Numerical simulation of computational mechanics

The mathematical simulation of mechanical phenomena usually comes down to solving ordinary differential equations, partial differential equations, integral equations or algebraic equations. There are two ways to solve these equations: one is to find an analytical solution, that is, a solution expressed by a formula; The other is to find a numerical solution, that is, a solution expressed by a number of numbers. Many mechanical problems are quite complicated, especially complex partial differential equations, and it is generally difficult to get their analytical solutions. However, the numerical method needs complex operation steps and consumes a lot of manpower, so it is necessary to use it before the emergence of electronic computers. Since 1950s, general digital computers with modern programming languages have appeared. The fast operation and large storage of computer make it possible to solve complex mechanical problems. In the past 30 years, with the improvement of computers, numerical methods have been widely used and greatly developed. The main idea is to calculate faster, more accurately and save money, and to construct algorithms for problems that could not be calculated before. There are many numerical methods, among which finite difference method and finite element method are the most widely used methods to solve partial differential equations. In addition, there are variational method, straight line method, characteristic line method and spectral method. The essence of these methods is to turn the partial differential equation problem into algebraic problem, and then use computer to find the numerical solution of unknown function. It is simple, flexible and universal. When using the difference method to solve the numerical solution, we must first discretize the domain of the independent variable, that is, we only try to calculate the approximation of the unknown function of a finite number of points in the domain of the independent variable. If there is only one independent variable, the interval to be calculated can be discretized into line segments. If there are two independent variables and the calculation area is a rectangle as shown in figure 1, the simplest discretization method is to divide the area into several small rectangles. The length and width of a small rectangle are called the direction and the step size of the direction respectively. The partial derivative (,) in differential equation is the limit of difference quotient in calculus, which is replaced by difference quotient in finite difference method. As shown in figure 1 [discretization of binary region], the midpoint can be replaced by difference quotient (()-()/2 in some cases, and (()-()/in other cases. If there is a second-order partial derivative, it can often be replaced by the second-order difference quotient (() -2 ()+()/2, where (). If every derivative of the differential equation is replaced by an appropriate difference quotient, we can get the corresponding.

For the difference equation of the original differential equation, how to choose the difference quotient is very important. In addition, partial differential equations always have boundary or initial conditions, and these conditions should also be expressed in difference form. In this way, for the unknown function value of each grid point, the algebraic equations of unknown quantity are formed. If the grid is closely divided, that is, the sum of steps is small, or the sum of steps is large, the unknown number of algebraic equations is large, but with the help of computer, the solution can be obtained quickly. Because the step size cannot be zero, the difference method can only get the approximate solution of the original differential equation. However, as long as the reasonable difference quotient and step size are selected, the calculation results can still be satisfactory, and sometimes high-precision solutions can be obtained. Finite element method

This method divides the calculation area into triangle (or other shape) units with different sizes, and then replaces the unknown function with an appropriate interpolation function on each unit. According to the variational principle, partial differential equations can be solved by transforming them into algebraic equations. This method has wide adaptability, especially suitable for solving problems with complex boundary shapes and physical conditions, and is easy to realize on computer. Since 1970, some general finite element programs suitable for a wide range of linear problems have been developed, which have played a great role in engineering design. According to the idea of finite element method, the automobile shell is divided into many triangular elements with different sizes, and the bending boundary can only be cut and straightened. Where the stress changes sharply and accurate calculation is needed, the element must be made smaller; Where the change is not drastic, the change can be greater. This method can not only adapt to complex areas, but also minimize the total number of cells, thus reducing the unknown quantity. If rectangular grids are used in the finite difference method, it is more difficult to deal with such a complex region.