What is Finite Element Method?

A discrete numerical method for stress analysis of mechanical parts or components. It divides the solved area into several interconnected subareas, these subareas are called finite elements. In solving, the value of the indeterminate function of the node is taken as the basic unknown quantity and the relation between each indeterminate function in the unit and the value of the node is established. Nodal values ​​can be parameters such as force, displacement, or temperature.

The essence of the solution process is to discretize the problem of finding a continuous function over the entire area in a system of algebraic equations containing indeterminate values ​​at a finite number of nodes. The lower the unit score, the closer the result will be to the correct continuous solution. At this time, hundreds or even thousands of algebraic equations will be obtained and solved by electronic computers.

The finite element method can use different shapes, sizes, and element element types (Figure 1) to simulate structures of any geometric shape, accommodate any support, loading conditions, and various complex material properties, and can be compiled into various general programs. Calculate on the computer.

??A brief history?B. Longifoss in 1952 and JH Aguiris in 1955 divided elements such as rods, beams, and columns into units, and used the structural matrix method to analyze the structure of the rod system. In 1956, MJ Turner, RW Clough, HC Martin, and LJ Top divided the complex structure and continuum into many triangular and rectangular units and successfully analyzed the airship structure. The development of electronic computers has provided reliable computational tools, allowing the application and rapid development of the finite element method.

In the 1970s, the finite element method has achieved research results in material nonlinearity and geometric nonlinearity, fracture mechanics, fluid mechanics, biomechanics, seepage analysis, pollution diffusion, and architectural acoustics. A recently developed boundary element method reduces the dimensionality of the problem and is easy to popularize on microcomputers. The calculation precision is high. Therefore, it has been developed to solve problems of infinite domain, stress concentration and related fracture mechanics.

The finite element method analysis process is to discretize the continuum into many elements and replace the original continuum with a combination of elements to meet the geometric approximation. In the parts where the tension changes are more complicated, the elements can be divided into smaller ones. For example, in the stress analysis of the finite element method for pipe joints, due to the stress concentration at the transition of the joints, the elements of this part are divided into smaller ones (Figure 2).

In structural stress analysis, if the node displacement is used as the basic unknown to be solved, the strain of each element is required to simulate the actual strain of the corresponding area in the original continuum. For this reason, a proper element offset function is required. The offset function is an assumption of the offset state in the element. According to the offset function, the offset of each point in the element can be obtained by interpolating the offset of the nodes of the element.

If the displacement function can make the displacement continuous on the element and on the common edge of adjacent elements, the smaller the element size, the finite element method solution can converge to the true solution.

??After establishing the relation between the displacement of any point of the element and the displacement of the node through the displacement function, the relation between the force of the element node and the displacement of the element node is further established. The number of nodes in various units is not the same, from a few to dozens. Use {?} to denote the set of nodal displacements of the elements; consequently, use { R } to denote the set of nodal forces of the elements. The relationship between the two is

In the formula, [
k ] is the stiffness matrix of the element, which is a numerical table organized by stiffness coefficients according to a certain rule. Each stiffness coefficient represents the amount of force that must be applied at that location or at another location to produce displacement of the element at a given location. Group each drive according to the drive node number position, and the complete structure can be obtained. For example, {
P } and {

} are used to denote the load and displacement set of the node of the structure respectively, and [
K ] denotes the total stiffness matrix of the structure, then

Based on the known boundary conditions, the above formula is modified as necessary, and it can be resolved that {

}, the nodal offset of the element and the nodal offset of the structure are equal at the same point. With nodal displacement, the displacement in the element can be calculated by interpolation, and the strain and stress in the element can be calculated according to the displacement using the formula of elastic mechanics. The entire process of the above analysis is compiled into a computer language program, and the original data of the structure is entered into the computer to calculate the required result.

??From a mathematical point of view, the finite element method consists of solving the indeterminate function of infinite degrees of freedom to solve the problem of the indeterminate value of the finite degrees of freedom.
??Program?The problem-solving process compiled in electronic computer language is called a program. The program can be stored in an electronic computer and the required results can be obtained by inputting the necessary data into the computer when solving the problem.

The structural analysis program can be compiled into a special or general-purpose large-scale program for a certain type of subject. The various shapes and types of units in the program accommodate various complex geometric shapes, forces, constraints, and material properties. Geometric figures, stress contour figures, and structure response figures before and after stress can be drawn on the computer. The structural analysis program systematizes, generalizes and automates the finite element method analysis process. Modular design is widely adopted in the program. The entire program is made up of many independent modules (subprograms). The applets can be replaced or added to make the program have enhanced and expanded functionality.
The program should be easy to use, have good documentation, have sub-structure capabilities, be able to perform multi-condition analysis, and have a wide range of post-processing programs.

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