TAILIEUCHUNG - Applied Structural Mechanics Fundamentals of Elasticity Part 12

Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 316 16 Algorithms of mathematical programming MP f x Min b f x Min gj 0 j Xn 0 i 1 2 Fig. Method of a exterior penalty functions b interior penalty functions Problems with constraints 317 Modified objective function p 1 f x - Rị 2 i 1 2 3 . . is to be minimized for a set of decreasing values of Rị . Rj 105 R2 104 R3 103 etc. . In this case the objective function f is penalized in the feasible domain by summation of the positive terms. For details on further procedures refer to . General nonlinear problems - dừect methods In the past a substantial number of direct algorithms of Mathematical Programming MP - algorithms have been developed for the solution of general nonlinear optimization problems. When applied to problems of component optimization these algorithms must be able to reduce the number of optimization steps during optimization since often extensive structural analyses have to be carried out at each iteration. In addition a sufficiently good convergence behaviour as well as reliability and robustness must be demanded of these procedures. These characteristics largely depend on the degree of nonlinearity of the posed problem. In the following two frequently used procedures shall be briefly described as typical algorithms a Sequential Linearization Procedure SLP The efficiency of linear methods can also be utilized for nonlinear design problems by successively solving linear substitute problems in the form of a so-called sequential linearization . By introducing upper and lower bounds hypercube move limits for all design variables GRIFFITH and STEWARD have augmented the range of application to problems where the solutions are not at the intersection of constraints but more general on a curved hypersurface . The objective function and constraints of the nonlinear scalar initial problem 15À are expanded in a TAYLOR-series in the vicinity of a point xk . By maintaining the linear terms only we obtain f xk Ax