7. INTRODUCTION TO THE FINITE ELEMENT METHOD
Engineers use a wide range of tools and techniques to ensure that the designs they create are safe. However, accidents sometimes happen and when they do, companies need to know if a product failed because the design was inadequate or if there is another cause, such as an user error. But they have to ensure that the product works well under a wide range of conditions, and try to avoid to the maximum a failure produced by any cause. One important tool to achieve this is the finite element method.
“The finite element method is one of the most powerful numerical techniques ever devised for solving differential (and integral) equations of initial and boundary-value problems in geometrically complicated regions.” (Reddy, 1988). There is some data that can not be ignored when analyzing an element by the finite element method. This input data is to define the domain, the boundary and initial conditions and also the physical properties. After knowing this data, if the analysis is done carefully, it will give satisfactory results. It can be said that the process to do this analysis is very methodical, and that it is why it is so popular, because that makes it easier to apply. “The finite element analysis of a problem is so systematic that it can be divided into a set of logical steps that can be implemented on a digital computer and can be utilized to solve a wide range of problems by merely changing the data input to the computer program.” (Reddy,1988).
The finite element analysis can be done for one, two and three-dimensional
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problems. But generally, the easier problems are those including one and two dimensions, and those can be solved without the aid of a computer, because even if they give a lot of equations, if they are handled with care, an exact result can be achieved. But if the analysis requires three-dimensional tools, then it would be a lot more complicated, because it will involve a lot of equations that are very difficult to solve without having an error. That is why engineers have developed softwares that can perform these analyses by computer, making everything easier. These softwares can make analysis of one, two and three dimensional problems with a very good accuracy.
A basic thing to understand how finite element works is to know that it divides the whole element into a finite number of small elements. “The domain of the problem is viewed as a collection of nonintersecting simple subdomains, called finite elements… The subdivision of a domain into elements is termed finite element discretization. The collection of the elements is called the finite element mesh of the domain.” (Reddy,1988). The advantage of dividing a big element into small ones is that it allows that every small element has a simpler shape, which leads to a good approximation for the analysis. Another advantage is that at every node (the intersection of the boundaries) arises an interpolant polynomial, which allows an accurate result at a specific point. Before the finite element method, engineers and physicians used a method that involved the use of differential equations, which is known as the finite difference method.
The method of the finite element is a numerical technique that solves or at least
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approximates enough to a solution of a system of differential equations related with a physics or engineering problem. As explained before, this method requires a completely defined geometrical space, and then it would be subdivided into small portions, which together will form a mesh. The difference between the method of the finite element and the method of the finite difference is that in the second one, the mesh consists of lines and rows of orthogonal lines, while in the method of the finite element the division does not necessarily involves orthogonal lines, and this results in a more accurate analysis (Figure 38).
Figure 38 (taken from Algor 13 ®)
The equations used for the finite element method are a lot, but they have the basison some single equations that describe a particular phenomenon. Those equations are:
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22202dxdyThe elliptic equation is described by
202tThe parabolic equation is described bydx
222012xThe hyperbolic equation is described byt
No matter which is the cause of the internal forces and the deformation that they cause, there are three basic conditions that allow the finite element analysis: the equilibrium of forces, the compatibility of displacements and the laws of material behavior. “The first condition merely requires that the internal forces balance the external applied loads.” (Rocket et. al., 1983). That is the most important condition, but the other two assure that the system will be a statically determinate problem. Another condition that must be taken into account is that there exists a relationship between the load applied and the deformation, and this is given by Hooke’s law, as explained in past chapters, but only in the elastic range.
In order to achieve a structural analysis by matrix methods, there might be three ways: stiffness (displacement) method, flexibility (force) method and mixed method. In the first two methods, two basic conditions of nodal equilibrium and compatibility must be reached. In the first method, the once the displacement compatibility conditions are reached, then an answer can be given. In the second method, once the conditions of nodal equilibrium are satisfied and then the compatibility of nodal
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displacement, forces are known in the members.
One of the principles that is the basis of the finite element method is the one known as principle of virtual work. “This principle is concerned with the relationship which exists between a set of external loads and the corresponding internal forces which together satisfy the equilibrium condition, and also with sets of joint (node) displacements and the corresponding member deformations which satisfy the conditions of compatibility.” This principle can be stated in terms of an equation of equilibrium of loads, where the work done by the external loads is equal to the internal virtual work absorbed by the element. Or expressed as an equation:
Fvd
where F are the external loads, the deflection, the system of internal forces, and the internal deformations.
A pin ended tie has similar characteristics to those of an elastic spring that is subjected on one end, looking downwards, suffering the effects of gravity and the effect of an external load. The direct relationship between the force and the displacement of the free end is:
Fk
the vale k is known as the stiffness of the spring. Once the value of the applied
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force and the value of the stiffness, the equation can be inverted to find the displacement:
1kF
This is a simple example for systems that imply only a few data. But when the problem implies more complex systems, then the equations become a little more complicated. When a number of simple members are interconnected at a number of nodes, the displacement caused by the load can only be described by simultaneous equations. Then, the simpler equation seen before becomes:
{F}[K]{}
where K is the stiffness of the whole structure.
For example, for a spring that has two pins, it generates two forces and two displacements. Therefore, the stiffness matrix would be of order 2 x 2:
F1k11k12u1uFkk221222
where u represents the displacement.
Boundary conditions are the limitations set for the problem. These limitations are necessary in order to solve it, because otherwise, the system would be taken as a rigid
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body. The limitations stated by these boundary conditions are like where does the element is likely to move, and were it is restricted. If there were not boundary conditions, the body would be floating in the space, and under the action of any load, it would not suffer any deformation, but it would move around the space as a rigid body. So, when assuming boundary conditions, it has to be assured that the element has enough of them in order to prevent moving as a rigid body. Once this is done, the values of the displacements are obtained and can be substituted in the last equation seen, which will give that the displacement is equal to zero, because the element can not move in any direction. Then, algebra is applied, and the values of the forces or of the stiffness are known.
7.1 Applications of the finite element method in engineering
Every designed made has to fulfill certain specifications, and among them is working under a variety of conditions: temperature, humidity, vibrations, etc. The job of the designer is to achieve this, and to assure that the product will work effectively, taking care of the user and of the element. An engineer has to follow certain steps in order to create a good product with high quality. First, the steps of the design flow chart must be followed:
1. recognize a need
2. specifications and requirements
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3. feasibility sturdy
4. creative design and development
5. detailed drawings
6. prototype building and testing
7. design for manufacture
These are the basic concepts of the design, but it involves a lot of other things to be able to assure a good performance of the product. After the designer has the calculations of the dimensions, tolerances, manufacturing parameters, etc, some other tests should be done. For example, an engineer has to know whether the product is going to support certain loads, or how it is going to behave with temperature variations, or what could happen if vibrations are present. This is where the finite element method enters in action in engineering.
In the last pages, it has been explained how the finite element works and which are the basis of it. Now it is time to explain what is it good for, the applications, the benefits, etc. For example, the first application of the method was introduced by Richard Courant to solve torsion on a cylinder. Then, in the middle 50’s, the method started to be applied for airframe and structural analysis, and then used in civil engineering. In general, finite element methods are used in a wide variety of
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engineering applications, like in computer graphics, heat transfer, electrical and magnetic fields, among others.
Use of finite element method in mechanical engineering is very wide. For example, it is used in mechanics of materials, for structures and trusses. It is used to understand and to prevent how some structures are going to behave under the action of some loads. For example, for a bridge, how is it going to behave with the vibrations, or with the effect of the air, or with the variations of temperature. The aircraft industry uses this method to determine the static and dynamic answer of planes and space crafts to the great variety of environments and conditions that can be found during their operation.
In the case of mechanics of fluids, the method is used to know how a wing of a plane is going to behave with the air flux passing through it, if it is going to resist, how much vibration is this going to cause, in order to avoid resonance (because of the vibrations caused by the turbines too). The finite element method allows calculating the drag and the flotation forces caused during operation.
For heat transfer, it allows to know how a turbine is going to behave, and how the material is going to be affected for the effect of the heat. It is known that heat also creates stresses, and this is a fundamental concept when talking about design of turbines. Another important thing to mention here is that when turbines are working, they reach very high temperatures, and they use a coolant to maintain the temperature under some point. This coolant, when touching the hot turbine, creates a
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thermal shock, which also produces stresses. These stresses can be measured through the finite element method. Something else that it is important is that some components of the turbine (like blades) have holes to let the air in to act as a coolant to avoid overheating. These holes can act as stress concentrators, and by this method, it can be analyzed which is the way they affect less the stress distribution.
One important area where this method is used is in design of mechanical elements. Generally, these elements have to support high loads, whether they are radial, tension, compression, tangential, etc. Sometimes these loads are combined, which makes the element designed more prone to failure. Most of the time, mechanical parts also have holes for assembly, for flow of a certain fluid, etc, and these holes create stress concentrations. A very specific and careful analysis has to be made. Here is where the finite element analysis plays an important role. For example, in the design of a crank, it is subjected to different loads, and it has specific boundary conditions. It is tested with total restriction on one side, and with a load applied tangentially on the other side. The results are like in Figure 39, where the areas that have bigger stresses can be identified according to the color specified on the right of the screen.
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Figure 39
Other application in design is in the design of a control arm. This is for a combination of mechanics with electronics, but it still has forces applied, and holes manufactured for assembly. The boundary conditions have to be established according to the allowable movements of the robot, and then the loads applied in the points where it carries something. The results are those in Figure 40
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Figure 40
The automotive industry uses this method to see how the cars behave under some conditions of load, and for crash tests. For this industry, it has a lot of advantages. After the car is modeled in a CAD software, it is exported to a finite element analysis software, and after all the conditions are set, the test begins. The method has the advantage of giving very accurate information. For example, the deformation suffered point by point (node by node) at a specific time. The fact of they doing these type of analyses represents a great advantage for them, because they save money, they save time, and they also save material, and yet they have very reliable information of the behavior of the car. For example, Figure 41, which is a model of a thesis done the last semester, where the student was analyzing the effect of the impact on the car, and also on a dummy. He could calculate the stress at any point he wanted to, and at any time of the collision, and he was able to determine which areas of the
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car and the dummy were the most affected with the impact. Another example is Figure 42.
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Figure 41 (from Alfredo Pérez Mitre)
Figure 42
7.2 Advantages
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The use of the finite element method has a lot of advantages, and most of them have already been commented in this thesis, but here it is going to be made a short condensation of them.
First, it is a very important tool for stress and strain analysis, not only because it provides accurate information, but because it also saves a lot of money and time by simulating the events in computer and not in real life. It is a really easy to use tool. Once some tutorials are followed, the only thing left to do is to explore a little bit of the software, and then to apply all the knowledge acquired. The principles of the method are easy to understand, even if when the model is complex, the analyses are also complex.
For every engineer it is a very reliable tool, because it is very specific for each occasion, and it is able to perform different analyses of the same model under different circumstances with simple changes in the boundary conditions, in the loads, in the material, or whatever the problem demands.
The method can help to modify each design in order to increase the service life of it as much as possible. That is the case of study of this thesis, where some analyses are run in the computer to see how the presence of a stress concentrator affects the behavior of the element. If the stress concentration is high, the element can be modified with ease and then subjected to analysis again, and depending on the results, a decision has to be taken to see if it needs more changes or if it has reached or if it is even close to its maximum service life condition.
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The combination of this as a software with other types of softwares is a very useful tool, because the program of finite element analysis allows the designer to import models from other CAD softwares, and that way it does not forces the engineer to make the solid in the FEA CAD. That way, complex models can be created and analyzed with the combination of those two powerful tools.
7.有限元方法的简介
工程师需要使用范围更广的工具和技术来保证他们创建的设计是安全的。但是，有时会发生意外，当这些意外发生时公司需要知道产品失败是否是因为设计不足，或者其他原因，如用户错误。但是，他们必须确保产品在大多数情况下能够正常工作，并能最大限度地避免任何原因的引起故障。有限元方法实现这一目标的一个重要工具。
有限元方法是解决复杂几何中微分和积分方程边值问题的最强大的数值方法之一(Reddy，1988年）。 有限元方法在分析单元时，有一些数据是不能忽视的。这些输入数据用来定义域，边界和初始条件，以及物理性质。输入这些数据后，如果分析过程很详细，那么就会得到满意的结果。它如此受欢迎是因为这种分析的过程非常有条理，并且这可以使它更容易应用。有限元分析的问题是那么系统，因此可以在数字计算机上实现划分逻辑步骤，并可以通过只更改输入到计算机程序的数据解决各种问题(Reddy，1988年）。
有限元分析可以解决一维，二维和三维的问题。但通常情况下，一维和二维的问题更容易解决，因为即使他们给大量的方程，如果他们小心处理，在不借助电脑的情况下也可以得到精确的结果。但如果需要分析三维的工具，那么便很复杂，因为它会涉及很多很难解决的方程，而且不能出错。这就是为什么工程师已经开发软件，可以执行这些分析的计算机，使这一切更
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容易。这些软件可以高精度的分析一维，二维和三维的问题。
了解有限元工程是如何工作的最基本的问题要知道它把整个单元分为有限数量的小单元。“问题的域被视为有简单的子域，称为有限元的集合，域划分的元素称为有限元离散化。被调用的元素集合称为域的有限元网格”(Reddy，1988年)。大单元分成小单元的优点是它允许小的每个单元都有一个简单的形状，从而导致好的近似分析。另一个优点是在每个节点 （边界的交集） 产生数插值多项式的情况下，允许在一个特定的点得到准确的结果。在有限元方法诞生之前, 工程师和医生使用一种涉及到微分方程的方法，它被称为有限差分法。
有限元方法是计算微分方程的一种数值方法，它能解决或至少足够接近得到与物理或工程的问题相关的系统解决方案。如前文所述，这种方法需要完全定义的几何空间，然后它会分为若干个小部分所构成的网。有限元方法和有限差分方法之间的区别是后者的网格包含行和列的正交线，而在有限元方法中不一定涉及正交的行，这将得到更准确的分析 （图 38）。
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图 38 (从算法 13 ® 摄)
求解有限元方法的方程有很多，但他们有一些能够描述一个特殊现象基础的方程。这些方程是：
22202dxdy描述椭圆的方程
202t描述抛物线的方程dx
222012x描述双曲线的方程t
无论引起内力及造成变形的原因三什么，有限元分析都需要三个基本条件： 平衡力、 位移的兼容性和材料性质。“第一个条件只需要内部力量平衡外部应用的负载”(Rocket et等，1983年)。这是最重要的条件，但另外两个条件保证系统是否是超静定问题。必须考虑的另一个条件是存在的载荷和变形之间的关系，这是在过去的章节阐述过的胡克定律，但只适用于弹性范围。
为了实现结构分析的矩阵方法可能有三种： (位移) 刚度法、 灵活性 (力) 方法和混合的方法。在前两个方法中，必须达到两个基本条件的节点的平衡性和兼容性。第一种方法，达到一次位移兼容性条件，即可得出结果。第二种方法，一旦结点平衡条件都满足，那么结点位移、结点力均为已知。
有限元方法基础的原理之一是虚功原理。这个原理是关于外载荷及其满足平衡条件的相应
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内力之间关系，也有关于节点位移及其满足兼容性条件的微元变形之间的关系。这个原理可以陈述为根据载荷平衡等式，外加载荷完成的工作与元素吸收的内部虚拟工作是相等的，或者表示为等式：
Fvd
F 可以是外加载荷、 变形、 系统内力和系统内部变形。
针结束领带具有与受重力影响和外部负荷的影响的弹性弹簧类似的特征。力和位移的自由端之间的直接关系为：
Fk
变量k是被称为弹簧的刚度。一旦知道施加的力和刚度值，就可以通过方程求解位移：
1kF
这是一个只有少量数据系统的简单例子。但是，对于更复杂的系统，方程将变得更复杂。解决一些简单的成员在多个节点相互关联的由负载引起的位移时，只能联立方程。然后，以前见过的简单公式变为：
{F}[K]{}
其中 K 是整个结构的刚度。
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例如，有两个针脚的弹簧，它生成两种力量和两个位移。因此，刚度矩阵将变为22：
F1k11k12u1F2k21k22u2
其中 u 表示位移。
边界条件的存在限制了问题的求解. 这些限制是对于解决问题是有必要的否则系统将会被视为刚体。这些边界条件所述的限制是单元可能被限制了移动。
如果没有边界条件，结构体会悬浮在空间内，在任何荷载作用下，它不会产生任何变形，但它会在空间内作为一个刚体进行移动。所以，必须假定边界条件，使它以确保单元的边界条件能够防止单元作为刚体移动。完成后得到的位移值可以在最后方程中看到被替换为零，因为该单元在任意方向上不产生位移。然后，运用代数计算，则可得到刚度的内力值。
7.1有限元法在工程中的应用
每个设计制造必须满足一定的规格，及其中各种条件下工作： 温度、 湿度、 振动等。设计者的任务是要达到这个目的，并保证产品会有效，考虑用户和元素。一名工程师不得不应用某些步骤以创建高质量的一个好的产品。首先，必须遵循设计流程图的步骤：
1.了解需求
2. 规格和要求
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3. 可行性
4. 创意设计和发展
5. 详解图
6. 建筑模型与测试
7. 制造设计
这些都是基本的概念设计，但它涉及到很多其他事情要能够保证产品的良好性能。设计者的尺寸、 公差、 制造参数等计算后，应该做一些其他测试。例如，工程师必须知道产品是否要支持特定的负荷，或如何对待温度的变化，如果存在振动，会发生什么。这就是有限元方法在工程中的用处。
在最后一页中，已经解释了有限元法是如何工作以及它的基本依据。现在是时候解释一下它有什么好处、 应用、 特点等。例如，Richard Courant首次应用有限元方法解决扭转圆柱上。之后，在五十年代中期，有限元法开始应用于机身和结构的分析，并随后将在土木工程。一般情况下，有限元方法用于多种工程应用中，如在计算机图形学、 换热、 电、 磁等各个领域。
有限元法在机械工程中的使用是很广泛的。例如，它用于材料力学、 结构和桁架。它用于了解，防止某些结构受荷载作用下的变化。例如，一座桥，它在震动或空气的影响下如何变化，或温度变化。飞机工业使用此方法来确定在工作环境和状态下的静态和动态分析。
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流体机械的情况下该方法用于了解在有气流能过时飞机机翼如何变化，如果它要抵抗，将引起多大的振动，为了避免共振 (因兴涡流引起的振动）。有限元方法允许计算工作状态下的牵引力和升力。
传热，它允许知道涡轮如何变化，以及材料如何受热的影响。众所周知，热还会产生压力，这是在涡轮机设计时的一个基本的概念，当谈。在这里谈谈的另一个重要的事情是，当涡轮机在非常高的温度在工作时，使用冷却剂维持在一定温度下。此冷却剂接触涡轮机时产生热的冲击，也会产生压力。可以通过有限元方法测量这些压力。涡轮上某些组件（如刀片）上的孔是很重要的，它让空气作为冷却剂，避免过热。这些孔可以产生集中应力，通过有限元方法，它可以分析哪种方法可以减少应力的分布影响。
在使用此方法的一个重要的方面是在机械零件设计。一般情况下，这些元素具有支持高负荷，无论它们是径向、 张力、 压缩，切等。有时这些负载相结合，使得设计失败的可能性增加。大多数情况下，机械零件装配也有孔，对于特定的液体，等等，这些孔上将产生集中应力。必须做出一个非常具体和详细的分析。这时有限元分析发挥了重要作用。例如，在曲柄的设计中，它受到不同的负荷，和特定的边界条件。它是经过一边有全部限制，另一边的切向载荷的测试。结果就如图39，根据指定的屏幕右侧的颜色，可以发现该区域压力的大小。
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沈阳航空航天大学毕业设计（外文翻译）
图39
在设计中的其他应用程序是控制机械臂设计。这是组合的力学与电子产品，但它仍然有力的应用和孔制造的程序集。在机器人允许的动作范围内建立边界条件，在它携带东西的点上加载载荷边界条件允许的动作的机器人，依法设立，然后加载应用中的点，在其携带的东西。在图40中可以看到结果：
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沈阳航空航天大学毕业设计（外文翻译）
图40
汽车工业常使用此方法来看看汽车碰撞试验及负载，某些情况下的行为方式。这一行业，它有很多优点。这辆车在CAD软件建模后，它将被导出到有限元分析软件中，并在设置所有条件后，测试开始。该方法的优点在于能给予非常准确的信息。例如，节点的节点变形的具体时间。事实上对他们来说做这些类型的分析，代有着很大的优势，因为它们节省金钱和时间，并且它们还保存材料，他们还有关于这辆车的非常可靠的信息。例如，图41是学生上学期做的模型分析，学生在模型的基础上写论文分析对这辆车的影响。它能够计算任意点的压力和任何时候的碰撞，它还能分析出汽车的哪一区域受碰撞影响最严重。另一个例子是图 42。
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沈阳航空航天大学毕业设计（外文翻译）
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沈阳航空航天大学毕业设计（外文翻译）
图 41 （阿尔弗雷多·佩雷斯)
图 42
7.2 优点
有限元方法有很多的优点，本文已提到了很多，在这里做一个总结。
首先，它是应力应变分析非常重要的工具，不只是因为它提供了准确的信息，而是因为通过在计算机中模拟还可以节省大量金钱和时间，而不用在现实现中做出来。它是一个非常简单易用的工具。一旦遵循一些教程，惟一剩下要做的是开发一点点的软件，，然后就可以应用所有获得的知识了。即使当模型非常复杂，分析也复杂，但方法的原理容易理解。
对每个工程师来说这是一种非常可靠的工具，因为每个场合它都是非常的具体，它是能够在不同的情况下执行的相同的模型，只是简单的更改下边界条件，如载荷，材料，或其它任
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沈阳航空航天大学毕业设计（外文翻译）
何要求的问题。
该方法可以帮助修改每种设计，以尽可能多增加它的使用寿命。这是这一论断研究中的例子，从计算机的一些分析中看到集中应力存在对该单元变化的影响。应力集中较高时，如果元素可以轻松地修改，然后根据结果再分析，决定一定要根据，看看是否需要更多的更改或是否它已经达到最优设计或是否它已接近其最大的工作条件。
作为与其他类型的软件的结合的软件,这是一个非常有用的工具，因为有限元分析的程序允许设计器，从其他 CAD 软件中导入模式，这种方式并不强制工程师使用固定的有限元分析的计算机辅助设计。这样一来，可以用两个功能强大的工具结合着创建和分析复杂的模型。
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