The basic concept of finite element analysis (FEA) is to replace complex problems with simpler ones before solving them. It regards the solution domain as composed of many interrelated subdomains called finite element, assumes a suitable (relatively simple) approximate solution for each element, and then deduces the general satisfaction conditions (such as structural equilibrium conditions) for solving this domain, thus obtaining the solution of the problem. This solution is not an exact solution, but an approximate solution, because the actual problem is replaced by a simpler one. Because most practical problems are difficult to get exact solutions, the finite element method is not only accurate, but also can adapt to various complex shapes, so it has become an effective means of engineering analysis.

English: The finite element method is a modern calculation method developed rapidly with the development of electronic computers. It is an effective numerical analysis method that was first applied to the field of continuum mechanics-static and dynamic characteristics analysis of aircraft structures in 1950s, and then it was widely used to solve continuity problems such as heat conduction, electromagnetic field and fluid mechanics. The thinking and practice of finite element analysis and calculation can be summarized as follows:

Edit this paragraph 1) object discretization

An engineering structure is discretized into a calculation model composed of various elements, which is called element division. Discrete units are connected with each other through the nodes of the units; The setting, nature and number of element nodes should depend on the nature of the problem, the need to describe the deformation form and the calculation progress (generally, the finer the element division, the more accurate the deformation description, that is, the closer to the actual deformation, but the greater the calculation amount). Therefore, the structure analyzed in the finite element method is not the original object or structure, but a discrete object composed of many units connected by new materials in a certain way. In this way, the results obtained by finite element analysis and calculation are only approximate. If the number of dividing units is large and reasonable, the results are consistent with the actual situation.

Edit this paragraph 2) Unit characteristic analysis

A. Selecting displacement mode In the finite element method, when the node displacement is selected as the basic unknown quantity, it is called displacement method; When the nodal force is selected as the basic unknown quantity, it is called the force method; Taking a part of nodal force and a part of nodal displacement as basic unknowns is called hybrid method. Displacement method is easy to realize calculation automation, so displacement method is the most widely used method in finite element method. When the displacement method is used, after the object or structure is discretized, some physical quantities of the element, such as displacement, strain and stress, can be expressed by node displacement. At this time, the displacement distribution in the element can be described by some approximate functions that can approximate the original function. Usually, in the finite element method, we express the displacement as a simple function of coordinate variables. This function is called displacement mode or displacement function. B, analyze the mechanical properties of the unit According to the material properties, shape, size, quantity, location and meaning of the unit, find out the relationship between node force and node displacement, which is a key step in the unit analysis. At this time, it is necessary to apply the geometric equations and physical equations in elasticity to establish the force and displacement equations, and then derive the element stiffness matrix, which is one of the basic steps of finite element method. C, after calculating the discretization of the equivalent nodal force object, it is assumed that the force is transferred from one element to another through the nodal point. However, for an actual continuum, the force is transferred from the male side of one unit to another. Therefore, the surface force, volume force and concentrated force acting on the boundary of the element need to be transmitted to the node equivalently, that is, all forces acting on the element should be replaced by equivalent node force.

Edit this paragraph 3) Cell Group Settings

Using the equilibrium conditions and boundary conditions of structural forces, the elements are reconnected according to the original structure to form a complete finite element equation (1- 1), where k is the stiffness matrix of the whole structure; Q is the node displacement array; F is the load array.

Edit this paragraph 4) Solve the unknown node displacement

Solve the finite element equation (1- 1) to get the displacement. Here, an appropriate calculation method can be selected according to the specific characteristics of the equation. From the above analysis, it can be seen that the basic idea of finite element method is "one point and one combination", in which one point is for element analysis and the other is for comprehensive analysis of the whole structure. Overview of finite element development 1943 courant takes piecewise continuous function defined in triangular domain in his paper and studies the torsion problem of Saint-Venant by using the principle of minimum potential energy. The name "finite element method" was used in Clough's paper on plane elasticity in 1960. 1965, Feng Kang published the article "Difference Scheme Based on Variational Principle", which is the main basis for the international academic circles to recognize the independent development of finite element method in China. 1970 with the development of computer and software, finite element has been developed. Contents involved: the theory on which finite element is based, the principle of element division, the selection and coordination of shape function. Finite element method involves: numerical calculation method and its error, convergence and stability. Scope of application: solid mechanics, fluid mechanics, heat conduction, electromagnetism, acoustics, biomechanics: elastic (linear and nonlinear), elastic-plastic or plastic problems (including static and dynamic problems) composed of rods, beams, plates, shells and blocks. It can solve all kinds of field distribution problems (steady and transient problems such as flow field, temperature field and electromagnetic field). ), water pipeline, circuit, lubrication, noise and the interaction of solid, fluid and temperature.

Edit this paragraph 5) The future of finite element is multi-physical field coupling.

5) The future of finite element is multi-physical field coupling. With the rapid development of computer technology, finite element analysis is more and more used in simulation to solve practical engineering problems. Over the years, more and more engineers, applied mathematicians and physicists have proved that many physical phenomena can be solved by solving partial differential equations, which can be used to describe flow, electromagnetic field, structural mechanics and so on. The finite element method is used to transform these well-known mathematical equations into approximate digital images. Early finite element mainly focused on a certain professional field, such as stress or fatigue, but generally speaking, physical phenomena did not exist alone. For example, as long as it moves, it will generate heat, which will affect some properties of materials, such as conductivity, chemical reaction rate, viscosity of fluid and so on. The coupling of this physical system is what we call multiple physical fields, which is much more complicated to analyze than analyzing a physical field alone. Obviously, we need a multi-physical field analysis tool now. Before 1990s, due to the lack of computer resources, the simulation of multiple physical fields only stayed in the theoretical stage, and the finite element modeling was limited to the simulation of a single physical field, the most common ones being the simulation of mechanics, heat transfer, fluid and electromagnetic field. It seems that the fate of finite element simulation is the simulation of a single physical field. Now this situation is beginning to change. After decades of efforts, the development of computational science has provided us with more agile, concise and fast algorithms and more powerful hardware configuration, which makes it possible to simulate multiple physical fields by finite element method. The emerging finite element method provides a new opportunity for multi-physical field analysis and meets the needs of engineers to solve practical physical systems. The future of finite element lies in solving multiple physical fields. There are countless words. The following examples can only illustrate some potential applications of multi-physical field finite element analysis in the future. Piezoelectric acoustic transducer can convert current into sound pressure field, and vice versa. This device is generally used for sound source devices in air or liquid, such as phased array microphone, ultrasonic biological imager, sonar sensor, acoustic biological therapeutic instrument and so on. It can also be used in some mechanical equipment such as inkjet printers and piezoelectric motors. Piezoelectric amplifier involves three different physical fields: structural field, electric field and sound field in fluid. Only software with multi-physical field analysis ability can solve this model. Piezoelectric material is PZT5-H crystal, which is widely used in piezoelectric sensors. At the interface between air and crystal, the boundary condition of sound field is set as that the pressure is equal to the normal acceleration of the structural field, so that the pressure can be transferred to the air. In addition, due to the influence of air pressure, the crystal domain will be deformed. After applying a current with an amplitude of 200V and an oscillation frequency of 300 KHz, the sound wave propagation generated by the simulated crystal is simulated. The description of this model and its perfect results show that under any complex model, we can express it with a series of mathematical models and then solve it. Another advantage of multi-physical field modeling is that in school, students intuitively obtain some phenomena that they could not see before, and the simple and easy-to-understand expression has won the favor of students. This is exactly what Dr. Krishan Kumar Bhatia felt when she introduced modeling and analysis tools to senior graduates at Rowan University in Glassboro, new york. His students' theme is how to cool the engine box of motorcycle. Dr bhatia taught them how to judge, find and solve problems with the concept of "design-manufacture-test". Without the application of computer simulation, it is unthinkable to popularize this method in class, because the cost is too high. COMSOL Multiphysics has an excellent user interface, which enables students to set up heat transfer problems conveniently and get the required results quickly. Dr. bhatia said: "My goal is to make every student understand partial differential equations, so that when they encounter such problems again, they will not worry." "It doesn't need to know too many analytical tools. On the whole, the students said,' This modeling tool is great' ". Many excellent high-tech engineering companies have seen that multi-physical field modeling can help them stay competitive. Multi-physical field modeling tools allow engineers to conduct more virtual analysis at a time, rather than physical tests. In this way, they can optimize their products quickly and economically. In Medrad Innovations Group in Indonesia, the research team led by Dr. John Kalafut used multi-physical field analysis tools to study the injection process of blood cells in slender syringes, which is a non-Newtonian fluid with high shear rate. Through this research, Medrad engineers made a new type of equipment called Vanguard Dx angiography catheter. Compared with the traditional sharp nozzle catheter, the new diffusion nozzle catheter makes the distribution of contrast agent more uniform. Contrast agent is a special material, which can show the diseased organs more clearly when shooting X-rays. Another problem is that the traditional catheter may cause the contrast agent to generate great velocity during use, which may damage blood vessels. Pioneer angiography catheter reduces the influence of contrast agent on blood vessels and minimizes the possibility of blood vessel injury. The key problem is how to design the nozzle shape of the catheter, which can not only optimize the fluid velocity but also reduce the structural deformation. Kalafut's research team used multi-physical field modeling method to couple the force generated by laminar flow into stress-strain analysis, and then carried out fluid-solid coupling analysis on the shapes and layouts of various nozzles. "One of our interns set up different nozzle layouts for different fluid areas and analyzed them," said Dr. Carafort. "We use these analysis results to evaluate the feasibility of these new ideas, thus reducing the number of times we make solid models." Friction stir welding (FSW) has been widely used in aluminum alloy welding since 199 1 was patented. The aviation industry first adopted these technologies, and now it is studying how to use them to reduce manufacturing costs. In the process of friction stir welding, a cylindrical cutter with a shoulder and a stirring head rotates and is inserted into the joint of two metals. The rotating shoulder and the stirring head are used to generate heat, but the heat is not enough to melt the metal. On the contrary, softening plastic metal will form a solid barrier to prevent oxygen from oxidizing metal and forming bubbles. The effects of crushing, stirring and extrusion can make the structure at the weld better than the original metal structure, and the strength can even be doubled. This welding equipment can even be used to weld different types of aluminum alloys. Airbus has funded a lot of research on friction stir welding. Before manufacturers invest and reorganize their production lines on a large scale, Dr. Paul Colegrove of Cranfield University used multi-physical field analysis tools to help them understand the processing process. The first research achievement is the mathematical model of friction stir welding, which allows Airbus engineers to "see through" the weld to check the temperature distribution and microstructure changes. Dr. Colegrove and his research team also wrote a simulation tool with a graphical interface, so that Airbus engineers can directly extract the thermal characteristics of materials and the ultimate strength of welds. In the simulation process of friction stir welding, three-dimensional heat transfer analysis and two-dimensional axisymmetric eddy current simulation are coupled. The heat transfer analysis calculates the heat distribution of the structure after the heat flux density is applied to the tool surface. Tool displacement, thermal boundary conditions and thermal properties of welding materials can be extracted. Next, the three-dimensional thermal distribution of the tool surface is mapped to a two-dimensional model. The coupling model can calculate the interaction between heat and fluid during machining. The electromagnetic, resistance and heat transfer behaviors of coupled substrates need real multi-physical field analysis tools. A typical application is that in the process of semiconductor processing and annealing, there is a hot fireplace using induction heating to grow semiconductor wafers, which is a key technology in the electronics industry. For example, emery can replace graphite receiver at high temperature of 2000℃, and the receiver is heated by RF device with power close to 10 kW. The design of the furnace cavity is very important for maintaining the uniformity of the temperature in the furnace at such a high temperature. Through the analysis of many physical field analysis tools, it is found that heat is mainly transmitted by radiation. In this model, we can see not only the temperature distribution on the wafer surface, but also the temperature distribution on the quartz tube of the furnace. In circuit design, the durability and service life of materials are important aspects that affect the selection of materials. The trend of miniaturization of electrical appliances makes the electronic components that can be installed on the circuit board develop rapidly. As we all know, resistors and other components installed on the circuit board will generate a lot of heat, which may lead to cracks at the solder feet of the components and eventually lead to the scrapping of the whole circuit board. Multi-physical field analysis tools can analyze the heat transfer on the whole circuit board, the stress change of the structure and the deformation caused by temperature rise. This can be used to improve the rationality of circuit board design and material selection. The improvement of computer ability makes the finite element analysis from single field analysis to multi-field analysis become a reality. In the next few years, multi-physical field analysis tools will bring shock to academic and engineering circles. The monotonous design method of "design-verification" will be gradually eliminated, and virtual modeling technology will make your thoughts go further and ignite the spark of innovation through simulation.