When there are few variables in the optimization problem, the following simple solutions can be used.

(1) graphic method. In the design space, draw the equivalent surface of the feasible region and the objective function, and then find out the position of the design point that minimizes the value of the objective function in the feasible region (or its boundary) from the diagram.

(2) Analysis method. When the problem is simple, it can be solved by analytical method.

2. Coding method

From the viewpoint of engineering and mechanics, the criterion method puts forward some criteria (such as synchronous failure criterion, full stress criterion, energy criterion, etc.). ) that is, what the structure should meet when it reaches the optimal design, and then the solution that meets these criteria is obtained by iterative method. The main characteristics of this method are fast convergence, the number of reanalysis is not directly related to the number of design variables, and the calculation amount is not large, but the application scope is limited, which is mainly suitable for the case of fixed structural layout and geometric shape. Although the criterion method has its shortcomings, from the perspective of engineering application, it is more convenient and acceptable, and its advantages are still the main ones. The simplest criterion methods are synchronous failure criterion and full stress criterion.

(1) synchronous failure criterion method. Its basic idea can be summarized as: under the action of load, the structure that can realize all possible failure modes at the same time is the optimal structure. The design of synchronous failure criterion has many obvious shortcomings. Because of algebraic operation with analytical expressions, synchronous failure design can only be used to deal with very simple component optimization; When the number of constraints is greater than the number of design variables, it is usually difficult to determine which failure modes should occur at the same time before the optimal design is given. When the number of constraints and the number of design variables are equal, there is no guarantee that the solution thus obtained is the optimal solution.

(2) Full stress criterion method. This method holds that giving full play to the potential of material strength can be regarded as a sign of structural optimization, and the full stress of components is taken as the criterion of optimal design. This method is widely used in the optimal design of truss and other bar systems. On this basis, the tooth line method combining ray step size and the full stress design of complex structures such as frames are developed.

3. Mathematical planning methods

The structural optimization problem is reduced to a mathematical programming problem, and then solved by mathematical programming method. The commonly used mathematical programming methods in structural optimization are nonlinear programming, sometimes linear programming, and dynamic programming, geometric programming, integer programming or stochastic programming may be used in special circumstances.

(1) linear programming. When the objective function and constraint equation are both linear functions of design variables, it is called linear programming problem. The solutions to this kind of problems are mature, among which simplex method is commonly used.

(2) Nonlinear programming. When the objective function or constraint equation is a nonlinear function of design variables, it is called nonlinear programming. Structural optimization design is mainly a constrained nonlinear programming problem. This kind of problem is much more complicated and difficult than linear programming problem. At present, there are the following methods: demand derivative analysis without conversion, such as gradient projection method and feasible direction method; A direct search method that does not require conversion or derivation, such as the composite method; Linear programming is used for one-to-one approximation, such as sequential linear programming; It is transformed into unconstrained extremum problems, such as penalty function method and multiplier method.

4. Mixing method

The mixed method adopts normative method and mathematical programming method.

5. Heuristic algorithm

Some heuristic algorithms have been developed in recent years. These algorithms include genetic algorithm, neural network algorithm and simulated annealing algorithm. They have been applied to the field of structural optimization.