎng)的奇異性，正如 節(jié)所討論的。種子網(wǎng)格點(diǎn)要在幾何模型創(chuàng)建后網(wǎng)格劃分前生成，用戶需要做的是在重要區(qū)域放置較密的種子網(wǎng)格點(diǎn)。 在建立問(wèn)題區(qū)域時(shí)對(duì)結(jié)果的要求是另一個(gè)重要因素，例如，在估計(jì)結(jié)果非常重要的區(qū)域，分析者通過(guò)會(huì)對(duì)幾何形狀進(jìn)行精細(xì)模擬。分析者首先應(yīng)該分析問(wèn)題，仔細(xì)審視問(wèn)題域的幾何形狀，對(duì)于滿足一維或二維單元假定的結(jié)構(gòu)區(qū)域或部分嘗試使用一維和二維單元。用戶需要做的是對(duì)問(wèn)題域劃分網(wǎng)格后利用這些工具使帶寬最小，這個(gè)簡(jiǎn)單的操作有時(shí)會(huì)大大的減少 CPU 時(shí)間。 學(xué)習(xí)這些模擬技術(shù) 的另一個(gè)原因是為了提高有限元結(jié)果的計(jì)算效率及精度，一個(gè)有經(jīng)驗(yàn)的分析者能夠用盡可能少的時(shí)間模擬和盡可能少地使用計(jì)算機(jī)資源的情況下得到精確的結(jié)果，有限元分析的效率是由花費(fèi)與精度的比值來(lái)衡量的，如圖 所示。在隨后的章節(jié)中，我們將討論當(dāng)進(jìn)行有限元分析時(shí)在計(jì)算機(jī)中到底進(jìn)行一些什么樣的運(yùn)行。通常，用戶還可以生成變量的等勢(shì)面或變量矢量場(chǎng)，也可以使用增強(qiáng)顯示效果的工具，如劃陰影線、光照和褶皺。 對(duì)于非線性問(wèn)題，還需要進(jìn)行另一層迭代循環(huán)，在迭代時(shí)，必須將非線性方程適當(dāng)?shù)暮?jiǎn)化為線性方程。求解系統(tǒng)方程選擇算法有兩個(gè)重要的考慮：一是存儲(chǔ)量的需求，另一個(gè)就是 CPU(中央處理器 )運(yùn)算時(shí)間。邊界、初始條件和加載情況對(duì)于各種問(wèn)題是不同的，并將在后續(xù)各章中詳細(xì)介紹。圖 給出了一些有限元網(wǎng)格的例子。 我們應(yīng)該進(jìn)行哪些工作呢？首先，在需要離散的區(qū)域內(nèi)離散控制微分方程時(shí)需要適當(dāng)?shù)睦碚撘?據(jù)，不同問(wèn)題所依據(jù)的理論也有所不同，本文對(duì)于各種問(wèn)題的理論將在后面章節(jié)中詳細(xì)討論。例如，一個(gè)具有三維幾何尺寸的板，在力學(xué)中的板的理論中，板是表示為二維物體（其原因?qū)⒃诘诙略敿?xì)介紹）。點(diǎn)、線、面、體都可以通過(guò)平移、旋轉(zhuǎn)、或反射來(lái)生成新的點(diǎn)、線、面、體。幾何形狀的曲線（面）部分可以使用曲線或曲面模擬，然而，必須注意幾何模型最終要由單元集合所表示，所以，如果使用線性單元，曲線或曲面是用分片的直線或平面來(lái)近似。對(duì)工程 系統(tǒng)中很多物理現(xiàn)象人們已經(jīng)建立了相應(yīng)的有限元法的數(shù)學(xué)模型。有限元法是一種數(shù)值解法，尋求對(duì)某些很難獲得解析結(jié)果的問(wèn)題的場(chǎng)變量分布的近似解。為確保最后產(chǎn)品的易加工性及較低的造價(jià)，在產(chǎn)品或系統(tǒng)制造之前需要做大量工作，其過(guò)程如圖 的所示。 本文主要討論的是模擬和仿真問(wèn)題，即圖 中下劃線部分。這樣一來(lái)，未知量就轉(zhuǎn)變成為場(chǎng)變量 在節(jié)點(diǎn)上的離散值。下節(jié)介紹使用有限元法進(jìn)行計(jì)算機(jī)模擬所包括的主要步驟。由于受到計(jì)算機(jī)硬件和軟件的限制，控制單元數(shù)量是很有必要的。通常，軟件模擬包可以直接讀入這些文件，這樣在生成模型的幾何形狀時(shí)就可以大大節(jié)省時(shí)間。 IDEAS軟件在 CAE中的應(yīng)用 (文獻(xiàn)翻譯 ) 5 網(wǎng)格劃分 網(wǎng)格劃分就是將幾何形狀離散成稱之為單元或網(wǎng)格的小塊。網(wǎng)格生成的目的在于將問(wèn)題區(qū)域劃分成合適形狀的單元，如三角形單元和四邊形單元，在劃分網(wǎng)格時(shí)必須形成單元連接信息，作為以后組建有限元方程時(shí)使用。例如，對(duì)于固體和結(jié)構(gòu)中的應(yīng)力分析，需要定義楊氏模量和剪切彈性模量，而對(duì)于熱分析就需要定義熱傳導(dǎo)系數(shù)，材料性質(zhì)常數(shù)通?？梢灾苯虞斎肭疤幚砥髦校治稣咝枰龅氖擎I入這些材料性質(zhì)數(shù)據(jù)并指定數(shù)據(jù)適用于幾何物體中的哪個(gè)區(qū)域或哪些單元。工程實(shí)踐表明：對(duì)固體和結(jié)構(gòu)前兩種方法用的最多，而對(duì)流體流動(dòng)模擬常常使用 其他兩種方法；而且